Xenharmonic Wiki - User contributions [en]

Kite Guitar Exercises and Techniques

2026-07-28T05:10:13Z

TallKite: remove link to empty page

Exercises and techniques for the [[Kite Guitar]].

==[[Kite_Guitar_Exercises_by_Kite_Giedraitis|Exercises and techniques by Kite Giedraitis]] ==
* [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Before%20You%20Get%20Your%20Kite%20Guitar|1 Before You Get Your Kite Guitar]]
* [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Exercises%20and%20Techniques%20for%20Players|2 Exercises and Techniques for Players]]
** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Vocal%20Exercises|2.1 Vocal Exercises]]
** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Parallel%20Thirds|2.2 Parallel Thirds]]
** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Half-fret%20Bends|2.3 Half-fret Bends]]
** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#The%20Circle%20of%205ths|2.4 The Circle of 5ths]]
** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Moving%20by%20Plain%20Minor%202nds|2.5 Moving by Plain Minor 2nds]]
** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Leaping%20by%205ths.2C%208ves%20and%20Unisons|2.6 Leaping by 5ths, 8ves and Unisons]]
** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Combination%20Exercises|2.7 Combination Exercises]]
** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Alternate%20Fingering%20Techniques|2.8 Alternate Fingering Techniques]]
*** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Cross-fret%20barre|2.8.1 Cross-fret barre]]
*** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Harmonics|2.8.2 Harmonics]]
** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Quarter-fret%20Bends.2C%20Sixth-fret%20Bends.2C%20etc.|2.9 Quarter-fret Bends, Sixth-fret Bends, etc.]]
*** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Primes%2011%20and%2013|2.9.1 Primes 11 and 13]]
* [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Exercises%20and%20Techniques%20for%20Composers%20and%20Arrangers|3 Exercises and Techniques for Composers and Arrangers]]
** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Interesting%20Root%20Movements|3.1 Interesting Root Movements]]
** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Harmonizing%20Chromatic%20Melodies|3.2 Harmonizing Chromatic Melodies]]
** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Rotations%20aka%20Inversions|3.3 Rotations aka Inversions]]
*** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Motivating%20Example|3.3.1 Motivating Example]]
*** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Rotating%20an%20Interval|3.3.2 Rotating an Interval]]
*** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Rotating%20a%20Melody|3.3.3 Rotating a Melody]]
*** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Rotating%20a%20Chord|3.3.4 Rotating a Chord]]
*** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Rotating%20a%20Chord%20Progression|3.3.5 Rotating a Chord Progression]]
*** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Rotating%20an%20Entire%20Song|3.3.6 Rotating an Entire Song]]
** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#The%20Big%20Switcheroo|3.4 The Big Switcheroo]]
** [[Kite Guitar Exercises and Techniques by Kite Giedraitis#Modulation%20via%20Dim7%20Chords|3.5 Modulation via Dim7 Chords]]

== Exercises and Techniques by Others ==
Guitar teachers, add your own exercises and/or techniques here, or make your own page of exercises!

[[Category:Kite Guitar]]

Kite's thoughts on hi-lo notation

2026-07-21T11:56:00Z

TallKite: optionally use "," for "add"

Hi-lo notation is for naming chord voicings. It applies to all tunings, even conventional ones like 12-edo or meantone. It allows arrangers, composers and theorists to discuss specific voicings much more accurately than the conventional 1st inversion, 2nd inversion, etc. nomenclature.

Hi-lo notation identifies every chord member by its interval from the chord root as 3rd, 5th, etc., written as 3, 5, etc. It then describes what needs to be done to the root position close voicing (known as the ''basic'' voicing) in order to change it into the given voicing.

* C E G = C (the basic voicing is the default voicing)
* C G E = Chi3 (hi3 because relative to the basic voicing, the 3rd is raised by an octave)
* G C E = Clo5 (aka 2nd inversion, the 5th is lowered from basic)
* E G C = ChiR (R stands for root) aka 1st inversion
* E C G = Clo3 (another example of 1st inversion)
* G E C = ChiRlo5

When there are multiple high or low notes, the terms ''hi'' and ''lo'' are only used once:

* C G E B = CM7hi37
* G Bb C E = C7lo57

"Add" is used for duplicated notes:

* C E G C = Cadd8
* G C E G = Caddlo5
* G C E G C = Cadd8lo5
* C E G E = Caddhi3 or Cadd10
* C C E G = CaddloR or possibly Chi35add8

This doesn't conflict with the usual use of "add" in chord names:

* C E G D = Cadd9
* C E G C D = Cadd89
* C D E G = Caddlo9 or Cadd2
* D C E G = Caddlo2

"No" is used as usual to omit notes, as in C9no5. The 4 words are used in this order: no - hi - lo - add - hi - lo. First omit notes, then move notes higher, then move notes lower, then add notes, then add high notes, then add low notes. Following this order removes any ambiguity:

* Chi3add8 = C G C E
* Cadd8hi3 = C E G C E
* Caddhi38 = C E G E C
"Add" can optionally be written more concisely with a comma: Chi3add8 becomes Chi3,8. This follows the comma usage of [[Kite's ups and downs notation|ups and downs notation]] and [[Kite's color notation|color notation]].

The common open chords of a guitar:

* C major: x32010 = C E G C E = C,8hi3
* D major: x00232 = A D A D F# = Dhi3,8lo5
* E major: 022100 = E B E G# B E = E,8loR5 or possibly Ehi3,8hi58
* G major: 320003 = G B D G B G = G,8hi38 or possibly Glo5,8loR3
* A major: 002220 = E A E A C# E = Ahi3,8hi5lo5
Use addhi8, not add15. No degrees higher than 13 are used, analogous to chord names.

The terms ''hi'' and ''lo'' and degrees R-8 cover voicings up to a 3-octave range, because every note is in either the lo octave, the central octave or the hi octave. Degrees 9-13 go even higher. To extend the range an extra octave both above and below, use hihi and lolo. For example, lolo7 is a note a 9th below the root.

To find the number of voices in a chord, start with the obvious: a triad has 3, a tetrad has 4, etc. Then subtract the "no" notes and add the "add" notes. For example, A7no5hi3,8 has 4 (A7 is a tetrad) - 1 (no 5th) + 1 (add 8ve) = 4 notes.

For more examples, see [[Kite Guitar Chord Shapes (downmajor tuning)]].

== Notating chord progressions ==
There are sometimes two possible names for a voicing, depending on what one considers the "home octave" to be. As we saw, C C E G can be either CaddloR or Chi35add8. This ambiguity can be removed by designating one note in a specific octave as "the" tonic, then naming all chord roots relative to that tonic as hiC, loF#, loVII, etc. Unlike [[wikipedia:Scientific_pitch_notation|octave numbers]] which only increase when going from B up to C, the hi-lo categories only change in the region of the tonic. This has the advantage that transposing to a new key won't change the hi-lo categories.
{| class="wikitable"
|+G major scale in both scientific pitch notation and hi-lo notation
!SPN names
|D2
|E2
|F#2
|G2
|A2
|B2
|C3
|D3
|E3
|F#3
|G3
|A3
|B3
|-
!hi-lo names
|loD
|loE
|loF#
|G
|A
|B
|C
|D
|E
|F#
|hiG
|hiA
|hiB
|}
Chord progressions can be written out as e.g. Cadd8 - loAmhi3add8 - Dm7 - loG7hi3. The entire 4-part harmony can be deduced note-for-note from this. Here are the notes, where the lower octave uses underlined letters and the upper octave uses lower-case letters:

* C E G c
* A E A c
* D F A c
* G D F B

This works with relative notation as well: Iadd8 - loVImhi3add8 - IIm7 - loV7hi3.

Naming the roots as hi or lo ''almost'' allows us to reduce a barbershop tag to a mere succession of chords. However voices sometimes coincide or cross. See [[Kite Guitar Translations by Kite Giedraitis#Barbershop tags]]

Hi-lo notation was invented by [[Kite Giedraitis]].

{{Navbox notation}}

Indian music

2026-07-17T04:09:00Z

TallKite:

'''Indian music''' uses pitch in many distinctive ways.
There are two main traditions, Hindustani (North Indian) and Carnatic (South Indian).<ref name="clarke2024rags" /><ref name="kiran1997appreciating" />

== Sargam notation ==
Both Indian traditions use a solfege system based on a heptatonic (7-tone) framework: Sa, Re (Ri in the south), Ga, Ma, Pa, Dha (Da in the south), and Ni.

Scales themselves are taken from a twelve-tone framework in both traditions, but the naming practices are a bit different. In North India, notes can be natural (shuddh), sharp (tivra), or flat (komal). The natural notes form a major scale (0 2 4 5 7 9 11). Sa and Pa are never sharp or flat; Ma can be sharp or natural; Re, Ga, Dha and Ni can be flat or natural.

In South India, things are a bit more complex, and it's best illustrated with a table. Sa and Pa have only one version, Ma has two, and Re, Ga, Dha, and Ni have three each. On the surface, it would appear to make a 16-tone framework, but some names refer to the same pitch category, much like enharmonics in Western music. The names are also more fanciful in Carnatic music.
{| class="wikitable"
|+
!Degree
!Hindustani name
!Carnatic name
!Western example (S = A)
|-
|0
|Sa
|Sa
|A
|-
|1
|Komal Re
|Shuddha Ri
|Bb
|-
| rowspan="2" |2
| rowspan="2" |Shuddh Re
|Chatusruti Ri
|B
|-
|Shuddha Ga
|Cb
|-
| rowspan="2" |3
| rowspan="2" |Komal Ga
|Shatsruti Ri
|B#
|-
|Sadharana Ga
|C
|-
|4
|Shuddh Ga
|Antara Ga
|C#
|-
|5
|Shuddh Ma
|Shuddha Ma
|D
|-
|6
|Tivra Ma
|Prati Ma
|D#
|-
|7
|Pa
|Pa
|E
|-
|8
|Komal Dha
|Shuddha Da
|F
|-
| rowspan="2" |9
| rowspan="2" |Shuddh Dha
|Chatusruti Da
|F#
|-
|Shuddha Ni
|Gb
|-
| rowspan="2" |10
| rowspan="2" |Komal Ni
|Shatsruti Da
|F##
|-
|Kaisiki Ni
|G
|-
|11
|Shuddh Ni
|Kakali Ni
|G#
|}

However, in Hindustani music, notes can be 'very flat' (ati komal), indicating a microtonal inflection.
See [[#clarke2024rags|''Rāgs Around the Clock'']] for a fuller discussion of notation in Hindustani music. In Carnatic music things are a bit different; speed and pitch profile of of ornaments, vibratos, and slides generally matter more than exact pitch in a raga.

== Ornamentation ==
Ornamentation in Indian music is very striking and characteristic of the music.
See [[#clarke2024rags|''Rāgs Around the Clock'']] for a detailed discussion and audio examples.

Meend are glides between different notes.<ref name="clarke2024rags" />
You can see meend graphically in the pitch contours shown at [https://autrimncpa.wordpress.com Music in Motion].

Andolan is a gentle oscillation in pitch within a given note.<ref name="clarke2024rags" />

In Hindustani music a gamaka is a 'shake' in pitch around a series of notes<ref name="clarke2024rags" />. In Carnatic terminology, gamaka is the general term for an ornament, and it includes various kinds such as:

* vibratos (both small, within a pitch class, and large, between pitch classes and possibly as wide as a minor third)
* slides
* quick grace notes (lower grace notes like A-G#-A or G#-A more common than upper ones like A-Bb-A or Bb-A)
* turns (like F#-E-D#-E, the first three being grace notes)
* more complex ones like tones sliding into vibratos (like A sliding up to a vibrato between E and F).

== The tanpura ==
The tanpura is a long necked fretless lute which provides a harmonic-rich drone.<ref name="clarke2024rags" />

The tanpura typically has four strings.
In the most common tuning, the middle two strings are tuned to the tonic (Sa),
and the first and fourth strings are tuned to the fifth (Pa) and tonic (Sa) in the octave below.<ref name="sambamoorthy1957drones" />
In ragas without a fifth (Pa), the first string is tuned to another note instead, most commonly the fourth (Ma) or seventh (Ni).<ref name="clarke2024rags" />

Sambamoorthy's [[#sambamoorthy1957drones|book on drones]] discusses the tanpura and other instruments used to provide drones.

== The harmonium ==
The harmonium is a portable reed organ.
It was introduced into Indian music in the 19th century.
Its use proved controversial, leading to a ban from All India Radio (AIR) from 1940-1971.<ref name="rahaim2011bane" />
The AIR seminar on the harmonium gives many arguments for and against the harmonium.<ref name="air1971harmonium" />
The use of equal temperament on the harmonium was one consideration; others included the inability to play meend and gamakas.

== Shruti ==
{{Wikipedia|Shruti (music)}}
In modern practice, shruti refer to microtonal inflections in general.<ref name="clarke2024rags" /><ref name="meer2010construction" />

There is a long history of specific systems of pitches called shruti, going back at least to Bharata's Natyashastra.<ref name="ghosh1961natyashastra" />
Bharata discusses a system of 22 notes and describes an experiment involving two vinas.
It is not obvious how to interpret Bharata's discussion in terms of exact pitches, and it has given rise to a long line of analyses and proposed interpretations, with different pitches and numbers of notes.<ref name="levy1982intonation" /><ref name="meer2010construction" /><ref name="sambamoorthy1963south" />
Given the complexity of the issue, it would be unwise to pick one system as ''the'' system of 22 shruti.
The relationship of systems of shruti to musical practice remains contentious.<ref name="levy1982intonation" /><ref name="meer2010construction" />

== References ==
<references>
<ref name="clarke2024rags">

David Clarke, [https://doi.org/10.11647/OBP.0313 Rāgs Around the Clock: A Handbook for North Indian Classical Music, with Online Recordings in the Khayāl Style]. Open Book Publishers, 2024.
</ref>
<ref name="kiran1997appreciating">

Ravi Kiran, [https://archive.org/details/appreciatingcarn0000ravi/ Appreciating Carnatic Music]. Ganesh & Co, 1997.
</ref>
<ref name="sambamoorthy1957drones">

P. Sambamoorthy, [https://archive.org/details/srutivadyasdrone00samb Sruti Vadyas (Drones)]. All India Handicrafts Board, 1957.
</ref>
<ref name="rahaim2011bane">
Matt Rahaim, [https://www.jstor.org/stable/41302388 That Ban(e) of Indian Music: Hearing Politics in the Harmonium]. The Journal of Asian Studies 70.3, 2011.
</ref>
<ref name="air1971harmonium">
[https://archive.org/details/dli.ministry.09019 AIR's seminar on the Harmonium]. Sangeet Natak Akademi, 1971.
</ref>
<ref name="ghosh1961natyashastra">
Bharata-Muni, [https://archive.org/details/NatyaShastra/natya_shastra_translation_volume_2_-_bharat_muni/page/n57/ The Nāṭyaśāstra], transl. Manomohan Ghosh. The Asiatic Society, 1961.
</ref>
<ref name="levy1982intonation">
Mark Levy, [https://archive.org/details/intonationinnort0000mark Intonation in North Indian Music]. Biblia Impex, 1982.
</ref>
<ref name="meer2010construction">
S. Rao and W. van der Meer, [https://pure.uva.nl/ws/files/4133925/76993_286708.pdf The Construction, Reconstruction and Deconstruction of Shruti]. Hindustani music: thirteenth to twentieth centuries, 2010.
</ref>
<ref name="sambamoorthy1963south">
P. Sambamoorthy, [https://archive.org/details/bwb_C0-AUR-484/page/100/ South Indian Music, Book IV]. The Indian Music Publishing House, 1963.
</ref>
</references>

== Further reading ==
* G.K. Koduri, S. Gulati, P. Rao and X. Serra, [https://www.researchgate.net/profile/Preeti-Rao/publication/263566239_Raga_Recognition_based_on_Pitch_Distribution_Methods/ Rāga Recognition based on Pitch Distribution Methods]. Journal of New Music Research 41.4, 2012.

== External links ==
* [https://digital.library.ucla.edu/catalog/ark:/21198/zz002gcntd Bake/Jairazbhoy Digital Archive of South Asian Traditional Music and Arts] - Archive of recordings and images
* [https://autrimncpa.wordpress.com Music in Motion] - Recordings synced with videos showing pitch contours
* [https://chandrakantha.com/music-and-dance/i-class-music/ www.chandrakantha.com - Indian classical music] - portal on Indian music, with detailed [https://chandrakantha.com/music-and-dance/i-class-music/index-of-rags/ descriptions of a number of North Indian rags]
* [http://musicresearchlibrary.net/omeka/ MusicResearchLibrary] - an Indian music research portal

[[Category:Indian music]]

Indian music

2026-07-17T03:22:57Z

TallKite: clarify no flat Pa, no sharp Re, etc.

'''Indian music''' uses pitch in many distinctive ways.
There are two main traditions, Hindustani (North Indian) and Carnatic (South Indian).<ref name="clarke2024rags" /><ref name="kiran1997appreciating" />

== Sargam notation ==
The seven notes (svara) of the major scale are called Sa, Re, Ga, Ma, Pa, Dha, and Ni.
Notes can be natural (shuddh), sharp (tivra), or flat (komal). Sa and Pa are always natural, Pa is never flat, and the others are never sharp. This gives, to begin with, a twelve note framework.
However, notes can be 'very flat' (ati komal), indicating a microtonal inflection.
See [[#clarke2024rags|''Rāgs Around the Clock'']] for a fuller discussion of notation in Hindustani music.
There is a comparison table of Hindustani and Carnatic notation in [[#kiran1997appreciating|''Appreciating Carnatic Music'']].

== Ornamentation ==
Ornamentation in Indian music is very striking and characteristic of the music.
See [[#clarke2024rags|''Rāgs Around the Clock'']] for a detailed discussion and audio examples.

Meend are glides between different notes.<ref name="clarke2024rags" />
You can see meend graphically in the pitch contours shown at [https://autrimncpa.wordpress.com Music in Motion].

Andolan is a gentle oscillation in pitch within a given note.<ref name="clarke2024rags" />

In Hindustani music a gamaka is a 'shake' in pitch around a series of notes
(in Carnatic terminology gamaka is the general term for an ornament).<ref name="clarke2024rags" />

== The tanpura ==
The tanpura is a long necked fretless lute which provides a harmonic-rich drone.<ref name="clarke2024rags" />

The tanpura typically has four strings.
In the most common tuning, the middle two strings are tuned to the tonic (Sa),
and the first and fourth strings are tuned to the fifth (Pa) and tonic (Sa) in the octave below.<ref name="sambamoorthy1957drones" />
In ragas without a fifth (Pa), the first string is tuned to another note instead, most commonly the fourth (Ma) or seventh (Ni).<ref name="clarke2024rags" />

Sambamoorthy's [[#sambamoorthy1957drones|book on drones]] discusses the tanpura and other instruments used to provide drones.

== The harmonium ==
The harmonium is a portable reed organ.
It was introduced into Indian music in the 19th century.
Its use proved controversial, leading to a ban from All India Radio (AIR) from 1940-1971.<ref name="rahaim2011bane" />
The AIR seminar on the harmonium gives many arguments for and against the harmonium.<ref name="air1971harmonium" />
The use of equal temperament on the harmonium was one consideration; others included the inability to play meend and gamakas.

== Shruti ==
{{Wikipedia|Shruti (music)}}
In modern practice, shruti refer to microtonal inflections in general.<ref name="clarke2024rags" /><ref name="meer2010construction" />

There is a long history of specific systems of pitches called shruti, going back at least to Bharata's Natyashastra.<ref name="ghosh1961natyashastra" />
Bharata discusses a system of 22 notes and describes an experiment involving two vinas.
It is not obvious how to interpret Bharata's discussion in terms of exact pitches, and it has given rise to a long line of analyses and proposed interpretations, with different pitches and numbers of notes.<ref name="levy1982intonation" /><ref name="meer2010construction" /><ref name="sambamoorthy1963south" />
Given the complexity of the issue, it would be unwise to pick one system as ''the'' system of 22 shruti.
The relationship of systems of shruti to musical practice remains contentious.<ref name="levy1982intonation" /><ref name="meer2010construction" />

== References ==
<references>
<ref name="clarke2024rags">

David Clarke, [https://doi.org/10.11647/OBP.0313 Rāgs Around the Clock: A Handbook for North Indian Classical Music, with Online Recordings in the Khayāl Style]. Open Book Publishers, 2024.
</ref>
<ref name="kiran1997appreciating">

Ravi Kiran, [https://archive.org/details/appreciatingcarn0000ravi/ Appreciating Carnatic Music]. Ganesh & Co, 1997.
</ref>
<ref name="sambamoorthy1957drones">

P. Sambamoorthy, [https://archive.org/details/srutivadyasdrone00samb Sruti Vadyas (Drones)]. All India Handicrafts Board, 1957.
</ref>
<ref name="rahaim2011bane">
Matt Rahaim, [https://www.jstor.org/stable/41302388 That Ban(e) of Indian Music: Hearing Politics in the Harmonium]. The Journal of Asian Studies 70.3, 2011.
</ref>
<ref name="air1971harmonium">
[https://archive.org/details/dli.ministry.09019 AIR's seminar on the Harmonium]. Sangeet Natak Akademi, 1971.
</ref>
<ref name="ghosh1961natyashastra">
Bharata-Muni, [https://archive.org/details/NatyaShastra/natya_shastra_translation_volume_2_-_bharat_muni/page/n57/ The Nāṭyaśāstra], transl. Manomohan Ghosh. The Asiatic Society, 1961.
</ref>
<ref name="levy1982intonation">
Mark Levy, [https://archive.org/details/intonationinnort0000mark Intonation in North Indian Music]. Biblia Impex, 1982.
</ref>
<ref name="meer2010construction">
S. Rao and W. van der Meer, [https://pure.uva.nl/ws/files/4133925/76993_286708.pdf The Construction, Reconstruction and Deconstruction of Shruti]. Hindustani music: thirteenth to twentieth centuries, 2010.
</ref>
<ref name="sambamoorthy1963south">
P. Sambamoorthy, [https://archive.org/details/bwb_C0-AUR-484/page/100/ South Indian Music, Book IV]. The Indian Music Publishing House, 1963.
</ref>
</references>

== Further reading ==
* G.K. Koduri, S. Gulati, P. Rao and X. Serra, [https://www.researchgate.net/profile/Preeti-Rao/publication/263566239_Raga_Recognition_based_on_Pitch_Distribution_Methods/ Rāga Recognition based on Pitch Distribution Methods]. Journal of New Music Research 41.4, 2012.

== External links ==
* [https://digital.library.ucla.edu/catalog/ark:/21198/zz002gcntd Bake/Jairazbhoy Digital Archive of South Asian Traditional Music and Arts] - Archive of recordings and images
* [https://autrimncpa.wordpress.com Music in Motion] - Recordings synced with videos showing pitch contours
* [https://chandrakantha.com/music-and-dance/i-class-music/ www.chandrakantha.com - Indian classical music] - portal on Indian music, with detailed [https://chandrakantha.com/music-and-dance/i-class-music/index-of-rags/ descriptions of a number of North Indian rags]
* [http://musicresearchlibrary.net/omeka/ MusicResearchLibrary] - an Indian music research portal

[[Category:Indian music]]

Kite's ups and downs notation

2026-07-02T21:07:20Z

TallKite:

{{Redirect|Ups and downs notation|the notation system with attached arrows|Stein–Zimmermann–Gould notation}}
'''Ups and downs notation''' (or ^v) is a [[musical notation]] system primarily designed for [[edo]]s. The up symbol "^" and the down symbol "v" indicate raising/lowering a note (or widening/narrowing an interval) by one EDOstep. The mid symbol, "~" is for intervals exactly midway between major and minor, e.g. 3\24 is a mid 2nd. The mid 4th (~4) is midway between perfect and augmented, i.e. halfway-augmented, and the mid 5th (~5) is a halfway-diminished 5th.

Ups and downs notation can be used with other tuning systems outside edos, by adjusting the definition of the arrows and perhaps also adding lifts (/) and drops (\) to accommodate finer tuning inflections. Lifts and drops can also be used with larger edos, especially [[Kite's thoughts on ring number|multi-ring]] edos where lifts and drops can be used to navigate between the rings of the edo.

Ups and downs notation was invented by [[Kite Giedraitis]] in early 2014.

{{Note|'''This page only discusses notation of edos.''' However, the notation of chords and chord progressions applies to all situations. For notation of rank-2 and rank-3 temperaments, see the [[pergen]]s article.}}

== Explanation (a 22edo example) ==
To understand the ups and downs notation, let's start with an edo that doesn't need it. 19edo is easy to notate because 7 fifths reduced by 4 octaves adds up to one EDOstep. C♯ is right next to C, and the keyboard runs {{nowrap|C, C♯, D♭, D, D♯, E♭, E}} etc. Conventional notation works perfectly with 19edo as long as you remember that C♯ and D♭ are different notes.

In contrast, 22edo is hard to notate because 7 fifths reduces to ''three'' EDOsteps, and the usual chain of fifths {{dash|E♭, B♭, F, C, G, D, A, E, B, F♯, C♯}} etc. creates the scale {{dash|C, D♭, B♯, C♯, D, E♭, F♭, D♯, E, F}}. That's very confusing because B♯–D♭ looks ascending on the page but sounds descending, and a 4:5:6 major chord is written {{dash|C, D♯, G}}, and the 5/4, usually a major third, becomes an augmented second. Some people forgo the chain of fifths for a maximally even scale like {{dash|C, D, E, F, G, A, B, C}}. But that's confusing because G–D and A–E are diminished 5ths. And if your piece is in G or A, that's really confusing. A notation system should work in every key!

The solution is to use the sharp symbol to mean "raised by 7 fifths", and to use the up-arrow symbol to mean "sharpened by one EDOstep". 22edo can be written {{dash|C, Db, ^Db, vD, D, Eb, ^Eb, vE, E, F}} etc. The notes are pronounced up-D-flat, down-D, etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp. The up or down comes before the note name to make naming chords easy.

The names change depending on the key, just like in conventional notation where F# in D major becomes Gb in Db major. So the B scale is {{dash|B, C, ^C, vC#, C#, D, ^D, vD#, D#, E}} etc.

The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible.

=== Relative notation and interval arithmetic ===
Ups and downs can be used not only for absolute notation (note names) but also for relative notation (intervals, chords and scales). Relative notation for 22edo intervals: {{dash|P1, m2, ^m2, vM2, M2, m3, ^m3, vM3, M3, P4, ^4/d5, vA4/^d5, A4/v5, P5}} etc. That's pronounced upminor 2nd, downmajor 3rd, etc. You can apply this pattern to any 22edo key. The '''plain''' notes (those without ups or downs) always form a chain of fifths.

A core principle of ups and downs notation is that '''interval arithmetic is always preserved'''. Ups and downs are simply added in:

{| class="wikitable" style="text-align: center;"
|-
!
! Interval between two notes
! Note plus an interval
! Sum of two intervals
|-
! conventional
| C to E = M3
| C + M3 = E
| M2 + M2 = M3
|-
! rowspan="2" | with ups and downs
| ^C to E = vM3
| ^C + M3 = ^E
| ^M2 + M2 = ^M3
|-
| C to ^E = ^M3
| C + ^M3 = ^E
| M2 + vM2 = vM3
|-
! (cancelling)
| ^C to ^E = M3
| ^C + vM3 = E
| ^M2 + vM2 = M3
|-
! (combining)
| ^C to vE = vvM3
| ^C + ^M3 = ^^E
| vM2 + vM2 = vvM3
|}

The same logic holds for a note minus an interval (C - vm3 = ^A) or one interval minus another interval (M3 - vM2 = ^M2).

=== "Arrow" as a term for EDOstep ===
Up and down are short for up-arrow and down-arrow, and arrow refers to both. Sometimes the name of a notation symbol comes to mean that which the symbol indicates. Just as "bar" (the vertical line that separates measures) has come to mean "measure", "[[arrow]]" has also come to mean "EDOstep".

=== Enharmonic unisons ===
Conventionally, in C you use D# instead of Eb when you have a Gaug chord. You have the freedom to spell your notes how you like, to make your chords look right. Likewise, in 22edo, D♭ can be spelled ^C or vB♯ or even ^^B (double-up B, or '''dup''' B for short, rhymes with "cup"). Respelling is done by adding or subtracting an [[Enharmonic unisons in ups and downs notation|enharmonic unison]], '''EU''' for short.

From the [[Pergen|pergens]] article: "Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. Enharmonic unisons are like vanishing commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of EUs needed always equals the difference between the notation's rank and the tuning's rank."

Since 22edo is rank-1, and conventional notation plus ups and downs is rank-3, two EUs are needed to define the notation: v3A1 and vm2. Either EU can be added to or subtracted from any note to respell the note. For example, ^C + vm2 = Db and ^^Eb + v3A1 = vE. Any combination of these two EUs is also an EU, for example their sum v4M2. Thus ^^F = ^^F + v4M2 = vvG (double-down G, or '''dud''' G for short, rhymes with "cud").

=== Larger EDOs ===
In larger edos, triple-arrows, quadruple-arrows, etc. can occur. Up, dup, trup and quup all rhyme, as do dud, trud and quud.

{| class="wikitable" style="text-align:center;"
|+ style="font-size: 105%;" | Symbols and words for multiple arrows
|-
! Written
! Spoken
! Etymology
!
! Written
! Spoken
! Etymology
|-
| ^
| up
|
! 1 arrow
| v
| down
|
|-
| ^^
| dup
| '''d'''ouble-'''up'''
! 2 arrows
| vv
| dud
| rowspan="4" | "-d" for down replaces "-p" for up
|-
| ^^^
| trup
| '''tr'''iple-'''up'''
! 3 arrows
| vvv
| trud
|-
| v>
| quup "kwup"
| '''qu'''adruple-'''up'''
! 4 arrows
| ^<
| quud "kwud"
|-
| >
| quip
| '''qui'''ntuple-u'''p'''
! 5 arrows
| <
| quid
|}

(In addition to dup, trup, etc. there is dub, trip, quad and quin, used for multiple sharps/flats and multiple lifts/drops, e.g. dubsharp or triplift.)

Very large edos can go well beyond 5 arrows. The sequence of names resembles tally counting I, II, III, IIII, <s>||||</s>. But the sequence of ''symbols'' resembles roman numerals I, II, III, IV, V. Thus 4 ups is spoken quup but written v>.

{| class="wikitable" style="text-align: center;"
|-
| up ^
| dup ^^
| trup ^^^
| quup v>
| quip >
| upquip ^>
| dupquip ^^>
| trupquip ^^^>
| quupquip v>>
| quipquip >>
| upquipquip ^>>
| dupquipquip ^^>>
| trupquipquip ^^^>>
| quupquipquip v>>>
| triplequip >>>
|-
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
! 14
! 15
|-
| down v
| dud vv
| trud vvv
| quud ^<
| quid <
| downquid v<
| dudquid vv<
| trudquid vvv<
| quudquid ^<<
| quidquid <<
| downquidquid v<<
| dudquidquid vv<<
| trudquidquid vvv<<
| quudquidquid ^<<<
| triplequid <<<
|}

Lifts and drops (/ and \) can be used for microinflections of less than an edostep, since they look like part of an arrow.
{| class="wikitable"
|+categories of pitch alterations
|^
|up
| rowspan="2" |arrow
| rowspan="4" |inflection
| rowspan="6" |alteration
|-
|v
|down
|-
|/
|lift
| rowspan="2" |slash
|-
|\
|drop
|-
|#
|sharp
| colspan="2" rowspan="2" |accidental
|-
|b
|flat
|}
For very large edos in which commas like 81/80 and 64/63 are many edosteps, the color notation accidental pairs yo/gu and zo/ru can be "edoized" to stand for a certain number of edosteps. For example, in [[311edo]], 81/80 is 6 edosteps. Thus g means ^> and y means v<. The colors can be combined with arrows as in upyo or dudgu (^y or vvg). Likewise, 64/63 is 7 edosteps, thus r means ^^> and z means vv<.

===Staff Notation===
For staff notation, put an arrow to the left of the note and any sharp or flat it might have. Like sharps and flats, an arrow applies to any similar note that follows in the measure. If C is upped, any other C in the same octave inherits the up. If an up-C is followed by a down-C, the down-arrow replaces the up-arrow.

But what happens when accidentals are mixed with arrows? What if the key signature makes that upped C be sharp? Or what if there is a C with a sharp just before the upped C? Does the up-arrow override or "cancel" the sharp? And what if an upped C is followed by a sharpened C?

There are several possible ways to handle this issue. The default is the simplest way, to explicitly specify both arrows and accidentals every time. Thus any accidental or arrow cancels any previous ones. An arrow by itself implies a natural sign.

{| class="wikitable" style="text-align:center;"
|-
! rowspan="2" | Start with this
! colspan="6" | Turn it into this
|-
! C
! ^C
! ^^C
! C#
! ^C#
! ^^C#
|-
! C
|        
| ^
| ^^
| #
| ^#
| ^^#
|-
! ^C
| <big>♮</big>
|        
| ^^
| #
| ^#
| ^^#
|-
! ^^C
| <big>♮</big>
| ^
|        
| #
| ^#
| ^^#
|-
! C#
| <big>♮</big>
| ^
| ^^
|        
| ^#
| ^^#
|-
! ^C#
| <big>♮</big>
| ^
| ^^
| #
|        
| ^^#
|-
! ^^C#
| <big>♮</big>
| ^
| ^^
| #
| ^#
|        
|}

See [[Kite Guitar originals#Cancelling rules]] for another way.

For more on staff notation, see the [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf Notation Guide for EDOs 5-72].

=== Key signatures ===
Key signatures follow the conventional practice, expanded to allow for double-sharps and double flats in some edos. For example, 19edo has the key of Bbb with a key signature of B𝄫 E𝄫 A♭ D♭ G♭ C♭ F♭. Some edos have upped/downed tonics, e.g. 24edo has the key of vD with a key signature of F♯ C♯ (v). The (v) is a "global down" that downs all 7 notes of the vD scale. See also [[Kite Guitar originals#Scales and key signatures]] for the use of '''arrow stacks'''.

=== Placement of the arrow ===
It might seem more natural to place the arrow after the note, for example B^ or Bb^. But the arrow must come first, to make chord names unambiguous. Otherwise B^m could mean either a minor chord rooted on B^ or an upminor chord rooted on B. (Chord names are explained fully below.)

The issue arises because while English normally places the adjective before the noun, it doesn't do so with sharps and flats. A flattened B should logically be called "flat B" not "B flat", and be written bB not Bb. If it were, then it would seem very natural to have the up come first, as in ^bB. This would be the typical English adjective-adjective-noun construction. Instead we must use ^Bb, an unnatural adjective-noun-adjective construction. This issue fortunately arises only for note names. On the staff, the flat comes before the note, so naturally the up comes before the flat. In relative notation, the quality comes before the interval, as in minor 3rd and augmented 4th, or in jazz terms flat 3rd and sharp 4th. So terms like upminor 3rd and downsharp 4th have a natural adjective-adjective-noun construction.

=== Further notes ===
Edo intervals are often written as 7\22. This can also be written as vM3\22. This is useful when comparing edos, e.g. vM3\22 vs. vM3\15.

== Examples: edos 12-24 ==
Sharp-1, flat-2, etc. refer to the [[sharpness]], the number of arrows made by seven 5ths minus four 8ves. All sharp-1 and flat-1 edos can be notated without ups and downs, because the up is exactly equivalent to a sharp or flat.

A ring is a circle of 5ths. In multi-ring (aka ringy) edos like 14, 15 and 24, a single ring doesn't contain all the edo's notes. In contrast, edos like 12, 19 and 22 are single-ring. It's possible to notate any single-ring edo with conventional notation if notes are permitted to be out of order (e.g. 22edo could have C Db B# C# D). But multi-ring edos absolutely require ups and downs.

13edo and 18edo aren't compatible with heptatonic notation, because the minor 2nd is descending. Thus the minor 3rd is flatter than the major 2nd, the 4th is flatter than the major 3rd, etc. These edos are best notated using the 2nd best fifth, as 13b and 18b.

There are four flat-N edos on this list. 16edo and 23edo are flat-1, 18b is flat-2 and 13b is flat-3. There are two ways to notate such edos: with sharp lowering the pitch, and major/aug narrower than minor/dim, or with sharp raising the pitch, and major/aug wider than minor/dim. Both notations are shown. In the 2nd notation, note that a fifth above B is Fb, not F#.

12edo is sharp-1, thus doesn't need ups and downs. Enharmonic unison: d2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[12-edo|12edo]] {{normal|sharp-1}}
| '''D'''
| D#/Eb
| '''E'''
| '''F'''
| F#/Gb
| '''G'''
| G#/Ab
| '''A'''
| A#/Bb
| '''B'''
| '''C'''
| C#/Db
| '''D'''
|-
| P1
| A1/m2
| M2
| m3
| M3
| P4
| A4/d5
| P5
| m6
| M6
| m7
| M7
| P8
|}

There are two ways to notate 13b-edo. The enharmonic unisons for the 1st notation are ^3A1 and vM2. For the 2nd they are v3A1 and vm2.

{| class="wikitable" style="text-align:center;"
|-
! rowspan="4" | [[13-edo|13b-edo]] {{normal|flat-3}}
! rowspan="2" | Sharp lowers the pitch, major narrower than minor
| '''D'''
| '''E'''
| ^E/F#
| vEb/^F#
| Eb/vF
| '''F'''
| '''G'''
| '''A'''
| '''B'''
| ^B/C#
| vBb/^C#
| Bb/vC
| '''C'''
| '''D'''
|-
| P1
| M2
| ^M2/M3
| vm2/^M3
| m2/vm3
| m3
| P4
| P5
| M6
| ^M6/M7
| vm6/^M7
| m6/vm7
| m7
| P8
|-
! rowspan="2" | Sharp raises the pitch, major wider than minor
| '''D'''
| '''E'''
| ^E/Fb
| vE#/^Fb
| E#/vF
| '''F'''
| '''G'''
| '''A'''
| '''B'''
| ^B/Cb
| vB#/^Cb
| B#/vC
| '''C'''
| '''D'''
|-
| P1
| m2
| ^m2/m3
| vM2/^m3
| M2/vM3
| M3
| P4
| P5
| m6
| ^m6/m7
| vM6/^m7
| M6/vM7
| M7
| P8
|}

Because every 14edo interval is perfect, the quality can be omitted. Sharps and flats can also be omitted. 14edo contains 2 rings of 7edo: an up/down-ring and a plain-ring. Enharmonic unisons: A1 and vvm2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[14-edo|14edo]] {{normal|sharp-0}}
| '''D'''
| ^D/vE
| '''E'''
| ^E/vF
| '''F'''
| ^F/vG
| '''G'''
| ^G/vA
| '''A'''
| ^A/vB
| '''B'''
| ^B/vC
| '''C'''
| ^C/vD
| '''D'''
|-
| 1
| ^1/v2
| 2
| ^2/v3
| 3
| ^3/v4
| 4
| ^4/v5
| 5
| ^5/v6
| 6
| ^6/v7
| 7
| ^7/v8
| 8
|}

15edo contains 3 rings of 5edo: an up-ring, a down-ring, and a plain-ring. Enharmonic unisons: v3A1 and m2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[15-edo|15edo]] {{normal|sharp-3}}
| '''D'''
| ^D
| vE
| '''E/F'''
| ^F
| vG
| '''G'''
| ^G
| vA
| '''A'''
| ^A
| vB
| '''B/C'''
| ^C
| vD
| '''D'''
|-
| P1
| ^m2
| vM2
| M2/m3
| ^m3
| vM3
| M3/P4
| ^4
| v5
| P5
| ^m6
| vM6
| M6/m7
| ^m7
| vM7
| P8
|}

16edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic unison: either AA2 or dd2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="4" | [[16-edo|16edo]] {{normal|flat-1}}
! rowspan="2" | Sharp lowers the pitch, major narrower than minor
| '''D'''
| Db/E#
| '''E'''
| Eb
| F#
| '''F'''
| Fb/G#
| '''G'''
| Gb/A#
| '''A'''
| Ab/B#
| '''B'''
| Bb
| C#
| '''C'''
| Cb/D#
| '''D'''
|-
| P1
| A2
| M2
| m2/A3
| M3
| m3
| d3/A4
| P4
| d4/A5
| P5
| d5/A6
| M6
| m6/A7
| M7
| m7
| d7
| P8
|-
! rowspan="2" | Sharp raises the pitch, major wider than minor
| '''D'''
| D#/Eb
| '''E'''
| E#
| Fb
| '''F'''
| F#/Gb
| '''G'''
| G#/Ab
| '''A'''
| A#/Bb
| '''B'''
| B#
| Cb
| '''C'''
| C#/Db
| '''D'''
|-
| P1
| d2
| m2
| M2
| m3
| M3
| A3
| P4
| A4/d5
| P5
| d6
| m6
| M6/d7
| m7
| M7
| A7
| P8
|}

17edo is sharp-2 and thus has mid intervals. Enharmonic unisons: vvA1 and vm2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[17edo]] {{normal|sharp-2}}
| '''D'''
| ^D/Eb
| D#/vE
| '''E'''
| '''F'''
| ^F/Gb
| F#/vG
| '''G'''
| ^G/Ab
| G#/vA
| '''A'''
| ^A/Bb
| A#/vB
| '''B'''
| '''C'''
| ^C/Db
| C#/vD
| '''D'''
|-
| P1
| ^1/m2
| A1/~2
| M2
| m3
| ~3
| M3
| P4
| ^4/~4/d5
| A4/v5/~5
| P5
| m6
| ~6
| M6
| m7
| ~7
| M7
| P8
|}

18b-edo contains 2 rings of 9edo: an up/down-ring and a plain-ring. There are two ways to notate it. Enharmonic unisons: either ^^A1 and vvM2, or vvA1 and vvm2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="4" | '''[[18-edo|18b-edo]]''' flat-2
! rowspan="2" | Sharp lowers, major is narrower
| '''D'''
| ^D/vE
| '''E'''
| ^E
| Eb/F#
| vF
| '''F'''
| ^F/vG
| '''G'''
| ^G/vA
| '''A'''
| ^A/vB
| '''B'''
| ^B
| Bb/C#
| vC
| '''C'''
| ^C/vD
| '''D'''
|-
| P1
| ^1/vM2
| M2
| ~2
| m2/M3
| ~3
| m3
| ^m3/v4
| P4
| ^4/v5
| P5
| ^5/vM6
| M6
| ~6
| m6/M7
| ~7
| m7
| ^m2/d8
| P8
|-
! rowspan="2" | Sharp raises, major is wider
| '''D'''
| ^D/vE
| '''E'''
| ^E
| E#/Fb
| vF
| '''F'''
| ^F/vG
| '''G'''
| ^G/vA
| '''A'''
| ^A/vB
| '''B'''
| ^B
| B#/Cb
| vC
| '''C'''
| ^C/vD
| '''D'''
|-
| P1
| ^1/vm2
| m2
| ~2
| M2/m3
| ~3
| M3
| ^M3/v4
| P4
| ^4/v5
| P5
| ^5/vm6
| m6
| ~6
| M6/m7
| ~7
| M7
| ^M7/d8
| P8
|}

19edo is sharp-1, thus doesn't need ups and downs. Enharmonic unison: dd2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[19-edo|19edo]] {{normal|sharp-1}}
| '''D'''
| D#
| Eb
| '''E'''
| E#/Fb
| '''F'''
| F#
| Gb
| '''G'''
| G#
| Ab
| '''A'''
| A#
| Bb
| '''B'''
| B#/Cb
| '''C'''
| C#
| Db
| '''D'''
|-
| P1
| d2
| m2
| M2
| d3
| m3
| M3
| A3
| P4
| A4
| d5
| P5
| A5
| m6
| M6
| d7
| m7
| M7
| A7
| P8
|}

20edo contains 4 rings of 5edo: an up-ring, a down-ring, a dup/dud-ring, and a plain-ring. Enharmonic unisons: v4A1 and m2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[20-edo|20edo]] {{normal|sharp-4}}
| '''D'''
| ^D
| ^^D/vvE
| vE
| '''E/F'''
| ^F
| ^^F/vvG
| vG
| '''G'''
| ^G
| ^^G/vvA
| vA
| '''A'''
| ^A
| ^^A/vvB
| vB
| '''B/C'''
| ^C
| ^^C/vvD
| vD
| '''D'''
|-
| P1/m2
| ^m2
| ~2
| vM2
| M2/m3
| ^m3
| ~3
| vM3
| M3/P4
| ^4
| ~4/~5
| v5
| P5/m6
| ^m6
| ~6
| vM6
| M6/m7
| ^m7
| ~7
| vM7
| P8
|}

Because every 21edo interval is perfect, the quality can be omitted. 21edo contains 3 rings of 7edo: an up-ring, a down-ring and a plain-ring. Enharmonic unisons: A1 and v3m2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[21-edo|21edo]] {{normal|sharp-0}}
| '''D'''
| ^D
| vE
| '''E'''
| ^E
| vF
| '''F'''
| ^F
| vG
| '''G'''
| ^G
| vA
| '''A'''
| ^A
| vB
| '''B'''
| ^B
| vC
| '''C'''
| ^C
| vD
| '''D'''
|-
| 1
| ^1
| v2
| 2
| ^2
| v3
| 3
| ^3
| v4
| 4
| ^4
| v5
| 5
| ^5
| v6
| 6
| ^6
| v7
| 7
| ^7
| v8
| 8
|}

22edo is sharp-3. Enharmonic unisons: v3A1 and vm2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[22-edo|22edo]] {{normal|sharp-3}}
| '''D'''
| ^D/Eb
| vD#/^Eb
| D#/vE
| '''E'''
| '''F'''
| ^F/Gb
| vF#/^Gb
| F#/vG
| '''G'''
| ^G/Ab
| vG#/^Ab
| G#/vA
| '''A'''
| etc.
|-
| P1
| ^1/m2
| vA1/^m2
| vM2
| M2
| m3
| ^m3
| vM3
| M3
| P4
| ^4/d5
| vA4/^d5
| A4/v5
| P5
| etc.
|}

23edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic unison: either A32 or d32.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="4" | [[23-edo|23edo]] {{normal|flat-1}}
! rowspan="2" | Sharp lowers, major is narrower
| '''D'''
| Db
| E#
| '''E'''
| Eb
| Ebb/Fx
| F#
| '''F'''
| Fb
| G#
| '''G'''
| Gb
| A#
| '''A'''
| Ab
| B#
| '''B'''
| Bb
| Bbb/Cx
| C#
| '''C'''
| Cb
| D#
| '''D'''
|-
| P1
| d1
| A2
| M2
| m2
| d2/A3
| M3
| m3
| d3
| A4
| P4
| d4
| A5
| P5
| d5
| A6
| M6
| m6
| d6/A7
| M7
| m7
| d7
| A8
| P8
|-
! rowspan="2" | Sharp raises, major is wider
| '''D'''
| D#
| Eb
| '''E'''
| E#
| Ex/Fbb
| Fb
| '''F'''
| F#
| Gb
| '''G'''
| G#
| Ab
| '''A'''
| A#
| Bb
| '''B'''
| B#
| Bx/Cbb
| Cb
| '''C'''
| C#
| Db
| '''D'''
|-
| P1
| A1
| d2
| m2
| M2
| A2/d3
| m3
| M3
| A3
| d4
| P4
| A4
| d5
| P5
| A5
| d6
| m6
| M6
| A6/d7
| m7
| M7
| A7
| d8
| P8
|}

24edo contains 2 rings of 12edo: an up/down-ring and a plain-ring. Enharmonic unisons: vvA1 and d2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[24-edo|24edo]] {{normal|sharp-2}}
| '''D'''
| ^D/vEb
| D#/Eb
| ^D#/vE
| '''E'''
| ^E/vF
| '''F'''
| ^F
| F#/Gb
| vG
| '''G'''
| ^G/vAb
| G#/Ab
| ^G#/vA
| '''A'''
| etc.
|-
| P1
| ^1/vm2
| A1/m2
| ~2
| M2
| ^M2/vm3
| m3
| ~3
| M3
| ^M3/v4
| P4
| ^4/~4
| A4/d5
| v5/~5
| P5
| etc.
|}

== Chords and chord progressions==
Chord names are based on jazz chord names. See Jim Aiken's book ''A Player's Guide to Chords & Harmony''. Alterations are enclosed in parentheses, additions never are. Alterations always come last in the chord name. Examples:

* [[19edo chords]]
* [[22edo chords]]
* [[24edo chord names]]
* [[31edo chord names]]
* [[41edo chord names]]
* [[Kite Guitar chord shapes (downmajor tuning)]]

In [[Sharpness|sharp-0]] edos aka perfect edos (7, 14, 21, 28 and 35), every interval is perfect, and there is no major or minor. In the following lists of chord names, omit major, minor, dim and aug. Substitute up for upmajor and upminor, and down for downmajor and downminor. The C-E-G chord is called "C perfect" or simply "C".

Chord progressions use ups/downs notation to name the roots, e.g. Cv - Gv - vA^m - F or Iv - Vv - vVI^m - IVv. In relative notation, '''never use lower case roman numerals''' for minor chords, because both vIIm and VIIm would be written vii.

=== Triads ===
The major chord and various alterations of it:
*C E G = C = "C" or "C major" (in perfect edos, "C" or "C perfect")
*C ^E G = C^ = "C up" or "C upmajor"
*C vE G = Cv = "C down" or "C downmajor" (in sharp-2 edos, C~ = "C mid")
* C vvE G = Cvv = "C dud" or "C dudmajor" (in sharp-4 edos, C~ = "C mid", in sharp-6 edos, C^~ = "C upmid")
This table shows how altering the 3rd or the 5th affects the name of the triad. The conventional abbreviations for aug and dim are + and o. These are rather cryptic, and can be replaced with the more obvious and intuitive a and d. Likewise the symbols Δ and − can be replaced with M and m.

{| class="wikitable" style="text-align:center;"
|-
!
! Major
! Minor
! sus4
! sus2
! colspan="2" | Augmented
! colspan="2" | Diminished
|-
! what's downed
! C E G
! C Eb G
! C F G
! C D G
! colspan="2" | C E G#
! colspan="2" | C Eb Gb
|-
! nothing
| C
| Cm
| C4
| C2
| Ca
| C+
| Cd
| Co
|-
! 3rd
| Cv
| Cvm
| Cv4
| Cv2
| Cva
| Cv+
| Cvd
| Cvo
|-
! 5th
| C(v5)
| Cm(v5)
| C4(v5)
| C2(v5)
| Ca(v5)
| C+(v5)
| Cd(v5)
| Co(v5)
|-
! 3rd, 5th
| Cv(v5)
| Cvm(v5)
| Cv4(v5)
| Cv2(v5)
| Cva(v5)
| Cv+(v5)
| Cvd(v5)
| Cvo(v5)
|}

Note that the dim chord is a triad, not a tetrad. A dim tetrad should always be written Co7, never Co. In jazz, the 7 is omitted because dim triads are so much rarer than dim tetrads. But ups and downs notation is meant to work for all genres, not just jazz. So the dim triad and the dim tetrad need different names.

Many edos have notes between the major 3rd and the perfect 4th, creating triads impossible in 12edo, such as:
*C Fb G = C(d4) or C(b4) = "C dim-four" or "C sus-flat-four"
*C E# G = C(a3) or C(#3) = "C aug-three" or "C sus-sharp-three"
*C Ebb G = C(d3) or C(bb3) = "C dim-three" or "C sus-double-flat-three"
*C D# G = C(a2) or C(#2) = "C aug-two" or "C sus-sharp-two"
The "sus" is needed so that C(#2) doesn't sound like C#2, which is C# D# G#.

=== Global arrows ===
A global arrow occurs between the chord root and the conventional chord type (e.g. C^m7). It raises or lowers the 3rd, and also the 6th, 7th or 11th, if present. Thus C down-nine is the usual C9 chord with the 3rd and 7th downed: Cv9 = C vE G vBb D. A global-mid chord has a mid 3rd, 6th, 7th, and/or 11th. Mnemonic: every other note of a stacked-3rds chord is affected: '''6th''' - root - '''3rd''' - 5th - '''7th''' - 9th - '''11th''' - 13th. Note that the 6th is affected, but the 13th is not.

The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd. Furthermore, in larger edos, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the 5th. By the same logic, if the 7th is upped or downed, the 11th would be too.

A 2nd or 4th in a sus chord is also affected: C4 = C F G but Cv4 = C vF G = "C down-four" or "C sus-down-four". But Cv7(4) = C F G vBb

Every conventional chord can accept a global arrow, with one exception: it's pointless for a C5 chord, because there is no 3rd, 6th or 7th to alter. Thus Cv5 is invalid. But C(v5) is valid, and if someone says "C down five", it means C(v5) = C E vG.

=== Sixth and seventh chords ===
If the 7th is not a perfect 5th or a dim 5th above the 3rd, the chord is named as a triad with an added 7th. An added 7th is usually preceded by a comma (the actual punctuation mark, not an interval), which is spoken as "add":
*C E G Bb = C7 = "C seven" (conventional chord)
*C vE G Bb = Cv,7 = "C down add-seven"
*C E G vBb = C,v7 = "C add down-seven"
*C vE G vBb = Cv7 = "C down seven" (global down)
All 7th chords follow this same pattern. Likewise, if the 6th is not a perfect 4th or aug 4th above the 3rd, it's an add-6 chord. Permitting add-7 chords has the added benefit that the wordy "minor-7 flat-5" and the illogical "half-dim" can both be replaced with "dim add-7", written Cd,7.

In the table below, if a chord is '''bolded''', the comma punctuation is not spoken as "add".
{| class="wikitable" style="text-align:center;"
|-
!
! maj7
! dom7
! min7
! colspan="3" | dim-add-7 or min7(b5) or half-dim
! colspan="2" | dim7
! maj6
! min6
|-
! what's downed
! C E G B
! C E G Bb
! C Eb G Bb
! colspan="3" | C Eb Gb Bb
! colspan="2" | C Eb Gb Bbb
! C E G A
! C Eb G A
|-
! nothing
| CM7
| C7
| Cm7
| Cd,7
| Cm7(b5)
| Cø
| Cd7
| Co7
| C6
| Cm6
|-
! 3rd
| Cv,M7
| Cv,7
| Cvm,7
| Cvd,7
| Cvm,7(b5)
| Cø(v3)
| '''Cvd,d7'''
| Cvod7
| Cv,6
| Cvm,6
|-
! 5th
| CM7(v5)
| C7(v5)
| Cm7(v5)
| Cd,7(v5)
| Cm7(vb5)
| Cø(v5)
| Cd7(v5)
| Co7(v5)
| C6(v5)
| Cm6(v5)
|-
! 6th/7th
| C,vM7
| C,v7
| Cmv7
| Cdv7
| Cmv7(b5)
| Cø(v7)
| Cdvd7
| Covd7
| C,v6
| Cmv6
|-
! 3rd, 5th
| Cv,M7(v5)
| Cv,7(v5)
| Cvm,7(v5)
| Cvd,7(v5)
| Cvm,7(vb5)
| Cø(v3v5)
| '''Cvd,d7(v5)'''
| Cvod7(v5)
| Cv,6(v5)
| Cvm,6(v5)
|-
! 3rd, 6th/7th
| CvM7
| Cv7
| Cvm7
| Cvdv7
| Cvm7(b5)
| Cvø
| Cvd7
| Cvo7
| Cv6
| Cvm6
|-
! 5th, 6th/7th
| C,vM7(v5)
| C,v7(v5)
| Cmv7(v5)
| Cdv7(v5)
| Cmv7(vb5)
| Cø(v5v7)
| Cdvd7(v5)
| Covd7(v5)
| C,v6(v5)
| Cm,v6(v5)
|-
! 3rd, 5th, 6th/7th
| CvM7(v5)
| Cv7(v5)
| Cvm7(v5)
| Cvdv7(v5)
| Cvm7(vb5)
| Cvø(v5)
| Cvd7(v5)
| Cvo7(v5)
| Cv6(v5)
| Cvm6(v5)
|}

Various unusual tetrads:
* {{nowrap|C vE G ^Bb {{=}} Cv^7 {{=}} "C down up-seven"}} (in sharp-2 edos 17, 24, 31, etc. {{nowrap|C~7 {{=}} "C mid-seven"}})
* {{nowrap|C E G A# {{=}} C,#6}} or {{nowrap|C,A6 {{=}} "C add sharp-six"}} or "C add aug-six"
* {{nowrap|C E G Ab {{=}} C,b6}} or {{nowrap|C,m6 {{=}} "C add flat-six"}} or "C add minor-six"
* {{nowrap|C E G Bbb {{=}} C,bb7}} or {{nowrap|C,d7 {{=}} "C add double-flat-seven"}} or "C add dim-seven" (19edo's 4:5:6:7 chord)
* {{nowrap|C E G B# {{=}} C,#7}} or {{nowrap|C,A7 {{=}} "C add sharp-seven"}} or "C add aug-seven"
* {{nowrap|C E G Cb {{=}} C,b8}} or {{nowrap|C,d8 {{=}} "C add flat-eight"}} or "C add dim-eight"

=== Ninth chords ===
As in conventional chord naming, a sharp-9 or flat-9 chord is always named as a 7th chord with an added 9th. Thus {{nowrap|B D# F# A C}} is named B7b9 (not Bb9 which would be {{nowrap|Bb D F A C}}). Likewise C#7b9 not C#b9, even thought the latter is clearly the same flat-9 chord as the former. Likewise Cm7b9 not Cmb9, etc.

Double alterations need only a single pair of parentheses, e.g. C E vG vB D is named CM9(v5v7). Double additions mostly need only a single comma, e.g. C E G vBb vD is named C,v7v9. But certain 6/9 chords require two commas. In '''bolded''' 6/9 chords, the comma between the 6 and the 9 is not spoken as "add". However any comma before "6" is, e.g. Cv,6,9 is "C down add six nine".

{| class="wikitable" style="text-align:center;"
|-
!
! add9
! maj9
! dom9
! min9
! dom7b9
! maj6/9
! min6/9
|-
! what's downed
! C E G D
! C E G B D
! C E G Bb D
! C Eb G Bb D
! C E G Bb Db
! C E G A D
! C Eb G A D
|-
! nothing
| C,9
| CM9
| C9
| Cm9
| C7b9
| '''C6,9'''
| '''Cm6,9'''
|-
! 3rd
| Cv,9
| CM9(v3)
| C9(v3)
| Cm9(v3)
| Cv,7b9
| '''Cv,6,9'''
| '''Cvm,6,9'''
|-
! 5th
| C,9(v5)
| CM9(v5)
| C9(v5)
| Cm9(v5)
| C7b9(v5)
| '''C6,9(v5)'''
| '''Cm6,9(v5)'''
|-
! 6th/7th
| ------
| CM9(v7)
| C9(v7)
| Cm9(v7)
| C,v7b9
| '''C,v6,9'''
| '''Cmv6,9'''
|-
! 9th
| C,v9
| CM7v9
| C7v9
| Cm7v9
| C7vb9
| C6v9
| Cm6v9
|-
! 3rd, 5th
| Cv,9(v5)
| CM9(v3v5)
| C9(v3v5)
| Cm9(v3v5)
| Cv,7b9(v5)
| '''Cv,6,9(v5)'''
| '''Cvm,6,9(v5)'''
|-
! 3rd, 6th/7th
| ------
| CvM9
| Cv9
| Cvm9
| Cv7b9
| '''Cv6,9'''
| '''Cvm6,9'''
|-
! 3rd, 9th
| Cv,v9
| Cv,M7v9 or CM7v9(v3)
| Cv,7v9 or C7v9(v3)
| Cvm,7v9 or Cm7v9(v3)
| Cv,7vb9 or C7vb9(v3)
| Cv,6v9 or C6v9(v3)
| Cvm,6v9 or Cm6v9(v3)
|-
! 5th, 6th/7th
| ------
| CM9(v5v7)
| C9(v5v7)
| Cm9(v5v7)
| C,v7b9(v5)
| '''C,v6,9(v5)'''
| '''Cm,v6,9(v5)'''
|-
! 5th, 9th
| C,v9(v5)
| CM7v9(v5)
| C7v9(v5)
| Cm7v9(v5)
| C7vb9(v5)
| C6v9(v5)
| Cm6v9(v5)
|-
! 6th/7th, 9th
| ------
| C,vM7v9
| C,v7v9
| Cmv7v9
| C,v7vb9
| C,v6v9
| Cmv6v9
|-
! 3rd, 5th, 6th/7th
| ------
| CvM9(v5)
| Cv9(v5)
| Cvm9(v5)
| Cv7b9(v5)
| '''Cv6,9(v5)'''
| '''Cvm6,9(v5)'''
|-
! 3rd, 5th, 9th
| Cv,v9(v5)
| Cv,M7v9(v5) or CM7v9(v3v5)
| Cv,7v9(v5) or C7v9(v3v5)
| Cvm,7v9(v5) or Cm7v9(v3v5)
| Cv,7vb9(v5) or C7vb9(v3v5)
| Cv,6v9(v5) or C6v9(v3v5)
| Cvm,6v9(v5) or Cm6v9(v3v5)
|-
! 3rd, 6th/7th, 9th
| ------
| CvM7v9
| Cv7v9
| Cvm7v9
| Cv7vb9
| Cv6v9
| Cvm6v9
|-
! 5th, 6th/7th, 9th
| ------
| C,vM7v9(v5)
| C,v7v9(v5)
| Cmv7v9(v5)
| C,v7vb9(v5)
| C,v6v9(v5)
| Cmv6v9(v5)
|-
! 3rd, 5th, 6th/7th, 9th
| ------
| CvM7v9(v5)
| Cv7v9(v5)
| Cvm7v9(v5)
| Cv7vb9(v5)
| Cv6v9(v5)
| Cvm6v9(v5)
|}

=== Rules for punctuation usage ===
Tetrads, pentads, etc. often require a comma (the actual punctuation mark) to ensure correct parsing of the chord name. Only use a comma when needed, to reduce clutter and standardize chord names. A comma is needed in {{nowrap|Cv,7 {{=}} C vE G Bb}} because omitting it makes {{nowrap|Cv7 {{=}} C vE G vBb}}, a different chord. But C7,v9 is incorrect because C7v9 is the same chord.

The rule is, omit the comma unless doing so changes the chord. This simple rule suffices in most situations. What follows is a detailed analysis, designed to aid in writing computer code that automates chord naming.

A comma separates an added note and prevents it from merging with what comes before it. The comma is unneeded in C7v9 because the 7 can't merge with the down to make a 7v. But Cm,7 is incorrect even though the m and the 7 can merge, because Cm7 is the same chord.

The various components of a chord name are either numbers (for the 6th, 7th, 9th, etc.) or adjectives (up, down, mid, sharp, flat, major, minor, aug and dim). These adjectives usually modify the following number, but they sometimes modify the preceding root, e.g. Caug or C#7. Up, down and mid can't modify the preceding root.

A comma is always needed to separate a number from a number (Cv6,9). It's usually needed to separate an adjective from a number (Cv,7). The only exception is for certain conventional chords like Cm7 where separation is unneeded. A comma is always needed to separate the root of a plain major chord from an adjective (D,v7) or a number (Eb,9). It's never needed to separate a number from an adjective (C7^9). It's needed to separate an adjective from an adjective only if the two adjectives could apply to a single noun. There are six types of such adjective pairs.

* up followed by any adjective except down (C^,^9 or C^,~7 or C^,#9 or C^,b9 or C^,M7 or C^,m6 or C^,a7 or C^,d7)
* down followed by any adjective except up
* sharp followed by sharp (C#,#9)
* flat followed by flat (Bb,b9)
* aug followed by aug (Ca,a7)
* dim followed by dim (Cd,d9)

No other adjective pair can apply to a single noun, thus the comma is omitted:

* {{nowrap|Cv^9 {{=}} C vE G ^D}} (an interval can't be both upped and downed)
* {{nowrap|CmM7 {{=}} C Eb G B}} (an interval can't be both minor and major) *
* {{nowrap|Cma7 {{=}} C Eb G B#}} (an interval can't be both minor and aug) **
* {{nowrap|Cm#11 {{=}} C Eb G F#}} (an interval can't be both minor and sharp)
* {{nowrap|Cvmm6 {{=}} C vEb G Ab}} (an interval can't be doubly minor)
* {{nowrap|Cmv7 {{=}} C Eb G vBb}} (an interval can be downminor, but it can't be minordown)
* {{nowrap|C~v7 {{=}} C vvE G vBb}} in a sharp-4 edo (an interval can be downmid, but it can't be middown)
* {{nowrap|C~~9 = C vvE G vvD}} in a sharp-4 edo (an interval can't be doubly mid)
<nowiki/>* But beware of the minor-major chord. CvmM7 means C vEb G vB and Cvm,M7 means C vEb G B.

<nowiki/>**But since Cma7 can be read as an alternate spelling of Cmaj7, adding a comma is wise: Cm,a7.

In the spoken name, a comma is almost always pronounced as "add". The only exceptions are:

* a comma separating two numbers: C6,9 is spoken as "C six nine"
* a comma separating two ups or two downs: Cv,v9 is spoken as "C-down down-nine", since Cvv9 would be "C dud-nine"
* a comma separating two sharps or two flats: C#,#9 is "C sharp sharp-nine" since C##9 would be "C double-sharp nine"
* a comma separating two augs or two dims: Cvd,d7 is "C down-dim dim-seven", since Cvdd7 would be "C down-double-dim-seven"

Of course, there's no great harm in saying "add" when it isn't strictly needed.
{| class="wikitable"
|+ style="font-size: 105%;" | When to write a comma or say "add"
|-
! colspan="2" rowspan="2" |
! colspan="2" | Component after the possible comma
|-
! adjective
! number
|-
! rowspan="3" | Component before the possible comma
! root
| comma always "add" always
| comma always "add" always
|-
! adjective
| comma sometimes "add" sometimes if comma, never if no comma
| comma usually "add" always if comma, never if no comma
|-
! number
| comma never "add" never
| comma always "add" never
|}

More examples, in which the comma is almost always spoken as "add":

*B9 = B D# F# AvC#
*B,9 = B D# F# C#
*Bb9 = Bb D F Ab C
* Bb,9 = Bb D F C
*B,b9 = B D# F# C
*B7b9 = B D# F# A C
* Bbb9 = Bbb Db Fb Abb Cb
*Bbb,9 = Bbb Db Fb Cb
*Bb,b9 = Bb D F Cb (no "add", "B flat flat-nine")
* B,bb9 = B D# F# Cbb

== Cross-edo considerations ==
In 22edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19edo, it's 0-6-11 = 0¢-379¢-695¢. The two chords sound quite different, because "major 3rd" is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound ru (7-under) and 19edo major chords sound yo (5-over).

A chord quality like "major" refers not to the sound but to the function of the chord. If you want to play a {{nowrap|{{dash|I, VIm, IIm, V, I}}}} progression without pitch shifts or tonic drift, you can do that in any edo, as long as you use only major and minor chords. The notation tells you what kind of chord can be used to play that progression. In 22edo, the chord that you need sounds like a ru chord.

In other words, {{nowrap|{{dash|I, VIm, IIm, V, I}}}} in just intonation implies {{nowrap|{{dash|Iy, VIg, IIg, Vy, Iy}}}}, but this implication only holds in those edos in which major sounds yo. Because 22edo's yo chord {{nowrap|{{dash|0, 7, 13}} {{=}} {{dash|0{{c}}, 382{{c}}, 709{{c}}}}}} is downmajor, it doesn't work in that progression.

Another example: {{nowrap|{{dash|I7, bVII7, IV7, I7}}}}. To play this progression without shifts or drifts, the 7th in the I7 chord must be a minor 7th. in 22edo, that 7th sounds zo (7-over, thus 7/4). In 19edo, it sounds gu (5-under, thus 9/5).

== Ups and downs solfege ==
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down. See [[Uniform solfege]].

== For other tuning systems ==
Ups and downs notation was primarily designed to notate edos, but it can also be used to notate other tuning systems.

=== Rank-2 temperaments ===
Ups and downs can also notate any [[Tour of Regular Temperaments|rank-2 temperament]], although some temperaments require an additional pair of accidentals, lifts and drops (/ and \). In this context, an up or a lift represents sharpening by a [[comma]] that has been tempered, but not tempered out. For example, in [[Porcupine|Triyoti aka Porcupine]], an up/down represents raising/lowering by a tempered 81/80, and lifts/drops aren't used. In practice, the two uses of the notation often coincide perfectly. Triyoti is supported by both 15edo and 22edo, and both edos map 81/80 to one EDOstep. Thus if Triyoti is tuned to 15edo, an up simultaneously means both a tempered 81/80 and 1\15. Likewise, if tuned to 22edo, the up means both 81/80 and 1\22. If not tuned to an edo at all, then the up only means 81/80. Thus a piece written in Triyoti can be converted to a piece written in 22edo by simply writing "22edo" on the top of the page.

=== Rank-3 JI ===
{{Main|Kite's thoughts on ups and downs notation for rank-3 JI}}
Ups and downs can also be used to notate rank-3 just intonation subgroups such as 2.3.5 or 2.3.7 or 2.3.11.

== See also ==
* [[Kite's thoughts on enharmonic unisons in ups and downs notation]]
* [[Lambda ups and downs notation]]

== Further reading ==
* [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf Notation guide for edos 5-72], which also covers chord names, slash chords, staff notation, key signatures, and scale trees

{{Navbox notation}}

[[Category:Ups and downs notation| ]] 
[[Category:Notation]]

Kite Guitar Originals by Kite Giedraitis

2026-06-21T03:50:39Z

TallKite: /* Downminor Etude */ added the bass guitar part

[[Kite Giedraitis]]'s originals for the [[The_Kite_Guitar|Kite Guitar]]. Originals by other people are at [[Kite Guitar originals|Kite Guitar originals.]]

All tabs are for a guitar tuned in downmajor 3rds. Some of these are more riffs than songs.

== Evening Rondo ==
See [https://www.tallkite.com/music/EveningRondo.html TallKite.com/music/EveningRondo.html] for score, tab, video and mp3s.

== Underground Revolution ==

Key: ^Bb downminor pentatonic
<tt>
verse:
Iv7 / IVv7 vbIII^9
Iv7 / IVv7 Vv7
Iv7 vbVII^m6 / Iv7
vbIII^m6 Vv7 Iv7 (short bar of 3 beats)
IVv7 / / /
/ / / /

chorus:
Iv7 / / /
vbIII^m6 / / /
IVv7 / / /
vbVI^m6 / Vv7 /

instrumental:
Iv7 N.C / /
vIII^m6 / / /
#IVvdv7 / / /
IVv7 / / /
Iv7 / / /
/ / / /
</tt>

The verse with rhythm, lead melody and high harmony (a comma in the rhythm means a skipped beat)
<tt>
Iv7 IVv7 vbIII^9 Iv7 IVv7 Vv7
It's the new Do Re Mi A fresh melody Harmony will be turned around
4 e & 1 & 2 & a, e a 4 1 & 2 & a, e a 4
8 8 vb7 5 vb7 8 8 8 5 5 vb7 vb7 8 5 vb7 8 8 vb7 5
v3 v3 v3 v3 vb3 vb3 vb3 2

Iv7 vbVII^m6 Iv7 vbIII^m6 Vv7 Iv7 IVv7
There's a new sound in this town Down a-round the underground scene Yeah
1 e & 2 3 & 4 1 & 2 & a, e & 1, , , 1, , ,
5 5 8 vb7 vb7 5 8 8 vb7 5 5 5 5 v3 4 vb10
b5 b5 v4 v4 v3 v3 v3 vb3
</tt>

The chorus with rhythm, high harmony, lead melody and low harmony
<tt>
Iv7 vbIII^m6 IVv7 vbVI^m6 Vv7
Welcome to the underground Re-vo-lu-tion New ways of thought giving us New so-lutions
1 &, & 3 a, e &, &, & 3 &, 1 &, & 3 a, e &, &, & 3 &,
5 v3 5 v3 2 2 v3 b5 vb3 1 vb3 4 vb3 4 vb3 1 1 vb3 4 vb3 v7 2
vb10 8 vb10 8 vb7 vb7 8 vb10 8 vb7 8 vb10 8 vb10 8 v6 v6 8 vb10 v7 5 v7
vb7 5 vb7 5 v3 v3 5 vb7 b5 vb3 b5 v6 4 v6 4 vb3 vb3 4 vb6 4 2 v4
</tt>

== I Hear Numbers ==

Based on the keyboard part, not the electric guitar part. I wrote this in 7-limit JI, but it works well in 41edo. The breakdown pumps the Bizozogu microcomma, which 41edo tempers out.

Full song: http://tallkite.com/music/IHearNumbers.mp3

Keyboard part with a guitar sound: http://tallkite.com/music/IHearNumbersKeyboardOnGuitar.mp3

Liner notes: http://tallkite.com/music/IHearNumbers.html

Chords:<tt>
intro: Vv7 vIII^d^7 Vv7 vIII^d^7
Vv7 vIII^d^7 Vv7 /

solos: Iv7 vbIII^m6 IVv7 /
Iv7 vbIII^m6 IVv7 /
bVIIv7 bVIIv7noR vbIII^m6 /
vbVI^ vbVI^7 Vv7 /

verse 1: Iv7 vbVII^m6 Iv7 vbVII^m6
Iv7 vbVII^m6 Iv7 vbVII^m6
Iv7 vbVII^m6 Iv7 vbVII^m6
Iv7 vbVII^m6 Iv7 /

chorus: vVI^m7 / vVII^d6 / (vVII^d6 = vIII^7b9noR)
/ / vVI^m7 /
/ / vVII^d6 /
vVI^7 vVI^d6 Vv7 /

breakdown: Iv7 vbVII^m6 bVv7 [vbbVII=^#V]^m6
^#IIv7 #IV^m6 ^VIv7 Vv7

verse 2: Iv7 vbVII^m6 Iv7 vbVII^m6
Iv7 vbVII^m6 Iv7 IIv7

bridge: Vv7 Vvdv7 IIv7 /
/ IIvdv7 Vv7 VIvm7
IIIv7 IIIvdv7 VIv7b9 VIv6
/ VIvdv7 IIv7 Vv7

coda: Iv7 / vbVII^m7 /
vbIII^7 / Vv7 /
</tt>

Guitar tab in ^C for a 6-string guitar. Uses [[KDF Fret Numbering|KDF fret numbering]]. Some chords had to be simplified, 8 strings are really needed for this song.

In the chorus, use the passing chords to get this bass line: 
(5 ⸰⸰0) (6 ⸰2) (5 ⸰⸰0) (5 ⸰1) 
(4 ⸰1) (5 ₒₒₒ1) (4 ⸰1) (5 ⸰3) etc.

The bridge has this bass line: 
(5 ⸰1) (6 ₒₒₒ3) (5 ⸰1) (5 ₒₒₒ1) 
(6 ₒₒₒ3) (4 ₒₒₒ2) (6 ₒₒₒ3) (5 ₒₒₒ3) 
(6 ₒₒₒ3) (4 ₒₒₒ2) (6 ₒₒₒ3) (5 ₒₒₒ1) 
(5,4,1) (6 ₒₒₒ3) (4 ₒₒₒ2) (4 ₒₒₒ2) 
(5 ₒₒₒ0) (6 ₒₒ2) (5 ₒₒₒ0) (5 ₒₒ0) etc.

<tt>
intro chords: Vv7 . 17 17 16 14 .
vIII^d^7 18 . 15 16 18 . (2nd time play 18 17 15 16)
walkdown (5,17) (5,13) (6,16) (6,15)

solo chords: Iv7 . 5 . 4 2 6
vbIII^m6 . 5 3 2 2 (4)
IVv7 . . 7 7 6 4

bVIIv7 8 . 7 5 9 .
bVIIv7noR . 8 7 5 9 .
vbIII^m6no5 . . 3 2 6 4 (fingering: 2 1 4 3)

vbVI^ 4 . 3 5 6 . (fingering: 2 1 3 4)
vbVI^9 . 18 19 17 16 16
Vv7 . 17 17 16 14 .

verse chords: Iv7 3 5 5 . 2 . (fingering: 2 3 4 1)
vbVII^m6 3 1 0 0 . . (fingering: 2 1)
vbVII^m6 3 1 0 0 2 1 (fingering: 4 1 3 2) (advanced version)

chorus chords: vVI^m7 18 20 19 . 18 . (see comments above for the bass line)
(passing chord) . 17 19 . 18 . (fingering: 1 3 2)
vVII^d6 . 13 17 16 20 .
(passing chord) . 19 . 16 20 . (fingering: 3 2 4)

vVI^9 . 20 21 19 18 18
vVI^d6 . 20 19 * . . (* riff between 17 and 23)
Vv7 . 17 17 16 14 .

breakdown: Iv7 . 17 * 19 . . (* riff between 15 and 19)
vbVII^m6 . 17 15 14 14 .
bVv7 . 15 15 14 12 .
^#V^m6 . 15 13 12 12 .

^#IIv7 . 11 11 10 8 .
#IV^m6 . 11 9 8 8 .
^VIv7 7 . 6 4 8 .
Vv7 3 . 2 0 4 .

bridge chords: Vv7 15 17 17 (16) 14 . (see comments above for the bass line)
Vvdv7 . 13 15 14 14 . (lacks the root unfortunately)
IIv7 15 . 14 12 16 . (use this for the pickup chord as well)
IIvdv7 . 13 12 12 14 . (lacks the root unfortunately)
VIvm7 . . 14 12 13 11

IIIv7 10 12 12 (11) 9 .
IIIvdv7 . 8 10 9 9 . (lacks the root unfortunately)
VIv7b9 . . 14 14 13 11 (lacks the 9th unfortunately)
VIv6 . . 14 14 16 15 (lacks the 5th unfortunately)
VIvdv7 . . 14 12 11 11
Vv7 . 17 17 16 14 .

coda chords: Iv7 . 5 . 4 2 6
vbVII^m7 . 1 0 0 2 1 (lacks the 7th unfortunately, I ran out of frets)
vbIII^7 . . 3 4 2 1
Vv7 3 . 2 0 4 (3)
</tt>

== Sweet Sweet Harmony (barbershop tag) ==
This is a collaboration with Aaron Wolf. He wrote a short piece for beginners that has lots of chromatic voice movement. Here's the most basic 3-voice version:
{| class="wikitable"
!guitar tab
! colspan="2" |chords
! colspan="2" |chord homonyms
|-
| - - 4 4 3 -
|Iv
|close
| colspan="2" |
|-
| - - 4 2 6 -
|vVI^d
|lo35
|lo IVv7noR
|hi3
|-
| - - 4 3 5 -
|^bVIv
|lo35
| colspan="2" |
|-
| - - 7 6 4 -
|vII^d
|close
|lo bVIIv7noR
|close
|-
| - - 4 4 3 -
|Iv
|close
| colspan="2" |
|}
The high voice neatly divides a vM2 into three equal steps. The middle voice divides an A1 into two equal steps.

The exercise commonly also included a high tonic drone (which adds a 9th to the penultimate chord). The drone can be on the 1st string 5th fret or come from another instrument or from singing.

Together, we turned it into a proper barbershop tag with lyrics and 4 parts. The tenor's P1 is everyone else's P8. All notes are half notes, except the 3 pickup notes ("Gimme that") are eighth notes and the final note is a whole note.
{| class="wikitable"
|+
! rowspan="2" |lyrics
! colspan="4" |vocal parts
! colspan="4" |[[41edo solfege|solfege]]
! colspan="2" |6-string guitar tab in vDb
! colspan="2" |6-string guitar tab in Eb
! colspan="2" rowspan="2" |chords
|-
!bass
!bari
!lead
!tenor
!Bs
!Br
!Ld
!Tn
![[KDF Fret Numbering|KDF frets]]
!frets
![[KDF Fret Numbering|KDF frets]]
!frets
|-
|Gim-
| ----
| ----
|P5
| ----
| ----
| ----
|Sa
| ----
| - - ₒₒ3 - - -
| - - 11 - - -
| - - ₒₒₒ3 - - -
| - - 15 - - -
| ----
| ----
|-
| -me
| ----
| ----
|M6
| ----
| ----
| ----
|La
| ----
| - - - ₒₒ0 - -
| - - - 8 - -
| - - - ₒₒₒ0 - -
| - - - 12 - -
| ----
| ----
|-
|That
| ----
| ----
|M7
| ----
| ----
| ----
|Ta
| ----
| - - ⸰0 - - -
| - - 16 - - -
| - - ⸰⸰0 - - -
| - - 20 - - -
| ----
| ----
|-
|Sweet
|P5
|vm7
| rowspan="5" |P8

|vM3
|Sa
|Tho
| rowspan="5" |↑Da
|Mo
| - - ₒₒ3 ₒₒ1 ₒₒₒ1 0
| - - 11 9 13 0
| - - ₒₒₒ3 ⸰1 ⸰1 0
| - - 15 17 17 0
|Iv7
|hiR3
|-
|Sweet
|P4
|vM6
|vm3
|Fa
|Lo
|No
| - ₒₒₒ2 ₒₒₒ2 ₒₒₒ1 ₒₒ3 -
| - 14 14 13 11 -
| - ⸰2 ⸰2 ⸰1 ₒₒₒ3 -
| - 18 18 17 15 -
|IVv7
|close
|-
|Har-
|^m3
|^m6
|b5
|Nu
|Flu
|Sha
| - ₒₒ3 ₒₒₒ1 ₒₒₒ1 - ₒₒ2
| - 11 13 13 - 10
| - ₒₒₒ3 ⸰1 ⸰1 - ₒₒₒ2
| - 15 17 17 - 14
|^bVIv7
|lo5
|-
| -mo-
|vM2
|vm6
|P4
|Ro
|Flo
|Fa
|ₒₒₒ3 - ₒₒₒ0 ₒₒₒ1 ₒₒₒ3 -
|15 - 12 13 15 -
|⸰3 - ⸰0 ⸰1 ⸰3 -
|19 - 16 17 19 -
|vII^d^7
|hi3
|-
| -ny
|P1
|P5
|vM3
|Da
|Sa
|Mo
|ₒₒₒ0 - ₒₒ3 ₒₒₒ1 ₒₒₒ1 -
|12 - 11 13 13 -
|⸰0 - ₒₒₒ3 ⸰1 ⸰1 -
|16 - 15 17 17 -
|Iv
|add8hi3
|}
The lead posts on "sweet" which turns into "-ny", so the lead ends up singing "swee". The bass uses the 41edo equi-dorian scale (P1 vM2 ^m3 P4 P5 vM6 ^m7 P8) which is reminiscent of [[250/243|Triyo/Porcupine]][7]. An optional 5th vocal part is a sub-bass which goes P8 P4 ^m6 m7 P8 (↑Da Fa Flu Tha ↑Da), with P8 being the same note as the bass's P1. This turns the vII^d^7 hi3 chord into a bVIIv9 hi5 chord.

The guitar tab uses an open string in the 1st chord to get the hiR3 voicing. In the vDb tab, that open string can optionally be added to all the other chords too.

Alternate lyrics: "Love My Kite Guitar, Love My Kite Guitar". The lead's melody is 5 5 6 v7 8 with a rhythm of 3 & 4 & 1. The other parts come in after the 1, with the same half-half-half-half-whole rhythm of 2 & 3 & 4.

The lead could improvise their pickup notes, e.g. "How I long for that sweet" with v7 8 9 8 v7 8 melody, 2 & 3 & 4 1 rhythm, or "Love love love my Kite guitar" with 5 v7 9 v7 v11 v7 8 melody, 1 & 2 & 3 4 1 rhythm.

JI version using [[color notation]]:
{| class="wikitable"
|+
! rowspan="2" |lyrics
! colspan="8" |vocal parts
! colspan="4" |[[41edo solfege]]
! colspan="3" rowspan="2" |chords
! colspan="3" rowspan="2" |chord
homonyms
|-
! colspan="2" |bass
! colspan="2" |bari
! colspan="2" |lead
! colspan="2" |tenor
!Bs
!Br
!Ld
!Tn
|-
|Gim-
| colspan="2" rowspan="3" |
| colspan="2" rowspan="3" |
|P5
|w5
| colspan="2" rowspan="3" |
| ----
| ----
|Sa
| ----
| colspan="3" rowspan="3" |
| colspan="3" rowspan="6" |
|-
| -me
|M6
|w6
| ----
| ----
|La
| ----
|-
|That
|M7
|Lw7
| ----
| ----
|Ta
| ----
|-
|Sweet
|P5
|w5
|vm7
|z7
| rowspan="5" |P8
| rowspan="5" |w8
|vM3
|y3
|Sa
|Tho
| rowspan="5" |↑Da
|Mo
|Iv7
|Ih7
|hiR3
|-
|Sweet
|P4
|w4
|vM6
|y6
|vm3
|z3
|Fa
|Lo
|No
|IVv7
|IVh7
|close
|-
|Har-
|^m3
|g3
|^m6
|g6
|b5
|zg5
|Nu
|Flu
|Sha
|^bVIv7
|gVIh7
|lo5
|-
| -mo-
|vM2
|y2
|vm6
|z6
|P4
|w4
|Ro
|Flo
|Fa
|vII^d^7
|yIIg7(zg5)
|hi3
|bVIIv9noR
|VIIh9noR
|hi5
|-
| -ny
|P1
|w1
|P5
|w5
|vM3
|y3
|Da
|Sa
|Mo
|Iv
|Iy
|add8hi3
| colspan="3" |
|}

== Downminor Nocturne==
Video: [https://youtu.be/hZ55MohHUb4 youtu.be/hZ55MohHUb4] (guitar & violin)

https://youtu.be/iTL_siVUnEs and https://youtu.be/uOsOQUDow6M (guitar & Linnstrument)

Video (backing part only): [http://tallkite.com/KiteGuitar/Downminor%20Etude.m4v TallKite.com/KiteGuitar/Downminor Etude.m4v]

Audio: [https://www.tallkite.com/music/DownminorNocturne.mp3 TallKite.com/music/DownminorNocturne.mp3] (acoustic & electric guitars)

Audio (backing part only): [http://tallkite.com/KiteGuitar/Downminor%20Etude.mp3 TallKite.com/KiteGuitar/Downminor Etude.mp3]

See also [https://www.tallkite.com/music/DownminorNocturne.html TallKite.com/music/DownminorNocturne.html]

Key: A downminor

Guitar part:
<tt>
vbVI^ Ivm vbVI^ Ivm
| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - |
| - - 5 - - - 5 - | - - 5 - - - 5 - | - - 5 - - - 5 - | - - 5 - - - 5 - | (3x)
| - 7 - 7 - 7 - 7 | - 7 - 7 - 7 - - | - 7 - 7 - 7 - 7 | - 7 - 7 - 7 - - |
5 | 6 - - - 6 - - - | 5 - - - - - - 5 | 6 - - - 6 - - - | 5 - - - - - - 5 |
| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - |
| - - - - - - - - | - - - - 6 - - - | - - - - - - - - | - - - - 6 - - - |

IVvm bVIIv IVv7
| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - |
| - - 9 - - - 9 - | - - 9 - - - 9 - | - - 5 - - - 5 - | - - 5 - - - 5 - |
| - 7 - 7 - 7 - 7 | - 10- 10- 10- 10| - 7 - 7 - 7 - 7 | - 7 - 7 - 7 - - |
| 6 - - - 6 - - - |10 - - - 10- - - | 8 - - - - - - - | 8 - - - - - - 5 |
| - - - - - - - - | - - - - - - - - | - - - - 8 - - - | - - - - 8 - - - |
| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - |

IVvm bVIIv Iv Ivd
| - - - - - - - - | - - - - - - - - | - - 6 - - - 6 - | - - 4 - - - 4 - |
| - - 9 - - - 9 - | - - 9 - - - 9 - | - 7 - 7 - 7 - 7 | - 5 - 5 - 5 - 5 |
| - 7 - 7 - 7 - 7 | - 10- 10- 10- 10| 7 - - - 7 - - - | 7 - - - 7 - - - |
| 6 - - - 6 - - - |10 - - - 10- - - | - - - - - - - - | - - - - - - - - |
| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - |
| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - |

vbVII^ vbVII^m Vvd vbIII^9
| - - 2 - - - 2 - | - - 2 - - - 2 - | - - - - - - - - | - - - - 2 - - - |
| - 4 - 4 - 4 - 4 | - 2 - 2 - 2 - - | - - 2 - - - 2 - | - - - 2 - 2 - - |
| 3 - - - 3 - - - | 3 - - - 3 - - 3 | - 3 - 3 - 3 - - | - - 3 - - - 3 - |
| - - - - - - - - | - - - - - - - - | 5 - - - 5 - - 5 | - 5 - - - - - 5 |
| - - - - - - - - | - - - - - - - - | - - - - - - - - | 4 - - - - - - - |
| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - |

vbVI^ Ivm vbVI^ Ivm
| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - |
| - - 5 - - - 5 - | - - 5 - - - 5 - | - - 5 - - - 5 - | - - 5 - - - - - | (2x)
| - 7 - 7 - 7 - 7 | - 7 - 7 - 7 - - | - 7 - 7 - 7 - 7 | - 7 - 7 - - - - |
| 6 - - - 6 - - - | 5 - - - - - - 5 | 6 - - - 6 - - - | 5 - - - - - - - |
| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - |
| - - - - - - - - | - - - - 6 - - - | - - - - - - - - | - - - - 6 - - - |
</tt>

Bass guitar: the tuning is 1 octave below the lowest 5 strings of the guitar. 
The pickup notes are written twice, for convenience.
<tt>
vbVI^ Ivm vbVI^ Ivm
|- - - - - - - -|- - - - - - - -|- - - - - - - -|- - - - - - - -|
|- - - - - - - -|- - - - - - - -|- - - 7 - - - -|- - - - - - - -|
|- - - - - - - -|- - - - - - - -|6 - - - - 6 - -|- - - 5 - - - -|
|- - - - - - - -|- - - - - - - 4|- - - - - - - 4|- - - - - - - 4|
|- - - - - - - -|- - - - - - - -|- - - - - - - -|6 - - - - - - -|

vbVI^ Ivm vbVI^ Ivm
|- - - - - - - -|- - - - - - - -|- - - - - - - -|- - - - - - - -| (2x)
|- - - 7 - - - -|- - - - - - - -|- - - 7 - - - -|- - - - - - - -|
|6 - - - - 6 - -|- - - 5 - - - -|6 - - - - 6 - -|- - - 5 - - - -|
4|- - - - - - - 4|- - - - - - - 4|- - - - - - - 4|- - - - - - - 4|
|- - - - - - - -|6 - - - - - - -|- - - - - - - -|6 - - - - - - -|

IVvm bVIIv IVv7
|- - - - - - - -|- - - - - - - -|- - - - - - - -|- - - - - - - -|
|- - - 7 - - - -|- - - - - - - -|- - - 7 - - - -|- - - - - - - -|
|6 - - - - 6 - -|- - - 10- - - -|8 - - - - 8 - -|- - - - - - - -|
4|- - - - - - - 8|- - - - - - - 8|- - - - - - - 8|- - - 8 - - - 4|
|- - - - - - - -|9 - - - - - - -|- - - - - - - -|6 - - - - - - -|

IVvm bVIIv Iv Ivd
|- - - - - - - -|- - - - - - - -|- - - - - - - -|- - - - - - - -|
|- - - 7 - - - -|- - - - - - - -|- - - 7 - - - -|- - - - - - - -|
|6 - - - - 6 - -|- - - 10- - - -|5 - - - - 5 - -|- - - 3 - - - -|
4|- - - - - - - 8|- - - - - - - -|- - - - - - - 6|- - - - - - - -|
|- - - - - - - -|9 - - - - - - 6|- - - - - - - -|6 - - - - - - 2|

vbVII^ vbVII^m Vvd vbIII^9
|- - - - - - - -|- - - - - - - -|- - - 2 - - - -|- - - - - - - -|
|- - - 3 - - - -|- - - - - - - -|3 - - - - 3 - -|- - - 3 - - - -|
|1 - - - - 1 - -|- - - 1 - - - -|- - - - - - - 5|- - - - - - - -|
|- - - - - - - 3|- - - - - - - 1|- - - - - - - -|4 - - - - - - 4|
2|- - - - - - - -|2 - - - - - - -|- - - - - - - -|- - - - - - - -|

vbVI^ Ivm vbVI^ Ivm
|- - - - - - - -|- - - - - - - -|- - - - - - - -|- - - - - - - -| (2x)
|- - - 7 - - - -|- - - - - - - -|- - - 7 - - - -|- - - - - - - -|
|6 - - - - 6 - -|- - - 5 - - - -|6 - - - - 6 - -|- - - 5 - - - -|
4|- - - - - - - 4|- - - - - - - 4|- - - - - - - 4|- - - - - - - 4|
|- - - - - - - -|6 - - - - - - -|- - - - - - - -|6 - - - - - - -|
</tt>

Bass line as notes (A' means A an octave higher, likewise for G' and vG' and Bb'):
<tt>
vF^ Avm vF^ Avm
vC vF A' vF vC A E

vF^ Avm vF^ Avm
vC vF A' vF vC A E vC vF A' vF vC A E (2x)

Dvm Gv Dv7
vC vF A' vF D vB G' D vF# A' vF# D A D

Dvm Gv Av Avd
vC vF A' vF D vB G' A E A' E vC# A Eb

vG^ vG^m Evd vC^9
vG vD vG' vD B vG vD Bb vG' Bb' vG' E vC vG'

vF^ Avm vF^ Avm
vC vF A' vF vC A E vC vF A' vF vC A E (2x)
</tt>

Chords and melody ("/" means bend up to the note):
<tt>
vF^ Avm vF^ Avm
| . . . . . . . . | . . . . . . . . | . . . . . . . . | . . . . . . . . |

vF^ Avm vF^ Avm
| . . vC D Eb D vC - | /vC - - A - - - - | . . vC D Eb D vC - | A - - E - . E - |

vF^ Avm vF^ Avm
| /vF - - - - . vF - | E - - - - vF E - | Eb - - - Eb - Eb - | D - - - - . . . |

Dvm Gv Dv7
| /vF - D - - - - . | /vF - D - - . vF - | E - - E E vC - A | - - - . . A vC E |

Dvm Gv Av Avd
| /vF - D - - - - . | /vF - D - - . vF - | E - A'- - . E - | vG - A' - - - - . |

vG^ vG^m Evd vC^9
| . . vG B'/vA'vG F - | E - Bb'- . . E vD | vC# - vG - - . /Bb'vG | - - - . /Bb'vG - Bb'|

vF^ Avm vF^ Avm
| A'- vF vC - - - . | vG A' - vG A'- vG - | /A'- vF vC - - A'- | vG - E vC - - - . |

vF^ Avm vF^ Avm
| . . vC D Eb D vC - | /vC - - A - - . . | . . vC D Eb D vC - | A - - - E - - - |
</tt>

== Blues in G==
Loosely based on Lance Vallis' solo guitar piece in G

http://lancevallis.tripod.com/webonmediacontents/Solo%20Blues%20in%20G%20II.pdf

https://www.youtube.com/watch?v=Q7a4yHGSiAI

But I Africanized it by syncopating the bass, and by changing chords early, before the bar line.
And I changed the melodies, because IMO simply transposing everything up a 4th on the IV chord is too predictable.
Warning: the repeat signs do NOT line up with the bar lines! The feel should be swingy, to make the syncopation obvious.

<tt>
& a 4 e & a 1 e & a 2 e & a 3 e
5 vb7 v6 b5 5
5 vb7 1 vb7 5 1 5 1 4 1 vb3 v3
// v3 \\
|| - - - - - - | - - - - - - - - | - - ||
|| - - - - - - | - - - 11* - - - - | - - || 4x
|| - - 13 - - - | - - - 13* - - 16 - |11 13* ||
|| - - - - - - | - - - 14* - - 16 - |12 14* ||
|| - 10 14 10 - - | - - - - - 14 - 14 | - - ||
||12 - - - 12 0 | - - - - - - - - | - - ||
\\ * slide up //

8 vb7 vb6 v6
5 v6 vb3 1 vb6 4 v6 1 5 1 ~2 vb3
// vb3 \\
|| - - - - - - | - - - - - - - - | - - ||
|| - - - - - - | - - - 15* - - 11 - | - - || 2x
|| - - - - - - | - - - 16* - - 13 - |14 16* ||
|| - - 12 - 0 - | - - - 12* - - - - |10 12* || (last 12 could be 16)
|| - - - 14 - 2 | - - - - - 14 - 14 | - - ||
||12 15 - - - - | - - - - - - - - | - - ||
\\ //

back to the 1st riff 2x

9 8 vb7 v7
b2 2 v4 2 vb7 5 v7 1 v6 1 b5 5
// v4 \\
|| - - - - - - | - - - 12* - - - - | - - ||
|| - - - - - - | - - - 13* - - 15 - | - - || 1x
|| - - 9 - - - | - - - 9* - - 16 - |11 13* ||
|| 9 11 - 11 - - | - - - - - - - - |11 13* ||
|| - - - - 10 - | - - - - - 14 - 14 | - - ||
|| - - - - - 12 | - - - - - - - - | - - ||
\\ //

2nd riff 1x

1st riff 2x

</tt>

==The Relative Song ==

So-called because it switches between relative major and minor. No roman numerals, because this song could be in either ^Bb downmajor or ^C upminor. The root movements are almost entirely by 4ths and 5ths. The descending m2 move leaps over to the fifthward side of the lattice, setting up the gradual fourthward progression.

Each column is a 16th note. It has nice long pauses after the chromatic bass runs, plenty of time to get your fingers in place for the strumming. Just for fun, you can start on the ^G chord or even the ^D chord.

<tt>
// ^C^m7 ^F^7 1,2-----\\ 3-------\\
(3x) || - - - - 5 - 5 5 | - 5 5 - - - - - | - - - - 5 - 5 5 | - 5 5 - - - - - || - - - - ||
|| - - - - 3 - 3 3 | - 3 3 - - - - - | - - - - 6 - 6 6 | - 6 6 - - - - - || - - - - ||
|| - - - - 4 - 4 4 | - 4 4 - - - - - | - - - - 8 - 8 8 | - 8 8 - - - - - || - - - - ||
|| - - - - - - - - | - - - - 5 - 6 - | 7 - - - 7 - 7 7 | - 7 7 - - - - - || 7 - - - ||
3 - 4 - || 5 - - - 5 - 5 5 | - 5 5 - - - - - | - - - - - - - - | - - - - 3 - 4 - || - - 8 - ||
|| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - || - - - - ||
\\ // //

// ^Bbv7 ^Ebv7 1,2-----\\ 3-------\\
(3x) || - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - || - - - - ||
|| - - - - 9 - 9 9 | - 9 9 - - - - - | - - - - 7 - 7 7 | - 7 7 - - - - - || - - - - ||
|| - - - - 5 - 5 5 | - 5 5 - - - - - | - - - - 9 - 9 9 | - 9 9 - - - - - || - - - - ||
|| - - - - 7 - 7 7 | - 7 7 - - - - - | - - - -10 -10 10| -10 10- - - - - || - - - - ||
|| - - - - - - - - | - - - - - - 9 - |10 - - -10 -10 10| -10 10- - - - - || 10- - - ||
|| 8 - - - 8 - 8 8 | - 8 8 - 8 - - - | - - - - - - - - | - - - - 10- 9 - || - - - - ||
\\ // //

^D^7 ^Gv7
(once) | - - - - 0 - 0 0 | - 0 0 - 0 - 0 - | - - 3 - 3 - 3 3 | - - - - - - - - |
| - - - - 1 - 1 1 | - 1 1 - 1 - 1 - | - - 4 - 4 - 4 4 | - - - - - - - - |
| - - - - 3 - 3 3 | - 3 3 - 3 - 3 - | - - 0 - 0 - 0 0 | - - - - - - - - |
| 2 - - - 2 - 2 2 | - 2 2 - 2 - 2 - | - - 2 - 2 - 2 2 | - - - - - - - - |
| - - - - 0 - 0 0 | - 0 0 - 0 - 0 - | - - - - - - - - | - - - - 3 - 4 - |
| - - - - - - - - | - - - - - - - - | 3 - 3 - 3 - 3 3 | - - - - - - - - |

</tt>

Chords:
<tt>
^C^m7 ^F^7 (3x)

^Bbv7 ^Ebv7 (3x)

^D^7 ^Gv7

</tt>

==The Tri-sub song==

In 12edo, a tritone substitution changes V7-IM7 to bII7-IM7. In 41edo, the bII can be translated to either ^bII or vbII. This song uses both. I used a ^dv7 chord instead of the more standard ^d7 chord because I liked how it led into the following chord. The final ^9 chord uses a very dissonant high-3 voicing which makes an extreme contrast with the very pleasant vM7 chord that it resolves to.

IvM7 vVI^m9
| - - 2 - 2 2 - - | - 2 - 2 - 2 2 - | - - 2 - 2 2 - - | - 2 - 2 - 2 2 - | (2x)
| - - 2 - 0 2 - - | - 0 - 2 - 0 2 - | - - 2 - 0 2 - - | - 0 - 2 - 0 2 - |
| - - 0 - 0 0 - - | - 0 - 0 - 0 0 - | - - 0 - 0 0 - - | - 0 - 0 - 0 0 - |
| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - |
| 1 - - - - - - 1 | - - - - - - - - | - - - - - - - - | - - - - - - - - |
| - - - - - - - - | - - - - - - - - | 2 - - - - - - 2 | - - - - - - - - |

vII^m9 IVv7 IV^dv7
| - - 4 - 2 4 - - | - 2 - 4 - 2 4 - | - - 0 - 0 0 - - | - 0 - 0 - - 0 - |
| - - 2 - 2 2 - - | - 2 - 2 - 2 2 - | - - 2 - 2 2 - - | - 0 - 0 - - 0 - |
| - - 3 - 3 3 - - | - 3 - 3 - 3 3 - | - - 3 - 3 3 - - | - 2 - 2 - - 2 - |
| - - - - - - - - | - - - - - - - - | 3 - - - - - - 3 | - - - - - 3 - - |
| 4 - - - - - - 4 | - - - - - - - - | - - - - - - - - | - - - - - - - - |
| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - |

^bIIv7 vbII^9
| - - 4 - 4 4 - - | - 0 - 4 - 0 4 - |
| - - 0 - 0 0 - - | - 0 - 0 - 0 0 - |
| - - 2 - 2 2 - - | - 1 - 1 - 1 1 - |
| - - - - - - - - | - - - - - - - - |
| 3 - - - - - - 2 | - - - - - - - - |
| - - - - - - - - | - - - - - - - - |

== The Walking Song==

This song pumps the Zo comma, which maps to an entire fret. Start in Ab (6th string of 6, 1st dot) and slowly march up the fretboard. When you reach the 4th dot, you can continue by jumping down to the 4th string, 3rd fret.

<tt>
Iv7 IVv7 vbIII^m7 ^^Vv7

^^Iv7 ~IVv7 ^bIII^m7 #Vv7

#Iv7 #IVv7 vIII^m7 ^^#Vv7

etc.

</tt>

Use this bass line to connect the chord roots. Each phrase has 2 pickup notes. Play the full chords during the "...".
<tt>
5 v3 1 ... 1 vb3 4 ... 4 v3 vb3 ... vb3 b5 ^^5 ... ^^5 ^3 ^^1 ...

</tt>

Some people simply don't hear the rapid upwards tonic drift until it's pointed out to them. Even after becoming aware of it, it still feels incredibly jarring to go from the ^^V chord to a I chord. The ear insists on hearing the ^^V chord as a V chord and not a vbVI chord, and it insists on the ^^I chord coming next.

The vocals are very close to the bass line:

Just a four-chord song (5 5 5 v3 1) (rhythm: 6 & 7 8 1) 
Won't you sing along (1 1 1 vb3 4) (6 & 7 8 1) 
But before too long (4 4 4 v3 vb3) (6 & 7 8 1) 
You'll hear that something's wro-ong (vb3 b5 vb3 vb7 vb7 vb7 ^^5) (& 6 & 7 8 1 &) 
With my four-chord song (^^5 ^^5 ^^5 ^3 ^^1) (6 & 7 8 1)

==II-V-I riff in A downmajor ==

Play with a Bossa Nova feel. When translating IIm7 - V7 - IM7, I don't like to use a Vv7 chord, because it sounds to me more resolved than the IvM7. The I chord doesn't feel like the final chord, and the progression feels incomplete. I much prefer V^7 when resolving to IvM7.

<tt>
II^m7 V^7 IvM7 Iv^7 Iv7
| - - 3 - - 3 - - | - 3 - - 3 - 3 - | - - 6 - 6 6 - 6 | - 6 - - 6 - 6 - |
| - - 1 - - 1 - - | - 4 - - 4 - 4 - | - - 7 - 7 7 - 7 | - 7 - - 7 - 7 - |
| - - 2 - - 2 - - | - 6 - - 6 - 6 - | - - 5 - 5 5 - 4 | - 4 - - 3 - 3 - |
| - - - - - - - 5 | - - - 5 - - - - | - - - - - - - - | - - - - - - - - |
| 3 - - - 3 - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - |
| - - - - - - - - | - - - - - - - - | 6 - - - - - - - | - - - 6 - - - - |

</tt>

==Latrizo Pump riff==

The Latrizo comma divides the 5th into three up-2nds. This riff walks up to the 5th and leaps back down. This song can't be translated to 12edo, as this video shows:

https://www.youtube.com/watch?v=x9K9HTHcRl8<nowiki/>

<tt>
Iv7 ^IIv7 vIVv7 Vv7

</tt>

Use this bass line to connect the chord roots. Play the full chords during the "...".
<tt>
1 ... 1 ^b2 ^2 ... ^2 ~3 v4 ... v4 b5 5 ... 5 --> 1 (slide down)

</tt>

==6-chord riff==

One of the first songs I wrote on the Kite guitar.

<tt>
Iv7 #Ivdv7 vVI^m7
| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - |
| 1 - - 1 - - - - | 3 - - 3 - - - - | 5 - - 5 - - - - | 5 - - 5 - - - - |
| - - 3 - - 3 - - | - - 3 - - 3 - - | - - 3 - - 3 - - | - - 3 - - 3 - - |
| - 4 - - - - 4 - | - 4 - - - - 4 - | - 4 - - - - 4 - | - 4 - - - - 4 - |
| 4 - - 4 - - - - | 6 - - 6 - - - - | - - - - - - - - | - - - - - - - - |
| - - - - - - - - | - - - - - - - - | 5 - - 5 - - - - | 5 - - 5 - - - - |

vII^7 vII^9 IVv7 V^9
| - - - - - - - - | - - - - 5 - - - | 3 - - 3 - - - - | - - - - - - - - |
| 5 - - 5 - - - - | 5 - - 5 - - - - | - - 5 - - 5 - - | 0 - - - 0 - - - |
| - - 6 - - 6 - - | - - 6 - - - - - | - 6 - - - - 6 - | - - - 0 - - - - |
| - 8 - - - - 8 - | - 8 - - - - - - | 6 - - 6 - - - - | - - 1 - - - - - |
| 7 - - 7 - - - - | 7 - - - - - - - | - - - - - - - - | - 3 - - - - - - |
| - - - - - - - - | - - - - - - - - | - - - - - - - - | 2 - - - - - - - |

</tt>

== Yin Yang riff ==

A nice demo of the two kinds of dom7 chords. Harmonic/subharmonic aka otonal/utonal are sort of like yin/yang. Or is that yang/yin?
<tt>

https://www.youtube.com/watch?v=p3ktvcCrSwc
| - - - - - - - 0 | - - - - - - - - | - - - - - - 1 5 | - 1 - - - - - - |
| - - - - - - 0 - | - 0 - - - - - - | - - - - 3 - - - | - - - 3 - - - - |
| - - - - 1 - - - | - - - 1 - - - - | - - 4 - - - - - | - - - - 4 - - - |
| - - 3 - - - - - | - - - - 3 - - - | 4 - - - - - - - | - - - - - - 4 - |
| 2 - - - - - - - | - - - - - - 2 - | - - - - - - - - | - - - - - - - - |
| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - |
</tt>
Which chord feels like the tonic chord to you? Is it I^9 - IVv7, or V^9 - Iv7?

==Im7 - V7 riff==

The Vv7 chord is fingered either xx4321 or x2x431. In other words, when the middle finger is playing the low bass note, the ring finger has to "cover" for it.
<tt>
I^m7 Vv7
5 | - - 5 - - - 5 - | - 5 - 5 - - - 1 | - - 1 - - - 1 - | - 1 - 1 - - - 1 |
5 | - - 5 - - - 5 - | - 5 - 5 - - - 3 | - - 3 - - - 3 - | - 3 - 3 - - - 3 |
3 | - - 3 - - - 3 - | - 3 - 3 - - - 4 | - - 4 - - - 4 - | - 4 - 4 - - - 4 |
4 | - - - - - 4 - - | - - - - - - 4 - | - - - 0 - 4 - - | - - - - 0 - 4 - |
| - - - 4 - - - - | - - - - 4 - - - | 2 - - - - - - - | 2 - - - - - - - |
| 5 - - - - - - - | 5 - - - - - - - | - - - - - - - - | - - - - - - - - |

Vv7 I^m7
| - - 1 - - - 1 - | - 1 - 1 - - - 5 | - - 5 - - - 5 - | - 5 - 5 - - - 5 |
| - - 3 - - - 3 - | - 3 - 3 - - - 5 | - - 5 - - - 5 - | - 5 - 5 - - - 5 |
| - - 4 - - - 4 - | - 4 - 4 - - - 3 | - - 3 - - - 3 - | - 3 - 3 - - - 3 |
| - - - 0 - 4 - - | - - - - 0 - 4 - | - - - - - 4 - - | - - - - - - 4 - |
| 2 - - - - - - - | 2 - - - - - - - | - - - 4 - - - - | - - - - 4 - - - |
| - - - - - - - - | - - - - - - - - | 5 - - - - - - - | 5 - - - - - - - |

</tt>

== Fun riff ==
Another one I wrote early on.
<tt>
// Dv7 1---------------\\ 2---------------\\
|| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - || - - - - - - - - ||
(2x) || - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - || - - - - - - - - ||
|| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - || - - - - - - - - ||
|| - - 8 - - - - - | - 6 - 8 6 - - - | - - 8 - - - - 8 | - - 6 - 5 - 4 - || - - - - - - - - ||
|| 8 - - - - - 4 8 | - - - - - - 4 - | 8 - - - - - 4 - | - - - - - - - - || - - 7 - 6 - 5 - ||
|| - - - - 6 - - - | - - - - - - - - | - - - - 6 - - - | - - - - - - - - || - - - - - - - - ||
\\ // //

// vC^m6 1---------------\\ 2---------------\\
|| - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - || - - - - - - - - ||
(2x) || - - - - - - - - | - - - - - - - - | - - - - - - - - | - - - - - - - - || - - - - - - - - ||
|| - - - - - - - 3 | - - - - - - - - | - - - - - - - 3 | - - - - - - - - || - - - - - - - - ||
|| - - 3 - - - 6 - | - 6 - 3 - - - - | - - 3 - - - 6 - | - - - - - - - - || - - - - - - - - ||
|| 4 - - - - - - - | - - - - 4 - - - | 4 - - - - - - - | - - 1 - 2 - 3 - || - - - - 6 - - - ||
|| - - - - 6 - - - | - - - - - - 6 - | - - - - 6 - - - | - - - - - - - - || - - - - - - - - ||
\\ // //
</tt>

== Cloud Chant ==
This one I wrote as a 7-limit JI vocal piece in 2014, long before the Kite Guitar. But it goes on guitar very well, if you use your thumb for the bass notes.
https://tallkite.bandcamp.com/track/cloud-chant

7-string tab for hi-7 tuning, in ^^C to match the bandcamp recording:
<tt>
Iv ^IIv7 ^IIvdv7 Iv7 Iv7 ^IIv7 ^IIvdv7 Iv
| 14 - 14 - | 14 - 10 - | 14 10 14 - | 14 - 14 - |
| 12 - - - | - - - - | 12 - - - | - - - - |
| 13 - 17 - | 15 - 13 - | 13 - 17 - | 15 - 13 - |
| 13 - 17 - | 13 - 13 - | 13 - 17 - | 13 - 13 - |
| - - 15 - | 13 - 11 - | - - 15 - | 13 - 11 - |
| - - - - | - - - - | - - - - | - - - - |
| 12 - 16 - | 16 - 12 - | 12 - 16 - | 16 - 12 - |
</tt>

==Mid-9th riff ==

Is it Iv - Vvm(~5), or just one big v7~9 chord? Who cares, it's the blues -- play it slow and bendy.

<tt>
| - - - 9 - - | 5 - 5 - - - | - - - 9 - - | 5 - - - - - |
| - - - 9 - - | 5 - 5 - - - | - - - 9 - - | 5 - - - - - |
| - - 7 - - 7 | - - - - - - | - - 7 - - 7 | - - - - - - |
| - - - - - - | - - - - - - | - - - - - - | - - - 6 7 8 | (slide up)
| 8 - - - - - | - - - - - - | 8 - - - - - | - - - - - - |
| - - - - - - | - - - - - - | - - - - - - | - - - - - - |

| - - - 9 - - | 5 - 5 - - - | - - - 9 - - | 5 - 5 - - - |
| - - - 9 - - | 5 - 5 - - - | - - - 9 - - | 5 - - 5 - - |
| - - 7 - - 7 | - - - - - - | - - 7 - - 7 | - - - - 7 - |
| - - - - - - | - - - - - - | - - - - - - | - - - - - 8 |
| 8 - - - - - | - - - - - - | 8 - - - - - | - - - - - - |
| - - - - - - | - - - - - - | - - - - - - | - - - - - - |

| - - - 9 - - | 5 - 5 - - - | - - - 9 - 5 | - - - - - - |
| - - - 9 - - | 5 - 5 - - - | - - - 9 - 5 | - - - - - - |
| - - 7 - - 7 | - - - - - - | - - 7 - - - | - - - - - - |
| - - - - - - | - - - - - - | - - - - - - | - - - 4 8 - | (slide up)
| 8 - - - - - | - - - - - - | 8 - - - - - | - - - 4 8 - |
| - - - - - - | - - - - - - | - - - - - - | - - - - - - |

| - - - 9 - - | 5 - 5 - - - | - - - - - - | - - - - - - |
| - - - 9 - - | 5 - 5 - - - | - - - - - - | - - - - - - |
| - - 7 - - 7 | - - - - - - | 5 - 5 - - 7 | - - - - - - |
| - - - - - - | - - - - - - | 6 - 6 - - 8 | - - - - - - |
| 8 - - - - - | - - - - - - | - - - - - - | - - - - - - |
| - - - - - - | - - - - - - | - - - - - - | - - - - - - |

etc.

</tt>

== James-Bond-ish riff ==

The cadence from the last chord back to the first takes a 12edo cliche -- a dim 5th resolving inward to a major 3rd -- and makes it new by having an updim 5th resolve to an upmajor 3rd.

<tt>
6 _ _ 6 _ _ _ _ | 6 _ _ 6 _ _ _ _ | 6 _ _ 6 _ _ _ _ | 6 _ _ 6 _ _ _ _
4 _ _ 4 _ _ _ _ | 6 _ _ 6 _ _ _ _ | 7 _ _ 7 _ _ _ _ | 6 _ _ 6 _ _ _ _
_ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _

6 _ _ 6 _ _ _ _ | 6 _ _ 6 _ _ _ _ | 6 _ _ 6 _ _ _ _ | 6 _ _ 6 _ _ _ _
4 _ _ 4 _ _ _ _ | 6 _ _ 6 _ _ _ _ | 7 _ _ 7 _ _ _ _ | 6 _ _ 6 _ _ _ _
3 _ _ 3 _ _ _ _ | 4 _ _ 4 _ _ _ _ | 7 _ _ 7 _ _ _ _ | 2 _ _ 2 _ _ _ _
_ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _

Ivm7 ^bVIv7 IVv7 V4^b9
6 _ _ 6 _ _ _ _ | 6 _ _ 6 _ _ _ _ | 6 _ _ 6 _ _ _ _ | 6 _ _ 6 _ _ _ _
4 _ _ 4 _ _ _ _ | 6 _ _ 6 _ _ _ _ | 7 _ _ 7 _ _ _ _ | 6 _ _ 6 _ _ _ _
3 _ _ 3 _ _ _ _ | 4 _ _ 4 _ _ _ _ | 7 _ _ 7 _ _ _ _ | 2 _ _ 2 _ _ _ _
_ _ 5 _ _ _ 5 _ | _ _ 5 _ _ _ 5 _ | _ _ 5 _ _ _ 5 _ | _ _ 5 _ _ _ 5 _
_ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _
</tt>

== Downminor riff ==

Beginning version. The fingering cues are low to high, so the initial "32" means 3rd finger on 6th string and 2nd finger on 2nd string. Note how the 4th string note is played in turn by both the 1st and 2nd fingers.
<tt>
32 2 31 1 etc.
_ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ 0 _ | _ _ _ _ _ _ _ _
3 _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | 3 _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _
_ _ _ 5 _ _ _ _ | 1 _ _ 5 _ _ _ _ | _ _ _ 5 _ _ _ _ | 1 _ _ 5 _ _ _ _
_ _ 3 _ _ _ 3 _ | _ _ 3 _ _ _ 3 _ | _ _ 3 _ _ _ 3 _ | _ _ 3 _ _ _ 3 _
_ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _
4 _ _ _ 4 _ _ _ | 4 _ _ _ 4 _ _ _ | 4 _ _ _ 4 _ _ _ | 4 _ _ _ 4 _ _ _
</tt>

Advanced version:
<tt>
_ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ 0 _ | _ _ _ _ _ _ _ _
3 _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | 3 _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _
_ _ _ 5 _ _ _ _ | 1 _ _ 5 _ _ _ _ | _ _ _ 5 _ _ _ _ | 1 _ _ 5 _ _ _ _
_ _ 3 _ _ _ 3 _ | _ _ 3 _ _ _ _ _ | _ _ 3 _ _ _ 3 _ | _ _ 3 _ _ _ _ _
_ _ _ _ _ _ _ _ | _ _ _ _ _ _ 2 _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ 2 _
4 _ _ _ 0 _ _ _ | 4 _ _ _ 0 _ _ _ | 4 _ _ _ 0 _ _ _ | 4 _ _ _ 0 _ _ _
</tt>

== Harmonic Blues ==
https://www.youtube.com/watch?v=pix_DruZZM4

No score yet, unfinished piece.

== Tiny Steps ==

The opposite of Coltrane's "Giant Steps"! A simple riff that modulates through 4 keys and demonstrates the freedom of an equal temperament.

The key goes up by smaller intervals each time. The bass line changes key a bit earlier than the high line does.

Play it slow and staccato, with lots of swing. "1--3", "9--11", etc. all indicate an optional slide up 2 frets.

Video: https://www.youtube.com/watch?v=s0QUqWhFMks <tt>
in B | _ _ 4 _ 1 _ _ _ | _ _ 2 _ _ _ _ _ | _ _ 4 _ 1 _ _ _ | _ _ 2 _ 1 _ 0 _ |
| _ _ 0 _ 0 _ 0 4 | _ _ _ _ 4 _ 0 _ | 4 _ 0 _ 0 _ 0 4 | _ _ _ _ _ _ _ _ |
_ _ 2 | _ _ _ _ _ _ 2 _ | _ _ 5 _ 5 _ _ 2 | _ _ _ _ _ _ 2 _ | _ 2 _ 2 _ _ _ _ |
1--3 _ | _ _ _ _ _ 3 _ _ | 5 _ _ _ _ 1--3 _ | _ _ _ _ _ 3 _ _ | _ _ _ _ _ _ _ 10 |
| 3 _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | 3 _ _ _ _ _ _ _ | 3 _ _ _ _ 9--11 _ |
| _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ |

in C | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ |
| _ _ 12 _ 9 _ _ _ | _ _ 10 _ _ _ _ _ | _ _ 12 _ 9 _ _ _ | _ _ 10 _ 9 _ 8 _ |
|12 _ 8 _ 8 _ 8 12 | _ _ _ _ 12 _ 8 _ |12 _ 8 _ 8 _ 8 12 | _ _ _ _ _ _ _ _ |
| _ _ _ _ _ _ 10 _ | _ _ 13 _ 13 _ _ 10 | _ _ _ _ _ _ 10 _ | _ 10 _ 10 _ _ _ 11 |
| _ _ _ _ _ 11 _ _ |13 _ _ _ _ 9--11 _ | _ _ _ _ _ 11 _ _ | _ _ _ _ _ 10--12 _ |
|11 _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ |11 _ _ _ _ _ _ _ |11 _ _ _ _ _ _ _ |

in ^^C | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ |
or vDb | _ _ 13 _ 10 _ _ _ | _ _ 11 _ _ _ _ _ | _ _ 13 _ 10 _ _ _ | _ _ 11 _ 10 _ 9 _ |
|13 _ 9 _ 9 _ 9 13 | _ _ _ _ 13 _ 9 _ |13 _ 9 _ 9 _ 9 13 | _ _ _ _ _ _ _ 5 |
| _ _ _ _ _ _ 11 _ | _ _ 14 _ 14 _ _ 11 | _ _ _ _ _ _ 11 _ | _ 11 _ 11 _ 4--6 _ |
| _ _ _ _ _ 12 _ _ |14 _ _ _ _ 10--12 _ | _ _ _ _ _ 12 _ _ | _ _ _ _ _ _ _ _ |
|12 _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ |12 _ _ _ _ _ _ _ |12 _ _ _ _ _ _ _ |

in Db | _ _ 7 _ 4 _ _ _ | _ _ 5 _ _ _ _ _ | _ _ _ _ 7 _ _ _ | 4 _ _ _ _ _ _ _ |
| 7 _ 3 _ 3 _ 3 7 | _ _ _ _ 7 _ 3 _ | 7 _ _ _ 3 _ _ _ | 3 _ _ _ _ _ _ _ |
| _ _ _ _ _ _ 5 _ | _ _ 8 _ 8 _ _ 5 | _ _ _ _ 5 _ _ _ | 5 _ _ _ _ _ _ _ |
| _ _ _ _ _ 6 _ _ | 8 _ _ _ _ 4--6 _ | _ _ _ _ _ _ _ _ | 6 _ _ _ _ _ _ _ |
| 6 _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | 6 _ _ _ 6 _ _ _ | - _ _ _ _ _ _ _ |
| _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ | _ _ _ _ _ _ _ _ |
</tt>

[[Category:Kite Guitar]]

File:Alt-pergenLister 15edo.png

2026-05-26T01:55:58Z

TallKite: TallKite uploaded a new version of File:Alt-pergenLister 15edo.png

== Summary ==
Imported file from Wikispaces. File was uploaded on 2018-03-25 03:46:00 by TallKite, and is 109295 bytes.

File:Alt-pergenLister 15edo.png

2026-05-26T01:54:24Z

TallKite: TallKite uploaded a new version of File:Alt-pergenLister 15edo.png

== Summary ==
Imported file from Wikispaces. File was uploaded on 2018-03-25 03:46:00 by TallKite, and is 109295 bytes.

Kite's thoughts on pergens

2026-05-26T01:53:53Z

TallKite: /* PergenLister */ update the screenshots

A '''pergen''' (pronounced "peer-jen", from '''per'''iod and '''gen'''erator) is a way of classifying a [[regular temperament]] solely by its [[periods and generators|period and generator(s)]]. For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. Every rank-2, rank-3, rank-4, etc. temperament has a pergen. Assuming the prime [[subgroup]] includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a fifth or some fraction of some 3-limit interval. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every [[strong extension]] of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but do not uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.

Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka Triyoti]] has the 5th sharpened by one-fifth of [[250/243]] ({{monzo| 1 -5 3 }}). Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Triguti tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name. (Note these uses of comma fractions are not convention universally; temperament pages tend to use comma fractions to indicate the damage to the generator rather than the fifth. This is analogous to indicating the amount of stretching of an edo by the damage to the edostep rather than to the octave. But the latter approach is much more common, because the damage to the octave is much more audible and thus much more musically relevant than the damage to an edostep, which often doesn't correspond to a simple ratio. Likewise for pergens: the damage to Triyoti/Porcupine's generator, which is both 10/9 and 11/10, is less relevant than the damage to Triyoti's 4th or 5th.)

Overwhelmed? See [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf ''Notation guide for rank-2 pergens''] for practical notation examples.

{{See also| Rank-2 temperaments by mapping of 3 }}

= Definition =
If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. Both fractions are always of the form 1/N, thus the octave and/or the 3-limit interval is '''split''' into N parts. The interval which is split into multiple generators is the '''multigen'''. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.

For example, the Srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. Dicot aka Yoyoti (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine aka Triyoti is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore aka Zozoti, a pun on "semi-fourth", is of course half-fourth.

Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both Srutal aka Saguguti and Injera aka Gu & Biruyoti sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using '''ups and downs''' (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and downs notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.

The largest category contains all single-comma rank-2 temperaments with a comma of the form 2x 3y P or 2x 3y P-1, where P is a prime > 3 (a '''higher prime'''), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called '''unsplit'''.

Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multigen, and as a tie-breaker, to minimize the size in cents of the multigen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.

For example, Srutal could be (P8/2, M2/2), but P5 is preferred because it is unsplit. Or it could be (P8/2, P12), but P5 is preferred because it is smaller. Or it could be (P8/2, P4), but P5 is always preferred over P4. Note that P5/2 is not preferred over P4/2. For example, Decimal is (P8/2, P4/2), not (P8/2, P5/2).

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! colspan="4" | example temperaments
|-
! | written
! | spoken
! | comma(s)
! | name
! colspan="2" | [[Color notation|color name]]
|-
| | (P8, P5)
| | unsplit
| | 81/80
| | Meantone
| | Guti
| | gT
|-
| | "
| | "
| | 64/63
| | Archy
| | Ruti
| | rT
|-
| | "
| | "
| | (-14,8,1)
| | Schismic
| | Layoti
| | LyT
|-
| | (P8/2, P5)
| | half-8ve
| | (11, -4, -2)
| | Srutal
| | Saguguti
| | sggT
|-
| | "
| | "
| | 81/80, 50/49
| | Injera
| | Gu & Biruyoti
| | g&rryyT
|-
| | (P8, P5/2)
| | half-5th
| | 25/24
| | Dicot
| | Yoyoti
| | yyT
|-
| | "
| | "
| | (-1,5,0,0,-2)
| | Mohajira
| | Luluti
| | 1uuT
|-
| | (P8, P4/2)
| | half-4th
| | 49/48
| | Semaphore
| | Zozoti
| | zzT
|-
| | (P8/2, P4/2)
| | half-everything
| | 25/24, 49/48
| | Decimal
| | Yoyo & Zozoti
| | yy&zzT
|-
| | (P8, P4/3)
| | third-4th
| | 250/243
| | Porcupine
| | Triyoti
| | y3T
|-
| | (P8, P11/3)
| | third-11th
| | (12,-1,0,0,-3)
| | Satrilu
| | Satriluti
| | s1u3T
|-
| | (P8/4, P5)
| | quarter-8ve
| | (3,4,-4)
| | Diminished
| | Quadguti
| | g4T
|-
| | (P8/2, M2/4)
| | half-8ve quarter-tone
| | (-17,2,0,0,4)
| | Laquadlo
| | Laquadloti
| | L1o4T
|-
| | (P8, P12/5)
| | fifth-12th
| | (-10,-1,5)
| | Magic
| | Laquinyoti
| | Ly5T
|}

(P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum/difference of octaves and fifths, every single 3-limit interval is also split in half.

The multigen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: bi- splits something into two parts, tri- into three parts, etc. For a comma with monzo (a,b,c,d...), the '''color depth''' is GCD (c,d...).

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[Kite's_color_notation|Color notation]] (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yo, (a,b,-1) = gu, (a,b,0,1) = zo, (a,b,0,-1) = ru, (a,b,0,0,1) = lo/ilo, (a,b,0,0,-1) = lu, and (a,b) = wa. Examples: 5/4 = y3 = yo 3rd, 7/5 = zg5 = zogu 5th, etc.

For example, Marvel aka Ruyoyoti (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). But colors can be replaced with ups and downs, to be higher-prime-agnostic. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.

More examples: Trizoguti (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Biruyoti (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, Biruyoti Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.

A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with microtonal accidentals: (P8, P5, ^1, /1,...).

=Derivation=

For any comma, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents, where GCD (0,x) = x. The comma will split the octave into m parts, and if n > m, it will split some 3-limit interval into n parts.

In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is '''dependent''' on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two '''independent''' primes.

For example, consider Sawa & Ruyoyoti (2.3.5.7 with commas 256/243 and 225/224). The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Notating multi-EDO pergens|multi-EDO pergens]] below).

To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [http://x31eq.com/temper/uv.html x31eq.com/temper/uv.html] that will find such a matrix, it's the reduced mapping. Next make a '''square mapping''' by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.

For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.

2/1 = P8 = x·P, thus P = P8/x

3/1 = P12 = y·P + z·G, thus G = [P12 - y·(P8/x)] / z = [-y·P8 + x·P12] / xz = (-y, x) / xz

M's 3-limit monzo is (-y, x), or (y, -x) if z is negative. To get alternate generators, add i periods to G, with i ranging from -x (subtracting a full octave) to +x (adding a full octave).

G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz

'''The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| <= x'''

A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen.

For example, Porcupine aka Triyoti (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 <= i <= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).

Rank-3 pergens are trickier to find. For example, Breedsmic aka Bizozoguti is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7 x31.com] gives us this matrix:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 5/1
! | 7/1
|-
! | period
| | 1
| | 1
| | 1
| | 2
|-
! | gen1
| | 0
| | 2
| | 1
| | 1
|-
! | gen2
| | 0
| | 0
| | 2
| | 1
|}

Thus 2/1 = P, 3/1 = P + 2·G1, 5/1 = P + G1 + 2·G2, and 7/1 = 2·P + G1 + G2. Discard the last column to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 5/1
|-
! | period
| | 1
| | 1
| | 1
|-
! | gen1
| | 0
| | 2
| | 1
|-
! | gen2
| | 0
| | 0
| | 2
|}

Use an [http://wims.unice.fr/wims/wims.cgi?session=GF84B8C7BF.1&lang=en&cmd=reply&module=tool%2Flinear%2Fmatmult.en&matA=1+1+1%0D%0A0+2+1%0D%0A0+0+2&matB=&show=A%5E-1 online tool] to invert it. Here "/4" means that each entry is to be divided by the determinant of the last matrix, which is 4.

{| class="wikitable" style="text-align:center;"
|-
! |
! | period
! | gen1
! | gen2
! |
|-
! | 2/1
| | 4
| | -2
| | -1
| |
|-
! | 3/1
| | 0
| | 2
| | -1
| |
|-
! | 5/1
| | 0
| | 0
| | 2
| | /4
|}

Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25:6)/4.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a double octave to the multigen. The alternate gens are P11/2 and P19/2, both of which are much larger, so the best gen1 is P5/2.

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96:25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128:75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675:512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.

Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 7/1
|-
! | period
| | 1
| | 1
| | 2
|-
! | gen1
| | 0
| | 2
| | 1
|-
! | gen2
| | 0
| | 0
| | 1
|}

This inverts to this matrix:

{| class="wikitable" style="text-align:center;"
|-
! |
! | period
! | gen1
! | gen2
! |
|-
! | 2/1
| | 2
| | -1
| | -3
| |
|-
! | 3/1
| | 0
| | 1
| | -1
| |
|-
! | 7/1
| | 0
| | 0
| | 2
| | /2
|}

Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, /1) with /1 = 64/63, rank-3 half-5th. This is far better than (P8, P5/2, (96:25)/4).

The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is > 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible. Using 7 instead of 5 in the pergen is very common for rank-3. See [[User:TallKite/Catalog of eleven-limit rank three temperaments with Color names]] for more examples.

=Applications=

Pergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources.

Another obvious application is to categorize regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), Semihemi is (P8/2, P4/2), Triforce is (P8/3, P4/2), both Tetracot and Semihemififths are (P8, P5/4), Fourfives is (P8/4, P5/5), Pental is (P8/5, P5), and Fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, either because it contains more than 2 primes, or because the split multigen isn't actually a generator. For example, Sensei, or Semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.

Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example Porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first Porcupine is Triyoti, and the second one is Triyo & Ruti. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.

Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See [[pergen#Further Discussion-Chord names and scale names|Chord names and scale names]] below.

The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.

All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and downs notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Luluti and Dicot aka Yoyoti are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.

One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, which is octave-equivalent, fifth-generated and heptatonic. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and all notation used here is backwards compatible.

Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, Schismic aka Layoti is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C vE G. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below.

Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multigen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a [[pergen#Further Discussion-Supplemental materials-Notaion guide PDF|notation guide]]. It even allows every pergen to be numbered.

The '''[[enharmonic unison]]''', or more briefly the '''EU''', can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's EU. The pergen and the EU together define the notation. (''Edited to add: not quite accurate, see the Addenda.'')

The '''genchain''' (chain of generators) in the table is only a short section of the full genchain.

C - G implies ...Eb Bb F C G D A E B F# C#...C - ^Eb=vE - G implies ...F -- ^Ab=vA -- C -- ^Eb=vE -- G -- ^Bb=vB -- D...

If the octave is split, the table has a '''perchain''' ("peer-chain", chain of periods) that shows the octave: C -- vF#=^Gb -- C. Genchains have a theoretically infinite length, but perchains have a finite length. The full rank-2 lattice has genchains running horizontally and perchains running vertically.

{| class="wikitable" style="text-align:center;"
|-
! |
! | pergen
! | enharmonic
unison(s)
! | equivalence(s)
! | split
interval(s)
! | perchain(s) and/or
genchains(s)
! | examples
|-
| | 1
| | (P8, P5)
unsplit
| | none
| | none
| | none
| | C - G
| | Pythagorean, Meantone, Dominant,
Schismic, Mavila, Archy, etc.
|-
! |
! | half-splits
! |
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5)
half-8ve
| | ^^d2 (if 5th
> 700¢
| | ^^C = B#
| | P8/2 = vA4 = ^d5
| | C - vF#=^Gb - C
| | Srutal aka Saguguti
^1 = 81/80
|-
| |
| | "
| | vvd2 (if 5th

< 700¢)
| | ^^C = Db
| | P8/2 = ^A4 = vd5
| | C - ^F#=vGb - C
| | Injera aka Gu & Biruyoti

^1 = 64/63
|-
| |
| | "
| | vvM2
| | ^^C = D
| | P8/2 = ^4 = v5
| | C - ^F=vG - C
| | Thothoti, if 13/8 = M6

^1 = 27/26
|-
| | 3
| | (P8, P4/2)

half-4th
| | vvm2
| | ^^C = Db
| | P4/2 = ^M2 = vm3
| | C - ^D=vEb - F
| | Semaphore aka Zozoti

^1 = 64/63
|-
| |
| | "
| | ^^dd2
| | ^^C = B##
| | P4/2 = vA2 = ^d3
| | C - vD#=^Ebb - F
| | Lala-yoyoti

^1 = 81/80
|-
| | 4
| | (P8, P5/2)

half-5th
| | vvA1
| | ^^C = C#
| | P5/2 = ^m3 = vM3
| | C - ^Eb=vE - G
| | Mohajira aka Luluti

^1 = 33/32
|-
| | 5
| | (P8/2, P4/2)

half-

everything
| | \\m2,

vvA1,

^^\\d2,

vv\\M2
| | //C = Db

^^C = C#

^^//C = D
| | P4/2 = /M2 = \m3

P5/2 = ^m3 = vM3

P8/2 = v/A4 = ^\d5

= ^/4 = v\5
| | C - /D=\Eb - F,

C - ^Eb=vE - G,

C - v/F#=^\Gb - C,

C - ^/F=v\G - C
| | Zozo & Luluti

^1 = 33/32

/1 = 64/63
|-
| |
| | "
| | ^^d2,

\\m2,

vv\\A1
| | ^^ C= B#

//C = Db

^^//C = C#
| | P8/2 = vA4 = ^d5

P4/2 = /M2 = \m3

P5/2 = ^/m3 = v\M3
| | C - vF#=^Gb - C,

C - /D=\Eb - F,

C - ^/Eb=v\E - G
| | Sagugu & Zozoti

^1 = 81/80

/1 = 64/63
|-
| |
| | "
| | ^^d2,

\\A1,

^^\\m2
| | ^^C = B#

//C = C#

^^\\C = B
| | P8/2 = vA4 = ^d5

P5/2 = /m3 = \M3

P4/2 =v/M2 = ^\m3
| | C - vF#=^Gb - C,

C - /Eb=\E - G,

C - v/D=^\Eb - F
| | Sagugu & Luluti

^1 = 81/80

/1 = 33/32
|-
! |
! | third-splits
! |
! |
! |
! |
! |
|-
| | 6
| | (P8/3, P5)

third-8ve
| | ^3d2
| | ^3C = B#
| | P8/3 = vM3 = ^^d4
| | C - vE - ^Ab - C
| | Augmented aka Triguti

^1 = 81/80
|-
| | 7
| | (P8, P4/3)

third-4th
| | v3A1
| | ^3C = C#
| | P4/3 = vM2 = ^^m2
| | C - vD - ^Eb - F
| | Porcupine aka Triyoti

^1 = 81/80
|-
| | 8
| | (P8, P5/3)

third-5th
| | v3m2
| | ^3C = Db
| | P5/3 = ^M2 = vvm3
| | C - ^D - vF - G
| | Slendric aka Latrizoti

^1 = 64/63
|-
| | 9
| | (P8, P11/3)

third-11th
| | ^3dd2
| | ^3C = B##
| | P11/3 = vA4 = ^^dd5
| | C - vF# - ^Cb - F
| | Satriluti, if 11/8 = A4

^1 = 729/704
|-
| |
| | "
| | v3M2
| | ^3C = D
| | P11/3 = ^4 = vv5
| | C - ^F - vC - F
| | Satriluti, if 11/8 = P4

^1 = 33/32
|-
| | 10
| | (P8/3, P4/2)

third-8ve, half-4th
| | v6A2
| | ^6C = D#
| | P8/3 = ^^m3 = v4A4

P4/2 = ^3m2 = v3M3
| | C - ^^Eb - vvA - C

C - ^3Db=v3E - F
| | Tribiloti, if 11/8 = P4

^1 = 33/32
|-
| |
| | "
| | ^3d2,

\\m2
| | ^3C = B#

//C = Db
| | P8/3 = vM3 = ^^d4

P4/2 = /M2 = \m3
| | C - vE - ^Ab - C

C - /D=\Eb - F
| | Triforce aka Trigu & Zozoti

^1 = 81/80, /1 = 64/63
|-
| | 11
| | (P8/3, P5/2)

third-8ve, half-5th
| | ^3d2

\\A1
| | ^3C = B#

//C = C#
| | P8/3 = vM3 = ^^d4

P5/2 = /m3 = \M3
| | C - vE - ^Ab - C

C - /Eb=\E - G
| | Satribizoti

^1 = 49/48, /1 = 343/324
|-
| | 12
| | (P8/2, P4/3)

half-8ve, third-4th
| | ^^d2

\3A1
| | ^^C = Dbb

/3C = C#
| | P8/2 = vA4 = ^d5

P4/3 = \M2 = //m2
| | C - vF#=^Gb - C

C - \D - /Eb - F
| | Latribiruti

^1 = 1029/1024, /1 = 49/48
|-
| | 13
| | (P8/2, P5/3)

half-8ve, third-5th
| | ^6d32
| | ^6C = B#3
| | P8/2 = v3AA4 = ^3dd5

P5/3 = vvA2 = ^4dd3
| | C - v3Fx=^3Gbb C

C - vvD# - ^^Fb - G
| | Latribiyoti

^1 = 81/80
|-
| |
| | "
| | ^^d2,

\3m2
| | ^^C = B#

/3C = Db
| | P8/2 = vA4 = ^d5

P5/3 = /M2 = \\m3
| | C - vF#=^Gb - C

C - /D - \F - G
| | Lemba aka Latrizo & Biruyoti

^1 = (10,-6,1,-1), /1 = 64/63
|-
| | 14
| | (P8/2, P11/3)

half-8ve, third-11th
| | v6M2
| | ^6C = D
| | P8/2 = ^34 = v35

P11/3 = ^^4 = v45
| | C - ^3F=v3G - C

C - ^^F - vvC - F
| | Latribiloti, if 11/8 = P4

^1 = 33/32
|-
| | 15
| | (P8/3, P4/3)

third-

everything
| | v3d2,

\3A1
| | ^3C = Dbb

/3C = C#
| | P8/3 = ^M3 = vvd4

P4/3 = \M2 = //m2

P5/3 = v/M2
| | C - ^E - vAb - C

C - \D - /Eb - F

C - v/D - ^\F - G
| | Triyo & Triguti

^1 = 64/63

/1 = 81/80
|-
| |
| | "
| | ^3d2,

\3m2
| | ^3C = B#

/3C = Db
| | P8/3 = vM3 = ^^d4

P5/3 = /M2 = \\m3

P4/3 = v\M2
| | C - vE - ^Ab - C

C - /D - \F - G

C - v\D - ^/Eb - F
| | Trigu & Latrizoti

^1 = 81/80

/1 = 64/63
|-
| |
| | "
| | v3A1,

\3m2
| | ^3C = C#

/3C = Db
| | P4/3 = vM2 = ^^m2

P5/3 = /M2 = \\m3

P8/3 = v/M3
| | C - vD - ^Eb - F

C - /D - \F - G

C - v/E - ^\Ab - C
| | Triyo & Latrizoti

^1 = 81/80

/1 = 64/63
|-
! |
! | quarter-splits
! |
! |
! |
! |
! |
|-
| | 16
| | (P8/4, P5)
| | ^4d2
| | ^4C = B#
| | P8/4 = vm3 = ^3A2
| | C vEb vvGb=^^F# ^A C
| | Diminished aka Quadguti
|-
| | 17
| | (P8, P4/4)
| | ^4dd2
| | ^4C = B##
| | P4/4 = ^m2 = v3AA1
| | C ^Db ^^Ebb=vvD# vE F
| | Negri aka Laquadyoti
|-
| | 18
| | (P8, P5/4)
| | v4A1
| | ^4C = C#
| | P5/4 = vM2 = ^3m2
| | C vD vvE=^^Eb ^F G
| | Tetracot aka Saquadyoti
|-
| | 19
| | (P8, P11/4)
| | v4dd3
| | ^4C = Eb3
| | P11/4 = ^M3 = v3dd5
| | C ^E ^^G# vDb F
| | Squares aka Laquadruti
|-
| | 20
| | (P8, P12/4)
| | v4m2
| | ^4C = Db
| | P12/4 = v4 = ^3M3
| | C vF vvBb=^^A ^D G
| | Vulture aka Sasa-quadyoti
|-
| |
| | etc.
| |
| |
| |
| |
| |
|}
The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably it isn't particularly complex.

Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢).

==Tipping points==

Removing the ups and downs from an EU makes an '''uninflected''' EU, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s EU is ^^d2, the uninflected EU is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the EU vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore '''up may need to be swapped with down, depending on the size of the 5th''' in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' EUs are upped or downed as if the 5th were just.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every uninflected EU, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the uninflected EU.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | uninflected EU
! | 3-exponent
! | tipping

point edo
! | edo's 5th
! | upping range
! | downing range
! | if the 5th is just
|-
| | M2
| | C - D
| | 2
| | 2-edo
| | 600¢
| | none
| | all
| | downed
|-
| | m3
| | C - Eb
| | -3
| | 3-edo
| | 800¢
| | none
| | all
| | downed
|-
| | m2
| | C - Db
| | -5
| | 5-edo
| | 720¢
| | none
| | all
| | downed
|-
| | A1
| | C - C#
| | 7
| | 7-edo
| | ~686¢
| | 600-686¢
| | 686¢-720¢
| | downed
|-
| | d2
| | C - Dbb
| | -12
| | 12-edo
| | 700¢
| | 700-720¢
| | 600-700¢
| | upped
|-
| | dd3
| | C - Eb3
| | -17
| | 17-edo
| | ~706¢
| | 706-720¢
| | 600-706¢
| | downed
|-
| | dd2
| | C - Db3
| | -19
| | 19-edo
| | ~695¢
| | 695-720¢
| | 600-695¢
| | upped
|-
| | d34
| | C - Fb3
| | -22
| | 22-edo
| | ~709¢
| | 709-720¢
| | 600-709¢
| | downed
|-
| | d32
| | C - Db4
| | -26
| | 26-edo
| | ~692¢
| | 692-720¢
| | 600-692¢
| | upped
|-
| | d44
| | C - Fb4
| | -29
| | 29-edo
| | ~703¢
| | 703-720¢
| | 600-703¢
| | downed
|-
| | d43
| | C - Eb5
| | -31
| | 31-edo
| | ~697¢
| | 697-720¢
| | 600-697¢
| | upped
|-
| | etc.
| |
| |
| |
| |
| |
| |
| |
|}

=Further Discussion=

==Naming very large intervals==

So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, adding an 8ve is indicated by "c" for '''compound''' (a conventional music theory term). Thus 10/3 = cM6 = compound major 6th, 9/2 = ccM2 or cM9, etc. For a pergen with an unsplit octave, the multigen is always some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen M is less than 3 octaves, and can be P4, P5, P11, P12, ccP4 or ccP5. The last one can be spoken as "coco-fifth". Tripe compound can be spoken as "trico".

==Secondary splits==

Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. Porcupine aka Triyoti) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with:

P4/3: C - vD - ^Eb - F

A4:/3 C - D - E - F# (the lack of ups and downs indicates that this interval was already split into 3 parts)

m7/3: C - ^Eb - vG - Bb (because m7 is already split into halves, we also have m7/6: C - vD - ^Eb - F - vG - ^Ab - Bb)

M7/3: C - vE - ^G - B

m10/3: C - F - Bb - Eb (m10 is already split into 3 parts, thus m10/9 also occurs)

M10/3: C - ^F - vB - E

Of course, an equal-step melody can span other intervals besides 3-limit ones. Simply extend or cut short the earlier examples to find other melodies:

^m3/2: C - vD - ^Eb (^m3 = 6/5)

^m6/5: C - vD - ^Eb - F - vG - ^Ab (^m6 = 8/5)

vm9/4: C - ^Eb - vG - Bb - ^Db (vm9 = 32/15)

vM7/2: C - ^F - vB (vM7 = 15/8, probably more harmonious than M7 = 243/128)

More remote intervals include A1, d4, d7 and d10. These unfortunately require a very long genchain. The most interesting melodically is A1: C - ^C - vC# - C#. From C to C# is seven 5ths, which equals 21 generators, so the genchain would have to contain 22 notes if it had no gaps. Note that A1 is the uninflected EU of third-4th. The uninflected EU is always a secondary split.

For a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occurring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.

Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof [[pergen#Further Discussion-Various proofs|below]]). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If only the 8ve is split, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the EU is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. "Every" means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.

The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occurring splits are listed too, under "all pergens". (P8/3, P4/2) has all the secondary splits that (P8/3, P5) and (P8, P4/2) have, plus additional ones.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! | secondary splits of a 12th or less
|-
| colspan="2" | all pergens
| | M3/2, A4/3, d5/2, A5/4, m7/2, M9/2, m10/3, A11/2
|-
! colspan="2" | half-splits
! |
|-
| | (P8/2, P5)
| | half-8ve
| | M2/2, M3/4, A4/6, A5/8, m6/2, M10/2, d12/2, A12/2
|-
| | (P8, P4/2)
| | half-4th
| | m2/2, M6/2, m7/4, A8/2, m10/6, A11/4, P12/2
|-
| | (P8, P5/2)
| | half-5th
| | A1/2, m3/2, M7/2, m9/2, P11/2
|-
| | (P8/2, P4/2)
| | half-everything
| | every 3-limit interval is split twice as much as before
|-
! colspan="2" | third-splits
! |
|-
| | (P8/3, P5)
| | third-8ve
| | m3/3, M6/3, d5/6, A11/3, d12/3
|-
| | (P8, P4/3)
| | third-4th
| | A1/3, m7/6, M7/3, m10/9, M10/3
|-
| | (P8, P5/3)
| | third-5th
| | m2/3, m6/3, M9/6, A8/3, A12/3
|-
| | (P8, P11/3)
| | third-11th
| | M2/3, M3/6, A4/9, A5/12, m9/3, P12/3
|-
| | (P8/3, P4/2)
| | third-8ve half-4th
| | third-8ve splits, half-4th splits, M6/6, m10/6, A11/12
|-
| | (P8/3, P5/2)
| | third-8ve half-5th
| | third-8ve splits, half-5th splits, m3/6, d5/12
|-
| | (P8/2, P4/3)
| | half-8ve third-4th
| | half-8ve splits, third-4th splits, A4/6, M10/6
|-
| | (P8/2, P5/3)
| | half-8ve third-5th
| | half-8ve splits, third-5th splits, m6/6, M9/6, A12/6
|-
| | (P8/2, P11/3)
| | half-8ve third-11th
| | half-8ve splits, third-11th splits, M2/6, M3/12, A4/18, A5/24
|-
| | (P8/3, P4/3)
| | third-everything
| | every 3-limit interval is split three times as much as before
|}

==Singles and doubles==

If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a '''single-split''' pergen. If it has two fractions, it's a '''double-split''' pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called '''single-pair''' notation because it adds only a single pair of accidentals to conventional notation. '''Double-pair''' notation uses both ups/downs and lifts/drops. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the EU for single-pair notation is a 3rd or larger, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs.

In this article, for double-pair notation, the period uses ups and downs, and the generator uses lifts and drops. But the choice of which pair of accidentals is used for what is arbitrary, and ups/downs could be exchanged with lifts/drops.

Every double-split pergen is either a '''true double''' or a '''false double'''. A true double, like third-everything (P8/3, P4/3) or half-8ve quarter-4th (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-8ve quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.

A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.

==Finding an example temperament==

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4⋅P and P8. If P is 6/5, the comma is 4⋅P - P8 = (6/5)4 ÷ (2/1) = 648/625, making the Diminished temperament aka Quadguti. If P is 7/6, the comma is P8 - 4⋅P = (2/1) · (7/6)-4, making the Quadruti temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.

Another method: if the generator's cents are known, look on the genchain for an interval that approximates a ratio of color depth ±1. Let the interval be I, and the genspan of this interval be x. Then n⋅x gens = n⋅I = x⋅M, where M is the multigen and M/n is the generator. The comma can be found from this equation, if n and x are coprime. For example, suppose (P8, P5/5) has G = 140¢. The genchain is all multiples of 140¢. Looking at the cents, 280¢ is about 7/6. Thus 2G = 7/6, and 10G = 5⋅(7/6) = 2⋅P5. Thus 2⋅P5 - 5⋅(7/6) = 0G = 0¢, and the comma is (3, 7, 0, -5). If the period is split and the generator isn't, use the perchain instead of the genchain. For example, (P8/7, P5) has a period of 141¢. 2 gens = 343¢, about 11/9. Thus 7⋅(11/9) = 2⋅P8, and the comma is (-2, -14, 0, 0, 7), making Saseploti.

If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.

Finding the comma(s) for a double-split pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is '''explicitly false'''. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4⋅G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).

If the pergen is not explicitly false, put the pergen in its '''unreduced''' form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n⋅P8 - m⋅M)/nm) = (P8/m, M'/n'). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P < G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2⋅P8 - 3⋅P5) / (3⋅2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This is explicitly false, thus the comma can be found from m3/6 alone. G' is about 50¢, and the comma is 6⋅G' - m3. The comma splits both the octave and the fifth.

This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit. To summarize:

<ul><li>'''A'''<nowiki/> '''double-split pergen is explicitly false if m = |b|, and not explicitly false if m > |b|.'''</li><li>'''A double-split pergen is a true double if and only if neither it nor its unreduced form is explicitly false'''''<nowiki/>'''''<nowiki/>.'''</li><li>'''A double-split pergen is a true double if''' '''GCD (m, n) > |b|,''' '''and a false double if GCD (m, n) = |b|.'''</li></ul>

A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.

Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an '''alternate''' generator. A generator or period plus or minus any number of EUs makes an '''equivalent''' generator or period. An equivalent generator is always the same size in cents, since the EU is always 0¢. An equivalent generator is the same interval, merely notated differently. For example: (P8, P5/2) has generator ^m3 and equivalent generator vM3. Another example, half-8ve (P8/2, P5) has period vA4 and equivalent period ^d5. It has generator P5 and alternate generators P4 and vA1. vA1 is equivalent to ^m2.

Of the two equivalent generators, the preferred generator is always the smaller one (smaller degree, or if the degrees are the same, more diminished). This is because the equivalent generator can be more easily found by adding the EU, rather than subtracting it. For example, ^m3 + vvA1 = vM3 is an easier calculation than vM3 - vvA1 = ^m3. This is particularly true with complex EUs like ^6dd2.

There are also alternate EUs, see below. For double-pair notation, there are also equivalent EUs.

==Ratio and cents of the accidentals==

The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2x · 3y · P±1, where P is a higher prime. They are called mapping commas because they equate or map the ratio P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for prime 5 include 81/80, 135/128, and the schisma aka Layoma = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 (besides 1/1 of course) are the currently employed commas and combinations of them.

If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in Lemba aka Latrizo & Biruyoti, where ^1 equals 64/63 minus 81/80.

Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also [[Pergen#Notating_multi-EDO_pergens|multi-EDO pergens]] pergens below.

Cents can be assigned to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + 7c. Since the EU = 0¢, we can derive the cents of the up symbol. If the EU is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its EU.

In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the EU is an A1.

This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, basic information for playing the score. Examples:
* 15-edo: # = 240¢, ^ = 80¢ (^ = third-sharp)
* 16-edo: # = -75¢
* 17-edo: # = 141¢, ^ = 71¢ (^ = half-sharp)
* 18b-edo: # = -133¢, ^ = 67¢ (^ = half-sharp)
* 19-edo: # = 63¢
* 21-edo: ^ = 57¢ (if used, # = 0¢)
* 22-edo: # = 164¢, ^ = 55¢ (^ = third-sharp)
* quarter-comma Meantone: # = 76¢
* fifth-comma Meantone: # = 84¢
* third-comma Archy aka Ruti: # = 177¢
* eighth-comma Porcupine aka Triyoti: # = 157¢, ^ = 52¢ (^ = third-sharp)
* seventh-comma Srutal aka Sagugu & Zoquadyoti: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #, seventh-comma means 1/7 of 2048/2025)
* third-comma Injera aka Gu & Biruyoti: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)
* eighth-comma Hedgehog aka Triyo & Biruyoti: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)
The last five examples are a generalization of the practice of naming meantone tunings as a fraction of a comma, e.g. quarter-comma. The 5th is sharpened or flattened by some fraction of the first comma in the color name. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both Srutal and Injera are half-8ve, but their optimal tunings are very different.

==Finding a notation for a pergen==

There are multiple notations for a given pergen, depending on the EU(s). Preferably, the EU's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:

<ul><li>For (P8/m, M/n), P8 = m⋅P + x⋅EU and M = n⋅G + y⋅EU', with 0 < |x| <= m/2 and 0 < |y| <= n/2</li><li>x is the count for EU, with EU occurring x times in one octave, and x⋅EU is the octave's '''multi-EU'''</li><li>y is the count for EU', with EU' occurring y times in one multigen, and y⋅EU' is the multigen's multi-EU</li><li>For false doubles using single-pair notation, EU = EU', but x and y are usually different, making different multi-EUs</li><li>The unreduced pergen is (P8/m, M'/n'), with a new enharmonic unison EU" and new counts, P8 = m⋅P + x'⋅EU", and M' = n'⋅G' + y'⋅EU"</li></ul>

The '''keyspan''' of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The '''[[stepspan]]''' of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.

Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a '''gedra''', analogous to a monzo, but written in brackets not parentheses: 3/2 = (-1,1) is a 7-semitone 5th, thus (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval with a monzo (a,b), there is a unique gedra [k,s], and vice versa:

<ul><li>k = 12a + 19b</li><li>s = 7a + 11b</li></ul>The matrix [(12,19) (7,11)] is unimodular, and can be inverted, and (a,b) can be derived from [k,s]:

<ul><li>a = -11k + 19s</li><li>b = 7k - 12s</li></ul>Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].

Gedras greatly facilitate finding a pergen's period, generator and EU(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an EU with the smallest possible keyspan and stepspan, which is the best EU for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up.

For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = m3. The EU can also be found using gedras: x⋅EU = M - n⋅G = P5 - 2⋅m3 = [7,4] - 2⋅[3,2] = [7,4] - [6,4] = [1,0] = A1.

Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. x⋅EU = P8 - m⋅P = P8 - 5⋅M2 = [12,7] - 5⋅[2,1] = [2,2] = 2⋅[1,1] = 2⋅m2. Because x = 2, EU will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2⋅m2 = d3). The EU's '''count''' is 2. The uninflected EU is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus EU = v5m2. Since P8 = 5⋅P + 2⋅EU, the period must be ^^M2, to make the ups and downs come out even. The number of the period's (or generator's) ups or downs always equals the count. Equipped with the period and the EU, the perchain is easily found:

P1 -- ^^M2=v3m3 -- v4 -- ^5 -- ^3M6=vvm7 -- P8
C -- ^^D=v3Eb -- vF -- ^G -- ^3A=vvBb -- C

Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has an uninflected generator [5,3]/5 = [1,1] = m2. The uninflected EU is P4 - 5⋅m2 = [5,3] - 5⋅[1,1] = [5,3] - [5,5] = [0,-2] = -2⋅[0,1] = two descending d2's. The d2 must be upped, and EU = ^5d2. Since P4 = 5⋅G - 2⋅EU, G must be ^^m2. The genchain is:

P1 -- ^^m2=v3A1 -- vM2 -- ^m3 -- ^3d4=vvM3 -- P4C -- ^^Db -- vD -- ^Eb -- vvE -- F

To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and EU from the fractional multigen as before. Then deduce the period from the EU. If the multigen was changed by unreducing, find the original generator from the period and the alternate generator.

For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The uninflected alternate generator G' is [1,1]/10 = [0,0] = P1. The uninflected EU is m2 - 10⋅P1 = m2. It must be downed, thus EU = v10m2. Since m2 = 10⋅G' + EU, G' is ^1. The octave plus or minus some number of EUs must equal 5 periods, thus (P8 + x⋅m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5⋅P + 2⋅EU, and P = ^4M2. Next, find the original half-4th generator. P = P8/5 = ~240¢, and G = P4/2 = ~250¢. Because P < G, G' is not P - G but G - P, and G is not P - G' but P + G', which equals ^4M2 + ^1 = ^5M2. While the alternate multigen is more complex than the original multigen, the alternate generator is usually simpler than the original generator.

P1- - ^4M2=v6m3 -- vv4 -- ^^5 -- ^6M6=v4m7 -- P8
C -- ^4D=v6Eb -- vvF -- ^^G -- ^6A=v4Bb -- C
P1 -- ^5M2=v5m3 -- P4
C -- ^5D=v5Eb -- F

To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. Each pair has its own enharmonic unison. For (P8/2, P4/2), the split octave implies P = vA4 and EU = ^^d2, and the split 4th implies G = /M2 and EU' = \\m2.

A false-double pergen can optionally use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair EU is not a unison or a 2nd, as with (P8/2, P4/3).

Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an EU that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So EU = v4dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The uninflected enharmonic unison is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic unison, we use the second pair of accidentals: EU' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic unisons is also an enharmonic unison: EU + EU' = v4\\d3 = 2·vv\m2, and EU - EU' = v4//ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent EUs, and v\4 and v/d4 are equivalent generators. Here is the genchain:

P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11
C -- ^E=v\F -- /Ab=\A -- ^/C=vDb -- F

One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and EU = v12m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the uninflected enharmonic unison: m3 - 3·m2 = [0,-1] = descending d2. Thus EU' = /3d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonic unisons are found from EU + EU' and EU - 2·EU'. Equivalent periods and generators are found from the many enharmonic unisons, which also allow much freedom in chord spelling. Enharmonic unison = v12m3 = /3d2 = v4/m2 = v4\\A1. Period = \M3 = v44 = //d4. Generator = ^\M3 = v34 = ^//d4.

P1 — \M3 — \\A5=/m6 — P8C — \E — /Ab — CP1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^38=v/m9 — P11
C — ^\E — ^^/Ab=vv\A — v/Db — F

It's not yet known if every pergen can avoid large EUs (those of a 3rd or more) with double-pair notation. One situation in which very large EUs occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the EU's stepspan to equal the multigen's stepspan.

Sixth-4th with single-pair notation has an awkward ^6d64 EU. This pergen might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore commas, aka Triyo & Zozoti), and its double-pair notation can also combine both. Third-4th has EU = v3A1 and G'= vM2 = ^^m2. Half-4th has EU' = \\m2 and G' = /M2 = \m3. G' - G = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G' - G = /M2 - vM2 = ^/1. Equivalent EUs are v3\\M2 and ^3\\d2.

P1 — ^/1=^\m2 — ^^m2=vM2 — /M2=\m3 — ^m3=vvM3 — v/M3=v\4 — P4
C — ^/C=^\Db — ^^Db=vD — /D=\Eb — ^Eb=vvE — v/E=v\F — F

When ups and downs are used to notate edos, a third symbol is used, a '''mid''' , written ~. The mid is only for relative notation (intervals), never for absolute notation (notes). For imperfect intervals, the mid interval is the interval exactly midway between major and minor. In addition, the mid-4th is midway between perfect and augmented, i.e. half-augmented, and the mid-5th is half-diminished. For example, 17edo's 2nds and 3rds run minor-mid-major. This avoids having to constantly choose between the equivalent terms upminor and downmajor. 72edo's 2nds and 3rds run minor, upminor, downmid, mid, upmid, downmajor, major. This makes the terms more concise. Mids can be used in pergen notation too. For single-pair, EU must be an A1, with an even number of downs. Using mids with double-pair notation is trickier. Mids never appear in the perchain. If one accidental pair appears only in the perchain and the other pair only in the genchain, then the mids appear only in the genchain, if at all.

==Alternate enharmonic unisons==

Sometimes the EU found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, ccM6/12), a false double. The uninflected alternate generator is ccM6/12 = [33,19]/12 = [3,2] = m3. The uninflected EU is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The EU becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v12A2, which is an improvement but still awkward. The period is ^4m3 and the generator is v3M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.

P1 -- ^4m3 -- v4M6 -- P8
C -- ^4Eb -- v4A -- C
P1 -- v3M2 -- v6M3=^6m2 -- ^3m3 -- P4
C -- v3D -- v6E=^6Db -- ^3Eb -- F

Because G is a M2 and EU is an A2, the equivalent generator G - EU is a descending A1. Ascending intervals that look descending can be confusing, one has to take into account the eighth-tone ups to see that v3D -- ^6Db is ascending. Double-pair notation may be preferable. This makes P = vM3, EU = ^3d2, G = /m2, and EU' = /4dd2.

P1 -- vM3 -- ^m6 -- P8
C -- vE -- ^Ab -- C
P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4
C -- /Db -- //Ebb=\\D# -- \E -- F

Because the EU found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans, like 5edo or 19edo. Heptatonic stepspans are best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the EU.

To search for an alternate EU, convert the EU to a gedra, then multiply it by the count to get the multi-EU. The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and EU is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-EU. If the stepspan becomes negative, or if it's zero and the keyspan becomes negative, invert the gedra. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new EU. Choose between ups and downs according to whether the 5th falls in the EU's upping or downing range, see [[pergen#Applications-Tipping points|tipping points]] above. Add n'''·'''count ups or downs to the new multi-EU. Add or subtract the new multi-EU from the multigen (or the octave) to get an interval which splits cleanly into n (or m) parts. Each part is the new generator (or period).

For example, (P8, P5/3) has n = 3, G = ^M2, and EU = v3m2 = [1,1]. G is upped only once, so the count is 1, and the multi-EU is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-EU [-2,1]. This can't be simplified, so the new EU is also [-2,1] = d32. Assuming a reasonably just 5th, EU needs to be upped, so EU = ^3d32. Add the multi-EU ^3[-2,1] to the multigen P5 = [7,4] to get ^3[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^3[-2,1] = v3[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·EU to check: 3·vA2 + 1·^3d32 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d32 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- vD# -- ^Fb -- G. Because d32 = -200¢ - 26·c, ^ = (-d32) / 3 = 67¢ + 8.67·c, about a third-tone.

Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one EU (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain EU. Specifically, the comma tempered out should map to the EU, or the multi-EU if the count is > 1. For example, consider Semaphore aka Zozoti (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 EU makes sense. G = ^M2 and the genchain is C -- ^D=vEb -- F. But consider Lala-yoyoti, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus EU = ^^dd2, G = vA2, and the genchain is C -- vD#=^Ebb -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.

Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 EU is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of EU.

For example, Paralimmal aka Satriluti tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the EU is v3M2, but if 11/8 is notated as a vA4, the EU is ^3dd2.

Sometimes the temperament implies an EU that isn't even a 2nd. For example, Liese aka Gu & Trizoguti (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and EU = 3·vd5 - P11 = v3dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- vGb -- ^B -- F.

This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an EU, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different EUs.

==Chord names and staff notation==

Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[Ups and downs notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.

In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G ^A and C E G vBb are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.

Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara aka Ru & Biruyoti (2.3.5.7 with 64/63 and 50/49) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and EU = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C vE G Bb = Cv,7.

A different temperament may result in the same pergen with the same EU, but may still produce a different name for the same chord. For example, Injera aka Gu & Biruyoti (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 EU is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and EU = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G vBb = C,v7.

Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes. This is in fact a requirement for MOS scales.

Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. Iv -- vIIIv -- vVI^m -- Iv. A Porcupine aka Triyoti (third-4th) comma pump can be written out like so: Cv -- vA^m -- vDv -- [vvB=^Bb]^m -- ^Ebv -- G^m -- Gv -- Cv. Brackets are used to show that vvB and ^Bb are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. vvB is written first to show that this root is a vM6 above the previous root, vD. ^Bb is second to show the P4 relationship to the next root, ^Eb. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either vvB^m or ^Bb^m, or possibly ^BbvvM = ^Bb vD ^F.

Lifts and drops, like ups and downs, precede the note head and any sharps or flats. Scores for melody instruments could instead have them above or below the staff. This score uses ups and downs, and has chord names. It's for the third-4th pergen, with EU = v3A1. As noted above, it can be played in EDOs 15, 22 or 29 as is, because ^1 maps to 1 edostep. But to be played in EDOs 30, 37 or 44, ^1 must be interpreted as two edosteps.

Mizarian Porcupine Overture by Herman Miller (P8, P4/3) ^3C = C#

[[File:Mizarian_Porcupine_Overture.png|alt=Mizarian Porcupine Overture.png|800x692px|Mizarian Porcupine Overture.png]]

==Tipping points and sweet spots==

The tipping point for half-octave with a d2 EU is 700¢, 12-edo's 5th. As noted above, the 5th of Pajara (half-8ve) tends to be sharp, thus it has EU = ^^d2. But Injera, also half-8ve, has a flat 5th, and thus EU = vvd2. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But if a single temperament "tips over", either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.

The tipping point depends on the choice of EU. It's not the temperament that tips, it's the notation. Half-8ve could be notated with an EU of vvM2. The tipping point becomes 600¢, a very unlikely 5th, and tipping is impossible. For single-comma temperaments, the EU usually equals the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of EU is a given. However, the mapping of primes 11 and 13 is not agreed on.

The notation's tipping point is determined by the uninflected EU, which is implied by the vanishing comma. For example, Porcupine aka Triyoti's 250/243 comma is an A1 = (-11,7), which implies an uninflected EU of A1, which implies 7-edo, and a 685.7¢ tipping point. Dicot aka Yoyoti's 25/24 comma is also an A1, and has the same tipping point. Semaphore aka Zozoti's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. See [[pergen#Applications-Tipping points|tipping points]] above for a more complete list.

Double-pair notation has two EUs, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two EUs can be any two of A1, m2 or d2. One can choose to use whichever two EUs best avoid tipping.

An example of a temperament that tips easily is Negri aka Laquadyoti, 2.3.5 and (-14,3,4). Because Negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with Negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and the EU could be either ^4dd2 or v4dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an EU of ^4dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma Negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.

Another "tippy" temperament is found by adding the mapping comma 81/80 to the Negri comma and getting the Latriyoma comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th.

==Notating unsplit pergens==

An unsplit pergen doesn't require ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a mapping comma, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (i.e. the monzo has a final exponent of ±1).

The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate. The up symbol's ratio is always the mapping comma, or its inverse.

{| class="wikitable" style="text-align:center;"
|-
! | 5-limit temperament
! | comma
! | sweet spot
! colspan="2" | no ups or downs
! colspan="3" | with ups and downs
! colspan="2" | up symbol
|-
! | (pergen is unsplit)
! |
! | (5th = 700¢ + c)
! | 5/4 is
! | 4:5:6 chord
! | 5/4 is
! | 4:5:6 chord
! | EU
! | ratio
! | cents
|-
| | Meantone aka Guti
| | 81/80 = P1
| | c = -3¢ to -5¢
| | M3
| | C E G
| | ---
| | ---
| | ---
| | ---
| | ---
|-
| | Mavila aka Layobiti
| | 135/128 = A1
| | c = -21¢ to -22¢
| | m3
| | C Eb G
| | ^M3
| | C ^E G
| | ^A1
| | 80/81 = d1
| | -100¢ - 7c = 47¢-54¢
|-
| | Laguti
| | (-15,11,-1) = A1
| | c = -10¢ to -12¢
| | A3
| | C E# G
| | ^M3
| | C ^E G
| | vA1
| | 80/81 = A1
| | 100¢ + 7c = 26¢-30¢
|-
| | Schismic aka Layoti
| | (-15,8,1) = -d2
| | c = 1.7¢ to 2.0¢
| | d4
| | C Fb G
| | vM3
| | C vE G
| | ^d2
| | 81/80 = -d2
| | 12c = 20¢-24¢
|-
| | Lalaguti
| | (-23,16,-1) = -d2
| | c = -0.9¢ to -1.2¢
| | AA2
| | C D## G
| | vM3
| | C vE G
| | vd2
| | 81/80 = d2
| | -12c = 10¢-15¢
|-
| | Father aka Gubiti
| | 16/15 = m2
| | c = 56¢ to 58¢
| | P4
| | C F G
| | vM3
| | C vE G
| | ^m2
| | 81/80 = -m2
| | -100¢ + 5c = 180-190¢
|-
| | Superpyth aka Sasayoti
| | (12,-9,1) = m2
| | c = 9¢ to 10¢
| | A2
| | C D# G
| | vM3
| | C vE G
| | vm2
| | 81/80 = m2
| | 100¢ - 5c = 50-55¢
|}
The Schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch. The Mavila temperament could perhaps be notated without ups and downs, because 5/4 is still a 3rd, and the 4:5:6 triad still looks like a triad.

For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the Schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-limit comma is the pythagorean comma (-19,12).

A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for Archy aka Ruti (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit Schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C vE G vBb.

==Notating rank-3 pergens==

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. Enharmonic unisons aka EUs are like vanishing commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of EUs needed always equals the difference between the notation's rank and the tuning's rank. Examples:

{| class="wikitable" style="text-align:center;"
|-
! | tuning
! | pergen
! | tuning's rank
! | notation
! | notation's rank without any EUs
! | # of EUs needed
! | EUs
|-
| | 12-edo
| | (P8/12)
| | rank-1
| | conventional
| | rank-2
| | 1
| | EU = d2
|-
| | 19-edo
| | (P8/19)
| | rank-1
| | conventional
| | rank-2
| | 1
| | EU = dd2
|-
| | 15-edo
| | (P8/15)
| | rank-1
| | single-pair
| | rank-3
| | 2
| | EU = m2, EU' = v3A1 = v3M2
|-
| | 24-edo
| | (P8/24)
| | rank-1
| | single-pair
| | rank-3
| | 2
| | EU = d2, EU' = vvA1 = vvm2
|-
| | 3-limit JI aka pythagorean
| | (P8, P5)
| | rank-2
| | conventional
| | rank-2
| | 0
| | ---
|-
| | Meantone aka Guti
| | (P8, P5)
| | rank-2
| | conventional
| | rank-2
| | 0
| | ---
|-
| | Diaschismic aka Saguguti
| | (P8/2, P5)
| | rank-2
| | single-pair
| | rank-3
| | 1
| | EU = ^^d2
|-
| | Semaphore aka Zozoti
| | (P8, P4/2)
| | rank-2
| | single-pair
| | rank-3
| | 1
| | EU = vvm2
|-
| | Decimal aka Yoyo & Zozoti
| | (P8/2, P4/2)
| | rank-2
| | double-pair
| | rank-4
| | 2
| | EU = vvd2, EU' = \\m2 = ^^\\A1
|-
| | 5-limit JI
| | (P8, P5, ^1)
| | rank-3
| | single-pair
| | rank-3
| | 0
| | ---
|-
| | Marvel aka Ruyoyoti
| | (P8, P5, ^1)
| | rank-3
| | single-pair
| | rank-3
| | 0
| | ---
|-
| | Breedsmic aka Bizozoguti
| | (P8, P5/2, ^1)
| | rank-3
| | double-pair
| | rank-4
| | 1
| | EU = \\dd3
|-
| | 7-limit JI
| | (P8, P5, ^1, /1)
| | rank-4
| | double-pair
| | rank-4
| | 0
| | ---
|}
When there is more than one EU, the first one can be added to or subtracted from the 2nd one, to make an equivalent 2nd EU.

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas. There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.

Even the unsplit rank-3 pergen requires single-pair notation, for the 2nd generator. Single-splits and false double-splits require double-pair, true doubles and false triples require triple-pair, and true triples require quadruple-pair. Some false triples and some true doubles may use quadruple-pair as well, to avoid awkward EUs of a 3rd or more. Rather than devising a third or fourth pair of symbols, and a third or fourth pair of adjectives to describe them, one might simply use colors.

A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.

Some examples of 7-limit rank-3 temperaments:

{| class="wikitable" style="text-align:center;"
|-
! | 7-limit temperament
! | comma
! | pergen
! | spoken pergen
! | notation
! | period
! | gen1
! | gen2
! | EU
|-
| | Marvel aka Ruyoyoti
| | 225/224
| | (P8, P5, ^1)
| | rank-3 unsplit
| | single-pair
| | P8
| | P5
| | ^1 = 81/80
| | ---
|-
| | "
| | "
| | "
| | "
| | double-pair
| | "
| | "
| | "
| | ^^\d2
|-
| | Biruyoti
| | 50/49
| | (P8/2, P5, ^1)
| | rank-3 half-8ve
| | double-pair
| | v/A4 = 10/7
| | P5
| | ^1 = 81/80
| | ^^\\d2
|-
| | Trizoguti
| | 1029/1000
| | (P8, P11/3, ^1)
| | rank-3 third-11th
| | double-pair
| | P8
| | ^\d5 = 7/5
| | ^1 = 81/80
| | ^^^\\\dd3
|-
| | Breedsmic aka Bizozoguti
| | 2401/2400
| | (P8, P5/2, ^1)
| | rank-3 half-5th
| | double-pair
| | P8
| | v//A2 = 60/49
| | /1 = 64/63
| | ^^\4dd3
|-
| | Demeter aka Trizo-aguguti
| | 686/675
| | (P8, P5, \m3/2)
| | half-downminor-3rd
| | double-pair
| | P8
| | P5
| | v/A1 = 15/14
| | ^^\\\dd3
|}
If using single-pair notation, Marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - vE - G - vvA#. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - vE - G - \Bb.

There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, Biruyoti is half-8ve, with a d2 comma, like Srutal. Using the same notation as Srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and EU = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as C vE G v/Bb. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C vE G \Bb.

With this standardization, the EU can be derived directly from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)-2 · (64/63)2 · (-19,12). This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the EU never is, therefore the EU = ^^\\d2. The period is found by adding/subtracting the EU from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both EU and P become more complex, but the ratio for /1 becomes simpler.

This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For Biruyoti, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore Biruyoti doesn't tip. Single-pair rank-3 notation has no EU, and thus no tipping point. Double-pair rank-3 notation has 1 EU, but two mapping commas. Rank-3 notations rarely tip.

Unlike the previous examples, Demeter aka Trizo-aguguti's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no EU. But the 4:5:6:7 chord would be spelled C -- ^^^Fbbb -- G -- ^^Bbb, very awkward! Standard double-pair notation is better. Gen2 = v/A1, EU = ^^\\\dd3, and ^^\\\C = A##. Genchain2 is C -- v/C# -- \Eb -- vE -- \\Gb -- v\G -- vvG#=\\\Bbb -- v\\Bb... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own EU, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own EU, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops only for the other kind of accidentals.

There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For Demeter, any combination of \m3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because there will always be a 3-limit comma small enough to be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the '''DOL''' ([[Odd limit|double odd limit]]) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred. And as a tie-breaker, in case of two ratios with the same DOL (such as 5/3 and 6/5), to minimize the size in cents of the multigen2. Since 5/3 = 884.4¢ and 6/5 = 315.6¢, 6/5 is preferred.

If ^1 = 81/80, possible half-split gen2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.

All possible rank-3 pergens can be listed, but the table is much longer than for rank-2 pergens. Here are all the half-split pergens:

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! colspan="4" | prime subgroup used by the pergen
|-
! | unsplit
! colspan="2" | 2.3.5 (^1 = 81/80)
! colspan="2" | 2.3.7 (^1 = 64/63)
|-
| | 1
| | (P8, P5, ^1)
| | rank-3 unsplit
| | same
| | same
|-
! | half-splits
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5, ^1)
| | rank-3 half-8ve
| | same
| | same
|-
| | 3
| | (P8, P4/2, ^1)
| | rank-3 half-4th
| | same
| | same
|-
| | 4
| | (P8, P5/2, ^1)
| | rank-3 half-5th
| | same
| | same
|-
| | 5
| | (P8/2, P4/2, ^1)
| | rank-3 half-everything
| | same
| | same
|-
| | 6
| | (P8, P5, ^m3/2)
| | half-upminor-3rd
| | (P8, P5, ^M2/2)
| | half-upmajor-2nd
|-
| | 7
| | (P8, P5, vM3/2)
| | half-downmajor-3rd
| | (P8, P5, vm3/2)
| | half-downminor-3rd
|-
| | 8
| | (P8, P5, ^m6/2)
| | half-upminor-6th
| | (P8, P5, ^M6/2)
| | half-upmajor-6th
|-
| | 9
| | (P8, P5, vM6/2)
| | half-downmajor-6th
| | (P8, P5, vm7/2)
| | half-downminor-7th
|-
| | 10
| | (P8/2, P5, ^m3/2)
| | half-8ve half-upminor-3rd
| | (P8/2, P5, ^M2/2)
| | half-8ve half-upmajor-2nd
|-
| | 11
| | (P8/2, P5, vM3/2)
| | half-8ve half-downmajor-3rd
| | (P8/2, P5, vm3/2)
| | half-8ve half-downminor-3rd
|-
| | 12
| | (P8, P4/2, vM3/2)
| | half-4th half-downmajor-3rd
| | (P8, P4/2, ^M2/2)
| | half-4th half-upmajor-2nd
|-
| | 13
| | (P8, P4/2, ^m6/2)
| | half-4th half-upminor-6th
| | (P8, P4/2, vm7/2)
| | half-4th half-downminor-7th
|-
| | 14
| | (P8, P5/2, vM3/2)
| | half-5th half-downmajor-3rd
| | (P8, P5/2, ^M2/2)
| | half-5th half-upmajor-2nd
|-
| | 15
| | (P8, P5/2, ^m6/2)
| | half-5th half-upminor-6th
| | (P8, P5/2, vm7/2)
| | half-5th half-downminor-7th
|-
| | 16
| | (P8/2, P4/2, vM3/2)
| | half-everything half-downmajor-3rd
| | (P8/2, P4/2, ^M2/2)
| | half-everything half-upmajor-2nd
|}
There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc.

==Notating multi-EDO pergens==

A multi-EDO pergen is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave, thus it doesn't appear in the pergen. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12. Pergens which imply an edo which doesn't have a decent 5th, e.g. P8/3, P8/4, P8/6, etc., are covered in the next section.

A multi-EDO pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the octave and removes the middle term from the pergen. For example, Blackwood aka Sawati plus Ya is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1). Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:

{| class="wikitable" style="text-align:center;"
|-
! | temperament
! | pergen
! | spoken name
! | enharmonic unisons
! | perchain
! | genchain
! | ^1 ratio
! | /1 ratio
|-
| | Blackwood aka Sawati+ya
| | (P8/5, ^1)
| | rank-2 5-edo
| | EU = m2
| | D E=F G A B=C D
| | D vF#=vG vvB...
| | 81/80 = 16/15
| | ---
|-
| | Whitewood aka Lawati+ya
| | (P8/7, ^1)
| | rank-2 7-edo
| | EU = A1
| | D E F G A B C D
| | D ^F ^^A...
| | 80/81 = 135/128
| | ---
|-
| | 10edo+ya
| | (P8/10, /1)
| | rank-2 10-edo
| | EU = m2, EU' = vvA1 = vvM2
| | D ^D=vE E=F ^F=vG G...
| | D \F#=\G \\B...
| | (see below)
| | 81/80
|-
| | 12edo+la
| | (P8/12, ^1)
| | rank-2 12-edo
| | EU = d2
| | D D#=Eb E F F#=Gb...
| | D ^G ^^C
| | 33/32
| | ---
|-
| | "
| | "
| | "
| | "
| | "
| | D vG#=vAb vvD...
| | 729/704
| | ---
|-
| | 17edo+ya
| | (P8/17, /1)
| | rank-2 17-edo
| | EU = dd3, EU' = vm2 = vvA1
| | D ^D=Eb D#=vE E F...
| | D \F# \\A#=v\\B...
| | 256/243
| | 81/80
|}
If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 10, 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15, or 12/11, or 13/12.

The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.

All multi-EDO pergens are of the form (P8/m, ^1) or (P8/m, /1), and they can be identified solely by their splitting fraction. Multi-EDO pergens are a small minority of rank-2 pergens.

It's possible to have a fifth-8ve pergen with an independent 5th, but there will be very small intervals of about 20¢. Here are two such:

{| class="wikitable" style="text-align:center;"
|-
! | temperament
! | subgroup
! | comma
! | pergen
! | spoken name
! | EU
! | perchain
! | genchain
! | ^1 ratio
|-
| | Laquinzoti
| | 2.3.7
| | (-14,0,0,5)
| | (P8/5, P5)
| | fifth-8ve
| | v5m2
| | D ^^E vG ^A vvC D
| | C G D A E...
| | 49/48
|-
| | Saquinruti
| | 2.3.7
| | (22,-5,0,-5)
| | "
| | "
| | "
| | "
| | "
| | 64/63
|}
Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C ^^D vF ^G vvBb C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section). A multi-EDO pergen should never have an uninflected genchain.

==Notating non-8ve and no-5ths pergens==

In multi-EDO pergens, the 5th is present but not independent. In no-5ths pergens, the 5th is not present, and the prime subgroup doesn't contain 3.

In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any EUs, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.

But in non-8ve and no-5ths pergens, not every name has a note. For example, Biruyoti Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - vF# - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a huge number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk.

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! colspan="6" | prime subgroup used by the pergen
|-
! | unsplit
! | 2.3
! | 2.5 (M3 = 5/4)
! | 2.7 (M2 = 8/7)
! | 3.5 (M6 = 5/3)
! | 3.7 (M3 = 9/7)
! | 5.7 (ccM3 = 5/1, d5 = 7/5)
|-
| | 1
| | (P8, P5)
| | (P8, M3)
| | (P8, M2)
| | (P12, M6)
| | (P12, M3)
| | (ccM3, d5)
|-
! | half-splits
! |
! |
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5)
| | (P8/2, M3)
| | (P8/2, M2)
| | (P12/2, M6)
| | (P12/2, M3)
| | (M9, d5)*
|-
| | 3
| | (P8, P4/2)
| | (P8, M2)*
| | (P8, M2/2)
| | (P12, M6/2)
| | (P12, M2)*
| | (ccM3, m3)*
|-
| | 4
| | (P8, P5/2)
| | (P8, m6/2)
| | (P8, P5)*
| | (P12, P4)*
| | (P12, m10/2)
| | (ccM3, M7)*
|-
| | 5
| | (P8/2, P4/2)
| | (P8/2, M2)*
| | (P8/2, M2/2)
| | (P12/2, M6/2)
| | (P12/2, M3/2)
| | (M9, m3)*
|-
! | third-splits
! |
! |
! |
! |
! |
! |
|-
| | 6
| | (P8/3, P5)
| | (P8/3, M3)
| | (P8/3, M2)
| | (P12/3, M6)
| | (P12/3, M3)
| | (ccM3/3, d5)
|-
| | 7
| | (P8, P4/3)
| | (P8, M3/3)
| | (P8, M2/3)
| | (P12, M6/3)
| | (P12, M3/3)
| | (ccM3, d5/3)
|-
| | 8
| | (P8, P5/3)
| | (P8, m6/3)
| | (P8, m7/3)
| | (P12, m7/3)
| | (P12, P4)*
| | (ccM3, cA6/3)
|-
| | 9
| | (P8, P11/3)
| | (P8, M10/3)
| | (P8, M9/3)
| | (P12, ccM3/3)
| | (P12, cM7/3)
| | (ccM3, ccm7/3)
|-
| | 10
| | (P8/3, P4/2)
| | (P8/3, M2)*
| | (P8/3, M2/2)
| | (P12/3, M6/2)
| | (P12/3, M2)*
| | (ccM3/3, m3)*
|-
| | 11
| | (P8/3, P5/2)
| | (P8/3. m6/2)
| | (P8/3, P5)*
| | (P12/3, P4)*
| | (P12/3, m10/2)
| | (ccM3/3, M7)*
|-
| | 12
| | (P8/2, P4/3)
| | (P8/2, M3/3)
| | (P8/2, M2/3)
| | (P12/2, M6/3)
| | (P12/2, M3/3)
| | (M9, d5/3)*
|-
| | 13
| | (P8/2, P5/3)
| | (P8/2, m6/3)
| | (P8/2, m7/3)
| | (P12/2, m7/3)
| | (P12/2, P4)*
| | (M9, cA6/3)*
|-
| | 14
| | (P8/2, P11/3)
| | (P8/2, M10/3)
| | (P8/2, M9/3)
| | (P12/2, ccM3/3)
| | (P12/2, cM7/3)
| | (M9, ccm7/3)*
|-
| | 15
| | (P8/3, P4/3)
| | (P8/3, M3/3)
| | (P8/3, M2/3)
| | (P12/3, M6/3)
| | (P12/3, P4)*
| | (ccM3/3, d5/3)
|}
For prime subgroup p.q, the unsplit pergen has period p/1. The unsplit pergen's generator is found by dividing q by p until it's less than p/1, and period-inverting if it's more than half of p/1.

Multi-EDO pergens also appear in this table. For example, in row #33, (P8/5, ^1) replaces both 2.5 (P8/5, M3) and 2.7 (P8/5, M2). This has the advantage of avoiding missing notes. Since one fifth of 5/1 is only 25¢ from 7/5, the 5.7 (ccM3/5, d5) can optionally be replaced too.

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! | 2.3
! | 2.5
! | 2.7
! | 3.5
! | 3.7
! | 5.7
|-
| | 33
| | (P8/5, P5)
| | (P8/5, ^1)
| | (P8/5, ^1)
| | (P12/5, M6)
| | (P12/5, M3)
| | (ccM3/5, ^1)
|}

Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the first 3.5 pergen, (P12, M6). The note one M6 above C1 is A1, and the note three P12's above C1 is A5. (There is no A2, A3 or A4.) Referring to a note as simply A is ambiguous.

Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table could be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2), with ^M2 = 8/7 and ^1 = √256/245 = √(81/80 * 64/63 * 64/63) = about 38¢. The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3), with ^M3 = 9/7 and ^1 = √6561/6125 = √[(81/80)3 * (64/63)2] = about 60¢.

==Pergen squares==

Pergen squares, which were discovered by Praveen Venkataramana, are a way to visualize pergens squares in a way that isn't specific to any primes at all. To understand them, let's assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.

For (P8, P5), the pergen square has 4 notes, shown here with octave numbers (ignore the periods).

C2 -- G2 
| . . . . | 
C1 -- G1

Splitting the 8ve or the multigen adds notes to the square. For (P8/2, P5), there are 6 notes. The new notes fall halfway between each 8ve:

C2 --- G2 
vF#1 vC#2 
C1 --- G1

A pergen square shows all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and vC#2 bisects it. vG#2 bisects the G2-A2 M2. Every "downed" note bisects an 8ve, a M2, a cm7 (e.g. D1 to C3), a cM9 (e.g. C1 to D3), and many other intervals.

C2 --- G2 --- D3 --- A3 
vF#1 vC#2 vG#2 vD#3 
C1 --- G1 --- D2 --- A2

Splitting the 5th adds notes to the horizontal edges of the square:

C2 vE2 G2 
| . . . . . . | 
C1 vE1 G1

Each downed note also bisects a P11 (e.g. G1-C3), as well as many other intervals.

C3 vE3 G3 
| . . . . . . | 
C2 vE2 G2 
| . . . . . . | 
C1 vE1 G1

From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square.

C2 ---- G2 
| . ^A1 . | 
C1 ---- G1

^A1 also bisects the P12 from C1 to G2.

Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Pierce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.

[[File:pergen_squares.png|alt=pergen squares.png|pergen squares.png]]

A similar chart could be made for all rank-3 pergens, using pergen cubes.

==Notating tunings with an arbitrary generator==

Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.

The next table lists all the ranges for all multigens up to seventh-splits. One can look up one's generator in the first column and find a possible multigen. Use the octave inverse if G > 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | primary choice
! colspan="4" | secondary choices
|-
! | generator
! | possible multigen
! | generator
! | multigen
! | generator
! | multigen
|-
| | 23-60¢
| | M2/4 (requires P8/2)
| |
| |
| |
| |
|-
| | 69-79¢
| | P4/7
| |
| |
| |
| |
|-
| | 80-92¢
| | P4/6
| |
| |
| |
| |
|-
| | 92-103¢
| | P5/7
| |
| |
| |
| |
|-
| | 96-111¢
| | P4/5
| |
| |
| |
| |
|-
| | 108-120¢
| | P5/6
| |
| |
| |
| |
|-
| | 120-138¢
| | P4/4
| |
| |
| |
| |
|-
| | 129-144¢
| | P5/5
| |
| |
| |
| |
|-
| | 160-185¢
| | P4/3
| | 162-180¢
| | P5/4
| |
| |
|-
| | 215-240¢
| | P5/3
| |
| |
| |
| |
|-
| | 240-277¢
| | P4/2
| | 240-251¢
| | P11/7
| | 264-274¢
| | P12/7
|-
| | 280-292¢
| | P11/6
| |
| |
| |
| |
|-
| | 308-320¢
| | P12/6
| |
| |
| |
| |
|-
| | 323-360¢
| | P5/2
| | 336-351¢
| | P11/5
| |
| |
|-
| | 369-384¢
| | P12/5
| |
| |
| |
| |
|-
| | 411-422¢
| | ccP4/7
| |
| |
| |
| |
|-
| | 420-438¢
| | P11/4
| |
| |
| |
| |
|-
| | 435-446¢
| | ccP5/7
| |
| |
| |
| |
|-
| | 462-480¢
| | P12/4
| | 462-480¢
| | M9/3
| |
| |
|-
| | 480-554¢
| | P4 = P5
| | 480-492¢
| | ccP4/6
| | 508-520¢
| | ccP5/6
|-
| | 560-585¢
| | P11/3
| |
| |
| |
| |
|-
| | 576-591¢
| | ccP4/5
| | 583-593¢
| | cccP4/7
| |
| |
|}
There are gaps in the table, especially near 150¢, 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation. But the total range of possible generators is mostly well covered, providing convenient notation options. In particular, if the tuning's generator is just over 720¢, to avoid descending 2nds, instead of calling the generator a 5th, one can call its inverse a quarter-12th. The generator is notated as an up-5th, and four of them make a ccP4.

The table assumes unsplit octaves. Splitting the octave creates alternate generators. For example, if P = P8/3 = 400¢ and G = 300¢, alternate generators are 100¢ and 500¢. Any of these can be used to find a convenient multigen.

See also the [[Map_of_rank-2_temperaments|map of rank-2 temperaments]].

==Pergens and MOS scales==

Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! colspan="3" | MOS scales of 5-12 notes
! |
! |
! |
|-
| | (P8, P5)
| | unsplit
| | 5 = 2L 3s
| | 7 = 5L 2s
| colspan="2" | 12 = 7L 5s (or 5L 7s)
| |
| |
|-
! colspan="2" | halves
! |
! |
! |
! |
! |
! |
|-
| | (P8/2, P5)
| | half-8ve
| | 6 = 2L 4s
| | 8 = 2L 6s
| | 10 = 2L 8s
| colspan="3" | 12 = 2L 10s (or 10L 2s)
|-
| | (P8, P4/2)
| | half-4th
| | 5 = 4L 1s
| | 9 = 5L 4s
| |
| |
| |
| |
|-
| | (P8, P5/2)
| | half-5th
| | 7 = 3L 4s
| | 10 = 7L 3s
| |
| |
| |
| |
|-
| | (P8/2, P4/2)
| | half-everything
| | 6 = 4L 2s
| | 10 = 4L 6s
| |
| |
| |
| |
|-
! colspan="2" | thirds
! |
! |
! |
! |
! |
! |
|-
| | (P8/3, P5)
| | third-8ve
| | 6 = 3L 3s
| | 9 = 3L 6s
| colspan="2" | 12 = 3L 9s (or 9L 3s)
| |
| |
|-
| | (P8, P4/3)
| | third-4th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 1L 6s
| | 8 = 7L 1s
| |
| |
|-
| | (P8, P5/3)
| | third-5th
| | 5 = 1L 4s
| | 6 = 5L 1s
| | 11 = 5L 6s
| |
| |
| |
|-
| | (P8, P11/3)
| | third-11th
| | 5 = 2L 3s
| | 7 = 2L 5s
| | 9 = 2L 7s
| | 11 = 2L 9s
| |
| |
|-
| | (P8/3, P4/2)
| | third-8ve, half-4th
| | 6 = 3L 3s *
| | 9 = 6L 3s
| |
| |
| |
| |
|-
| | (P8/3, P5/2)
| | third-8ve, half-5th
| | 6 = 3L 3s *
| | 9 = 3L 6s
| | 12 = 3L 9s
| |
| |
| |
|-
| | (P8/2, P4/3)
| | half-8ve, third-4th
| | 6 = 2L 4s *
| | 8 = 6L 2s
| |
| |
| |
| |
|-
| | (P8/2, P5/3)
| | half-8ve, third-5th
| | 6 = 4L 2s *
| | 10 = 6L 4s
| |
| |
| |
| |
|-
| | (P8/2, P11/3)
| | half-8ve, third-11th
| | 6 = 2L 4s *
| | 8 = 2L 6s
| | 10 = 2L 8s
| | 12 = 2L 10s
| |
| |
|-
| | (P8/3, P4/3)
| | third-everything
| | 6 = 3L 3s *
| | 9 = 6L 3s *
| |
| |
| |
| |
|-
! colspan="2" | quarters
! |
! |
! |
! |
! |
! |
|-
| | (P8/4, P5)
| | quarter-8ve
| | 8 = 4L 4s
| colspan="2" | 12 = 4L 8s (or 8L 4s)
| |
| |
| |
|-
| | (P8, P4/4)
| | quarter-4th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 1L 6s
| | 8 = 1L 7s
| | 9 = 1L 8s
| | 10 = 9L 1s
|-
| | (P8, P5/4)
| | quarter-5th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 6L 1s
| |
| |
| |
|-
| | (P8, P11/4)
| | quarter-11th
| | 5 = 3L 2s
| | 8 = 3L 5s
| | 11 = 3L 8s
| |
| |
| |
|-
| | (P8, P12/4)
| | quarter-12th
| | 5 = 3L 2s
| | 8 = 5L 3s
| |
| |
| |
| |
|-
| | (P8/4, P4/2)
| | quarter-8ve half-4th
| | 8 = 4L 4s *
| | 12 = 4L 8s
| |
| |
| |
| |
|-
| | (P8/2, M2/4)
| | half-8ve quarter-tone
| | 6 = 2L 4s *
| | 8 = 2L 6s *
| | 10 = 2L 8s
| | 12 = 2L 10s
| |
| |
|-
| | (P8/2, P4/4)
| | half-8ve quarter-4th
| | 6 = 2L 4s *
| | 8 = 2L 6s *
| | 10 = 8L 2s
| |
| |
| |
|-
| | (P8/2, P5/4)
| | half-8ve quarter-5th
| | 6 = 2L 4s *
| | 8 = 6L 2s *
| |
| |
| |
| |
|-
| | (P8/4, P4/3)
| | quarter-8ve third-4th
| | 8 = 4L 4s *
| | 12 = 8L 4s *
| |
| |
| |
| |
|-
| | (P8/4, P5/3)
| | quarter-8ve third-5th
| | 8 = 4L 4s *
| | 12 = 4L 8s *
| |
| |
| |
| |
|-
| | (P8/4, P11/3)
| | quarter-8ve third-11th
| | 8 = 4L 4s *
| | 12 = 4L 8s *
| |
| |
| |
| |
|-
| | (P8/3, P4/4)
| | third-8ve quarter-4th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 9L 3s *
| |
| |
| |
|-
| | (P8/3, P5/4)
| | third-8ve quarter-5th
| | 6 = 3L 3s *
| | 9 = 6L 3s *
| |
| |
| |
| |
|-
| | (P8/3, P11/4)
| | third-8ve quarter-11th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 3L 9s *
| |
| |
| |
|-
| | (P8/3, P12/4)
| | third-8ve quarter-12th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 3L 9s *
| |
| |
| |
|-
| | (P8/4, P4/4)
| | quarter-everything
| | 8 = 4L 4s *
| | 12 = 8L 4s *
| |
| |
| |
| |
|}

A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the simplest, and also one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided.

{| class="wikitable" style="text-align:center;"
|-
! | MOS scale
! colspan="2" | primary example
! | secondary examples
|-
! | Pentatonic
! |
! |
! |
|-
| | 1L 4s
| | (P8, P5/3) [5]
| | third-5th pentatonic
| | third-4th, quarter-4th, quarter-5th
|-
| | 2L 3s
| | (P8, P5) [5]
| | unsplit pentatonic
| | third-11th
|-
| | 3L 2s
| | (P8, P12/4) [5]
| | quarter-12th pentatonic
| | quarter-11th
|-
| | 4L 1s
| | (P8, P4/2) [5]
| | half-4th pentatonic
| |
|-
! | Hexatonic
! |
! |
! |
|-
| | 1L 5s
| | (P8, P4/3) [6]
| | third-4th hexatonic
| | quarter-4th, quarter-5th, fifth-4th, fifth-5th
|-
| | 2L 4s
| | (P8/2, P5) [6]
| | half-8ve hexatonic
| |
|-
| | 3L 3s
| | (P8/3, P5) [6]
| | third-8ve hexatonic
| |
|-
| | 4L 2s
| | (P8/2, P4/2) [6]
| | half-everything hexatonic
| |
|-
| | 5L 1s
| | (P8, P5/3) [6]
| | third-5th hexatonic
| |
|-
! | Heptatonic
! |
! |
! |
|-
| | 1L 6s
| | (P8, P4/3) [7]
| | third-4th heptatonic
| | quarter-4th, fifth-4th, fifth-5th, sixth-4th, sixth-5th
|-
| | 2L 5s
| | (P8, P11/3) [7]
| | third-11th heptatonic
| | fifth-double-compound-4th, sixth-double-compound-5th
|-
| | 3L 4s
| | (P8, P5/2) [7]
| | half-5th heptatonic
| | fifth-12th
|-
| | 4L 3s
| | (P8, P11/5) [7]
| | fifth-11th heptatonic
| | sixth-12th
|-
| | 5L 2s
| | (P8, P5) [7]
| | unsplit heptatonic
| | sixth-double-compound-4th
|-
| | 6L 1s
| | (P8, P5/4) [7]
| | quarter-5th heptatonic
| |
|-
! | Octotonic
! |
! |
! |
|-
| | 1L 7s
| | (P8, P4/4) [8]
| | quarter-4th octotonic
| | fifth-4th, fifth-5th, sixth-4th, sixth-5th, seventh-4th, seventh-5th
|-
| | 2L 6s
| | (P8/2, P5) [8]
| | half-8ve octotonic
| |
|-
| | 3L 5s
| | (P8, P11/4) [8]
| | quarter-11th octotonic
| | seventh-cc4th, seventh-cc5th
|-
| | 4L 4s
| | (P8/4, P5) [8]
| | quarter-8ve octotonic
| |
|-
| | 5L 3s
| | (P8, P12/4) [8]
| | quarter-12th octotonic
| | (very lopsided, unless 5th is quite flat)
|-
| | 6L 2s
| | (P8/2, P4/3) [8]
| | half-8ve third-4th octotonic
| |
|-
| | 7L 1s
| | (P8, P4/3) [8]
| | third-4th octotonic
| |
|-
! | Nonatonic
! |
! |
! |
|-
| | 1L 8s
| | (P8, P4/4) [9]
| | quarter-4th nonatonic
| | fifth-4th, sixth-4th, sixth-5th, seventh-4th/5th, eighth-4th/5th
|-
| | 2L 7s
| | (P8, c3P5/8) [9]
| | eighth-c35th nonatonic
| | third-11th, fifth-cc4th
|-
| | 3L 6s
| | (P8/3, P5) [9]
| | third-8ve nonatonic
| | third-8ve half-5th
|-
| | 4L 5s
| | (P8, P12/7) [9]
| | seventh-12th nonatonic
| | sixth-11th
|-
| | 5L 4s
| | (P8, P4/2) [9]
| | half-4th nonatonic
| | (lopsided unless 4th is sharp), seventh-11th
|-
| | 6L 3s
| | (P8/3, P4/2) [9]
| | third-8ve half-4th nonatonic
| |
|-
| | 7L 2s
| | (P8, ccP5/6)[9]
| | sixth-cc5th nonatonic
| | (lopsided unless 5th is sharp)
|-
| | 8L 1s
| | (P8, P5/5) [9]
| | fifth-5th nonatonic
| |
|-
! | Decatonic
! |
! |
! |
|-
| | 1L 9s
| | (P8, P5/6) [10]
| | sixth-5th decatonic
| | fifth-4th, sixth-4th, seventh-4th/5th, eighth-4th/5th, ninth-4th/5th
|-
| | 2L 8s
| | (P8/2, P5) [10]
| | half-8ve decatonic
| | half-8ve quartertone, half-8ve third-11th
|-
| | 3L 7s
| | (P8, P12/5) [10]
| | fifth-12th decatonic
| | eighth-cc4th, eighth-cc5th
|-
| | 4L 6s
| | (P8/2, P4/2) [10]
| | half-everything decatonic
| |
|-
| | 5L 5s
| | (P8/5, P5) [10]
| | fifth-8ve decatonic
| | (lopsided unless 5th is quite flat)
|-
| | 6L 4s
| | (P8/2, P5/3) [10]
| | half-8ve third-5th decatonic
| |
|-
| | 7L 3s
| | (P8, P5/2) [10]
| | half-5th decatonic
| | ninth-cc5th
|-
| | 8L 2s
| | (P8/2, P4/4) [10]
| | half-8ve quarter-4th decatonic
| | half-8ve quarter-12th
|-
| | 9L 1s
| | (P8, P4/2) [10]
| | quarter-4th decatonic
| |
|}

The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the Sensei aka Sepgu & Ruyoyoti generator would be (P8, ccP5/7) [5]. The rationale would be that two Sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be Roulette [7] aka Zozoquinguti Nowa [7], with a 2.5.7 pergen (P8, (5/4)/2) = (P8, M3/2) = (P8, M2), where 5·G = A6 = 7/4.

==Pergens and EDOs==

Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored.

Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset.

How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinitely many possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, ccP5/31),... (P8, (i-1,1)/n), where n = 12i+7.

How many edos support a given pergen? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be a multiple of m, and k must be divisible by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.

Given an edo, a period, and a generator, what is the pergen? There is usually more than one right answer. For 10edo with P = 5\10 and G = 2\10, it could be either (P8/2, P4/2) or (P8/2, P5/3). Every coprime period/generator pair results in a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.

'''EDOs Supporting A Pergen'''

This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Both of these edos are incompatible with heptatonic notation, and 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. 6-edo, 11-edo and 23-edo could also be considered ambiguous.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! | supporting edos (12-31 only)
|-
| | (P8, P5)
| | unsplit
| | 12, 13, 13b, 14*, 15*, 16, 17, 18, 18b*, 19, 20*,

21*, 22, 23, 24*, 25*, 26, 27, 28*, 29, 30*, 31
|-
! | halves
! |
! |
|-
| | (P8/2, P5)
| | half-8ve
| | 12, 14, 16, 18, 18b, 20*, 22, 24*, 26, 28*, 30*
|-
| | (P8, P4/2)
| | half-4th
| | 13b, 14, 15*, 18b*, 19, 20*, 23, 24, 25*, 28*, 29, 30*
|-
| | (P8, P5/2)
| | half-5th
| | 13, 14*, 17, 18b, 20*, 21*, 24, 27, 28*, 30*, 31
|-
| | (P8/2, P4/2)
| | half-everything
| | 14, 18b, 20*, 24, 28*, 30*
|-
! | thirds
! |
! |
|-
| | (P8/3, P5)
| | third-8ve
| | 12, 15, 18, 18b*, 21, 24*, 27, 30*
|-
| | (P8, P4/3)
| | third-4th
| | 13b, 14*, 15, 21*, 22, 28*, 29, 30*
|-
| | (P8, P5/3)
| | third-5th
| | 15*, 16, 20*, 21, 25*, 26, 30*, 31
|-
| | (P8, P11/3)
| | third-11th
| | 13, 15, 17, 21, 23, 30*
|-
| | (P8/3, P4/2)
| | third-8ve, half-4th
| | 15, 18b*, 24, 30*
|-
| | (P8/3, P5/2)
| | third-8ve, half-5th
| | 18b, 21, 24, 27, 30
|-
| | (P8/2, P4/3)
| | half-8ve, third-4th
| | 14, 22, 28*, 30
|-
| | (P8/2, P5/3)
| | half-8ve, third-5th
| | 16, 20*, 26, 30*
|-
| | (P8/2, P11/3)
| | half-8ve, third-11th
| | 19, 30
|-
| | (P8/3, P4/3)
| | third-everything
| | 15, 21, 30*
|-
! | quarters
! |
! |
|-
| | (P8/4, P5)
| | quarter-8ve
| | 12, 16, 20, 24*, 28
|-
| | (P8, P4/4)
| | quarter-4th
| | 18b*, 19, 20*, 28, 29, 30*
|-
| | (P8, P5/4)
| | quarter-5th
| | 13, 14*, 20, 21*, 27, 28*
|-
| | (P8, P11/4)
| | quarter-11th
| | 14, 17, 20, 28*, 31
|-
| | (P8, P12/4)
| | quarter-12th
| | 13b, 15*, 18b, 20*, 23, 25*, 28, 30*
|-
| | (P8/4, P4/2)
| | quarter-8ve, half-4th
| | 20, 24, 28
|-
| | (P8/2, M2/4)
| | half-8ve, quarter-tone
| | 18, 20, 22, 24, 26, 28
|-
| | (P8/2, P4/4)
| | half-8ve, quarter-4th
| | 18b, 20*, 28, 30*
|-
| | (P8/2, P5/4)
| | half-8ve, quarter-5th
| | 14, 20, 28*
|-
| | (P8/4, P4/3)
| | quarter-8ve, third-4th
| | 28
|-
| | (P8/4, P5/3)
| | quarter-8ve, third-5th
| | 16, 20
|-
| | (P8/4, P11/3)
| | quarter-8ve, third-11th
| |
|-
| | (P8/3, P4/4)
| | third-8ve, quarter-4th
| | 18b*, 30
|-
| | (P8/3, P5/4)
| | third-8ve, quarter-5th
| | 21, 27
|-
| | (P8/3, P11/4)
| | third-8ve, quarter-11th
| |
|-
| | (P8/3, P12/4)
| | third-8ve, quarter-12th
| | 15, 18b, 30*
|-
| | (P8/4, P4/4)
| | quarter-everything
| | 20, 28
|}
The edos that support the fewest pergens are prime-number edos like 11edo or 13edo. The most "pergen-friendly" edos tend to be ones in which the circle of 5ths doesn't reach every edostep. For example, 24edo supports all half-split pergens, since both P8 and P5 map to an even number of edosteps. 72edo supports all half-splits and all third-splits. 15, 21 and 36 edo support many but not all third-splits (not those with m = 2 or n = 2).

'''Notating a pergen tuned to an EDO'''

If both the pergen and the EDO are notated with ups and downs, what is the relationship between the two kinds of up? If the edo supports the pergen, fully or partially, then the pergen's up equals some multiple of the EDO's up, i.e. some number of edosteps. For third-4th in 22edo or 29edo, the pergen's up = 1 edostep. But in 37edo or 44edo, ^1 = 2 edosteps. For half-8ve in 12edo, ^1 = 0 edosteps, and the ups and downs in the score can simply be ignored. In fact, it seems every pergen in 5edo, 7edo and 12edo has ^1 = 0 edosteps. It's not yet known why.

When notating a piece in a specific pergen meant to be played in a specific EDO, either the pergen notation or the EDO notation can be used. For small or mid-sized EDOs, they're usually identical. If one has to choose, the pergen notation is generally preferred. It's less cluttered. Also, it's easier to mentally double every up and down in larger EDOs than it is to halve them in smaller EDOs.

Half-8ve in 22edo has P = vA4. But in 16edo, P = A4 + 2 edosteps, and ^1 = -2 edosteps. Negative edosteps means up is down, and should be avoided. The notation has tipped, and the period should be notated as ^A4, making ^1 = 2 edostep. Even better would be P = ^4, making ^1 = 1 edostep.

Half-5th has EU = vvA1, hence any EDO in which A1 = 4 edosteps will have ^1 = 2 edosteps. These "doubled EDOs" are 20, 27, 34, 41, 48, 55, etc. The "tripled EDOs" with A1 = 6 edosteps and ^1 = 3 edosteps are every 7th EDO from 30 to 72.

Half-4th has EU = vvm2. Doubled EDOs have m2 = 4 edosteps. These are 23, 28, 33, 38, 43, 48, 53, etc. Tripled EDOs are every 5th one from 47 to 72.

Third-4th has EU = v3A1. Doubled EDOs are the same ones as half-5th's tripled EDOs. Third-5th has EU = v3m2. Doubled EDOs are the same as half-4th's tripled EDOs.

The relationship between a pergen's up and an EDO's up when the pergen uses double-pair notation is complex. Consider half-everything, notated with ups/downs in the perchain and lifts/drops in the genchain. 10edo has ^1 = 1 edostep and /1 = 0 edosteps, thus one simply ignores lifts and drops. But for 24edo, ^1 = 0 edosteps and /1 = 1 edostep. One must ignore ups and downs and convert lifts/drops to edosteps.

'''Pergens Within An EDO'''

See the screenshots in the Supplemental Materials section for a complete list of which pergens are supported by 12, 15 and 19 edo. The list includes multiple pergens per period/generator combination. The next table shows only the simplest pergen for each combination. Combinations are excluded if the period and generator are not coprime, because the scale is contained in a smaller edo. For example, 15edo has no unsplit pergen, because those scales are contained in 5edo. Periods of 1 or 2 edosteps are excluded as too trivial, because the scale is the entire edo. Every step of the edo appears in the genchains, even if the chains are only one step long.

To find the pergen, find the edo, then find the row that corresponds to the period, then find the column that corresponds to the generator. It appears, but has yet to be formally proven, that the number of P/G combinations that N-edo supports is (N-1)/2 rounded down. If we include the trivial combination where P = 1 edostep, the number of combinations would be N/2 rounded up.

{| class="wikitable" style="text-align:center;"
|-
! | EDO
! | Period
! colspan="11" | Generator in edosteps
|-
! |
! | in edosteps
! | 1
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
|-
! | 5
! | 5 = P8
| | P4/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 6
! | 6 = P8
| | P4/2
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 7
! | 7 = P8
| | P4/3
| | P5/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 8
! | 8 = P8
| | P4/3
| | -
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/2
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 9
! | 9 = P8
| | P4/4
| | P4/2
| | -
| | P5
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/3
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 10
! | 10 = P8
| | P4/4
| | -
| | P5/2
| | -
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 5 = P8/2
| | P5
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 11
! | 11 = P8
| | P4/5
| | P5/3
| | P5/2
| | P11/4
| | P5
| |
| |
| |
| |
| |
| |
|-
! | 12
! | 12 = P8
| | P4/5
| | -
| | -
| | -
| | P5
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/2
| | P5
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/3
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/4
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 13b
! | 13 = P8
| | P4/6
| | P4/3
| | P4/2
| | P12/5
| | P12/4
| | P5
| |
| |
| |
| |
| |
|-
! | 14
! | 14 = P8
| | P4/6
| | -
| | P4/2
| | -
| | P11/4
| | -
| |
| |
| |
| |
| |
|-
! | "
! | 7 = P8/2
| | P5
| | P4/3
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 15
! | 15 = P8
| | P4/6
| | P4/3
| | -
| | P12/6
| | -
| | -
| | P11/3
| |
| |
| |
| |
|-
! | "
! | 5 = P8/3
| | P5
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/5
| | P4/3
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 16
! | 16 = P8
| | P4/7
| | -
| | P5/3
| | -
| | P12/5
| | -
| | P5
| |
| |
| |
| |
|-
! | "
! | 8 = P8/2
| | P5
| | -
| | P5/3
| | -
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/4
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 17
! | 17 = P8
| | P4/7
| | P5/5
| | P11/8
| | P11/6
| | P5/2
| | P11/4
| | P5
| | P11/3
| |
| |
| |
|-
! | 18b
! | 18 = P8
| | P4/8
| | -
| | -
| | -
| | P5/2
| | -
| | P12/4
| | -
| |
| |
| |
|-
! | "
! | 9 = P8/2
| | P5
| | P4/4
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/3
| | P5/2
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/6
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 19
! | 19 = P8
| | P4/8
| | P4/4
| | P11/9
| | P4/2
| | P12/6
| | P12/5
| | ccP5/7
| | P5
| | P11/3
| |
| |
|-
! | 20
! | 20 = P8
| | P4/8
| | -
| | P5/4
| | -
| | -
| | -
| | P11/4
| | -
| | c3P5/8
| |
| |
|-
! | "
! | 10 = P8/2
| | M2/4
| | -
| | P5/4
| | -
| | -
| |
| |
| |
| |
| |
| |
|-
! | "
! | 5 = P8/4
| | P4/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/5
| | P5/4
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 21
! | 21 = P8
| | P4/9
| | P5/6
| | -
| | P5/3
| | P11/6
| | -
| | -
| | c3P4/9
| | -
| | P11/3
| |
|-
! | "
! | 7 = P8/3
| | P5/2
| | P5
| | P4/3
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/7
| | P5/3
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 22
! | 22 = P8
| | P4/9
| | -
| | P4/3
| | -
| | P12/7
| | -
| | P12/5
| | -
| | P5
| | -
| |
|-
! | "
! | 11 = P8/2
| | M2/4
| | P5
| | P4/3
| | P12/5
| | P12/7
| |
| |
| |
| |
| |
| |
|-
! | 23
! | 23 = P8
| | P4/10
| | P4/5
| | P11/11
| | P12/9
| | P4/2
| | P12/6
| | ccP4/8
| | ccP4/7
| | P12/4
| | P5
| | P11/3
|-
! | 24
! | 24 = P8
| | P4/10
| | -
| | -
| | -
| | P4/2
| | -
| | P5/2
| | -
| | -
| | -
| | c4P5/10
|-
! | "
! | 12 = P8/2
| | M2/4
| | -
| | -
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
|-
! | "
! | 8 = P8/3
| | P5/2
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/4
| | P4/2
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/6
| | P4/2
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/8
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! |
! |
! | 1
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
|}
Larger edos can have very awkward pergens. For example, 31edo with G = 14/31 is (P8, c5P4/12). It's much simpler to think of the generator as the downfifth = 17\31, and the pergen as the pseudopergen (P8, v5).

'''EDO-pair names'''

Just as a pair of edos and a prime subgroup can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo & N'-edo, m = GCD (N,N'). The period P equals both (N/m)\N and (N'/m)\N'. For example, for 12edo and 16edo, m = 4, and the period is both 3\12 and 4\16.

For each edo, find the nearest '''edomapping''' (patent val) for the 2.3 subgroup. If the edo has a "b" wart, e.g. 13b-edo, use the second-nearest edomapping. Form a 2x2 matrix from these edomappings. Let d be the determinant of this matrix. If d = m, -m or 0, the generator is the 5th, and the pergen is simply (P8/m, P5).

For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.

If |d| ≠ m or 0, P5 is not the generator, and we must find the multigen. Take the ratio of the two edos N/N' and reduce it by m. In the scale tree ([http://tallkite.com/misc_files/Scale-Tree-Complete.pdf pdf] or [http://tallkite.com/misc_files/Scale-Tree-Complete.jpg jpeg]), let g/g' be the smallest ancestor of this ratio. The generator G maps to both g\N and g'\N'. The two edos come closest to coinciding here.

For example, for 7-edo and 17-edo, m = 1 but d = 2, so G ≠ P5. The smallest ancestor of 7/17 is 2/5. G maps to both 2\7 and 5\17, which are both neutral 3rds. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths. 2\7 = 343¢ and 5\17 = 353¢, and their difference is only 1\N", where N" = LCM (N, N'). This 10¢ difference is the least difference between any 7edo note and any 17edo note, except that the two neutral 6ths differ by the same 10¢, and of course some note pairs coincide exactly, such as 0\7 and 0\17.

Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found from the scale tree, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this new vector with either edomapping is zero. Treating this new vector as a monzo, since the dot product is zero, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for. If the comma is (a, b, c), then a·P8 + b·P5 + c·G = 0, and G = (b-a, -b) / c.

For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2).

To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 (Dicot aka Yoyo). 11/9 also works, it yields 243/242 (Mohajira aka Lulu).

If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a multi-EDO pergen.

If the 1st edo is 7edo, the pergen indicates the natural heptatonic notation for the 2nd edo, e.g. 12edo = unsplit, 15edo = third-4th and 17edo = half-5th.

The closer two edos are in the scale tree, the smaller the splitting fractions in the pergen they make. Examples:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 12-edo
! | 13b-edo
! | 14-edo
! | 15-edo
! | 16-edo
! | 17-edo
! | 18b-edo
! | 19-edo
! | 20-edo
|-
! | 13b-edo
| | (P8, P5/7)
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 14-edo
| | (P8/2, P5)
| | (P8, P4/6)
| |
| |
| |
| |
| |
| |
| |
|-
! | 15-edo
| | (P8/3, P5)
| | (P8, c5P5/12)
| | (P8, P4/6)
| |
| |
| |
| |
| |
| |
|-
! | 16-edo
| | (P8/4, P5)
| | (P8, P12/5)
| | (P8/2, P5)
| | (P8, P5/9)
| |
| |
| |
| |
| |
|-
! | 17-edo
| | (P8, P5)
| | (P8, ccP5/11)
| | (P8, P11/4)
| | (P8, P11/3)
| | (P8/2, P4/7)
| |
| |
| |
| |
|-
! | 18b-edo
| | (P8/6, P5)
| | (P8, P12/4)
| | (P8/2, P4/2)
| | (P8/3, P12/4)
| | (P8/2, P5)
| | (P8, P5/10)
| |
| |
| |
|-
! | 19-edo
| | (P8, P5)
| | (P8, P12/10)
| | (P8, P4/2)
| | (P8, P12/6)
| | (P8, P12/5)
| | (P8, P11/3)
| | (P8, P4/8)
| |
| |
|-
! | 20-edo
| | (P8/4, P5)
| | (P8, ccP4/16)
| | (P8/2, P5/4)
| | (P8/5, ^1)
| | (P8/4, P5/3)
| | (P8, P11/4)
| | (P8/2, P4/8)
| | (P8, P4/8)
| |
|-
! | 21-edo
| | (P8/3, P5)
| | (P8, c3P4/9)
| | (P8/7, ^1)
| | (P8/3, P4/3)
| | (P8, P5/3)
| | (P8, P11/6)
| | (P8/3, P5/2)
| | (P8, P11/3)
| | (P8, P5/12)
|-
! | 22-edo
| | (P8/2, P5)
| | (P8, c3P4/15)
| | (P8/2, P4/3)
| | (P8, P4/3)
| | (P8/2, P12/5)
| | (P8, P5)
| | (P8/2, P12/7)
| | (P8, P12/5)
| | (P8/2, M2/4)
|-
! | 23-edo
| | (P8, P4/5)
| | (P8, ccP4/8)
| | (P8, P4/2)
| | (P8, P12/12)
| | (P8, P5)
| | (P8, P12/9)
| | (P8, P12/4)
| | (P8, P12/6)
| | (P8, c5P5/16)
|-
! | 24-edo
| | (P8/12, ^1)
| | (P8, c6P4/14)
| | (P8/2, P4/2)
| | (P8/3, P4/2)
| | (P8/8, P5)
| | (P8, P5/2)
| | (P8/6, P4/2)
| | (P8, P4/2)
| | (P8/4, P4/2)
|}

A specific pergen can be converted to an edo pair by finding the range of its generator cents in the [[pergen#Further Discussion-Notating tunings with an arbitrary generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of edos 7, 10 and 17 defines (P8, P5/2).

==Array Keyboards (unfinished)==

Array keyboards have a 2-dimensional layout of keys, and are very appropriate for rank-2 tunings. A good layout can be found from the tuning's pergen. First find an edo N-edo that is compatible with the pergen, then arrange the keys in N columns to the 8ve, with each row usually containing the multigen interval. The unsplit pergen in 7 columns:

{| class="wikitable" style="text-align:center;"
|-
| | D#
| |
| |
| |
| |
| |
| |
| |
|-
| | D
| | E
| | F#
| | G#
| | A#
| |
| |
| |
|-
| | Db
| | Eb
| | F
| | G
| | A
| | B
| | C#
| | D#
|-
| |
| |
| |
| | Gb
| | Ab
| | Bb
| | C
| | D
|-
| |
| |
| |
| |
| |
| |
| |
| | Db
|}
Higher notes are at the top of each column. The rows would actually be angled so that the two D's are at the same horizontal level. The vertical steps are A1 and the horizontal steps are M2, and the keyboard is defined as 7(A1, M2).

The third-4th keyboard is 7(A1/3 = ^1 = vvA1, P4/3 = vM2 = ^^m2).

{| class="wikitable" style="text-align:center;"
|-
| | vD#
| | ^E
| | F#
| | vG#
| | ^A
| | B
| | vC#
| | ^D
|-
| | ^D
| | E
| | vF#
| | ^G
| | A
| | vB
| | ^C
| | D
|-
| | D
| | vE
| | ^F
| | G
| | vA
| | ^B
| | C
| | vD
|-
| | vD
| | ^Eb
| | F
| | vG
| | ^Ab
| | Bb
| | vC
| | ^Db
|}

''Hypothesis: Let the 5th's keyspan (i.e. column-span) be F. In order for the keyboard to have the pitches in order, the fifth must fall between the two Stern-Brocot ancestors of F\N (simplified if possible). For example, an 8-column keyboard has F = 5, the ancestors of 5\8 are 3\5 and 2\3, and the 5th must be between 720¢ and 800¢. Thus the most musically useful N values are 5, 7, 10, 12 and 14.''

(more to come)

==Supplemental materials==

===Notation guide PDF===

This PDF is a rank-2 notation guide that shows the full lattice for the first 32 pergens, up through the quarter-splits block. It also includes the single-split pergens from the fifth-split, sixth-split and seventh-split blocks. It includes alternate EUs for many pergens.

[http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf '''<big>TallKite.com/misc_files/notation guide for rank-2 pergens.pdf</big>''']

{| class="wikitable" style="text-align:center;"
|+Table of contents for the N'''otation Guide for Rank-2 Pergens''' (* indicates a true double)
|-
! colspan="3" |unsplit
! colspan="3" |quarter-splits
! colspan="3" |single-split fifth-splits
! colspan="3" |single-split seventh-splits
|-
!1
|(P8, P5)
|unsplit
!16
|(P8/4, P5)
|quarter-8ve
!33
|(P8/5, P5)
|fifth-8ve
!96
|(P8/7, P5)
|seventh-8ve
|-
! colspan="3" |half-splits
!17
|(P8, P4/4)
|quarter-4th
!34
|(P8, P4/5)
|fifth-4th
!97
|(P8, P4/7)
|seventh-4th
|-
!2
|(P8/2, P5)
|half-8ve
!18
|(P8, P5/4)
|quarter-5th
!35
|(P8, P5/5)
|fifth-5th
!98
|(P8, P5/7)
|seventh-5th
|-
!3
|(P8, P4/2)
|half-4th
!19
|(P8, P11/4)
|quarter-11th
!36
|(P8, P11/5)
|fifth-11th
!99
|(P8, P11/7)
|seventh-11th
|-
!4
|(P8, P5/2)
|half-5th
!20
|(P8, P12/4)
|quarter-12th
!37
|(P8, P12/5)
|fifth-12th
!100
|(P8, P12/7)
|seventh-12th
|-
!5
|(P8/2, P4/2) *
|half-everything *
!21
|(P8/4, P4/2) *
|quarter-8ve, half-4th *
!38
|(P8, ccP4/5)
|fifth-coco-4th
!101
|(P8, ccP4/7)
|seventh-coco-4th
|-
! colspan="3" |third-splits
!22
|(P8/2, M2/4)
|half-8ve, quarter-tone
! colspan="3" |single-split sixth-splits
!102
|(P8, ccP5/7)
|seventh-coco-5th
|-
!6
|(P8/3, P5)
|third-8ve
!23
|(P8/2, P4/4) *
|half-8ve, quarter-4th *
!64
|(P8/6, P5)
|sixth-8ve
!103
|(P8, c3P4/7)
|seventh-trico-4th
|-
!7
|(P8, P4/3)
|third-4th
!24
|(P8/2, P5/4) *
|half-8ve, quarter-5th *
!65
|(P8, P4/6)
|sixth-4th
| colspan="3" rowspan="9" |
|-
!8
|(P8, P5/3)
|third-5th
!25
|(P8/4, P4/3)
|quarter-8ve, third-4th
!66
|(P8, P5/6)
|sixth-5th
|-
!9
|(P8, P11/3)
|third-11th
!26
|(P8/4, P5/3)
|quarter-8ve, third-5th
!67
|(P8, P11/6)
|sixth-11th
|-
!10
|(P8/3, P4/2)
|third-8ve, half-4th
!27
|(P8/4, P11/3)
|quarter-8ve, third-11th
!68
|(P8, P12/6)
|sixth-12th
|-
!11
|(P8/3, P5/2)
|third-8ve, half-5th
!28
|(P8/3, P4/4)
|third-8ve, quarter-4th
!69
|(P8, ccP4/6)
|sixth-coco-4th
|-
!12
|(P8/2, P4/3)
|half-8ve, third-4th
!29
|(P8/3, P5/4)
|third-8ve, quarter-5th
!70
|(P8, ccP5/6)
|sixth-coco-5th
|-
!13
|(P8/2, P5/3)
|half-8ve, third-5th
!30
|(P8/3, P11/4)
|third-8ve, quarter-11th
| colspan="3" rowspan="3" |
|-
!14
|(P8/2, P11/3)
|half-8ve, third-11th
!31
|(P8/3, P12/4)
|third-8ve, quarter-12th
|-
!15
|(P8/3, P4/3) *
|third-everything *
!32
|(P8/4, P4/4) *
|quarter-everything *
|}Screenshots of the first 2 pages:

[[File:pergens_1.png|alt=pergens 1.png|704x948px|pergens 1.png]]

[[File:pergens_2.png|alt=pergens 2.png|704x760px|pergens 2.png]]

===PergenLister===

PergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonic unisons for each one. Alternate EUs are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair.

http://www.tallkite.com/apps/pergenLister.html (written in Jesusonic, runs inside Reaper or ReaJS)

The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. EUs of a 3rd or more are in red.

Screenshots of the first 170 pergens:

[[File:alt-pergenLister_1.png|852x852px|alt-pergenLister 1.png]]

[[File:Alt-pergenLister 2a.png|frameless|852x852px]]

[[File:Alt-pergenLister 3.png|frameless|854x854px]]

The first 39 pergens supported by 12edo:

[[File:alt-pergenLister_12edo.png|857x857px|alt-pergenLister 12edo.png]]

Some of the pergens supported by 15edo. A red asterisk means partial support, e.g. (P8, P5) only uses a 5edo subset of 15edo.

[[File:alt-pergenLister_15edo.png|854x854px|alt-pergenLister 15edo.png]]

Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.

[[File:alt-pergenLister_19edo.png|857x857px|alt-pergenLister 19edo.png]]

The first 54 imperfect pergens:

[[File:Imperfect pergens.png|frameless|863x863px]]

Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:

If z > 0, then y is at least ceiling (x·(3/2 - z/2)) and at most floor (x·5/3)

If z < 0, then y is at least ceiling (x·3/2) and at most floor (x·(5/3 - z/2))

Next we loop through all combinations of x and z in such a way that larger values of x and z come last:

i = 1; loop (maxFraction,

<ul><li>j = 1; loop (i - 1,<ul><li>makeMapping (i, j); makeMapping (i, -j);</li><li>makeMapping (j, i); makeMapping (j, -i);</li><li>j += 1;</li></ul></li><li>);</li><li>makeMapping (i, i); makeMapping (i, -i);</li><li>i += 1;</li></ul>);

The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.

==Various proofs (unfinished)==

Although not yet rigorously proven, the two false-double tests have been empirically verified by pergenLister.

The interval P8/2 has a "ratio" of the square root of 2, which equals 21/2, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the '''pergen matrix''' [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.

Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise a, b, and n could all be reduced by GCD (a,b), and the multigen could be simplified. Since GCD (a, b) = 1 and -am/b is an integer, it follows that m must be a multiple of |b| as well.

Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|) = |b| · r. If r = 1, then GCD (m, n) = |b|, and vice versa, which is the proposed test for a false double.

If m = |b|, is the pergen explicitly false? Does splitting (a,b) into n generators also split P8 into m periods?

(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5

Because n is a multiple of b, n/b is an integer

M/b = (n/b)·M/n = (n/b)·G 
(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)

Let c and d be the bezout pair of a+b and b, with c·(a+b) + d·b = 1

Since the pergen is a double-split, m > 1, therefore |b| > 1, therefore c ≠ 0

c·(a+b)·P8 = c·b·((n/b)·G - P5) 
(1 - d·b)·P8 = c·b·((n/b)·G - P5) 
P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5) 
P8/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)

Therefore P8 is split into m periods 
Therefore if m = |b|, the pergen is explicitly false

Assume the pergen is a false double, and there's a comma C that splits both P8 and (a,b) appropriately. Can we prove r = 1? Let Q = the higher prime that C uses. Express P, G and C as monzos of the prime subgroup 2.3.Q, by expanding the 2x2 pergen matrix to a 3x3 matrix A:

P = (1/m, 0, 0) 
G = (a/n, b/n, 0) 
C = (u, v, w)

Here u, v and w are integers. If GCD (u, v, w) > 1, simplify C so that it = 1. The inverse of A expresses 2, 3 and Q in terms of P, G and C. If C is tempered out, the C column can be discarded, making the usual 3x2 period-generator mapping. However, if C is not tempered out, the inverse of A is a 3x3 period-generator-comma mapping, which is simply a change of basis. For example, 5-limit JI can be generated by 2/1, 3/2 and 81/80. Here is the inverse of A:

2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) 
3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C) 
Q = Q/1 = ((av-bu)m/wb, -vn/wb, 1/w) · (P, G, C)

Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. Every pergen except the unsplit one requires |w| > 1, so the last column almost always has a fraction. To avoid fractions in the first two columns, A must be unimodular '''''[I think, not sure]''''', and we have wb/mn = ±1, and w = ±mn/b. Substituting for w, we have:

2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) 
3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C) 
Q = Q/1 = (±(av-bu)/n, ±(-v)/m, ±b/mn) · (P, G, C)

For v/m to be an integer, v must equal i·m for some integer i. Likewise, av-bu must equal j·n for some integer j. Thus bu = av - jn = iam - jn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb. Substituting, we get bu = iaprb - jqrb, and u = r(iap - jq). Furthermore, v = im = iprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r > 1, this contradicts the requirement that GCD (u, v, w) = 1, therefore r must be 1, and GCD (m, n) = |b|, and all false doubles pass the false-double test.

Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q?

Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b.

a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)

Thus a·P8 splits into b parts, and since m = |b|, a·P8 splits into m parts. Proceed as before with a bezout pair to find the monzo for P8/m.

Next, assume the pergen isn't explicitly false. The unreduced form is (P8/m, (n - am, -bm) / mn). Substituting in m = pb and n = qb, with p and q coprime, we get (P8/m, (q - ap, -pb) / pqb).

Assume the pergen is a true double, and r > 1. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.

Given a pergen (P8/m, (a,b)/n), how many parts is an arbitrary interval (a',b') split into?

(a',b') = (a'·b, b'·b) / b = (a'·b - a·b', 0) / b + (a·b', b'·b) / b = (a'·b - a·b')·P8 / b + b'·(a,b) / b = (a'·b - a·b')·(m/b)·P + b'·(n/b)·G

Therefore (a',b') is split into GCD (a'·b - a·b')·(m/b), b'·(n/b)) parts.

If m = 1, then b = ±1, and we have GCD (a' ± a·b', b'·n)

If n = 1, then a = -1 and b = 1, and we have GCD (a'·m + b'·m, b') = GCD (a'·m, b')

If m = 1 and n = 1, we have GCD (a', b') = the naturally occurring split.

If m = n (nth-everything), we have n · GCD (a', b')

The multigen and the arbitrary interval can be expressed as gedras:

(a,b) = [k,s] = (-11k+19s, 7k-12s)

(a',b') = [k',s'] = (-11k'+19s', 7k'-12s')

a'·b - a·b' works out to be k·s' - k'·s, and we have GCD ((k·s' - k'·s)·m/b, b'·n/b)

If s is a multiple of n (happens when EU is an A1) and s' is a multiple of n, let s = x·n and s' = y·n

GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')

Thus every such interval is split, e.g. the half-5th pergen splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. This is an easy way to determine if an interval is split or not by the pergen.

To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts

if r > 1, it's a true double

a·P8 = (a,0) = (a,b) - (0,b) = M - b·P12

M = n·G = qrb·G

a·P8 = qrb·G - b·P12 = b·(qr·G - P12)

Let c and d be the bezout pair of a and b, with c·a + d·b = 1

If |b| = 1, let c = 1 and d = ±a, to avoid c = 0

ca·P8 = cb·(qr·G - P12)

(1 - d·b)·P8 = c·b·(qr·G - P12)

P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) = b · (d,-c) + bcqr·G

== Glossary ==
to find the definition, use control-F to search for the first (bolded) occurrence of the word on this page.

pergen 
split 
multigen 
ups and downs (the ^ and v symbols) 
higher prime (any prime > 3) 
color depth 
dependent/independent 
square mapping 
lifts and drops (the / and \ symbols) 
enharmonic unison, EU 
uninflected 
genchain 
perchain 
compound (increased by an octave) 
single-split, double-split 
single-pair, double-pair (number of new accidentals in the notation) 
true double, false double 
explicitly false 
unreduced 
alternate vs. equivalent (generator or period) 
mapping comma 
keyspan 
stepspan 
gedra 
count 
mid 
edomapping 
upspan 
liftspan

chain number 
single-chain 
multi-chain 
arrow comma

==Miscellaneous Notes==

=== Combining pergens ===
Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). If adding a comma to a temperament doesn't change the pergen, it's a strong extension, otherwise it's a weak extension.

General rules for combining pergens:

<ul><li>(P8/m, M/n) + (P8, P5) = (P8/m, M/n)</li>
<li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li>
<li>(P8/m, P5) + (P8/m', P5) = (P8/m", P5), where m" = LCM (m,m')</li>
<li>(P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n')</li></ul>

However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.

If a false double pergen can be broken down into two simpler ones, that may help with finding a double-pair notation with an EU of a 2nd or less. For example, sixth-4th's single pair notation has an EU of a 4th. But since sixth-4th is half-4th plus third-4th, and those two have a good EU, sixth-4th can be notated with one pair from half-4th and another from third-4th.

=== Expanding gedras ===
Gedras can be expanded to 5-limit or higher by including another keyspan that is compatible with 7 and 12, such as 9 or 16. But a more useful approach is for the third number to be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]:

k = 12a + 19b + 28c + 34d 
s = 7a + 11b + 14c + 20d 
g = -c 
r = -d

a = -11k + 19s - 4g + 6r 
b = 7k - 12s + 4g - 2r 
c = -g 
d = -r

Gedras can also be expanded by adding an entry for ups/downs. The entry shows the '''upspan''', which is the number of ups the interval has. Downed intervals have a negative upspan. An entry for '''liftspan''' can also be added. For example, vM3 = [4,2,-1], and ^^\4 = [5,3,2,-1].

=== Height of a pergen ===
The LCM of the pergen's two splitting fractions could be called the '''height''' of the pergen. For example, (P8, P5) has height 1, and (P8/2, M2/4) has height 4. In single-pair notation, the EU's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.

=== Generalizing the pergen ===
See [[User:AthiTrydhen/Abstract pergens]]

=== Credits ===
Pergens were discovered by [[KiteGiedraitis|Kite Giedraitis]] in 2017, and developed with the help of [[Praveen Venkataramana]].

== Addenda (late 2023) ==
=== New terminology===
All temperaments have a '''chain number''', which is the number of fifthchains in the temperament's lattice. Any (P8, P5) temperament has a chain number of 1, and is '''single-chain'''. All other pergens are '''multi-chain'''. For example, Porcupine/Triyoti has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Saguguti has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m * n / f, where M is the multigen and f is the absolute value of M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.

===The EU(s) define the pergen===
The pergen can be derived directly from the EU(s). Thus the EU(s) define both the pergen and the notation. An EU can be thought of as a comma in the 2.3.^ subgroup. One derives a mapping from this comma as one would for any 3-prime comma, and one derives the pergen by inverting the first 2 columns of the mapping.

For example, if the EU is vvd2, the 2.3.^ monzo is [19 -12 -2]. There are many possible mappings, but only one that gives a canonical pergen: [(2 2 7) (0 1 -6)]. Discarding the last column and inverting gives us [(1/2 0) (-1 1)] = (P8/2, P5). Another example: vvA1 = [-11 7 -2]. The mapping [(1 1 -2) (0 2 7)] reduces to [(1 1) (0 2)], which inverts to [(1 0) (-1/2 1/2)] = (P8, P5/2).

One can explore the universe of possible EUs, and thus possible pergen notations, more easily if using the gedra format, expressing the EU as a combination of A1's, d2's and arrows. Thus vvA1 = [1 0 -2], v3m2 = [1 1 -3], etc. Unlike a conventional 2.3.^ monzo, the first two numbers in a A1.d2.^1 monzo are fairly small, and the second number is never negative (since it's an EU). And we can require that the first two numbers be coprime (see the next section). All this facilitates one's search.

===Simplifying a "squared" EU===
Consider an uninflected EU of AA1. AA1 is "squared" in the sense that AA1 = A1 + A1, thus both numbers in its ratio are square numbers. If it had an even upspan (the number of ups), it would obviously be an invalid EU, for if v4AA1 = 0¢, then so does vvA1, and v4AA1 could be replaced with vvA1. So the upspan must be odd.

Consider an EU of v3AA1. The pergen is (P8, P4/3). Here are the [[twin squares]].

<math>
\begin{array} {rrr}
\text{P8} \\
\text{^m2} \\
\text{vvvAA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & {\color {Green}0} \\
8 & -5 & {\color {Green}1} \\
\hline
{\color {Red}-22} & {\color {Red}14} & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & {\color {Red}2} \\
0 & -3 & {\color {Red}-14} \\
\hline
{\color {Green}0} & {\color {Green}-1} & -5 \\
\end{array} \right]
</math>

The two red numbers in the lower left are both even, because AA1 is a squared ratio. As a result, the two red numbers in the upper right must both be even to ensure their rows' dot products with the EU are zero, which is an even number. If we halve all the red numbers and double all the green numbers, all the various row dot products will be unchanged, and the twin squares will remain valid:

<math>
\begin{array} {rrr}
\text{P8} \\
\text{^^m2} \\
\text{vvvA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & {\color {Green}0} \\
8 & -5 & {\color {Green}2} \\
\hline
{\color {Red}-11} & {\color {Red}7} & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & {\color {Red}1} \\
0 & -3 & {\color {Red}-7} \\
\hline
{\color {Green}0} & {\color {Green}-2} & -5 \\
\end{array} \right]
</math>

But while the EU has become simpler, the generator has become more complex in that it is dup not up. This is remedied by adding the EU to it. Changes are in red:

<math>
\begin{array} {rrr}
\text{P8} \\
\text{vM2} \\
\text{vvvA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & 0 \\
{\color {Red}-3} & {\color {Red}2} & {\color {Red}-1} \\
\hline
-11 & 7 & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & 1 \\
0 & -3 & -7 \\
\hline
{\color {Red}0} & {\color {Red}1} & {\color {Red}2} \\
\end{array} \right]
</math>

Following this procedure, it's always possible to simplify a squared (or cubed, etc.) EU.

===Arrow commas===
The '''[[arrow]] comma''' is the ratio that equals the up [[arrow]]. This term overlaps a lot with the term mapping comma, but it isn't quite identical to it. For 5-limit temperaments the arrow comma is almost always 81/80, and for 7-limit it's almost always 64/63. But other commas can occur.

Consider Triyoti/Porcupine which is (P8, P4/3). The vanishing comma or '''VC''' is 250/243, which has a 2.3.5 monzo of [2 -5 3]. The EU is v3A1, which has a 2.3.^ monzo of [-11 7 -3]. The arrow comma or '''AC''' equals an up, therefore it vanishes when downed. The downed AC (or '''vAC''') can be expressed as a 2.3.5.^ monzo. For Triyoti/Porcupine with an EU of v3A1, the vAC is v(81/80) or [-4 4 -1 -1].

===The three commas ===
Thus when we consider a single-comma temperament along with its notation, there are three commas of interest, the VC, the vAC and the EU. In a 5-limit rank-2 temperament, they can all be expressed as 2.3.5.^ monzos.

Any two of these 3 commas determines the third comma. Thus we can approach temperaments and notations from 6 different angles. We can start with any one of these three commas and give it a specific monzo. Then we can choose one of the other two commas, experiment with a variety of specific monzos and examine the results. Let's start with specifying the VC, experimenting with the vAC and seeing what the EU turns out to be.

The EU always equals the VC (possibly inverted) plus or minus some number of vAC's. That number is whatever is needed to eliminate the higher prime. The VC must be inverted if the resulting EU would otherwise have a negative stepspan, or is a diminished unison.

In our Triyoti example, 250/243 plus 3 downed syntonic commas = v3A1. As 2.3.5.^ monzos, we have [1 -5 3 0] + 3·[-4 4 -1 -1] = [-11 7 0 -3]. Note the zeros. The VC always has a zero arrow-count and the EU always has a zero prime-5-count.

Visualizing the three commas in the lattice, one starts at the VC and heads towards the row of 3-limit intervals via the vAC. Wherever one lands is the uninflected EU. One can try other AC's besides 81/80. The AC's prime-5-count must be ±1, so [[135/128|Layobi]] (135/128) or [[Schisma|Layo]] (the schisma) are possibilities. But either of these would lead to a very remote EU (v3AA1 and v3d44 respectively), making a very awkward notation.

Next let's specify the AC, experiment with the VC and see what the EU turns out to be. For 5-limit temperaments, one can require that the AC always be 81/80, and derive the EU (and thus the notation) from the VC. For example, Saguguti/Diaschismic has VC = [11 -4 -2 0]. Subtracting two vAC's makes [19 -12 0 2] = ^^d2. This is in fact the recommended EU for (P8/2, P5).

More examples: Laquinyoti/Magic is (P8, P12/5) and has VC = [-10 -1 5 0]. Adding five vAC's makes [-30 19 0 -5]. This appears to be a AA7 but is actually a negative 2nd. We invert to get [30 -19 0 5] = ^5dd2. Guguti/Bug has VC = 27/25 = [0 3 -2 0]. Subtracting two vAC's makes [8 -5 0 2] = ^^m2. Again, this is the recommended EU.

Let's try a 7-limit temperament with the obvious vAC of 64/63 = [6 -2 -1 -1]. Zozoti/Semaphore has VC = 49/48 = [-4 -1 2 0]. Adding two vAC's makes [8 -5 0 -2] = vvm2.{{todo|complete section|comment=apply to multi-comma temperaments|inline=1}}

== Addenda (late 2024) ==

=== Chord names ===
When naming chords, it's very convenient to have the freedom to rename an aug 4th as a dim 5th, or a minor 10th as an aug ninth. Thus for some pergens, an extra pair of accidentals is used. Some examples:

* [[Chords of meantone]] (P8, P5) (^1 = -d2 = pythagorean comma)
* [[Chords of diaschismic]] (P8/2, P5)
* [[Chords of hemififths]] (P8, P5/2) (/1 = vm2 = ~81/80 = ~64/63)
* [[Chords of porcupine]] (P8, P4/3)
* [[Chords of magic]] (P8, P12/5) (/1 = ^^d2)

=== Frequency of imperfect pergens ===
Imperfect pergens occur when there are multiple genchains (i.e. the octave is split), and the fifth is on a different genchain than the tonic, and also on a different perchain. How often do they occur? In order to answer that, we need to survey all pergens in order. But the question of how to do that depends on how they are sorted. The pergenLister app sorts them by the size of the larger denominator. Using this order, pergenLister finds about 4% of all pergens are imperfect. But they can also be sorted by their canonical mappings [(a b) (0 c)]. If sorted by a (octave fraction), and then by |c| (perfect multigen's fraction), more complex pergens appear sooner, and the percentage rises to about 25%.

This table lists all pergens with an unsplit octave up to the fifth-splits. In each column, the pergens are sorted by the size of the generator. The generator is listed followed by a, b and c from its mapping. All pergens with an unsplit octave are perfect.
{| class="wikitable"
|+Pergens of the form (P8, x), showing generator and mapping (a = 1)
!unsplit
!half-splits
!third-splits
!quarter-splits
!fifth-splits
!sixth-splits
|-
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (1 1 1)
|P4/2 (1 2 -2)
|P4/3 (1 2 -3)
|P4/4 (1 2 -4)
|P4/5 (1 2 -5)
|P4/6 (1 2 -6)
|-
|
|P5/2 (1 1 2)
|P5/3 (1 1 3)
|P5/4 (1 1 4)
|P5/5 (1 1 5)
|P5/6 (1 1 6)
|-
|
|
|P11/3 (1 3 -3)
|P11/4 (1 3 -4)
|P11/5 (1 3 -5)
|P11/6 (1 3 -6)
|-
|
|
|
|P12/4 (1 0 4)
|P12/5 (1 0 5)
|P12/6 (1 0 6)
|-
|
|
|
|
|ccP4/5 (1 4 -5)
|ccP4/6 (1 4 -6)
|-
|
|
|
|
|
|ccP5/6 (1 -1 6)
|}
Of all the half-octave pergens, half of every other column (i.e. 25%) are imperfect. Imperfect pergens occur whenever b is not a multiple of a.
{| class="wikitable"
|+Pergens of the form (P8/2, x), showing generator and mapping (a = 2)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (2 2 1)
|'''M2/4 (2 3 2)'''
|P4/3 (2 4 -3)
|'''M2/8 (2 3 4)'''
|P4/5 (2 4 -5)
|'''M2/12 (2 3 6)'''
|-
|
|P4/2 (2 4 -2)
|P5/3 (2 2 3)
|P4/4 (2 4 -4)
|P5/5 (2 2 5)
|P4/6 (2 4 -6)
|-
|
|
|P11/3 (2 6 -3)
|P5/4 (2 2 4)
|P11/5 (2 6 -5)
|P5/6 (2 2 6)
|-
|
|
|
|'''cm7/8 (2 5 -4)'''
|P12/5 (2 0 5)
|P11/6 (2 6 -6)
|-
|
|
|
|
|ccP4/5 (2 8 -5)
|'''cm7/12 (2 5 -6)'''
|-
|
|
|
|
|
|'''cM9/12 (2 1 6)'''
|}
Note that some of these pergens, when put in mingen form, become imperfect. For example, (P8/2, P11/3) becomes (P8/2, M2/6). Also note that for many of these pergens, the generators are comma-sized, and MOS scales will either be very "hard" (L/s very large) or else will contain very many notes per octave. For example, to bring the L/s ratio down to about 5, (P8/2, M2/4) needs a 16 note scale, and (P8/2, P11/3) needs a 28 note scale!

Of all the third-octave pergens, two-thirds of every third column (2/9 or 22%) are imperfect:
{| class="wikitable"
|+Pergens of the form (P8/3, x), showing generator and mapping (a = 3)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (3 3 1)
|P4/2 (3 6 -2)
|'''m3/9 (3 5 -3)'''
|P4/4 (3 6 -4)
|P4/5 (3 6 -5)
|'''m3/18 (3 5 -6)'''
|-
|
|P5/2 (3 3 2)
|'''M6/9 (3 4 3)'''
|P5/4 (3 3 4)
|P5/5 (3 3 5)
|'''M6/18 (3 4 6)'''
|-
|
|
|P4/3 (3 6 -3)
|P11/4 (3 9 -4)
|P11/5 (3 9 -5)
|P4/6 (3 6 -6)
|-
|
|
|
|P12/4 (3 0 4)
|P12/5 (5 0 5)
|P5/6 (3 3 6)
|-
|
|
|
|
|ccP4/5 (3 12 -5)
|'''ccm3/18 (3 7 -6)'''
|-
|
|
|
|
|
|'''ccM6/18 (3 2 6)'''
|}
Of all the quarter-octave pergens, imperfection occurs in half of every 4th column and 3/4 of every 4th column (5/16 or 31.25%).
{| class="wikitable"
|+Pergens of the form (P8/4, x), showing generator and mapping (a = 4)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
!c = ±7
!c = ±8
|-
|P5 (4 4 1)
|'''m6/8 (4 7 -2)'''
|P4/3 (4 8 -3)
|'''M2/8 (4 6 4)'''
|P4/5 (4 8 -5)
|P4/6 (4 8 -6)
|P4/7 (4 8 -7)
|'''M2/16 (4 6 8)'''
|-
|
|P4/2 (4 8 -2)
|P5/3 (4 4 3)
|'''m6/16 (4 7 -4)'''
|P5/5 (4 4 5)
|P5/6 (4 4 6)
|P5/7 (4 4 7)
|'''m6/32 (4 7 -8)'''
|-
|
|
|P11/3 (4 12 -3)
|'''M10/16 (4 5 4)'''
|P11/5 (4 12 -5)
|P11/6 (4 12 -6)
|P11/7 (4 12 -7)
|'''M10/32 (4 5 8)'''
|-
|
|
|
|P4/4 (4 8 -4)
|P12/5 (4 0 5)
|'''m6/24 (4 7 -6)'''
|P12/7 (4 0 7)
|P4/8 (4 8 -8)
|-
|
|
|
|
|ccP4/5 (4 16 -5)
|'''M10/24 (4 5 6)'''
|ccP4/7 (4 16 -7)
|P5/8 (4 4 8)
|-
|
|
|
|
|
|'''ccm6/24 (4 9 -6)'''
|ccP5/7 (4 -4 7)
|'''ccm6/32 (4 9 -8)'''
|-
|
|
|
|
|
|
|c3P4/7 (4 20 -7)
|'''cm7/16 (4 10 -8)'''
|-
|
|
|
|
|
|
|
|'''c3M3/32 (4 3 8)'''
|}
Percentage of imperfect pergens in each category:
{| class="wikitable"
|+
!(P8, x)
!(P8/2, x)
!(P8/3, x)
!(P8/4, x)
!(P8/5, x)
!(P8/6, x)
!(P8/7, x)
|-
|none
|1/4
|2/9
|5/16
|4/25
|5/12
|6/49
|-
|0%
|25%
|22.22%
|31.25%
|16%
|41.67%
|12.24%
|}

== Addenda (Spring 2026) ==

=== Initial commas ===
As one stacks generators and octave-reduces, at some point one overshoots or undershoots the octave by an interval of about a quartertone or less. This small interval is the pergen's initial comma. For example, (P8, P5)'s initial comma is the pythagorean comma, its next comma is Mercator's comma, etc. Each comma has a certain genspan, here 12 and 53. The genspan of the initial comma limits the size of a scale one can construct, assuming one wants to avoid overly-small steps. Thus one can have a pythagorean scale of up to 12 notes, but a 13-note scale will have a very small step. Note that the genspan gives the maximum notes per period, not per octave. Assuming one also wants to avoid extreme L/s step ratios also limits the maximum notes per octave.

The table below lists the initial comma of various pergens. "±" indicates a tippy pergen. "c" is the difference between the fifth and 7\12. "abs(6c)" means the absolute value of 6c. The dim 2nd is a pythagorean comma.
{| class="wikitable sortable"
|+Initial comma of each pergen
!#
!pergen
!interval
!cents
!genspan
!notes per octave
|-
!1
!(P8, P5)
|±d2
|abs(12c)
|±12G
|12
|-
!2
!(P8/2, P5)
|±d2/2
|abs(6c)
|±6G
|12
|-
!3
!(P8, P4/2)
|m2/2
|50¢ - 2.5c
|5G
|5
|-
!4
!(P8, P5/2)
|A1/2
|50¢ + 3.5c
|7G
|7
|-
!5
!(P8/2, P4/2)
| colspan="2" | ''same as #3 (P8, P4/2)''
|5G
|10
|-
!6
!(P8/3, P5)
|±d2/3
|abs(4c)
|±4G
|12
|-
!7
!(P8, P4/3)
|A1/3
|33.3¢ + 2.33c
| -7G
|7
|-
!8
!(P8, P5/3)
|m2/3
|33.3¢ - 1.67c
| -5G
|5
|-
!9
!(P8, P11/3)
|M2/3
|66.7¢ + 0.67c
|2G
|2 (or >= 14)
|-
!10
!(P8/3, P4/2)
|A2/6
|50¢ + 1.5c
|3G
|9
|-
!11
!(P8/3, P5/2)
|m3/6
|50¢ - 0.5c
|1G
|3
|-
!12
!(P8/2, P4/3)
| colspan="2" | ''same as #7 (P8, P4/3)''
| -7G
|14
|-
!13
!(P8/2, P5/3)
| colspan="2" | ''same as #8 (P8, P5/3)''
| -5G
|10
|-
!14
!(P8/2, P11/3)
|M2/6
|33.3¢ + 0.33c
|1G
|2
|-
!15
!(P8/3, P4/3)
| colspan="2" | ''same as #7 (P8, P4/3)''
| -7G
|21
|-
!16
!(P8/4, P5)
|±d2/4
|abs(3c)
|±3G
|12
|-
!17
!(P8, P4/4)
| colspan="2" | ''same as #3 (P8, P4/2)''
|10G
|10
|-
!18
!(P8, P5/4)
|A1/4
|25¢ + 1.75c
|7G
|7
|-
!19
!(P8, P11/4)
|dd3/4
|25¢ - 4.25c
| -17G
|17
|-
!20
!(P8, P12/4)
|m2/4
|25¢ - 1.25c
| -5G
|5
|-
!21
!(P8/4, P4/2)
|M2/4
|50¢ + c/2
|G
|4
|-
!22
!(P8/2, M2/4)
|M2/4
|50¢ + c/2
|G
|2
|-
!23
!(P8/2, P4/4)
|m2/4
|25¢ - 1.25c
|5G
|10
|-
!24
!(P8/2, P5/4)
| colspan="2" |''same as #18 (P8, P5/4)''
|7G
|14
|-
!25
!(P8/4, P4/3)
|d4/12
|33.3¢ - 0.67c
|2G
|8
|}
The initial comma of #9 (P8, P11/3) is about 67¢, which is not too small to be a scale step. But if there are more than 2 notes per 8ve, the L/s ratio becomes enormous. The ratio only becomes reasonable (roughly 3) when there are at least 14 notes per octave.

Note the initial comma is often equivalent to the uninflected EU divided by the height. For example, (P8, P4/2) has comma m2/2 and EU vvm2. In other words, the up-arrow is often the initial comma.

This suggests a new algorithm for finding a good EU for a pergen. Search the cents table (in the Notation Guide For rank-2 Pergens pdf, the first table of each pergen) for a small step. The search can be easily done by computer. Then derive the EU from that small step.

For example, pergen #25 (P8/4, P4/3) has a 33¢ step at 2G - P. Thus ^1 = 2G - P. Multiplying 2G by 3 gives us unsplit 4ths, and multiplying P by 4 gives us unsplit octaves. Thus we must multiply the up-arrow by 12 to get an unsplit 3-limit interval. 12 ups = 24G - 12P = 8P4 - 3P8 = dim 4th. Thus the EU is v12d4, and ^12C = Fb. But the pergenLister program lists the single-pair notation for this pergen as having an EU of ^12d94. Thus the pergenLister algorithm missed a much simpler EU, and hence a much simpler notation.

Not all initial commas imply a valid notation. For pergen #66 (P8, P5/6), the initial comma is P - 10G = m2/3 = 33.3¢ - 1.67c. The implied EU is v3m2. But this notation is incomplete because it only notates 3 of the 6 fifthchains. There must be 6 arrows in the EU, not 3. Or else there must be a second EU with 2 arrows. Fortunately, the next comma on the genchain is 21G - 2P = A1/2 = 50¢ - 3.5c, which implies a double-pair notation with EU' = \\A1. This is the notation found by pergenLister.

True doubles require double-pair notation and thus require finding two commas.

[[Category:Regular temperament theory]]
[[Category:Notation]]
[[Category:Pages with proofs]]

File:Alt-pergenLister 2a.png

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pergenLister screenshot

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pergenLister screenshot

File:Imperfect pergens.png

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pergenLister screenshot

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Comma and diesis

2026-05-20T02:51:27Z

TallKite:

: ''This article is about "comma" and "diesis" as interval regions. For other senses of these two words, see [[comma]] and [[diesis]].''
{{Infobox interval region
| Name=Comma, diesis
| Cents lower=0
| Cents upper=40
| Cents upper wide=60
| JI intervals=81/80, 128/125
| Complement=(Imperfect) [[octave]]
| Lower region=[[Unison]]
| Higher region=[[Semitone (interval region)|Semitone]]
}}

"Comma" and "diesis" are two terms used to refer to intervals that are less than about 60{{cent}} in size. In terms of [[interval region]]s, "comma" refers to an interval flatter than about 30{{cent}}, and "diesis" refers to an interval between about 30 and 60{{cent}}. In [[Sagittal notation]], a comma is specifically defined as between half of the [[Pythagorean comma]] {{monzo| -19 12}} and half of the Pythagorean 17-fifths diesis {{monzo| 27 -17}}, about 11.7{{c}} to 33.4{{c}}, and a diesis is defined as between the comma upper bound and half of the Pythagorean 19-fifths apotome-plus-comma {{monzo| -30 19}}, about 68.6{{c}}.

"[[Comma]]" also refers to an interval that is tempered out by any given [[temperament]].

The range of dieses largely overlaps with the range of [[quartertone]]s (between 40 and 60{{c}}, reasonably mapped to 1/24edo), which, according to systems that determine consonance in terms of proximity to simple just ratios, is one of the most dissonant interval regions. This also corresponds to an [[interseptimal]] interval range. However, quarter tones are still covered here to provide a resource for them in the same format as the other interval region pages.

In the diatonic scale, the analogous concepts are '''subchromatic''' and '''enharmonic''' steps. A subchromatic step (a "comma") does not change the interval category (for example, in most just notation systems, if you flatten the major third [[81/64]] by an [[81/80]] comma to produce [[5/4]], the latter is still considered a major third). Diatonically, subchromatic steps are '''perfect unisons (P1)''', and there are none that are not a unison in a rank-2 diatonic tuning. An enharmonic step (a "diesis", although this is controversial) changes the interval category to an enharmonic interval (for example, a major third to a diminished fourth, or a chromatic semitone to a diatonic semitone). Similarly, enharmonic steps are ascending or descending '''diminished seconds (d2)'''.

== In just intonation ==
=== By prime limit ===
In just intonation, commas are often seen as the difference between two similar intervals, so it is hard to find intervals within this range that are treated as steps in their own right. The 3-limit interval in this range is the Pythagorean comma of [[531441/524288]], which can be considered an augmented seventh (octave-reduced), and is about 23{{c}}.

For the remainder of this list, intervals are provided that are ''not'' mostly treated as commas (in the temperament sense). Higher-limit intervals in the comma and diesis range are:

* The 5-limit '''augmented diesis''' is a ratio of 128/125, and is about 41{{c}}.
** There is also the 5-limit '''magic comma''' of 3125/3072, which is about 30{{c}}.
* The 7-limit '''slendro diesis''' is a ratio of 49/48, and is about 36{{c}}.
* The 11-limit '''quarter tone''' is a ratio of 33/32, and is about 53{{c}}.
* The 13-limit '''minor diesis''' is a ratio of 40/39, and is about 43{{c}}.

=== By delta ===
As comma and diesis is the smallest interval class, it may be represented by:

* Any delta-1 (i.e. superparticular) interval smaller than 29/28
* Any delta-2 interval smaller than 57/55
* Any delta-3 interval smaller than 88/85

== In EDOs ==
The following table lists the best tuning of 128/125, and other dieses or commas if present, in various significant [[edos|EDOs]]. Not included are EDOs (i.e. those smaller than 15) where the best tuning is the unison, or 0{{c}}, or those where the best tuning is sharper than 60{{c}} (i.e. not a diesis or comma). Note that this does not depend on how each EDO tunes the intervals that 128/125 might be derived from, only on which edostep is closest to 128/125's size.

{| class="wikitable"
|-
! EDO
! 128/125
! Other commas and dieses
|-
| 22
| 54{{c}}
|
|-
| 24
| 50{{c}}
| 50¢ ≈ 33/32
|-
| 25
| 48{{c}}
|
|-
| 26
| 46{{c}}
|
|-
| 27
| 44{{c}}
|
|-
| 29
| 41{{c}}
|
|-
| 31
| 39{{c}}
|
|-
| 34
| 35{{c}}
|
|-
| 41
| 29{{c}}
| {{nowrap|59{{c}} ≈ 33/32}}
|-
| 53
| 45{{c}}
| {{nowrap|22{{c}} ≈ 81/80}}
|}

== In regular temperaments ==
The role of commas and dieses in regular temperaments is often as the intervals that are tempered out (i.e. equated to 0 cents). Discussing that is not within the scope of this article; you may learn more at [[Regular temperament]].

However, there are, rarely, temperaments generated by commas. One example is [[slender]], where a stack of ten [[49/48]]'s equals [[5/4]].

== In MOS scales ==
Intervals less than 100{{c}} generate the following [[mos|MOS]] scales:

These tables start from the last monolarge MOS generated by the interval range.

Scales with more than 12 notes are not included.

{| class="wikitable"
|-
! Range
! MOS
|-
| 0–100{{c}}
| [[1L 11s]]
|}

{{Navbox intervals}}

Kite's thoughts on pergens

2026-05-14T23:41:05Z

TallKite: /* Addenda (Spring 2026) */

A '''pergen''' (pronounced "peer-jen", from '''per'''iod and '''gen'''erator) is a way of classifying a [[regular temperament]] solely by its [[periods and generators|period and generator(s)]]. For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. Every rank-2, rank-3, rank-4, etc. temperament has a pergen. Assuming the prime [[subgroup]] includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a fifth or some fraction of some 3-limit interval. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every [[strong extension]] of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but do not uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.

Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka Triyoti]] has the 5th sharpened by one-fifth of [[250/243]] ({{monzo| 1 -5 3 }}). Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Triguti tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name. (Note these uses of comma fractions are not convention universally; temperament pages tend to use comma fractions to indicate the damage to the generator rather than the fifth. This is analogous to indicating the amount of stretching of an edo by the damage to the edostep rather than to the octave. But the latter approach is much more common, because the damage to the octave is much more audible and thus much more musically relevant than the damage to an edostep, which often doesn't correspond to a simple ratio. Likewise for pergens: the damage to Triyoti/Porcupine's generator, which is both 10/9 and 11/10, is less relevant than the damage to Triyoti's 4th or 5th.)

Overwhelmed? See [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf ''Notation guide for rank-2 pergens''] for practical notation examples.

{{See also| Rank-2 temperaments by mapping of 3 }}

= Definition =
If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. Both fractions are always of the form 1/N, thus the octave and/or the 3-limit interval is '''split''' into N parts. The interval which is split into multiple generators is the '''multigen'''. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.

For example, the Srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. Dicot aka Yoyoti (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine aka Triyoti is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore aka Zozoti, a pun on "semi-fourth", is of course half-fourth.

Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both Srutal aka Saguguti and Injera aka Gu & Biruyoti sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using '''ups and downs''' (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and downs notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.

The largest category contains all single-comma rank-2 temperaments with a comma of the form 2x 3y P or 2x 3y P-1, where P is a prime > 3 (a '''higher prime'''), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called '''unsplit'''.

Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multigen, and as a tie-breaker, to minimize the size in cents of the multigen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.

For example, Srutal could be (P8/2, M2/2), but P5 is preferred because it is unsplit. Or it could be (P8/2, P12), but P5 is preferred because it is smaller. Or it could be (P8/2, P4), but P5 is always preferred over P4. Note that P5/2 is not preferred over P4/2. For example, Decimal is (P8/2, P4/2), not (P8/2, P5/2).

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! colspan="4" | example temperaments
|-
! | written
! | spoken
! | comma(s)
! | name
! colspan="2" | [[Color notation|color name]]
|-
| | (P8, P5)
| | unsplit
| | 81/80
| | Meantone
| | Guti
| | gT
|-
| | "
| | "
| | 64/63
| | Archy
| | Ruti
| | rT
|-
| | "
| | "
| | (-14,8,1)
| | Schismic
| | Layoti
| | LyT
|-
| | (P8/2, P5)
| | half-8ve
| | (11, -4, -2)
| | Srutal
| | Saguguti
| | sggT
|-
| | "
| | "
| | 81/80, 50/49
| | Injera
| | Gu & Biruyoti
| | g&rryyT
|-
| | (P8, P5/2)
| | half-5th
| | 25/24
| | Dicot
| | Yoyoti
| | yyT
|-
| | "
| | "
| | (-1,5,0,0,-2)
| | Mohajira
| | Luluti
| | 1uuT
|-
| | (P8, P4/2)
| | half-4th
| | 49/48
| | Semaphore
| | Zozoti
| | zzT
|-
| | (P8/2, P4/2)
| | half-everything
| | 25/24, 49/48
| | Decimal
| | Yoyo & Zozoti
| | yy&zzT
|-
| | (P8, P4/3)
| | third-4th
| | 250/243
| | Porcupine
| | Triyoti
| | y3T
|-
| | (P8, P11/3)
| | third-11th
| | (12,-1,0,0,-3)
| | Satrilu
| | Satriluti
| | s1u3T
|-
| | (P8/4, P5)
| | quarter-8ve
| | (3,4,-4)
| | Diminished
| | Quadguti
| | g4T
|-
| | (P8/2, M2/4)
| | half-8ve quarter-tone
| | (-17,2,0,0,4)
| | Laquadlo
| | Laquadloti
| | L1o4T
|-
| | (P8, P12/5)
| | fifth-12th
| | (-10,-1,5)
| | Magic
| | Laquinyoti
| | Ly5T
|}

(P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum/difference of octaves and fifths, every single 3-limit interval is also split in half.

The multigen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: bi- splits something into two parts, tri- into three parts, etc. For a comma with monzo (a,b,c,d...), the '''color depth''' is GCD (c,d...).

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[Kite's_color_notation|Color notation]] (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yo, (a,b,-1) = gu, (a,b,0,1) = zo, (a,b,0,-1) = ru, (a,b,0,0,1) = lo/ilo, (a,b,0,0,-1) = lu, and (a,b) = wa. Examples: 5/4 = y3 = yo 3rd, 7/5 = zg5 = zogu 5th, etc.

For example, Marvel aka Ruyoyoti (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). But colors can be replaced with ups and downs, to be higher-prime-agnostic. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.

More examples: Trizoguti (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Biruyoti (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, Biruyoti Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.

A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with microtonal accidentals: (P8, P5, ^1, /1,...).

=Derivation=

For any comma, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents, where GCD (0,x) = x. The comma will split the octave into m parts, and if n > m, it will split some 3-limit interval into n parts.

In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is '''dependent''' on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two '''independent''' primes.

For example, consider Sawa & Ruyoyoti (2.3.5.7 with commas 256/243 and 225/224). The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Notating multi-EDO pergens|multi-EDO pergens]] below).

To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [http://x31eq.com/temper/uv.html x31eq.com/temper/uv.html] that will find such a matrix, it's the reduced mapping. Next make a '''square mapping''' by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.

For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.

2/1 = P8 = x·P, thus P = P8/x

3/1 = P12 = y·P + z·G, thus G = [P12 - y·(P8/x)] / z = [-y·P8 + x·P12] / xz = (-y, x) / xz

M's 3-limit monzo is (-y, x), or (y, -x) if z is negative. To get alternate generators, add i periods to G, with i ranging from -x (subtracting a full octave) to +x (adding a full octave).

G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz

'''The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| <= x'''

A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen.

For example, Porcupine aka Triyoti (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 <= i <= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).

Rank-3 pergens are trickier to find. For example, Breedsmic aka Bizozoguti is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7 x31.com] gives us this matrix:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 5/1
! | 7/1
|-
! | period
| | 1
| | 1
| | 1
| | 2
|-
! | gen1
| | 0
| | 2
| | 1
| | 1
|-
! | gen2
| | 0
| | 0
| | 2
| | 1
|}

Thus 2/1 = P, 3/1 = P + 2·G1, 5/1 = P + G1 + 2·G2, and 7/1 = 2·P + G1 + G2. Discard the last column to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 5/1
|-
! | period
| | 1
| | 1
| | 1
|-
! | gen1
| | 0
| | 2
| | 1
|-
! | gen2
| | 0
| | 0
| | 2
|}

Use an [http://wims.unice.fr/wims/wims.cgi?session=GF84B8C7BF.1&lang=en&cmd=reply&module=tool%2Flinear%2Fmatmult.en&matA=1+1+1%0D%0A0+2+1%0D%0A0+0+2&matB=&show=A%5E-1 online tool] to invert it. Here "/4" means that each entry is to be divided by the determinant of the last matrix, which is 4.

{| class="wikitable" style="text-align:center;"
|-
! |
! | period
! | gen1
! | gen2
! |
|-
! | 2/1
| | 4
| | -2
| | -1
| |
|-
! | 3/1
| | 0
| | 2
| | -1
| |
|-
! | 5/1
| | 0
| | 0
| | 2
| | /4
|}

Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25:6)/4.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a double octave to the multigen. The alternate gens are P11/2 and P19/2, both of which are much larger, so the best gen1 is P5/2.

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96:25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128:75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675:512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.

Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 7/1
|-
! | period
| | 1
| | 1
| | 2
|-
! | gen1
| | 0
| | 2
| | 1
|-
! | gen2
| | 0
| | 0
| | 1
|}

This inverts to this matrix:

{| class="wikitable" style="text-align:center;"
|-
! |
! | period
! | gen1
! | gen2
! |
|-
! | 2/1
| | 2
| | -1
| | -3
| |
|-
! | 3/1
| | 0
| | 1
| | -1
| |
|-
! | 7/1
| | 0
| | 0
| | 2
| | /2
|}

Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, /1) with /1 = 64/63, rank-3 half-5th. This is far better than (P8, P5/2, (96:25)/4).

The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is > 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible. Using 7 instead of 5 in the pergen is very common for rank-3. See [[User:TallKite/Catalog of eleven-limit rank three temperaments with Color names]] for more examples.

=Applications=

Pergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources.

Another obvious application is to categorize regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), Semihemi is (P8/2, P4/2), Triforce is (P8/3, P4/2), both Tetracot and Semihemififths are (P8, P5/4), Fourfives is (P8/4, P5/5), Pental is (P8/5, P5), and Fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, either because it contains more than 2 primes, or because the split multigen isn't actually a generator. For example, Sensei, or Semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.

Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example Porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first Porcupine is Triyoti, and the second one is Triyo & Ruti. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.

Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See [[pergen#Further Discussion-Chord names and scale names|Chord names and scale names]] below.

The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.

All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and downs notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Luluti and Dicot aka Yoyoti are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.

One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, which is octave-equivalent, fifth-generated and heptatonic. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and all notation used here is backwards compatible.

Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, Schismic aka Layoti is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C vE G. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below.

Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multigen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a [[pergen#Further Discussion-Supplemental materials-Notaion guide PDF|notation guide]]. It even allows every pergen to be numbered.

The '''[[enharmonic unison]]''', or more briefly the '''EU''', can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's EU. The pergen and the EU together define the notation. (''Edited to add: not quite accurate, see the Addenda.'')

The '''genchain''' (chain of generators) in the table is only a short section of the full genchain.

C - G implies ...Eb Bb F C G D A E B F# C#...C - ^Eb=vE - G implies ...F -- ^Ab=vA -- C -- ^Eb=vE -- G -- ^Bb=vB -- D...

If the octave is split, the table has a '''perchain''' ("peer-chain", chain of periods) that shows the octave: C -- vF#=^Gb -- C. Genchains have a theoretically infinite length, but perchains have a finite length. The full rank-2 lattice has genchains running horizontally and perchains running vertically.

{| class="wikitable" style="text-align:center;"
|-
! |
! | pergen
! | enharmonic
unison(s)
! | equivalence(s)
! | split
interval(s)
! | perchain(s) and/or
genchains(s)
! | examples
|-
| | 1
| | (P8, P5)
unsplit
| | none
| | none
| | none
| | C - G
| | Pythagorean, Meantone, Dominant,
Schismic, Mavila, Archy, etc.
|-
! |
! | half-splits
! |
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5)
half-8ve
| | ^^d2 (if 5th
> 700¢
| | ^^C = B#
| | P8/2 = vA4 = ^d5
| | C - vF#=^Gb - C
| | Srutal aka Saguguti
^1 = 81/80
|-
| |
| | "
| | vvd2 (if 5th

< 700¢)
| | ^^C = Db
| | P8/2 = ^A4 = vd5
| | C - ^F#=vGb - C
| | Injera aka Gu & Biruyoti

^1 = 64/63
|-
| |
| | "
| | vvM2
| | ^^C = D
| | P8/2 = ^4 = v5
| | C - ^F=vG - C
| | Thothoti, if 13/8 = M6

^1 = 27/26
|-
| | 3
| | (P8, P4/2)

half-4th
| | vvm2
| | ^^C = Db
| | P4/2 = ^M2 = vm3
| | C - ^D=vEb - F
| | Semaphore aka Zozoti

^1 = 64/63
|-
| |
| | "
| | ^^dd2
| | ^^C = B##
| | P4/2 = vA2 = ^d3
| | C - vD#=^Ebb - F
| | Lala-yoyoti

^1 = 81/80
|-
| | 4
| | (P8, P5/2)

half-5th
| | vvA1
| | ^^C = C#
| | P5/2 = ^m3 = vM3
| | C - ^Eb=vE - G
| | Mohajira aka Luluti

^1 = 33/32
|-
| | 5
| | (P8/2, P4/2)

half-

everything
| | \\m2,

vvA1,

^^\\d2,

vv\\M2
| | //C = Db

^^C = C#

^^//C = D
| | P4/2 = /M2 = \m3

P5/2 = ^m3 = vM3

P8/2 = v/A4 = ^\d5

= ^/4 = v\5
| | C - /D=\Eb - F,

C - ^Eb=vE - G,

C - v/F#=^\Gb - C,

C - ^/F=v\G - C
| | Zozo & Luluti

^1 = 33/32

/1 = 64/63
|-
| |
| | "
| | ^^d2,

\\m2,

vv\\A1
| | ^^ C= B#

//C = Db

^^//C = C#
| | P8/2 = vA4 = ^d5

P4/2 = /M2 = \m3

P5/2 = ^/m3 = v\M3
| | C - vF#=^Gb - C,

C - /D=\Eb - F,

C - ^/Eb=v\E - G
| | Sagugu & Zozoti

^1 = 81/80

/1 = 64/63
|-
| |
| | "
| | ^^d2,

\\A1,

^^\\m2
| | ^^C = B#

//C = C#

^^\\C = B
| | P8/2 = vA4 = ^d5

P5/2 = /m3 = \M3

P4/2 =v/M2 = ^\m3
| | C - vF#=^Gb - C,

C - /Eb=\E - G,

C - v/D=^\Eb - F
| | Sagugu & Luluti

^1 = 81/80

/1 = 33/32
|-
! |
! | third-splits
! |
! |
! |
! |
! |
|-
| | 6
| | (P8/3, P5)

third-8ve
| | ^3d2
| | ^3C = B#
| | P8/3 = vM3 = ^^d4
| | C - vE - ^Ab - C
| | Augmented aka Triguti

^1 = 81/80
|-
| | 7
| | (P8, P4/3)

third-4th
| | v3A1
| | ^3C = C#
| | P4/3 = vM2 = ^^m2
| | C - vD - ^Eb - F
| | Porcupine aka Triyoti

^1 = 81/80
|-
| | 8
| | (P8, P5/3)

third-5th
| | v3m2
| | ^3C = Db
| | P5/3 = ^M2 = vvm3
| | C - ^D - vF - G
| | Slendric aka Latrizoti

^1 = 64/63
|-
| | 9
| | (P8, P11/3)

third-11th
| | ^3dd2
| | ^3C = B##
| | P11/3 = vA4 = ^^dd5
| | C - vF# - ^Cb - F
| | Satriluti, if 11/8 = A4

^1 = 729/704
|-
| |
| | "
| | v3M2
| | ^3C = D
| | P11/3 = ^4 = vv5
| | C - ^F - vC - F
| | Satriluti, if 11/8 = P4

^1 = 33/32
|-
| | 10
| | (P8/3, P4/2)

third-8ve, half-4th
| | v6A2
| | ^6C = D#
| | P8/3 = ^^m3 = v4A4

P4/2 = ^3m2 = v3M3
| | C - ^^Eb - vvA - C

C - ^3Db=v3E - F
| | Tribiloti, if 11/8 = P4

^1 = 33/32
|-
| |
| | "
| | ^3d2,

\\m2
| | ^3C = B#

//C = Db
| | P8/3 = vM3 = ^^d4

P4/2 = /M2 = \m3
| | C - vE - ^Ab - C

C - /D=\Eb - F
| | Triforce aka Trigu & Zozoti

^1 = 81/80, /1 = 64/63
|-
| | 11
| | (P8/3, P5/2)

third-8ve, half-5th
| | ^3d2

\\A1
| | ^3C = B#

//C = C#
| | P8/3 = vM3 = ^^d4

P5/2 = /m3 = \M3
| | C - vE - ^Ab - C

C - /Eb=\E - G
| | Satribizoti

^1 = 49/48, /1 = 343/324
|-
| | 12
| | (P8/2, P4/3)

half-8ve, third-4th
| | ^^d2

\3A1
| | ^^C = Dbb

/3C = C#
| | P8/2 = vA4 = ^d5

P4/3 = \M2 = //m2
| | C - vF#=^Gb - C

C - \D - /Eb - F
| | Latribiruti

^1 = 1029/1024, /1 = 49/48
|-
| | 13
| | (P8/2, P5/3)

half-8ve, third-5th
| | ^6d32
| | ^6C = B#3
| | P8/2 = v3AA4 = ^3dd5

P5/3 = vvA2 = ^4dd3
| | C - v3Fx=^3Gbb C

C - vvD# - ^^Fb - G
| | Latribiyoti

^1 = 81/80
|-
| |
| | "
| | ^^d2,

\3m2
| | ^^C = B#

/3C = Db
| | P8/2 = vA4 = ^d5

P5/3 = /M2 = \\m3
| | C - vF#=^Gb - C

C - /D - \F - G
| | Lemba aka Latrizo & Biruyoti

^1 = (10,-6,1,-1), /1 = 64/63
|-
| | 14
| | (P8/2, P11/3)

half-8ve, third-11th
| | v6M2
| | ^6C = D
| | P8/2 = ^34 = v35

P11/3 = ^^4 = v45
| | C - ^3F=v3G - C

C - ^^F - vvC - F
| | Latribiloti, if 11/8 = P4

^1 = 33/32
|-
| | 15
| | (P8/3, P4/3)

third-

everything
| | v3d2,

\3A1
| | ^3C = Dbb

/3C = C#
| | P8/3 = ^M3 = vvd4

P4/3 = \M2 = //m2

P5/3 = v/M2
| | C - ^E - vAb - C

C - \D - /Eb - F

C - v/D - ^\F - G
| | Triyo & Triguti

^1 = 64/63

/1 = 81/80
|-
| |
| | "
| | ^3d2,

\3m2
| | ^3C = B#

/3C = Db
| | P8/3 = vM3 = ^^d4

P5/3 = /M2 = \\m3

P4/3 = v\M2
| | C - vE - ^Ab - C

C - /D - \F - G

C - v\D - ^/Eb - F
| | Trigu & Latrizoti

^1 = 81/80

/1 = 64/63
|-
| |
| | "
| | v3A1,

\3m2
| | ^3C = C#

/3C = Db
| | P4/3 = vM2 = ^^m2

P5/3 = /M2 = \\m3

P8/3 = v/M3
| | C - vD - ^Eb - F

C - /D - \F - G

C - v/E - ^\Ab - C
| | Triyo & Latrizoti

^1 = 81/80

/1 = 64/63
|-
! |
! | quarter-splits
! |
! |
! |
! |
! |
|-
| | 16
| | (P8/4, P5)
| | ^4d2
| | ^4C = B#
| | P8/4 = vm3 = ^3A2
| | C vEb vvGb=^^F# ^A C
| | Diminished aka Quadguti
|-
| | 17
| | (P8, P4/4)
| | ^4dd2
| | ^4C = B##
| | P4/4 = ^m2 = v3AA1
| | C ^Db ^^Ebb=vvD# vE F
| | Negri aka Laquadyoti
|-
| | 18
| | (P8, P5/4)
| | v4A1
| | ^4C = C#
| | P5/4 = vM2 = ^3m2
| | C vD vvE=^^Eb ^F G
| | Tetracot aka Saquadyoti
|-
| | 19
| | (P8, P11/4)
| | v4dd3
| | ^4C = Eb3
| | P11/4 = ^M3 = v3dd5
| | C ^E ^^G# vDb F
| | Squares aka Laquadruti
|-
| | 20
| | (P8, P12/4)
| | v4m2
| | ^4C = Db
| | P12/4 = v4 = ^3M3
| | C vF vvBb=^^A ^D G
| | Vulture aka Sasa-quadyoti
|-
| |
| | etc.
| |
| |
| |
| |
| |
|}
The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably it isn't particularly complex.

Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢).

==Tipping points==

Removing the ups and downs from an EU makes an '''uninflected''' EU, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s EU is ^^d2, the uninflected EU is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the EU vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore '''up may need to be swapped with down, depending on the size of the 5th''' in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' EUs are upped or downed as if the 5th were just.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every uninflected EU, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the uninflected EU.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | uninflected EU
! | 3-exponent
! | tipping

point edo
! | edo's 5th
! | upping range
! | downing range
! | if the 5th is just
|-
| | M2
| | C - D
| | 2
| | 2-edo
| | 600¢
| | none
| | all
| | downed
|-
| | m3
| | C - Eb
| | -3
| | 3-edo
| | 800¢
| | none
| | all
| | downed
|-
| | m2
| | C - Db
| | -5
| | 5-edo
| | 720¢
| | none
| | all
| | downed
|-
| | A1
| | C - C#
| | 7
| | 7-edo
| | ~686¢
| | 600-686¢
| | 686¢-720¢
| | downed
|-
| | d2
| | C - Dbb
| | -12
| | 12-edo
| | 700¢
| | 700-720¢
| | 600-700¢
| | upped
|-
| | dd3
| | C - Eb3
| | -17
| | 17-edo
| | ~706¢
| | 706-720¢
| | 600-706¢
| | downed
|-
| | dd2
| | C - Db3
| | -19
| | 19-edo
| | ~695¢
| | 695-720¢
| | 600-695¢
| | upped
|-
| | d34
| | C - Fb3
| | -22
| | 22-edo
| | ~709¢
| | 709-720¢
| | 600-709¢
| | downed
|-
| | d32
| | C - Db4
| | -26
| | 26-edo
| | ~692¢
| | 692-720¢
| | 600-692¢
| | upped
|-
| | d44
| | C - Fb4
| | -29
| | 29-edo
| | ~703¢
| | 703-720¢
| | 600-703¢
| | downed
|-
| | d43
| | C - Eb5
| | -31
| | 31-edo
| | ~697¢
| | 697-720¢
| | 600-697¢
| | upped
|-
| | etc.
| |
| |
| |
| |
| |
| |
| |
|}

=Further Discussion=

==Naming very large intervals==

So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, adding an 8ve is indicated by "c" for '''compound''' (a conventional music theory term). Thus 10/3 = cM6 = compound major 6th, 9/2 = ccM2 or cM9, etc. For a pergen with an unsplit octave, the multigen is always some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen M is less than 3 octaves, and can be P4, P5, P11, P12, ccP4 or ccP5. The last one can be spoken as "coco-fifth". Tripe compound can be spoken as "trico".

==Secondary splits==

Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. Porcupine aka Triyoti) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with:

P4/3: C - vD - ^Eb - F

A4:/3 C - D - E - F# (the lack of ups and downs indicates that this interval was already split into 3 parts)

m7/3: C - ^Eb - vG - Bb (because m7 is already split into halves, we also have m7/6: C - vD - ^Eb - F - vG - ^Ab - Bb)

M7/3: C - vE - ^G - B

m10/3: C - F - Bb - Eb (m10 is already split into 3 parts, thus m10/9 also occurs)

M10/3: C - ^F - vB - E

Of course, an equal-step melody can span other intervals besides 3-limit ones. Simply extend or cut short the earlier examples to find other melodies:

^m3/2: C - vD - ^Eb (^m3 = 6/5)

^m6/5: C - vD - ^Eb - F - vG - ^Ab (^m6 = 8/5)

vm9/4: C - ^Eb - vG - Bb - ^Db (vm9 = 32/15)

vM7/2: C - ^F - vB (vM7 = 15/8, probably more harmonious than M7 = 243/128)

More remote intervals include A1, d4, d7 and d10. These unfortunately require a very long genchain. The most interesting melodically is A1: C - ^C - vC# - C#. From C to C# is seven 5ths, which equals 21 generators, so the genchain would have to contain 22 notes if it had no gaps. Note that A1 is the uninflected EU of third-4th. The uninflected EU is always a secondary split.

For a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occurring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.

Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof [[pergen#Further Discussion-Various proofs|below]]). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If only the 8ve is split, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the EU is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. "Every" means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.

The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occurring splits are listed too, under "all pergens". (P8/3, P4/2) has all the secondary splits that (P8/3, P5) and (P8, P4/2) have, plus additional ones.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! | secondary splits of a 12th or less
|-
| colspan="2" | all pergens
| | M3/2, A4/3, d5/2, A5/4, m7/2, M9/2, m10/3, A11/2
|-
! colspan="2" | half-splits
! |
|-
| | (P8/2, P5)
| | half-8ve
| | M2/2, M3/4, A4/6, A5/8, m6/2, M10/2, d12/2, A12/2
|-
| | (P8, P4/2)
| | half-4th
| | m2/2, M6/2, m7/4, A8/2, m10/6, A11/4, P12/2
|-
| | (P8, P5/2)
| | half-5th
| | A1/2, m3/2, M7/2, m9/2, P11/2
|-
| | (P8/2, P4/2)
| | half-everything
| | every 3-limit interval is split twice as much as before
|-
! colspan="2" | third-splits
! |
|-
| | (P8/3, P5)
| | third-8ve
| | m3/3, M6/3, d5/6, A11/3, d12/3
|-
| | (P8, P4/3)
| | third-4th
| | A1/3, m7/6, M7/3, m10/9, M10/3
|-
| | (P8, P5/3)
| | third-5th
| | m2/3, m6/3, M9/6, A8/3, A12/3
|-
| | (P8, P11/3)
| | third-11th
| | M2/3, M3/6, A4/9, A5/12, m9/3, P12/3
|-
| | (P8/3, P4/2)
| | third-8ve half-4th
| | third-8ve splits, half-4th splits, M6/6, m10/6, A11/12
|-
| | (P8/3, P5/2)
| | third-8ve half-5th
| | third-8ve splits, half-5th splits, m3/6, d5/12
|-
| | (P8/2, P4/3)
| | half-8ve third-4th
| | half-8ve splits, third-4th splits, A4/6, M10/6
|-
| | (P8/2, P5/3)
| | half-8ve third-5th
| | half-8ve splits, third-5th splits, m6/6, M9/6, A12/6
|-
| | (P8/2, P11/3)
| | half-8ve third-11th
| | half-8ve splits, third-11th splits, M2/6, M3/12, A4/18, A5/24
|-
| | (P8/3, P4/3)
| | third-everything
| | every 3-limit interval is split three times as much as before
|}

==Singles and doubles==

If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a '''single-split''' pergen. If it has two fractions, it's a '''double-split''' pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called '''single-pair''' notation because it adds only a single pair of accidentals to conventional notation. '''Double-pair''' notation uses both ups/downs and lifts/drops. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the EU for single-pair notation is a 3rd or larger, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs.

In this article, for double-pair notation, the period uses ups and downs, and the generator uses lifts and drops. But the choice of which pair of accidentals is used for what is arbitrary, and ups/downs could be exchanged with lifts/drops.

Every double-split pergen is either a '''true double''' or a '''false double'''. A true double, like third-everything (P8/3, P4/3) or half-8ve quarter-4th (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-8ve quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.

A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.

==Finding an example temperament==

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4⋅P and P8. If P is 6/5, the comma is 4⋅P - P8 = (6/5)4 ÷ (2/1) = 648/625, making the Diminished temperament aka Quadguti. If P is 7/6, the comma is P8 - 4⋅P = (2/1) · (7/6)-4, making the Quadruti temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.

Another method: if the generator's cents are known, look on the genchain for an interval that approximates a ratio of color depth ±1. Let the interval be I, and the genspan of this interval be x. Then n⋅x gens = n⋅I = x⋅M, where M is the multigen and M/n is the generator. The comma can be found from this equation, if n and x are coprime. For example, suppose (P8, P5/5) has G = 140¢. The genchain is all multiples of 140¢. Looking at the cents, 280¢ is about 7/6. Thus 2G = 7/6, and 10G = 5⋅(7/6) = 2⋅P5. Thus 2⋅P5 - 5⋅(7/6) = 0G = 0¢, and the comma is (3, 7, 0, -5). If the period is split and the generator isn't, use the perchain instead of the genchain. For example, (P8/7, P5) has a period of 141¢. 2 gens = 343¢, about 11/9. Thus 7⋅(11/9) = 2⋅P8, and the comma is (-2, -14, 0, 0, 7), making Saseploti.

If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.

Finding the comma(s) for a double-split pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is '''explicitly false'''. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4⋅G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).

If the pergen is not explicitly false, put the pergen in its '''unreduced''' form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n⋅P8 - m⋅M)/nm) = (P8/m, M'/n'). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P < G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2⋅P8 - 3⋅P5) / (3⋅2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This is explicitly false, thus the comma can be found from m3/6 alone. G' is about 50¢, and the comma is 6⋅G' - m3. The comma splits both the octave and the fifth.

This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit. To summarize:

<ul><li>'''A'''<nowiki/> '''double-split pergen is explicitly false if m = |b|, and not explicitly false if m > |b|.'''</li><li>'''A double-split pergen is a true double if and only if neither it nor its unreduced form is explicitly false'''''<nowiki/>'''''<nowiki/>.'''</li><li>'''A double-split pergen is a true double if''' '''GCD (m, n) > |b|,''' '''and a false double if GCD (m, n) = |b|.'''</li></ul>

A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.

Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an '''alternate''' generator. A generator or period plus or minus any number of EUs makes an '''equivalent''' generator or period. An equivalent generator is always the same size in cents, since the EU is always 0¢. An equivalent generator is the same interval, merely notated differently. For example: (P8, P5/2) has generator ^m3 and equivalent generator vM3. Another example, half-8ve (P8/2, P5) has period vA4 and equivalent period ^d5. It has generator P5 and alternate generators P4 and vA1. vA1 is equivalent to ^m2.

Of the two equivalent generators, the preferred generator is always the smaller one (smaller degree, or if the degrees are the same, more diminished). This is because the equivalent generator can be more easily found by adding the EU, rather than subtracting it. For example, ^m3 + vvA1 = vM3 is an easier calculation than vM3 - vvA1 = ^m3. This is particularly true with complex EUs like ^6dd2.

There are also alternate EUs, see below. For double-pair notation, there are also equivalent EUs.

==Ratio and cents of the accidentals==

The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2x · 3y · P±1, where P is a higher prime. They are called mapping commas because they equate or map the ratio P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for prime 5 include 81/80, 135/128, and the schisma aka Layoma = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 (besides 1/1 of course) are the currently employed commas and combinations of them.

If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in Lemba aka Latrizo & Biruyoti, where ^1 equals 64/63 minus 81/80.

Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also [[Pergen#Notating_multi-EDO_pergens|multi-EDO pergens]] pergens below.

Cents can be assigned to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + 7c. Since the EU = 0¢, we can derive the cents of the up symbol. If the EU is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its EU.

In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the EU is an A1.

This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, basic information for playing the score. Examples:
* 15-edo: # = 240¢, ^ = 80¢ (^ = third-sharp)
* 16-edo: # = -75¢
* 17-edo: # = 141¢, ^ = 71¢ (^ = half-sharp)
* 18b-edo: # = -133¢, ^ = 67¢ (^ = half-sharp)
* 19-edo: # = 63¢
* 21-edo: ^ = 57¢ (if used, # = 0¢)
* 22-edo: # = 164¢, ^ = 55¢ (^ = third-sharp)
* quarter-comma Meantone: # = 76¢
* fifth-comma Meantone: # = 84¢
* third-comma Archy aka Ruti: # = 177¢
* eighth-comma Porcupine aka Triyoti: # = 157¢, ^ = 52¢ (^ = third-sharp)
* seventh-comma Srutal aka Sagugu & Zoquadyoti: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #, seventh-comma means 1/7 of 2048/2025)
* third-comma Injera aka Gu & Biruyoti: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)
* eighth-comma Hedgehog aka Triyo & Biruyoti: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)
The last five examples are a generalization of the practice of naming meantone tunings as a fraction of a comma, e.g. quarter-comma. The 5th is sharpened or flattened by some fraction of the first comma in the color name. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both Srutal and Injera are half-8ve, but their optimal tunings are very different.

==Finding a notation for a pergen==

There are multiple notations for a given pergen, depending on the EU(s). Preferably, the EU's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:

<ul><li>For (P8/m, M/n), P8 = m⋅P + x⋅EU and M = n⋅G + y⋅EU', with 0 < |x| <= m/2 and 0 < |y| <= n/2</li><li>x is the count for EU, with EU occurring x times in one octave, and x⋅EU is the octave's '''multi-EU'''</li><li>y is the count for EU', with EU' occurring y times in one multigen, and y⋅EU' is the multigen's multi-EU</li><li>For false doubles using single-pair notation, EU = EU', but x and y are usually different, making different multi-EUs</li><li>The unreduced pergen is (P8/m, M'/n'), with a new enharmonic unison EU" and new counts, P8 = m⋅P + x'⋅EU", and M' = n'⋅G' + y'⋅EU"</li></ul>

The '''keyspan''' of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The '''[[stepspan]]''' of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.

Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a '''gedra''', analogous to a monzo, but written in brackets not parentheses: 3/2 = (-1,1) is a 7-semitone 5th, thus (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval with a monzo (a,b), there is a unique gedra [k,s], and vice versa:

<ul><li>k = 12a + 19b</li><li>s = 7a + 11b</li></ul>The matrix [(12,19) (7,11)] is unimodular, and can be inverted, and (a,b) can be derived from [k,s]:

<ul><li>a = -11k + 19s</li><li>b = 7k - 12s</li></ul>Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].

Gedras greatly facilitate finding a pergen's period, generator and EU(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an EU with the smallest possible keyspan and stepspan, which is the best EU for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up.

For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = m3. The EU can also be found using gedras: x⋅EU = M - n⋅G = P5 - 2⋅m3 = [7,4] - 2⋅[3,2] = [7,4] - [6,4] = [1,0] = A1.

Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. x⋅EU = P8 - m⋅P = P8 - 5⋅M2 = [12,7] - 5⋅[2,1] = [2,2] = 2⋅[1,1] = 2⋅m2. Because x = 2, EU will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2⋅m2 = d3). The EU's '''count''' is 2. The uninflected EU is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus EU = v5m2. Since P8 = 5⋅P + 2⋅EU, the period must be ^^M2, to make the ups and downs come out even. The number of the period's (or generator's) ups or downs always equals the count. Equipped with the period and the EU, the perchain is easily found:

P1 -- ^^M2=v3m3 -- v4 -- ^5 -- ^3M6=vvm7 -- P8
C -- ^^D=v3Eb -- vF -- ^G -- ^3A=vvBb -- C

Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has an uninflected generator [5,3]/5 = [1,1] = m2. The uninflected EU is P4 - 5⋅m2 = [5,3] - 5⋅[1,1] = [5,3] - [5,5] = [0,-2] = -2⋅[0,1] = two descending d2's. The d2 must be upped, and EU = ^5d2. Since P4 = 5⋅G - 2⋅EU, G must be ^^m2. The genchain is:

P1 -- ^^m2=v3A1 -- vM2 -- ^m3 -- ^3d4=vvM3 -- P4C -- ^^Db -- vD -- ^Eb -- vvE -- F

To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and EU from the fractional multigen as before. Then deduce the period from the EU. If the multigen was changed by unreducing, find the original generator from the period and the alternate generator.

For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The uninflected alternate generator G' is [1,1]/10 = [0,0] = P1. The uninflected EU is m2 - 10⋅P1 = m2. It must be downed, thus EU = v10m2. Since m2 = 10⋅G' + EU, G' is ^1. The octave plus or minus some number of EUs must equal 5 periods, thus (P8 + x⋅m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5⋅P + 2⋅EU, and P = ^4M2. Next, find the original half-4th generator. P = P8/5 = ~240¢, and G = P4/2 = ~250¢. Because P < G, G' is not P - G but G - P, and G is not P - G' but P + G', which equals ^4M2 + ^1 = ^5M2. While the alternate multigen is more complex than the original multigen, the alternate generator is usually simpler than the original generator.

P1- - ^4M2=v6m3 -- vv4 -- ^^5 -- ^6M6=v4m7 -- P8
C -- ^4D=v6Eb -- vvF -- ^^G -- ^6A=v4Bb -- C
P1 -- ^5M2=v5m3 -- P4
C -- ^5D=v5Eb -- F

To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. Each pair has its own enharmonic unison. For (P8/2, P4/2), the split octave implies P = vA4 and EU = ^^d2, and the split 4th implies G = /M2 and EU' = \\m2.

A false-double pergen can optionally use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair EU is not a unison or a 2nd, as with (P8/2, P4/3).

Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an EU that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So EU = v4dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The uninflected enharmonic unison is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic unison, we use the second pair of accidentals: EU' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic unisons is also an enharmonic unison: EU + EU' = v4\\d3 = 2·vv\m2, and EU - EU' = v4//ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent EUs, and v\4 and v/d4 are equivalent generators. Here is the genchain:

P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11
C -- ^E=v\F -- /Ab=\A -- ^/C=vDb -- F

One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and EU = v12m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the uninflected enharmonic unison: m3 - 3·m2 = [0,-1] = descending d2. Thus EU' = /3d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonic unisons are found from EU + EU' and EU - 2·EU'. Equivalent periods and generators are found from the many enharmonic unisons, which also allow much freedom in chord spelling. Enharmonic unison = v12m3 = /3d2 = v4/m2 = v4\\A1. Period = \M3 = v44 = //d4. Generator = ^\M3 = v34 = ^//d4.

P1 — \M3 — \\A5=/m6 — P8C — \E — /Ab — CP1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^38=v/m9 — P11
C — ^\E — ^^/Ab=vv\A — v/Db — F

It's not yet known if every pergen can avoid large EUs (those of a 3rd or more) with double-pair notation. One situation in which very large EUs occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the EU's stepspan to equal the multigen's stepspan.

Sixth-4th with single-pair notation has an awkward ^6d64 EU. This pergen might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore commas, aka Triyo & Zozoti), and its double-pair notation can also combine both. Third-4th has EU = v3A1 and G'= vM2 = ^^m2. Half-4th has EU' = \\m2 and G' = /M2 = \m3. G' - G = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G' - G = /M2 - vM2 = ^/1. Equivalent EUs are v3\\M2 and ^3\\d2.

P1 — ^/1=^\m2 — ^^m2=vM2 — /M2=\m3 — ^m3=vvM3 — v/M3=v\4 — P4
C — ^/C=^\Db — ^^Db=vD — /D=\Eb — ^Eb=vvE — v/E=v\F — F

When ups and downs are used to notate edos, a third symbol is used, a '''mid''' , written ~. The mid is only for relative notation (intervals), never for absolute notation (notes). For imperfect intervals, the mid interval is the interval exactly midway between major and minor. In addition, the mid-4th is midway between perfect and augmented, i.e. half-augmented, and the mid-5th is half-diminished. For example, 17edo's 2nds and 3rds run minor-mid-major. This avoids having to constantly choose between the equivalent terms upminor and downmajor. 72edo's 2nds and 3rds run minor, upminor, downmid, mid, upmid, downmajor, major. This makes the terms more concise. Mids can be used in pergen notation too. For single-pair, EU must be an A1, with an even number of downs. Using mids with double-pair notation is trickier. Mids never appear in the perchain. If one accidental pair appears only in the perchain and the other pair only in the genchain, then the mids appear only in the genchain, if at all.

==Alternate enharmonic unisons==

Sometimes the EU found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, ccM6/12), a false double. The uninflected alternate generator is ccM6/12 = [33,19]/12 = [3,2] = m3. The uninflected EU is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The EU becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v12A2, which is an improvement but still awkward. The period is ^4m3 and the generator is v3M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.

P1 -- ^4m3 -- v4M6 -- P8
C -- ^4Eb -- v4A -- C
P1 -- v3M2 -- v6M3=^6m2 -- ^3m3 -- P4
C -- v3D -- v6E=^6Db -- ^3Eb -- F

Because G is a M2 and EU is an A2, the equivalent generator G - EU is a descending A1. Ascending intervals that look descending can be confusing, one has to take into account the eighth-tone ups to see that v3D -- ^6Db is ascending. Double-pair notation may be preferable. This makes P = vM3, EU = ^3d2, G = /m2, and EU' = /4dd2.

P1 -- vM3 -- ^m6 -- P8
C -- vE -- ^Ab -- C
P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4
C -- /Db -- //Ebb=\\D# -- \E -- F

Because the EU found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans, like 5edo or 19edo. Heptatonic stepspans are best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the EU.

To search for an alternate EU, convert the EU to a gedra, then multiply it by the count to get the multi-EU. The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and EU is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-EU. If the stepspan becomes negative, or if it's zero and the keyspan becomes negative, invert the gedra. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new EU. Choose between ups and downs according to whether the 5th falls in the EU's upping or downing range, see [[pergen#Applications-Tipping points|tipping points]] above. Add n'''·'''count ups or downs to the new multi-EU. Add or subtract the new multi-EU from the multigen (or the octave) to get an interval which splits cleanly into n (or m) parts. Each part is the new generator (or period).

For example, (P8, P5/3) has n = 3, G = ^M2, and EU = v3m2 = [1,1]. G is upped only once, so the count is 1, and the multi-EU is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-EU [-2,1]. This can't be simplified, so the new EU is also [-2,1] = d32. Assuming a reasonably just 5th, EU needs to be upped, so EU = ^3d32. Add the multi-EU ^3[-2,1] to the multigen P5 = [7,4] to get ^3[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^3[-2,1] = v3[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·EU to check: 3·vA2 + 1·^3d32 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d32 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- vD# -- ^Fb -- G. Because d32 = -200¢ - 26·c, ^ = (-d32) / 3 = 67¢ + 8.67·c, about a third-tone.

Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one EU (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain EU. Specifically, the comma tempered out should map to the EU, or the multi-EU if the count is > 1. For example, consider Semaphore aka Zozoti (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 EU makes sense. G = ^M2 and the genchain is C -- ^D=vEb -- F. But consider Lala-yoyoti, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus EU = ^^dd2, G = vA2, and the genchain is C -- vD#=^Ebb -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.

Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 EU is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of EU.

For example, Paralimmal aka Satriluti tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the EU is v3M2, but if 11/8 is notated as a vA4, the EU is ^3dd2.

Sometimes the temperament implies an EU that isn't even a 2nd. For example, Liese aka Gu & Trizoguti (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and EU = 3·vd5 - P11 = v3dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- vGb -- ^B -- F.

This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an EU, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different EUs.

==Chord names and staff notation==

Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[Ups and downs notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.

In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G ^A and C E G vBb are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.

Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara aka Ru & Biruyoti (2.3.5.7 with 64/63 and 50/49) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and EU = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C vE G Bb = Cv,7.

A different temperament may result in the same pergen with the same EU, but may still produce a different name for the same chord. For example, Injera aka Gu & Biruyoti (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 EU is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and EU = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G vBb = C,v7.

Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes. This is in fact a requirement for MOS scales.

Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. Iv -- vIIIv -- vVI^m -- Iv. A Porcupine aka Triyoti (third-4th) comma pump can be written out like so: Cv -- vA^m -- vDv -- [vvB=^Bb]^m -- ^Ebv -- G^m -- Gv -- Cv. Brackets are used to show that vvB and ^Bb are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. vvB is written first to show that this root is a vM6 above the previous root, vD. ^Bb is second to show the P4 relationship to the next root, ^Eb. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either vvB^m or ^Bb^m, or possibly ^BbvvM = ^Bb vD ^F.

Lifts and drops, like ups and downs, precede the note head and any sharps or flats. Scores for melody instruments could instead have them above or below the staff. This score uses ups and downs, and has chord names. It's for the third-4th pergen, with EU = v3A1. As noted above, it can be played in EDOs 15, 22 or 29 as is, because ^1 maps to 1 edostep. But to be played in EDOs 30, 37 or 44, ^1 must be interpreted as two edosteps.

Mizarian Porcupine Overture by Herman Miller (P8, P4/3) ^3C = C#

[[File:Mizarian_Porcupine_Overture.png|alt=Mizarian Porcupine Overture.png|800x692px|Mizarian Porcupine Overture.png]]

==Tipping points and sweet spots==

The tipping point for half-octave with a d2 EU is 700¢, 12-edo's 5th. As noted above, the 5th of Pajara (half-8ve) tends to be sharp, thus it has EU = ^^d2. But Injera, also half-8ve, has a flat 5th, and thus EU = vvd2. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But if a single temperament "tips over", either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.

The tipping point depends on the choice of EU. It's not the temperament that tips, it's the notation. Half-8ve could be notated with an EU of vvM2. The tipping point becomes 600¢, a very unlikely 5th, and tipping is impossible. For single-comma temperaments, the EU usually equals the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of EU is a given. However, the mapping of primes 11 and 13 is not agreed on.

The notation's tipping point is determined by the uninflected EU, which is implied by the vanishing comma. For example, Porcupine aka Triyoti's 250/243 comma is an A1 = (-11,7), which implies an uninflected EU of A1, which implies 7-edo, and a 685.7¢ tipping point. Dicot aka Yoyoti's 25/24 comma is also an A1, and has the same tipping point. Semaphore aka Zozoti's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. See [[pergen#Applications-Tipping points|tipping points]] above for a more complete list.

Double-pair notation has two EUs, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two EUs can be any two of A1, m2 or d2. One can choose to use whichever two EUs best avoid tipping.

An example of a temperament that tips easily is Negri aka Laquadyoti, 2.3.5 and (-14,3,4). Because Negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with Negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and the EU could be either ^4dd2 or v4dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an EU of ^4dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma Negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.

Another "tippy" temperament is found by adding the mapping comma 81/80 to the Negri comma and getting the Latriyoma comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th.

==Notating unsplit pergens==

An unsplit pergen doesn't require ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a mapping comma, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (i.e. the monzo has a final exponent of ±1).

The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate. The up symbol's ratio is always the mapping comma, or its inverse.

{| class="wikitable" style="text-align:center;"
|-
! | 5-limit temperament
! | comma
! | sweet spot
! colspan="2" | no ups or downs
! colspan="3" | with ups and downs
! colspan="2" | up symbol
|-
! | (pergen is unsplit)
! |
! | (5th = 700¢ + c)
! | 5/4 is
! | 4:5:6 chord
! | 5/4 is
! | 4:5:6 chord
! | EU
! | ratio
! | cents
|-
| | Meantone aka Guti
| | 81/80 = P1
| | c = -3¢ to -5¢
| | M3
| | C E G
| | ---
| | ---
| | ---
| | ---
| | ---
|-
| | Mavila aka Layobiti
| | 135/128 = A1
| | c = -21¢ to -22¢
| | m3
| | C Eb G
| | ^M3
| | C ^E G
| | ^A1
| | 80/81 = d1
| | -100¢ - 7c = 47¢-54¢
|-
| | Laguti
| | (-15,11,-1) = A1
| | c = -10¢ to -12¢
| | A3
| | C E# G
| | ^M3
| | C ^E G
| | vA1
| | 80/81 = A1
| | 100¢ + 7c = 26¢-30¢
|-
| | Schismic aka Layoti
| | (-15,8,1) = -d2
| | c = 1.7¢ to 2.0¢
| | d4
| | C Fb G
| | vM3
| | C vE G
| | ^d2
| | 81/80 = -d2
| | 12c = 20¢-24¢
|-
| | Lalaguti
| | (-23,16,-1) = -d2
| | c = -0.9¢ to -1.2¢
| | AA2
| | C D## G
| | vM3
| | C vE G
| | vd2
| | 81/80 = d2
| | -12c = 10¢-15¢
|-
| | Father aka Gubiti
| | 16/15 = m2
| | c = 56¢ to 58¢
| | P4
| | C F G
| | vM3
| | C vE G
| | ^m2
| | 81/80 = -m2
| | -100¢ + 5c = 180-190¢
|-
| | Superpyth aka Sasayoti
| | (12,-9,1) = m2
| | c = 9¢ to 10¢
| | A2
| | C D# G
| | vM3
| | C vE G
| | vm2
| | 81/80 = m2
| | 100¢ - 5c = 50-55¢
|}
The Schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch. The Mavila temperament could perhaps be notated without ups and downs, because 5/4 is still a 3rd, and the 4:5:6 triad still looks like a triad.

For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the Schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-limit comma is the pythagorean comma (-19,12).

A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for Archy aka Ruti (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit Schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C vE G vBb.

==Notating rank-3 pergens==

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. Enharmonic unisons aka EUs are like vanishing commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of EUs needed always equals the difference between the notation's rank and the tuning's rank. Examples:

{| class="wikitable" style="text-align:center;"
|-
! | tuning
! | pergen
! | tuning's rank
! | notation
! | notation's rank without any EUs
! | # of EUs needed
! | EUs
|-
| | 12-edo
| | (P8/12)
| | rank-1
| | conventional
| | rank-2
| | 1
| | EU = d2
|-
| | 19-edo
| | (P8/19)
| | rank-1
| | conventional
| | rank-2
| | 1
| | EU = dd2
|-
| | 15-edo
| | (P8/15)
| | rank-1
| | single-pair
| | rank-3
| | 2
| | EU = m2, EU' = v3A1 = v3M2
|-
| | 24-edo
| | (P8/24)
| | rank-1
| | single-pair
| | rank-3
| | 2
| | EU = d2, EU' = vvA1 = vvm2
|-
| | 3-limit JI aka pythagorean
| | (P8, P5)
| | rank-2
| | conventional
| | rank-2
| | 0
| | ---
|-
| | Meantone aka Guti
| | (P8, P5)
| | rank-2
| | conventional
| | rank-2
| | 0
| | ---
|-
| | Diaschismic aka Saguguti
| | (P8/2, P5)
| | rank-2
| | single-pair
| | rank-3
| | 1
| | EU = ^^d2
|-
| | Semaphore aka Zozoti
| | (P8, P4/2)
| | rank-2
| | single-pair
| | rank-3
| | 1
| | EU = vvm2
|-
| | Decimal aka Yoyo & Zozoti
| | (P8/2, P4/2)
| | rank-2
| | double-pair
| | rank-4
| | 2
| | EU = vvd2, EU' = \\m2 = ^^\\A1
|-
| | 5-limit JI
| | (P8, P5, ^1)
| | rank-3
| | single-pair
| | rank-3
| | 0
| | ---
|-
| | Marvel aka Ruyoyoti
| | (P8, P5, ^1)
| | rank-3
| | single-pair
| | rank-3
| | 0
| | ---
|-
| | Breedsmic aka Bizozoguti
| | (P8, P5/2, ^1)
| | rank-3
| | double-pair
| | rank-4
| | 1
| | EU = \\dd3
|-
| | 7-limit JI
| | (P8, P5, ^1, /1)
| | rank-4
| | double-pair
| | rank-4
| | 0
| | ---
|}
When there is more than one EU, the first one can be added to or subtracted from the 2nd one, to make an equivalent 2nd EU.

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas. There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.

Even the unsplit rank-3 pergen requires single-pair notation, for the 2nd generator. Single-splits and false double-splits require double-pair, true doubles and false triples require triple-pair, and true triples require quadruple-pair. Some false triples and some true doubles may use quadruple-pair as well, to avoid awkward EUs of a 3rd or more. Rather than devising a third or fourth pair of symbols, and a third or fourth pair of adjectives to describe them, one might simply use colors.

A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.

Some examples of 7-limit rank-3 temperaments:

{| class="wikitable" style="text-align:center;"
|-
! | 7-limit temperament
! | comma
! | pergen
! | spoken pergen
! | notation
! | period
! | gen1
! | gen2
! | EU
|-
| | Marvel aka Ruyoyoti
| | 225/224
| | (P8, P5, ^1)
| | rank-3 unsplit
| | single-pair
| | P8
| | P5
| | ^1 = 81/80
| | ---
|-
| | "
| | "
| | "
| | "
| | double-pair
| | "
| | "
| | "
| | ^^\d2
|-
| | Biruyoti
| | 50/49
| | (P8/2, P5, ^1)
| | rank-3 half-8ve
| | double-pair
| | v/A4 = 10/7
| | P5
| | ^1 = 81/80
| | ^^\\d2
|-
| | Trizoguti
| | 1029/1000
| | (P8, P11/3, ^1)
| | rank-3 third-11th
| | double-pair
| | P8
| | ^\d5 = 7/5
| | ^1 = 81/80
| | ^^^\\\dd3
|-
| | Breedsmic aka Bizozoguti
| | 2401/2400
| | (P8, P5/2, ^1)
| | rank-3 half-5th
| | double-pair
| | P8
| | v//A2 = 60/49
| | /1 = 64/63
| | ^^\4dd3
|-
| | Demeter aka Trizo-aguguti
| | 686/675
| | (P8, P5, \m3/2)
| | half-downminor-3rd
| | double-pair
| | P8
| | P5
| | v/A1 = 15/14
| | ^^\\\dd3
|}
If using single-pair notation, Marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - vE - G - vvA#. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - vE - G - \Bb.

There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, Biruyoti is half-8ve, with a d2 comma, like Srutal. Using the same notation as Srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and EU = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as C vE G v/Bb. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C vE G \Bb.

With this standardization, the EU can be derived directly from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)-2 · (64/63)2 · (-19,12). This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the EU never is, therefore the EU = ^^\\d2. The period is found by adding/subtracting the EU from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both EU and P become more complex, but the ratio for /1 becomes simpler.

This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For Biruyoti, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore Biruyoti doesn't tip. Single-pair rank-3 notation has no EU, and thus no tipping point. Double-pair rank-3 notation has 1 EU, but two mapping commas. Rank-3 notations rarely tip.

Unlike the previous examples, Demeter aka Trizo-aguguti's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no EU. But the 4:5:6:7 chord would be spelled C -- ^^^Fbbb -- G -- ^^Bbb, very awkward! Standard double-pair notation is better. Gen2 = v/A1, EU = ^^\\\dd3, and ^^\\\C = A##. Genchain2 is C -- v/C# -- \Eb -- vE -- \\Gb -- v\G -- vvG#=\\\Bbb -- v\\Bb... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own EU, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own EU, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops only for the other kind of accidentals.

There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For Demeter, any combination of \m3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because there will always be a 3-limit comma small enough to be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the '''DOL''' ([[Odd limit|double odd limit]]) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred. And as a tie-breaker, in case of two ratios with the same DOL (such as 5/3 and 6/5), to minimize the size in cents of the multigen2. Since 5/3 = 884.4¢ and 6/5 = 315.6¢, 6/5 is preferred.

If ^1 = 81/80, possible half-split gen2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.

All possible rank-3 pergens can be listed, but the table is much longer than for rank-2 pergens. Here are all the half-split pergens:

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! colspan="4" | prime subgroup used by the pergen
|-
! | unsplit
! colspan="2" | 2.3.5 (^1 = 81/80)
! colspan="2" | 2.3.7 (^1 = 64/63)
|-
| | 1
| | (P8, P5, ^1)
| | rank-3 unsplit
| | same
| | same
|-
! | half-splits
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5, ^1)
| | rank-3 half-8ve
| | same
| | same
|-
| | 3
| | (P8, P4/2, ^1)
| | rank-3 half-4th
| | same
| | same
|-
| | 4
| | (P8, P5/2, ^1)
| | rank-3 half-5th
| | same
| | same
|-
| | 5
| | (P8/2, P4/2, ^1)
| | rank-3 half-everything
| | same
| | same
|-
| | 6
| | (P8, P5, ^m3/2)
| | half-upminor-3rd
| | (P8, P5, ^M2/2)
| | half-upmajor-2nd
|-
| | 7
| | (P8, P5, vM3/2)
| | half-downmajor-3rd
| | (P8, P5, vm3/2)
| | half-downminor-3rd
|-
| | 8
| | (P8, P5, ^m6/2)
| | half-upminor-6th
| | (P8, P5, ^M6/2)
| | half-upmajor-6th
|-
| | 9
| | (P8, P5, vM6/2)
| | half-downmajor-6th
| | (P8, P5, vm7/2)
| | half-downminor-7th
|-
| | 10
| | (P8/2, P5, ^m3/2)
| | half-8ve half-upminor-3rd
| | (P8/2, P5, ^M2/2)
| | half-8ve half-upmajor-2nd
|-
| | 11
| | (P8/2, P5, vM3/2)
| | half-8ve half-downmajor-3rd
| | (P8/2, P5, vm3/2)
| | half-8ve half-downminor-3rd
|-
| | 12
| | (P8, P4/2, vM3/2)
| | half-4th half-downmajor-3rd
| | (P8, P4/2, ^M2/2)
| | half-4th half-upmajor-2nd
|-
| | 13
| | (P8, P4/2, ^m6/2)
| | half-4th half-upminor-6th
| | (P8, P4/2, vm7/2)
| | half-4th half-downminor-7th
|-
| | 14
| | (P8, P5/2, vM3/2)
| | half-5th half-downmajor-3rd
| | (P8, P5/2, ^M2/2)
| | half-5th half-upmajor-2nd
|-
| | 15
| | (P8, P5/2, ^m6/2)
| | half-5th half-upminor-6th
| | (P8, P5/2, vm7/2)
| | half-5th half-downminor-7th
|-
| | 16
| | (P8/2, P4/2, vM3/2)
| | half-everything half-downmajor-3rd
| | (P8/2, P4/2, ^M2/2)
| | half-everything half-upmajor-2nd
|}
There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc.

==Notating multi-EDO pergens==

A multi-EDO pergen is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave, thus it doesn't appear in the pergen. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12. Pergens which imply an edo which doesn't have a decent 5th, e.g. P8/3, P8/4, P8/6, etc., are covered in the next section.

A multi-EDO pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the octave and removes the middle term from the pergen. For example, Blackwood aka Sawati plus Ya is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1). Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:

{| class="wikitable" style="text-align:center;"
|-
! | temperament
! | pergen
! | spoken name
! | enharmonic unisons
! | perchain
! | genchain
! | ^1 ratio
! | /1 ratio
|-
| | Blackwood aka Sawati+ya
| | (P8/5, ^1)
| | rank-2 5-edo
| | EU = m2
| | D E=F G A B=C D
| | D vF#=vG vvB...
| | 81/80 = 16/15
| | ---
|-
| | Whitewood aka Lawati+ya
| | (P8/7, ^1)
| | rank-2 7-edo
| | EU = A1
| | D E F G A B C D
| | D ^F ^^A...
| | 80/81 = 135/128
| | ---
|-
| | 10edo+ya
| | (P8/10, /1)
| | rank-2 10-edo
| | EU = m2, EU' = vvA1 = vvM2
| | D ^D=vE E=F ^F=vG G...
| | D \F#=\G \\B...
| | (see below)
| | 81/80
|-
| | 12edo+la
| | (P8/12, ^1)
| | rank-2 12-edo
| | EU = d2
| | D D#=Eb E F F#=Gb...
| | D ^G ^^C
| | 33/32
| | ---
|-
| | "
| | "
| | "
| | "
| | "
| | D vG#=vAb vvD...
| | 729/704
| | ---
|-
| | 17edo+ya
| | (P8/17, /1)
| | rank-2 17-edo
| | EU = dd3, EU' = vm2 = vvA1
| | D ^D=Eb D#=vE E F...
| | D \F# \\A#=v\\B...
| | 256/243
| | 81/80
|}
If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 10, 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15, or 12/11, or 13/12.

The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.

All multi-EDO pergens are of the form (P8/m, ^1) or (P8/m, /1), and they can be identified solely by their splitting fraction. Multi-EDO pergens are a small minority of rank-2 pergens.

It's possible to have a fifth-8ve pergen with an independent 5th, but there will be very small intervals of about 20¢. Here are two such:

{| class="wikitable" style="text-align:center;"
|-
! | temperament
! | subgroup
! | comma
! | pergen
! | spoken name
! | EU
! | perchain
! | genchain
! | ^1 ratio
|-
| | Laquinzoti
| | 2.3.7
| | (-14,0,0,5)
| | (P8/5, P5)
| | fifth-8ve
| | v5m2
| | D ^^E vG ^A vvC D
| | C G D A E...
| | 49/48
|-
| | Saquinruti
| | 2.3.7
| | (22,-5,0,-5)
| | "
| | "
| | "
| | "
| | "
| | 64/63
|}
Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C ^^D vF ^G vvBb C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section). A multi-EDO pergen should never have an uninflected genchain.

==Notating non-8ve and no-5ths pergens==

In multi-EDO pergens, the 5th is present but not independent. In no-5ths pergens, the 5th is not present, and the prime subgroup doesn't contain 3.

In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any EUs, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.

But in non-8ve and no-5ths pergens, not every name has a note. For example, Biruyoti Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - vF# - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a huge number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk.

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! colspan="6" | prime subgroup used by the pergen
|-
! | unsplit
! | 2.3
! | 2.5 (M3 = 5/4)
! | 2.7 (M2 = 8/7)
! | 3.5 (M6 = 5/3)
! | 3.7 (M3 = 9/7)
! | 5.7 (ccM3 = 5/1, d5 = 7/5)
|-
| | 1
| | (P8, P5)
| | (P8, M3)
| | (P8, M2)
| | (P12, M6)
| | (P12, M3)
| | (ccM3, d5)
|-
! | half-splits
! |
! |
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5)
| | (P8/2, M3)
| | (P8/2, M2)
| | (P12/2, M6)
| | (P12/2, M3)
| | (M9, d5)*
|-
| | 3
| | (P8, P4/2)
| | (P8, M2)*
| | (P8, M2/2)
| | (P12, M6/2)
| | (P12, M2)*
| | (ccM3, m3)*
|-
| | 4
| | (P8, P5/2)
| | (P8, m6/2)
| | (P8, P5)*
| | (P12, P4)*
| | (P12, m10/2)
| | (ccM3, M7)*
|-
| | 5
| | (P8/2, P4/2)
| | (P8/2, M2)*
| | (P8/2, M2/2)
| | (P12/2, M6/2)
| | (P12/2, M3/2)
| | (M9, m3)*
|-
! | third-splits
! |
! |
! |
! |
! |
! |
|-
| | 6
| | (P8/3, P5)
| | (P8/3, M3)
| | (P8/3, M2)
| | (P12/3, M6)
| | (P12/3, M3)
| | (ccM3/3, d5)
|-
| | 7
| | (P8, P4/3)
| | (P8, M3/3)
| | (P8, M2/3)
| | (P12, M6/3)
| | (P12, M3/3)
| | (ccM3, d5/3)
|-
| | 8
| | (P8, P5/3)
| | (P8, m6/3)
| | (P8, m7/3)
| | (P12, m7/3)
| | (P12, P4)*
| | (ccM3, cA6/3)
|-
| | 9
| | (P8, P11/3)
| | (P8, M10/3)
| | (P8, M9/3)
| | (P12, ccM3/3)
| | (P12, cM7/3)
| | (ccM3, ccm7/3)
|-
| | 10
| | (P8/3, P4/2)
| | (P8/3, M2)*
| | (P8/3, M2/2)
| | (P12/3, M6/2)
| | (P12/3, M2)*
| | (ccM3/3, m3)*
|-
| | 11
| | (P8/3, P5/2)
| | (P8/3. m6/2)
| | (P8/3, P5)*
| | (P12/3, P4)*
| | (P12/3, m10/2)
| | (ccM3/3, M7)*
|-
| | 12
| | (P8/2, P4/3)
| | (P8/2, M3/3)
| | (P8/2, M2/3)
| | (P12/2, M6/3)
| | (P12/2, M3/3)
| | (M9, d5/3)*
|-
| | 13
| | (P8/2, P5/3)
| | (P8/2, m6/3)
| | (P8/2, m7/3)
| | (P12/2, m7/3)
| | (P12/2, P4)*
| | (M9, cA6/3)*
|-
| | 14
| | (P8/2, P11/3)
| | (P8/2, M10/3)
| | (P8/2, M9/3)
| | (P12/2, ccM3/3)
| | (P12/2, cM7/3)
| | (M9, ccm7/3)*
|-
| | 15
| | (P8/3, P4/3)
| | (P8/3, M3/3)
| | (P8/3, M2/3)
| | (P12/3, M6/3)
| | (P12/3, P4)*
| | (ccM3/3, d5/3)
|}
For prime subgroup p.q, the unsplit pergen has period p/1. The unsplit pergen's generator is found by dividing q by p until it's less than p/1, and period-inverting if it's more than half of p/1.

Multi-EDO pergens also appear in this table. For example, in row #33, (P8/5, ^1) replaces both 2.5 (P8/5, M3) and 2.7 (P8/5, M2). This has the advantage of avoiding missing notes. Since one fifth of 5/1 is only 25¢ from 7/5, the 5.7 (ccM3/5, d5) can optionally be replaced too.

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! | 2.3
! | 2.5
! | 2.7
! | 3.5
! | 3.7
! | 5.7
|-
| | 33
| | (P8/5, P5)
| | (P8/5, ^1)
| | (P8/5, ^1)
| | (P12/5, M6)
| | (P12/5, M3)
| | (ccM3/5, ^1)
|}

Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the first 3.5 pergen, (P12, M6). The note one M6 above C1 is A1, and the note three P12's above C1 is A5. (There is no A2, A3 or A4.) Referring to a note as simply A is ambiguous.

Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table could be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2), with ^M2 = 8/7 and ^1 = √256/245 = √(81/80 * 64/63 * 64/63) = about 38¢. The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3), with ^M3 = 9/7 and ^1 = √6561/6125 = √[(81/80)3 * (64/63)2] = about 60¢.

==Pergen squares==

Pergen squares, which were discovered by Praveen Venkataramana, are a way to visualize pergens squares in a way that isn't specific to any primes at all. To understand them, let's assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.

For (P8, P5), the pergen square has 4 notes, shown here with octave numbers (ignore the periods).

C2 -- G2 
| . . . . | 
C1 -- G1

Splitting the 8ve or the multigen adds notes to the square. For (P8/2, P5), there are 6 notes. The new notes fall halfway between each 8ve:

C2 --- G2 
vF#1 vC#2 
C1 --- G1

A pergen square shows all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and vC#2 bisects it. vG#2 bisects the G2-A2 M2. Every "downed" note bisects an 8ve, a M2, a cm7 (e.g. D1 to C3), a cM9 (e.g. C1 to D3), and many other intervals.

C2 --- G2 --- D3 --- A3 
vF#1 vC#2 vG#2 vD#3 
C1 --- G1 --- D2 --- A2

Splitting the 5th adds notes to the horizontal edges of the square:

C2 vE2 G2 
| . . . . . . | 
C1 vE1 G1

Each downed note also bisects a P11 (e.g. G1-C3), as well as many other intervals.

C3 vE3 G3 
| . . . . . . | 
C2 vE2 G2 
| . . . . . . | 
C1 vE1 G1

From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square.

C2 ---- G2 
| . ^A1 . | 
C1 ---- G1

^A1 also bisects the P12 from C1 to G2.

Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Pierce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.

[[File:pergen_squares.png|alt=pergen squares.png|pergen squares.png]]

A similar chart could be made for all rank-3 pergens, using pergen cubes.

==Notating tunings with an arbitrary generator==

Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.

The next table lists all the ranges for all multigens up to seventh-splits. One can look up one's generator in the first column and find a possible multigen. Use the octave inverse if G > 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | primary choice
! colspan="4" | secondary choices
|-
! | generator
! | possible multigen
! | generator
! | multigen
! | generator
! | multigen
|-
| | 23-60¢
| | M2/4 (requires P8/2)
| |
| |
| |
| |
|-
| | 69-79¢
| | P4/7
| |
| |
| |
| |
|-
| | 80-92¢
| | P4/6
| |
| |
| |
| |
|-
| | 92-103¢
| | P5/7
| |
| |
| |
| |
|-
| | 96-111¢
| | P4/5
| |
| |
| |
| |
|-
| | 108-120¢
| | P5/6
| |
| |
| |
| |
|-
| | 120-138¢
| | P4/4
| |
| |
| |
| |
|-
| | 129-144¢
| | P5/5
| |
| |
| |
| |
|-
| | 160-185¢
| | P4/3
| | 162-180¢
| | P5/4
| |
| |
|-
| | 215-240¢
| | P5/3
| |
| |
| |
| |
|-
| | 240-277¢
| | P4/2
| | 240-251¢
| | P11/7
| | 264-274¢
| | P12/7
|-
| | 280-292¢
| | P11/6
| |
| |
| |
| |
|-
| | 308-320¢
| | P12/6
| |
| |
| |
| |
|-
| | 323-360¢
| | P5/2
| | 336-351¢
| | P11/5
| |
| |
|-
| | 369-384¢
| | P12/5
| |
| |
| |
| |
|-
| | 411-422¢
| | ccP4/7
| |
| |
| |
| |
|-
| | 420-438¢
| | P11/4
| |
| |
| |
| |
|-
| | 435-446¢
| | ccP5/7
| |
| |
| |
| |
|-
| | 462-480¢
| | P12/4
| | 462-480¢
| | M9/3
| |
| |
|-
| | 480-554¢
| | P4 = P5
| | 480-492¢
| | ccP4/6
| | 508-520¢
| | ccP5/6
|-
| | 560-585¢
| | P11/3
| |
| |
| |
| |
|-
| | 576-591¢
| | ccP4/5
| | 583-593¢
| | cccP4/7
| |
| |
|}
There are gaps in the table, especially near 150¢, 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation. But the total range of possible generators is mostly well covered, providing convenient notation options. In particular, if the tuning's generator is just over 720¢, to avoid descending 2nds, instead of calling the generator a 5th, one can call its inverse a quarter-12th. The generator is notated as an up-5th, and four of them make a ccP4.

The table assumes unsplit octaves. Splitting the octave creates alternate generators. For example, if P = P8/3 = 400¢ and G = 300¢, alternate generators are 100¢ and 500¢. Any of these can be used to find a convenient multigen.

See also the [[Map_of_rank-2_temperaments|map of rank-2 temperaments]].

==Pergens and MOS scales==

Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! colspan="3" | MOS scales of 5-12 notes
! |
! |
! |
|-
| | (P8, P5)
| | unsplit
| | 5 = 2L 3s
| | 7 = 5L 2s
| colspan="2" | 12 = 7L 5s (or 5L 7s)
| |
| |
|-
! colspan="2" | halves
! |
! |
! |
! |
! |
! |
|-
| | (P8/2, P5)
| | half-8ve
| | 6 = 2L 4s
| | 8 = 2L 6s
| | 10 = 2L 8s
| colspan="3" | 12 = 2L 10s (or 10L 2s)
|-
| | (P8, P4/2)
| | half-4th
| | 5 = 4L 1s
| | 9 = 5L 4s
| |
| |
| |
| |
|-
| | (P8, P5/2)
| | half-5th
| | 7 = 3L 4s
| | 10 = 7L 3s
| |
| |
| |
| |
|-
| | (P8/2, P4/2)
| | half-everything
| | 6 = 4L 2s
| | 10 = 4L 6s
| |
| |
| |
| |
|-
! colspan="2" | thirds
! |
! |
! |
! |
! |
! |
|-
| | (P8/3, P5)
| | third-8ve
| | 6 = 3L 3s
| | 9 = 3L 6s
| colspan="2" | 12 = 3L 9s (or 9L 3s)
| |
| |
|-
| | (P8, P4/3)
| | third-4th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 1L 6s
| | 8 = 7L 1s
| |
| |
|-
| | (P8, P5/3)
| | third-5th
| | 5 = 1L 4s
| | 6 = 5L 1s
| | 11 = 5L 6s
| |
| |
| |
|-
| | (P8, P11/3)
| | third-11th
| | 5 = 2L 3s
| | 7 = 2L 5s
| | 9 = 2L 7s
| | 11 = 2L 9s
| |
| |
|-
| | (P8/3, P4/2)
| | third-8ve, half-4th
| | 6 = 3L 3s *
| | 9 = 6L 3s
| |
| |
| |
| |
|-
| | (P8/3, P5/2)
| | third-8ve, half-5th
| | 6 = 3L 3s *
| | 9 = 3L 6s
| | 12 = 3L 9s
| |
| |
| |
|-
| | (P8/2, P4/3)
| | half-8ve, third-4th
| | 6 = 2L 4s *
| | 8 = 6L 2s
| |
| |
| |
| |
|-
| | (P8/2, P5/3)
| | half-8ve, third-5th
| | 6 = 4L 2s *
| | 10 = 6L 4s
| |
| |
| |
| |
|-
| | (P8/2, P11/3)
| | half-8ve, third-11th
| | 6 = 2L 4s *
| | 8 = 2L 6s
| | 10 = 2L 8s
| | 12 = 2L 10s
| |
| |
|-
| | (P8/3, P4/3)
| | third-everything
| | 6 = 3L 3s *
| | 9 = 6L 3s *
| |
| |
| |
| |
|-
! colspan="2" | quarters
! |
! |
! |
! |
! |
! |
|-
| | (P8/4, P5)
| | quarter-8ve
| | 8 = 4L 4s
| colspan="2" | 12 = 4L 8s (or 8L 4s)
| |
| |
| |
|-
| | (P8, P4/4)
| | quarter-4th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 1L 6s
| | 8 = 1L 7s
| | 9 = 1L 8s
| | 10 = 9L 1s
|-
| | (P8, P5/4)
| | quarter-5th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 6L 1s
| |
| |
| |
|-
| | (P8, P11/4)
| | quarter-11th
| | 5 = 3L 2s
| | 8 = 3L 5s
| | 11 = 3L 8s
| |
| |
| |
|-
| | (P8, P12/4)
| | quarter-12th
| | 5 = 3L 2s
| | 8 = 5L 3s
| |
| |
| |
| |
|-
| | (P8/4, P4/2)
| | quarter-8ve half-4th
| | 8 = 4L 4s *
| | 12 = 4L 8s
| |
| |
| |
| |
|-
| | (P8/2, M2/4)
| | half-8ve quarter-tone
| | 6 = 2L 4s *
| | 8 = 2L 6s *
| | 10 = 2L 8s
| | 12 = 2L 10s
| |
| |
|-
| | (P8/2, P4/4)
| | half-8ve quarter-4th
| | 6 = 2L 4s *
| | 8 = 2L 6s *
| | 10 = 8L 2s
| |
| |
| |
|-
| | (P8/2, P5/4)
| | half-8ve quarter-5th
| | 6 = 2L 4s *
| | 8 = 6L 2s *
| |
| |
| |
| |
|-
| | (P8/4, P4/3)
| | quarter-8ve third-4th
| | 8 = 4L 4s *
| | 12 = 8L 4s *
| |
| |
| |
| |
|-
| | (P8/4, P5/3)
| | quarter-8ve third-5th
| | 8 = 4L 4s *
| | 12 = 4L 8s *
| |
| |
| |
| |
|-
| | (P8/4, P11/3)
| | quarter-8ve third-11th
| | 8 = 4L 4s *
| | 12 = 4L 8s *
| |
| |
| |
| |
|-
| | (P8/3, P4/4)
| | third-8ve quarter-4th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 9L 3s *
| |
| |
| |
|-
| | (P8/3, P5/4)
| | third-8ve quarter-5th
| | 6 = 3L 3s *
| | 9 = 6L 3s *
| |
| |
| |
| |
|-
| | (P8/3, P11/4)
| | third-8ve quarter-11th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 3L 9s *
| |
| |
| |
|-
| | (P8/3, P12/4)
| | third-8ve quarter-12th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 3L 9s *
| |
| |
| |
|-
| | (P8/4, P4/4)
| | quarter-everything
| | 8 = 4L 4s *
| | 12 = 8L 4s *
| |
| |
| |
| |
|}

A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the simplest, and also one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided.

{| class="wikitable" style="text-align:center;"
|-
! | MOS scale
! colspan="2" | primary example
! | secondary examples
|-
! | Pentatonic
! |
! |
! |
|-
| | 1L 4s
| | (P8, P5/3) [5]
| | third-5th pentatonic
| | third-4th, quarter-4th, quarter-5th
|-
| | 2L 3s
| | (P8, P5) [5]
| | unsplit pentatonic
| | third-11th
|-
| | 3L 2s
| | (P8, P12/4) [5]
| | quarter-12th pentatonic
| | quarter-11th
|-
| | 4L 1s
| | (P8, P4/2) [5]
| | half-4th pentatonic
| |
|-
! | Hexatonic
! |
! |
! |
|-
| | 1L 5s
| | (P8, P4/3) [6]
| | third-4th hexatonic
| | quarter-4th, quarter-5th, fifth-4th, fifth-5th
|-
| | 2L 4s
| | (P8/2, P5) [6]
| | half-8ve hexatonic
| |
|-
| | 3L 3s
| | (P8/3, P5) [6]
| | third-8ve hexatonic
| |
|-
| | 4L 2s
| | (P8/2, P4/2) [6]
| | half-everything hexatonic
| |
|-
| | 5L 1s
| | (P8, P5/3) [6]
| | third-5th hexatonic
| |
|-
! | Heptatonic
! |
! |
! |
|-
| | 1L 6s
| | (P8, P4/3) [7]
| | third-4th heptatonic
| | quarter-4th, fifth-4th, fifth-5th, sixth-4th, sixth-5th
|-
| | 2L 5s
| | (P8, P11/3) [7]
| | third-11th heptatonic
| | fifth-double-compound-4th, sixth-double-compound-5th
|-
| | 3L 4s
| | (P8, P5/2) [7]
| | half-5th heptatonic
| | fifth-12th
|-
| | 4L 3s
| | (P8, P11/5) [7]
| | fifth-11th heptatonic
| | sixth-12th
|-
| | 5L 2s
| | (P8, P5) [7]
| | unsplit heptatonic
| | sixth-double-compound-4th
|-
| | 6L 1s
| | (P8, P5/4) [7]
| | quarter-5th heptatonic
| |
|-
! | Octotonic
! |
! |
! |
|-
| | 1L 7s
| | (P8, P4/4) [8]
| | quarter-4th octotonic
| | fifth-4th, fifth-5th, sixth-4th, sixth-5th, seventh-4th, seventh-5th
|-
| | 2L 6s
| | (P8/2, P5) [8]
| | half-8ve octotonic
| |
|-
| | 3L 5s
| | (P8, P11/4) [8]
| | quarter-11th octotonic
| | seventh-cc4th, seventh-cc5th
|-
| | 4L 4s
| | (P8/4, P5) [8]
| | quarter-8ve octotonic
| |
|-
| | 5L 3s
| | (P8, P12/4) [8]
| | quarter-12th octotonic
| | (very lopsided, unless 5th is quite flat)
|-
| | 6L 2s
| | (P8/2, P4/3) [8]
| | half-8ve third-4th octotonic
| |
|-
| | 7L 1s
| | (P8, P4/3) [8]
| | third-4th octotonic
| |
|-
! | Nonatonic
! |
! |
! |
|-
| | 1L 8s
| | (P8, P4/4) [9]
| | quarter-4th nonatonic
| | fifth-4th, sixth-4th, sixth-5th, seventh-4th/5th, eighth-4th/5th
|-
| | 2L 7s
| | (P8, c3P5/8) [9]
| | eighth-c35th nonatonic
| | third-11th, fifth-cc4th
|-
| | 3L 6s
| | (P8/3, P5) [9]
| | third-8ve nonatonic
| | third-8ve half-5th
|-
| | 4L 5s
| | (P8, P12/7) [9]
| | seventh-12th nonatonic
| | sixth-11th
|-
| | 5L 4s
| | (P8, P4/2) [9]
| | half-4th nonatonic
| | (lopsided unless 4th is sharp), seventh-11th
|-
| | 6L 3s
| | (P8/3, P4/2) [9]
| | third-8ve half-4th nonatonic
| |
|-
| | 7L 2s
| | (P8, ccP5/6)[9]
| | sixth-cc5th nonatonic
| | (lopsided unless 5th is sharp)
|-
| | 8L 1s
| | (P8, P5/5) [9]
| | fifth-5th nonatonic
| |
|-
! | Decatonic
! |
! |
! |
|-
| | 1L 9s
| | (P8, P5/6) [10]
| | sixth-5th decatonic
| | fifth-4th, sixth-4th, seventh-4th/5th, eighth-4th/5th, ninth-4th/5th
|-
| | 2L 8s
| | (P8/2, P5) [10]
| | half-8ve decatonic
| | half-8ve quartertone, half-8ve third-11th
|-
| | 3L 7s
| | (P8, P12/5) [10]
| | fifth-12th decatonic
| | eighth-cc4th, eighth-cc5th
|-
| | 4L 6s
| | (P8/2, P4/2) [10]
| | half-everything decatonic
| |
|-
| | 5L 5s
| | (P8/5, P5) [10]
| | fifth-8ve decatonic
| | (lopsided unless 5th is quite flat)
|-
| | 6L 4s
| | (P8/2, P5/3) [10]
| | half-8ve third-5th decatonic
| |
|-
| | 7L 3s
| | (P8, P5/2) [10]
| | half-5th decatonic
| | ninth-cc5th
|-
| | 8L 2s
| | (P8/2, P4/4) [10]
| | half-8ve quarter-4th decatonic
| | half-8ve quarter-12th
|-
| | 9L 1s
| | (P8, P4/2) [10]
| | quarter-4th decatonic
| |
|}

The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the Sensei aka Sepgu & Ruyoyoti generator would be (P8, ccP5/7) [5]. The rationale would be that two Sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be Roulette [7] aka Zozoquinguti Nowa [7], with a 2.5.7 pergen (P8, (5/4)/2) = (P8, M3/2) = (P8, M2), where 5·G = A6 = 7/4.

==Pergens and EDOs==

Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored.

Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset.

How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinitely many possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, ccP5/31),... (P8, (i-1,1)/n), where n = 12i+7.

How many edos support a given pergen? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be a multiple of m, and k must be divisible by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.

Given an edo, a period, and a generator, what is the pergen? There is usually more than one right answer. For 10edo with P = 5\10 and G = 2\10, it could be either (P8/2, P4/2) or (P8/2, P5/3). Every coprime period/generator pair results in a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.

'''EDOs Supporting A Pergen'''

This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Both of these edos are incompatible with heptatonic notation, and 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. 6-edo, 11-edo and 23-edo could also be considered ambiguous.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! | supporting edos (12-31 only)
|-
| | (P8, P5)
| | unsplit
| | 12, 13, 13b, 14*, 15*, 16, 17, 18, 18b*, 19, 20*,

21*, 22, 23, 24*, 25*, 26, 27, 28*, 29, 30*, 31
|-
! | halves
! |
! |
|-
| | (P8/2, P5)
| | half-8ve
| | 12, 14, 16, 18, 18b, 20*, 22, 24*, 26, 28*, 30*
|-
| | (P8, P4/2)
| | half-4th
| | 13b, 14, 15*, 18b*, 19, 20*, 23, 24, 25*, 28*, 29, 30*
|-
| | (P8, P5/2)
| | half-5th
| | 13, 14*, 17, 18b, 20*, 21*, 24, 27, 28*, 30*, 31
|-
| | (P8/2, P4/2)
| | half-everything
| | 14, 18b, 20*, 24, 28*, 30*
|-
! | thirds
! |
! |
|-
| | (P8/3, P5)
| | third-8ve
| | 12, 15, 18, 18b*, 21, 24*, 27, 30*
|-
| | (P8, P4/3)
| | third-4th
| | 13b, 14*, 15, 21*, 22, 28*, 29, 30*
|-
| | (P8, P5/3)
| | third-5th
| | 15*, 16, 20*, 21, 25*, 26, 30*, 31
|-
| | (P8, P11/3)
| | third-11th
| | 13, 15, 17, 21, 23, 30*
|-
| | (P8/3, P4/2)
| | third-8ve, half-4th
| | 15, 18b*, 24, 30*
|-
| | (P8/3, P5/2)
| | third-8ve, half-5th
| | 18b, 21, 24, 27, 30
|-
| | (P8/2, P4/3)
| | half-8ve, third-4th
| | 14, 22, 28*, 30
|-
| | (P8/2, P5/3)
| | half-8ve, third-5th
| | 16, 20*, 26, 30*
|-
| | (P8/2, P11/3)
| | half-8ve, third-11th
| | 19, 30
|-
| | (P8/3, P4/3)
| | third-everything
| | 15, 21, 30*
|-
! | quarters
! |
! |
|-
| | (P8/4, P5)
| | quarter-8ve
| | 12, 16, 20, 24*, 28
|-
| | (P8, P4/4)
| | quarter-4th
| | 18b*, 19, 20*, 28, 29, 30*
|-
| | (P8, P5/4)
| | quarter-5th
| | 13, 14*, 20, 21*, 27, 28*
|-
| | (P8, P11/4)
| | quarter-11th
| | 14, 17, 20, 28*, 31
|-
| | (P8, P12/4)
| | quarter-12th
| | 13b, 15*, 18b, 20*, 23, 25*, 28, 30*
|-
| | (P8/4, P4/2)
| | quarter-8ve, half-4th
| | 20, 24, 28
|-
| | (P8/2, M2/4)
| | half-8ve, quarter-tone
| | 18, 20, 22, 24, 26, 28
|-
| | (P8/2, P4/4)
| | half-8ve, quarter-4th
| | 18b, 20*, 28, 30*
|-
| | (P8/2, P5/4)
| | half-8ve, quarter-5th
| | 14, 20, 28*
|-
| | (P8/4, P4/3)
| | quarter-8ve, third-4th
| | 28
|-
| | (P8/4, P5/3)
| | quarter-8ve, third-5th
| | 16, 20
|-
| | (P8/4, P11/3)
| | quarter-8ve, third-11th
| |
|-
| | (P8/3, P4/4)
| | third-8ve, quarter-4th
| | 18b*, 30
|-
| | (P8/3, P5/4)
| | third-8ve, quarter-5th
| | 21, 27
|-
| | (P8/3, P11/4)
| | third-8ve, quarter-11th
| |
|-
| | (P8/3, P12/4)
| | third-8ve, quarter-12th
| | 15, 18b, 30*
|-
| | (P8/4, P4/4)
| | quarter-everything
| | 20, 28
|}
The edos that support the fewest pergens are prime-number edos like 11edo or 13edo. The most "pergen-friendly" edos tend to be ones in which the circle of 5ths doesn't reach every edostep. For example, 24edo supports all half-split pergens, since both P8 and P5 map to an even number of edosteps. 72edo supports all half-splits and all third-splits. 15, 21 and 36 edo support many but not all third-splits (not those with m = 2 or n = 2).

'''Notating a pergen tuned to an EDO'''

If both the pergen and the EDO are notated with ups and downs, what is the relationship between the two kinds of up? If the edo supports the pergen, fully or partially, then the pergen's up equals some multiple of the EDO's up, i.e. some number of edosteps. For third-4th in 22edo or 29edo, the pergen's up = 1 edostep. But in 37edo or 44edo, ^1 = 2 edosteps. For half-8ve in 12edo, ^1 = 0 edosteps, and the ups and downs in the score can simply be ignored. In fact, it seems every pergen in 5edo, 7edo and 12edo has ^1 = 0 edosteps. It's not yet known why.

When notating a piece in a specific pergen meant to be played in a specific EDO, either the pergen notation or the EDO notation can be used. For small or mid-sized EDOs, they're usually identical. If one has to choose, the pergen notation is generally preferred. It's less cluttered. Also, it's easier to mentally double every up and down in larger EDOs than it is to halve them in smaller EDOs.

Half-8ve in 22edo has P = vA4. But in 16edo, P = A4 + 2 edosteps, and ^1 = -2 edosteps. Negative edosteps means up is down, and should be avoided. The notation has tipped, and the period should be notated as ^A4, making ^1 = 2 edostep. Even better would be P = ^4, making ^1 = 1 edostep.

Half-5th has EU = vvA1, hence any EDO in which A1 = 4 edosteps will have ^1 = 2 edosteps. These "doubled EDOs" are 20, 27, 34, 41, 48, 55, etc. The "tripled EDOs" with A1 = 6 edosteps and ^1 = 3 edosteps are every 7th EDO from 30 to 72.

Half-4th has EU = vvm2. Doubled EDOs have m2 = 4 edosteps. These are 23, 28, 33, 38, 43, 48, 53, etc. Tripled EDOs are every 5th one from 47 to 72.

Third-4th has EU = v3A1. Doubled EDOs are the same ones as half-5th's tripled EDOs. Third-5th has EU = v3m2. Doubled EDOs are the same as half-4th's tripled EDOs.

The relationship between a pergen's up and an EDO's up when the pergen uses double-pair notation is complex. Consider half-everything, notated with ups/downs in the perchain and lifts/drops in the genchain. 10edo has ^1 = 1 edostep and /1 = 0 edosteps, thus one simply ignores lifts and drops. But for 24edo, ^1 = 0 edosteps and /1 = 1 edostep. One must ignore ups and downs and convert lifts/drops to edosteps.

'''Pergens Within An EDO'''

See the screenshots in the Supplemental Materials section for a complete list of which pergens are supported by 12, 15 and 19 edo. The list includes multiple pergens per period/generator combination. The next table shows only the simplest pergen for each combination. Combinations are excluded if the period and generator are not coprime, because the scale is contained in a smaller edo. For example, 15edo has no unsplit pergen, because those scales are contained in 5edo. Periods of 1 or 2 edosteps are excluded as too trivial, because the scale is the entire edo. Every step of the edo appears in the genchains, even if the chains are only one step long.

To find the pergen, find the edo, then find the row that corresponds to the period, then find the column that corresponds to the generator. It appears, but has yet to be formally proven, that the number of P/G combinations that N-edo supports is (N-1)/2 rounded down. If we include the trivial combination where P = 1 edostep, the number of combinations would be N/2 rounded up.

{| class="wikitable" style="text-align:center;"
|-
! | EDO
! | Period
! colspan="11" | Generator in edosteps
|-
! |
! | in edosteps
! | 1
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
|-
! | 5
! | 5 = P8
| | P4/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 6
! | 6 = P8
| | P4/2
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 7
! | 7 = P8
| | P4/3
| | P5/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 8
! | 8 = P8
| | P4/3
| | -
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/2
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 9
! | 9 = P8
| | P4/4
| | P4/2
| | -
| | P5
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/3
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 10
! | 10 = P8
| | P4/4
| | -
| | P5/2
| | -
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 5 = P8/2
| | P5
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 11
! | 11 = P8
| | P4/5
| | P5/3
| | P5/2
| | P11/4
| | P5
| |
| |
| |
| |
| |
| |
|-
! | 12
! | 12 = P8
| | P4/5
| | -
| | -
| | -
| | P5
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/2
| | P5
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/3
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/4
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 13b
! | 13 = P8
| | P4/6
| | P4/3
| | P4/2
| | P12/5
| | P12/4
| | P5
| |
| |
| |
| |
| |
|-
! | 14
! | 14 = P8
| | P4/6
| | -
| | P4/2
| | -
| | P11/4
| | -
| |
| |
| |
| |
| |
|-
! | "
! | 7 = P8/2
| | P5
| | P4/3
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 15
! | 15 = P8
| | P4/6
| | P4/3
| | -
| | P12/6
| | -
| | -
| | P11/3
| |
| |
| |
| |
|-
! | "
! | 5 = P8/3
| | P5
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/5
| | P4/3
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 16
! | 16 = P8
| | P4/7
| | -
| | P5/3
| | -
| | P12/5
| | -
| | P5
| |
| |
| |
| |
|-
! | "
! | 8 = P8/2
| | P5
| | -
| | P5/3
| | -
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/4
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 17
! | 17 = P8
| | P4/7
| | P5/5
| | P11/8
| | P11/6
| | P5/2
| | P11/4
| | P5
| | P11/3
| |
| |
| |
|-
! | 18b
! | 18 = P8
| | P4/8
| | -
| | -
| | -
| | P5/2
| | -
| | P12/4
| | -
| |
| |
| |
|-
! | "
! | 9 = P8/2
| | P5
| | P4/4
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/3
| | P5/2
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/6
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 19
! | 19 = P8
| | P4/8
| | P4/4
| | P11/9
| | P4/2
| | P12/6
| | P12/5
| | ccP5/7
| | P5
| | P11/3
| |
| |
|-
! | 20
! | 20 = P8
| | P4/8
| | -
| | P5/4
| | -
| | -
| | -
| | P11/4
| | -
| | c3P5/8
| |
| |
|-
! | "
! | 10 = P8/2
| | M2/4
| | -
| | P5/4
| | -
| | -
| |
| |
| |
| |
| |
| |
|-
! | "
! | 5 = P8/4
| | P4/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/5
| | P5/4
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 21
! | 21 = P8
| | P4/9
| | P5/6
| | -
| | P5/3
| | P11/6
| | -
| | -
| | c3P4/9
| | -
| | P11/3
| |
|-
! | "
! | 7 = P8/3
| | P5/2
| | P5
| | P4/3
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/7
| | P5/3
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 22
! | 22 = P8
| | P4/9
| | -
| | P4/3
| | -
| | P12/7
| | -
| | P12/5
| | -
| | P5
| | -
| |
|-
! | "
! | 11 = P8/2
| | M2/4
| | P5
| | P4/3
| | P12/5
| | P12/7
| |
| |
| |
| |
| |
| |
|-
! | 23
! | 23 = P8
| | P4/10
| | P4/5
| | P11/11
| | P12/9
| | P4/2
| | P12/6
| | ccP4/8
| | ccP4/7
| | P12/4
| | P5
| | P11/3
|-
! | 24
! | 24 = P8
| | P4/10
| | -
| | -
| | -
| | P4/2
| | -
| | P5/2
| | -
| | -
| | -
| | c4P5/10
|-
! | "
! | 12 = P8/2
| | M2/4
| | -
| | -
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
|-
! | "
! | 8 = P8/3
| | P5/2
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/4
| | P4/2
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/6
| | P4/2
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/8
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! |
! |
! | 1
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
|}
Larger edos can have very awkward pergens. For example, 31edo with G = 14/31 is (P8, c5P4/12). It's much simpler to think of the generator as the downfifth = 17\31, and the pergen as the pseudopergen (P8, v5).

'''EDO-pair names'''

Just as a pair of edos and a prime subgroup can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo & N'-edo, m = GCD (N,N'). The period P equals both (N/m)\N and (N'/m)\N'. For example, for 12edo and 16edo, m = 4, and the period is both 3\12 and 4\16.

For each edo, find the nearest '''edomapping''' (patent val) for the 2.3 subgroup. If the edo has a "b" wart, e.g. 13b-edo, use the second-nearest edomapping. Form a 2x2 matrix from these edomappings. Let d be the determinant of this matrix. If d = m, -m or 0, the generator is the 5th, and the pergen is simply (P8/m, P5).

For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.

If |d| ≠ m or 0, P5 is not the generator, and we must find the multigen. Take the ratio of the two edos N/N' and reduce it by m. In the scale tree ([http://tallkite.com/misc_files/Scale-Tree-Complete.pdf pdf] or [http://tallkite.com/misc_files/Scale-Tree-Complete.jpg jpeg]), let g/g' be the smallest ancestor of this ratio. The generator G maps to both g\N and g'\N'. The two edos come closest to coinciding here.

For example, for 7-edo and 17-edo, m = 1 but d = 2, so G ≠ P5. The smallest ancestor of 7/17 is 2/5. G maps to both 2\7 and 5\17, which are both neutral 3rds. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths. 2\7 = 343¢ and 5\17 = 353¢, and their difference is only 1\N", where N" = LCM (N, N'). This 10¢ difference is the least difference between any 7edo note and any 17edo note, except that the two neutral 6ths differ by the same 10¢, and of course some note pairs coincide exactly, such as 0\7 and 0\17.

Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found from the scale tree, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this new vector with either edomapping is zero. Treating this new vector as a monzo, since the dot product is zero, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for. If the comma is (a, b, c), then a·P8 + b·P5 + c·G = 0, and G = (b-a, -b) / c.

For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2).

To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 (Dicot aka Yoyo). 11/9 also works, it yields 243/242 (Mohajira aka Lulu).

If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a multi-EDO pergen.

If the 1st edo is 7edo, the pergen indicates the natural heptatonic notation for the 2nd edo, e.g. 12edo = unsplit, 15edo = third-4th and 17edo = half-5th.

The closer two edos are in the scale tree, the smaller the splitting fractions in the pergen they make. Examples:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 12-edo
! | 13b-edo
! | 14-edo
! | 15-edo
! | 16-edo
! | 17-edo
! | 18b-edo
! | 19-edo
! | 20-edo
|-
! | 13b-edo
| | (P8, P5/7)
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 14-edo
| | (P8/2, P5)
| | (P8, P4/6)
| |
| |
| |
| |
| |
| |
| |
|-
! | 15-edo
| | (P8/3, P5)
| | (P8, c5P5/12)
| | (P8, P4/6)
| |
| |
| |
| |
| |
| |
|-
! | 16-edo
| | (P8/4, P5)
| | (P8, P12/5)
| | (P8/2, P5)
| | (P8, P5/9)
| |
| |
| |
| |
| |
|-
! | 17-edo
| | (P8, P5)
| | (P8, ccP5/11)
| | (P8, P11/4)
| | (P8, P11/3)
| | (P8/2, P4/7)
| |
| |
| |
| |
|-
! | 18b-edo
| | (P8/6, P5)
| | (P8, P12/4)
| | (P8/2, P4/2)
| | (P8/3, P12/4)
| | (P8/2, P5)
| | (P8, P5/10)
| |
| |
| |
|-
! | 19-edo
| | (P8, P5)
| | (P8, P12/10)
| | (P8, P4/2)
| | (P8, P12/6)
| | (P8, P12/5)
| | (P8, P11/3)
| | (P8, P4/8)
| |
| |
|-
! | 20-edo
| | (P8/4, P5)
| | (P8, ccP4/16)
| | (P8/2, P5/4)
| | (P8/5, ^1)
| | (P8/4, P5/3)
| | (P8, P11/4)
| | (P8/2, P4/8)
| | (P8, P4/8)
| |
|-
! | 21-edo
| | (P8/3, P5)
| | (P8, c3P4/9)
| | (P8/7, ^1)
| | (P8/3, P4/3)
| | (P8, P5/3)
| | (P8, P11/6)
| | (P8/3, P5/2)
| | (P8, P11/3)
| | (P8, P5/12)
|-
! | 22-edo
| | (P8/2, P5)
| | (P8, c3P4/15)
| | (P8/2, P4/3)
| | (P8, P4/3)
| | (P8/2, P12/5)
| | (P8, P5)
| | (P8/2, P12/7)
| | (P8, P12/5)
| | (P8/2, M2/4)
|-
! | 23-edo
| | (P8, P4/5)
| | (P8, ccP4/8)
| | (P8, P4/2)
| | (P8, P12/12)
| | (P8, P5)
| | (P8, P12/9)
| | (P8, P12/4)
| | (P8, P12/6)
| | (P8, c5P5/16)
|-
! | 24-edo
| | (P8/12, ^1)
| | (P8, c6P4/14)
| | (P8/2, P4/2)
| | (P8/3, P4/2)
| | (P8/8, P5)
| | (P8, P5/2)
| | (P8/6, P4/2)
| | (P8, P4/2)
| | (P8/4, P4/2)
|}

A specific pergen can be converted to an edo pair by finding the range of its generator cents in the [[pergen#Further Discussion-Notating tunings with an arbitrary generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of edos 7, 10 and 17 defines (P8, P5/2).

==Array Keyboards (unfinished)==

Array keyboards have a 2-dimensional layout of keys, and are very appropriate for rank-2 tunings. A good layout can be found from the tuning's pergen. First find an edo N-edo that is compatible with the pergen, then arrange the keys in N columns to the 8ve, with each row usually containing the multigen interval. The unsplit pergen in 7 columns:

{| class="wikitable" style="text-align:center;"
|-
| | D#
| |
| |
| |
| |
| |
| |
| |
|-
| | D
| | E
| | F#
| | G#
| | A#
| |
| |
| |
|-
| | Db
| | Eb
| | F
| | G
| | A
| | B
| | C#
| | D#
|-
| |
| |
| |
| | Gb
| | Ab
| | Bb
| | C
| | D
|-
| |
| |
| |
| |
| |
| |
| |
| | Db
|}
Higher notes are at the top of each column. The rows would actually be angled so that the two D's are at the same horizontal level. The vertical steps are A1 and the horizontal steps are M2, and the keyboard is defined as 7(A1, M2).

The third-4th keyboard is 7(A1/3 = ^1 = vvA1, P4/3 = vM2 = ^^m2).

{| class="wikitable" style="text-align:center;"
|-
| | vD#
| | ^E
| | F#
| | vG#
| | ^A
| | B
| | vC#
| | ^D
|-
| | ^D
| | E
| | vF#
| | ^G
| | A
| | vB
| | ^C
| | D
|-
| | D
| | vE
| | ^F
| | G
| | vA
| | ^B
| | C
| | vD
|-
| | vD
| | ^Eb
| | F
| | vG
| | ^Ab
| | Bb
| | vC
| | ^Db
|}

''Hypothesis: Let the 5th's keyspan (i.e. column-span) be F. In order for the keyboard to have the pitches in order, the fifth must fall between the two Stern-Brocot ancestors of F\N (simplified if possible). For example, an 8-column keyboard has F = 5, the ancestors of 5\8 are 3\5 and 2\3, and the 5th must be between 720¢ and 800¢. Thus the most musically useful N values are 5, 7, 10, 12 and 14.''

(more to come)

==Supplemental materials==

===Notation guide PDF===

This PDF is a rank-2 notation guide that shows the full lattice for the first 32 pergens, up through the quarter-splits block. It also includes the single-split pergens from the fifth-split, sixth-split and seventh-split blocks. It includes alternate EUs for many pergens.

[http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf '''<big>TallKite.com/misc_files/notation guide for rank-2 pergens.pdf</big>''']

{| class="wikitable" style="text-align:center;"
|+Table of contents for the N'''otation Guide for Rank-2 Pergens''' (* indicates a true double)
|-
! colspan="3" |unsplit
! colspan="3" |quarter-splits
! colspan="3" |single-split fifth-splits
! colspan="3" |single-split seventh-splits
|-
!1
|(P8, P5)
|unsplit
!16
|(P8/4, P5)
|quarter-8ve
!33
|(P8/5, P5)
|fifth-8ve
!96
|(P8/7, P5)
|seventh-8ve
|-
! colspan="3" |half-splits
!17
|(P8, P4/4)
|quarter-4th
!34
|(P8, P4/5)
|fifth-4th
!97
|(P8, P4/7)
|seventh-4th
|-
!2
|(P8/2, P5)
|half-8ve
!18
|(P8, P5/4)
|quarter-5th
!35
|(P8, P5/5)
|fifth-5th
!98
|(P8, P5/7)
|seventh-5th
|-
!3
|(P8, P4/2)
|half-4th
!19
|(P8, P11/4)
|quarter-11th
!36
|(P8, P11/5)
|fifth-11th
!99
|(P8, P11/7)
|seventh-11th
|-
!4
|(P8, P5/2)
|half-5th
!20
|(P8, P12/4)
|quarter-12th
!37
|(P8, P12/5)
|fifth-12th
!100
|(P8, P12/7)
|seventh-12th
|-
!5
|(P8/2, P4/2) *
|half-everything *
!21
|(P8/4, P4/2) *
|quarter-8ve, half-4th *
!38
|(P8, ccP4/5)
|fifth-coco-4th
!101
|(P8, ccP4/7)
|seventh-coco-4th
|-
! colspan="3" |third-splits
!22
|(P8/2, M2/4)
|half-8ve, quarter-tone
! colspan="3" |single-split sixth-splits
!102
|(P8, ccP5/7)
|seventh-coco-5th
|-
!6
|(P8/3, P5)
|third-8ve
!23
|(P8/2, P4/4) *
|half-8ve, quarter-4th *
!64
|(P8/6, P5)
|sixth-8ve
!103
|(P8, c3P4/7)
|seventh-trico-4th
|-
!7
|(P8, P4/3)
|third-4th
!24
|(P8/2, P5/4) *
|half-8ve, quarter-5th *
!65
|(P8, P4/6)
|sixth-4th
| colspan="3" rowspan="9" |
|-
!8
|(P8, P5/3)
|third-5th
!25
|(P8/4, P4/3)
|quarter-8ve, third-4th
!66
|(P8, P5/6)
|sixth-5th
|-
!9
|(P8, P11/3)
|third-11th
!26
|(P8/4, P5/3)
|quarter-8ve, third-5th
!67
|(P8, P11/6)
|sixth-11th
|-
!10
|(P8/3, P4/2)
|third-8ve, half-4th
!27
|(P8/4, P11/3)
|quarter-8ve, third-11th
!68
|(P8, P12/6)
|sixth-12th
|-
!11
|(P8/3, P5/2)
|third-8ve, half-5th
!28
|(P8/3, P4/4)
|third-8ve, quarter-4th
!69
|(P8, ccP4/6)
|sixth-coco-4th
|-
!12
|(P8/2, P4/3)
|half-8ve, third-4th
!29
|(P8/3, P5/4)
|third-8ve, quarter-5th
!70
|(P8, ccP5/6)
|sixth-coco-5th
|-
!13
|(P8/2, P5/3)
|half-8ve, third-5th
!30
|(P8/3, P11/4)
|third-8ve, quarter-11th
| colspan="3" rowspan="3" |
|-
!14
|(P8/2, P11/3)
|half-8ve, third-11th
!31
|(P8/3, P12/4)
|third-8ve, quarter-12th
|-
!15
|(P8/3, P4/3) *
|third-everything *
!32
|(P8/4, P4/4) *
|quarter-everything *
|}Screenshots of the first 2 pages:

[[File:pergens_1.png|alt=pergens 1.png|704x948px|pergens 1.png]]

[[File:pergens_2.png|alt=pergens 2.png|704x760px|pergens 2.png]]

===PergenLister===

PergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonic unisons for each one. Alternate EUs are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair.

http://www.tallkite.com/apps/pergenLister.html (written in Jesusonic, runs inside Reaper or ReaJS)

The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. EUs of a 3rd or more are in red.

Screenshots of the first 69 pergens:

[[File:alt-pergenLister_1.png|alt=alt-pergenLister 1.png|800x427px|alt-pergenLister 1.png]]

[[File:alt-pergenLister_2.png|alt=alt-pergenLister 2.png|800x455px|alt-pergenLister 2.png]]

The first 29 pergens supported by 12edo:

[[File:alt-pergenLister_12edo.png|alt=alt-pergenLister 12edo.png|800x449px|alt-pergenLister 12edo.png]]

Some of the pergens supported by 15edo. A red asterisk means partial support.

[[File:alt-pergenLister_15edo.png|alt=alt-pergenLister 15edo.png|800x493px|alt-pergenLister 15edo.png]]

Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.

[[File:alt-pergenLister_19edo.png|alt=alt-pergenLister 19edo.png|800x459px|alt-pergenLister 19edo.png]]

Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:

If z > 0, then y is at least ceiling (x·(3/2 - z/2)) and at most floor (x·5/3)

If z < 0, then y is at least ceiling (x·3/2) and at most floor (x·(5/3 - z/2))

Next we loop through all combinations of x and z in such a way that larger values of x and z come last:

i = 1; loop (maxFraction,

<ul><li>j = 1; loop (i - 1,<ul><li>makeMapping (i, j); makeMapping (i, -j);</li><li>makeMapping (j, i); makeMapping (j, -i);</li><li>j += 1;</li></ul></li><li>);</li><li>makeMapping (i, i); makeMapping (i, -i);</li><li>i += 1;</li></ul>);

The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.

==Various proofs (unfinished)==

Although not yet rigorously proven, the two false-double tests have been empirically verified by pergenLister.

The interval P8/2 has a "ratio" of the square root of 2, which equals 21/2, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the '''pergen matrix''' [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.

Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise a, b, and n could all be reduced by GCD (a,b), and the multigen could be simplified. Since GCD (a, b) = 1 and -am/b is an integer, it follows that m must be a multiple of |b| as well.

Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|) = |b| · r. If r = 1, then GCD (m, n) = |b|, and vice versa, which is the proposed test for a false double.

If m = |b|, is the pergen explicitly false? Does splitting (a,b) into n generators also split P8 into m periods?

(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5

Because n is a multiple of b, n/b is an integer

M/b = (n/b)·M/n = (n/b)·G 
(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)

Let c and d be the bezout pair of a+b and b, with c·(a+b) + d·b = 1

Since the pergen is a double-split, m > 1, therefore |b| > 1, therefore c ≠ 0

c·(a+b)·P8 = c·b·((n/b)·G - P5) 
(1 - d·b)·P8 = c·b·((n/b)·G - P5) 
P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5) 
P8/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)

Therefore P8 is split into m periods 
Therefore if m = |b|, the pergen is explicitly false

Assume the pergen is a false double, and there's a comma C that splits both P8 and (a,b) appropriately. Can we prove r = 1? Let Q = the higher prime that C uses. Express P, G and C as monzos of the prime subgroup 2.3.Q, by expanding the 2x2 pergen matrix to a 3x3 matrix A:

P = (1/m, 0, 0) 
G = (a/n, b/n, 0) 
C = (u, v, w)

Here u, v and w are integers. If GCD (u, v, w) > 1, simplify C so that it = 1. The inverse of A expresses 2, 3 and Q in terms of P, G and C. If C is tempered out, the C column can be discarded, making the usual 3x2 period-generator mapping. However, if C is not tempered out, the inverse of A is a 3x3 period-generator-comma mapping, which is simply a change of basis. For example, 5-limit JI can be generated by 2/1, 3/2 and 81/80. Here is the inverse of A:

2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) 
3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C) 
Q = Q/1 = ((av-bu)m/wb, -vn/wb, 1/w) · (P, G, C)

Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. Every pergen except the unsplit one requires |w| > 1, so the last column almost always has a fraction. To avoid fractions in the first two columns, A must be unimodular '''''[I think, not sure]''''', and we have wb/mn = ±1, and w = ±mn/b. Substituting for w, we have:

2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) 
3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C) 
Q = Q/1 = (±(av-bu)/n, ±(-v)/m, ±b/mn) · (P, G, C)

For v/m to be an integer, v must equal i·m for some integer i. Likewise, av-bu must equal j·n for some integer j. Thus bu = av - jn = iam - jn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb. Substituting, we get bu = iaprb - jqrb, and u = r(iap - jq). Furthermore, v = im = iprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r > 1, this contradicts the requirement that GCD (u, v, w) = 1, therefore r must be 1, and GCD (m, n) = |b|, and all false doubles pass the false-double test.

Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q?

Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b.

a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)

Thus a·P8 splits into b parts, and since m = |b|, a·P8 splits into m parts. Proceed as before with a bezout pair to find the monzo for P8/m.

Next, assume the pergen isn't explicitly false. The unreduced form is (P8/m, (n - am, -bm) / mn). Substituting in m = pb and n = qb, with p and q coprime, we get (P8/m, (q - ap, -pb) / pqb).

Assume the pergen is a true double, and r > 1. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.

Given a pergen (P8/m, (a,b)/n), how many parts is an arbitrary interval (a',b') split into?

(a',b') = (a'·b, b'·b) / b = (a'·b - a·b', 0) / b + (a·b', b'·b) / b = (a'·b - a·b')·P8 / b + b'·(a,b) / b = (a'·b - a·b')·(m/b)·P + b'·(n/b)·G

Therefore (a',b') is split into GCD (a'·b - a·b')·(m/b), b'·(n/b)) parts.

If m = 1, then b = ±1, and we have GCD (a' ± a·b', b'·n)

If n = 1, then a = -1 and b = 1, and we have GCD (a'·m + b'·m, b') = GCD (a'·m, b')

If m = 1 and n = 1, we have GCD (a', b') = the naturally occurring split.

If m = n (nth-everything), we have n · GCD (a', b')

The multigen and the arbitrary interval can be expressed as gedras:

(a,b) = [k,s] = (-11k+19s, 7k-12s)

(a',b') = [k',s'] = (-11k'+19s', 7k'-12s')

a'·b - a·b' works out to be k·s' - k'·s, and we have GCD ((k·s' - k'·s)·m/b, b'·n/b)

If s is a multiple of n (happens when EU is an A1) and s' is a multiple of n, let s = x·n and s' = y·n

GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')

Thus every such interval is split, e.g. the half-5th pergen splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. This is an easy way to determine if an interval is split or not by the pergen.

To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts

if r > 1, it's a true double

a·P8 = (a,0) = (a,b) - (0,b) = M - b·P12

M = n·G = qrb·G

a·P8 = qrb·G - b·P12 = b·(qr·G - P12)

Let c and d be the bezout pair of a and b, with c·a + d·b = 1

If |b| = 1, let c = 1 and d = ±a, to avoid c = 0

ca·P8 = cb·(qr·G - P12)

(1 - d·b)·P8 = c·b·(qr·G - P12)

P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) = b · (d,-c) + bcqr·G

== Glossary ==
to find the definition, use control-F to search for the first (bolded) occurrence of the word on this page.

pergen 
split 
multigen 
ups and downs (the ^ and v symbols) 
higher prime (any prime > 3) 
color depth 
dependent/independent 
square mapping 
lifts and drops (the / and \ symbols) 
enharmonic unison, EU 
uninflected 
genchain 
perchain 
compound (increased by an octave) 
single-split, double-split 
single-pair, double-pair (number of new accidentals in the notation) 
true double, false double 
explicitly false 
unreduced 
alternate vs. equivalent (generator or period) 
mapping comma 
keyspan 
stepspan 
gedra 
count 
mid 
edomapping 
upspan 
liftspan

chain number 
single-chain 
multi-chain 
arrow comma

==Miscellaneous Notes==

=== Combining pergens ===
Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). If adding a comma to a temperament doesn't change the pergen, it's a strong extension, otherwise it's a weak extension.

General rules for combining pergens:

<ul><li>(P8/m, M/n) + (P8, P5) = (P8/m, M/n)</li>
<li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li>
<li>(P8/m, P5) + (P8/m', P5) = (P8/m", P5), where m" = LCM (m,m')</li>
<li>(P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n')</li></ul>

However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.

If a false double pergen can be broken down into two simpler ones, that may help with finding a double-pair notation with an EU of a 2nd or less. For example, sixth-4th's single pair notation has an EU of a 4th. But since sixth-4th is half-4th plus third-4th, and those two have a good EU, sixth-4th can be notated with one pair from half-4th and another from third-4th.

=== Expanding gedras ===
Gedras can be expanded to 5-limit or higher by including another keyspan that is compatible with 7 and 12, such as 9 or 16. But a more useful approach is for the third number to be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]:

k = 12a + 19b + 28c + 34d 
s = 7a + 11b + 14c + 20d 
g = -c 
r = -d

a = -11k + 19s - 4g + 6r 
b = 7k - 12s + 4g - 2r 
c = -g 
d = -r

Gedras can also be expanded by adding an entry for ups/downs. The entry shows the '''upspan''', which is the number of ups the interval has. Downed intervals have a negative upspan. An entry for '''liftspan''' can also be added. For example, vM3 = [4,2,-1], and ^^\4 = [5,3,2,-1].

=== Height of a pergen ===
The LCM of the pergen's two splitting fractions could be called the '''height''' of the pergen. For example, (P8, P5) has height 1, and (P8/2, M2/4) has height 4. In single-pair notation, the EU's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.

=== Generalizing the pergen ===
See [[User:AthiTrydhen/Abstract pergens]]

=== Credits ===
Pergens were discovered by [[KiteGiedraitis|Kite Giedraitis]] in 2017, and developed with the help of [[Praveen Venkataramana]].

== Addenda (late 2023) ==
=== New terminology===
All temperaments have a '''chain number''', which is the number of fifthchains in the temperament's lattice. Any (P8, P5) temperament has a chain number of 1, and is '''single-chain'''. All other pergens are '''multi-chain'''. For example, Porcupine/Triyoti has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Saguguti has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m * n / f, where M is the multigen and f is the absolute value of M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.

===The EU(s) define the pergen===
The pergen can be derived directly from the EU(s). Thus the EU(s) define both the pergen and the notation. An EU can be thought of as a comma in the 2.3.^ subgroup. One derives a mapping from this comma as one would for any 3-prime comma, and one derives the pergen by inverting the first 2 columns of the mapping.

For example, if the EU is vvd2, the 2.3.^ monzo is [19 -12 -2]. There are many possible mappings, but only one that gives a canonical pergen: [(2 2 7) (0 1 -6)]. Discarding the last column and inverting gives us [(1/2 0) (-1 1)] = (P8/2, P5). Another example: vvA1 = [-11 7 -2]. The mapping [(1 1 -2) (0 2 7)] reduces to [(1 1) (0 2)], which inverts to [(1 0) (-1/2 1/2)] = (P8, P5/2).

One can explore the universe of possible EUs, and thus possible pergen notations, more easily if using the gedra format, expressing the EU as a combination of A1's, d2's and arrows. Thus vvA1 = [1 0 -2], v3m2 = [1 1 -3], etc. Unlike a conventional 2.3.^ monzo, the first two numbers in a A1.d2.^1 monzo are fairly small, and the second number is never negative (since it's an EU). And we can require that the first two numbers be coprime (see the next section). All this facilitates one's search.

===Simplifying a "squared" EU===
Consider an uninflected EU of AA1. AA1 is "squared" in the sense that AA1 = A1 + A1, thus both numbers in its ratio are square numbers. If it had an even upspan (the number of ups), it would obviously be an invalid EU, for if v4AA1 = 0¢, then so does vvA1, and v4AA1 could be replaced with vvA1. So the upspan must be odd.

Consider an EU of v3AA1. The pergen is (P8, P4/3). Here are the [[twin squares]].

<math>
\begin{array} {rrr}
\text{P8} \\
\text{^m2} \\
\text{vvvAA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & {\color {Green}0} \\
8 & -5 & {\color {Green}1} \\
\hline
{\color {Red}-22} & {\color {Red}14} & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & {\color {Red}2} \\
0 & -3 & {\color {Red}-14} \\
\hline
{\color {Green}0} & {\color {Green}-1} & -5 \\
\end{array} \right]
</math>

The two red numbers in the lower left are both even, because AA1 is a squared ratio. As a result, the two red numbers in the upper right must both be even to ensure their rows' dot products with the EU are zero, which is an even number. If we halve all the red numbers and double all the green numbers, all the various row dot products will be unchanged, and the twin squares will remain valid:

<math>
\begin{array} {rrr}
\text{P8} \\
\text{^^m2} \\
\text{vvvA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & {\color {Green}0} \\
8 & -5 & {\color {Green}2} \\
\hline
{\color {Red}-11} & {\color {Red}7} & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & {\color {Red}1} \\
0 & -3 & {\color {Red}-7} \\
\hline
{\color {Green}0} & {\color {Green}-2} & -5 \\
\end{array} \right]
</math>

But while the EU has become simpler, the generator has become more complex in that it is dup not up. This is remedied by adding the EU to it. Changes are in red:

<math>
\begin{array} {rrr}
\text{P8} \\
\text{vM2} \\
\text{vvvA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & 0 \\
{\color {Red}-3} & {\color {Red}2} & {\color {Red}-1} \\
\hline
-11 & 7 & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & 1 \\
0 & -3 & -7 \\
\hline
{\color {Red}0} & {\color {Red}1} & {\color {Red}2} \\
\end{array} \right]
</math>

Following this procedure, it's always possible to simplify a squared (or cubed, etc.) EU.

===Arrow commas===
The '''[[arrow]] comma''' is the ratio that equals the up [[arrow]]. This term overlaps a lot with the term mapping comma, but it isn't quite identical to it. For 5-limit temperaments the arrow comma is almost always 81/80, and for 7-limit it's almost always 64/63. But other commas can occur.

Consider Triyoti/Porcupine which is (P8, P4/3). The vanishing comma or '''VC''' is 250/243, which has a 2.3.5 monzo of [2 -5 3]. The EU is v3A1, which has a 2.3.^ monzo of [-11 7 -3]. The arrow comma or '''AC''' equals an up, therefore it vanishes when downed. The downed AC (or '''vAC''') can be expressed as a 2.3.5.^ monzo. For Triyoti/Porcupine with an EU of v3A1, the vAC is v(81/80) or [-4 4 -1 -1].

===The three commas ===
Thus when we consider a single-comma temperament along with its notation, there are three commas of interest, the VC, the vAC and the EU. In a 5-limit rank-2 temperament, they can all be expressed as 2.3.5.^ monzos.

Any two of these 3 commas determines the third comma. Thus we can approach temperaments and notations from 6 different angles. We can start with any one of these three commas and give it a specific monzo. Then we can choose one of the other two commas, experiment with a variety of specific monzos and examine the results. Let's start with specifying the VC, experimenting with the vAC and seeing what the EU turns out to be.

The EU always equals the VC (possibly inverted) plus or minus some number of vAC's. That number is whatever is needed to eliminate the higher prime. The VC must be inverted if the resulting EU would otherwise have a negative stepspan, or is a diminished unison.

In our Triyoti example, 250/243 plus 3 downed syntonic commas = v3A1. As 2.3.5.^ monzos, we have [1 -5 3 0] + 3·[-4 4 -1 -1] = [-11 7 0 -3]. Note the zeros. The VC always has a zero arrow-count and the EU always has a zero prime-5-count.

Visualizing the three commas in the lattice, one starts at the VC and heads towards the row of 3-limit intervals via the vAC. Wherever one lands is the uninflected EU. One can try other AC's besides 81/80. The AC's prime-5-count must be ±1, so [[135/128|Layobi]] (135/128) or [[Schisma|Layo]] (the schisma) are possibilities. But either of these would lead to a very remote EU (v3AA1 and v3d44 respectively), making a very awkward notation.

Next let's specify the AC, experiment with the VC and see what the EU turns out to be. For 5-limit temperaments, one can require that the AC always be 81/80, and derive the EU (and thus the notation) from the VC. For example, Saguguti/Diaschismic has VC = [11 -4 -2 0]. Subtracting two vAC's makes [19 -12 0 2] = ^^d2. This is in fact the recommended EU for (P8/2, P5).

More examples: Laquinyoti/Magic is (P8, P12/5) and has VC = [-10 -1 5 0]. Adding five vAC's makes [-30 19 0 -5]. This appears to be a AA7 but is actually a negative 2nd. We invert to get [30 -19 0 5] = ^5dd2. Guguti/Bug has VC = 27/25 = [0 3 -2 0]. Subtracting two vAC's makes [8 -5 0 2] = ^^m2. Again, this is the recommended EU.

Let's try a 7-limit temperament with the obvious vAC of 64/63 = [6 -2 -1 -1]. Zozoti/Semaphore has VC = 49/48 = [-4 -1 2 0]. Adding two vAC's makes [8 -5 0 -2] = vvm2.{{todo|complete section|comment=apply to multi-comma temperaments|inline=1}}

== Addenda (late 2024) ==

=== Chord names ===
When naming chords, it's very convenient to have the freedom to rename an aug 4th as a dim 5th, or a minor 10th as an aug ninth. Thus for some pergens, an extra pair of accidentals is used. Some examples:

* [[Chords of meantone]] (P8, P5) (^1 = -d2 = pythagorean comma)
* [[Chords of diaschismic]] (P8/2, P5)
* [[Chords of hemififths]] (P8, P5/2) (/1 = vm2 = ~81/80 = ~64/63)
* [[Chords of porcupine]] (P8, P4/3)
* [[Chords of magic]] (P8, P12/5) (/1 = ^^d2)

=== Frequency of imperfect pergens ===
Imperfect pergens occur when there are multiple genchains (i.e. the octave is split), and the fifth is on a different genchain than the tonic, and also on a different perchain. How often do they occur? In order to answer that, we need to survey all pergens in order. But the question of how to do that depends on how they are sorted. The pergenLister app sorts them by the size of the larger denominator. Using this order, pergenLister finds about 4% of all pergens are imperfect. But they can also be sorted by their canonical mappings [(a b) (0 c)]. If sorted by a (octave fraction), and then by |c| (perfect multigen's fraction), more complex pergens appear sooner, and the percentage rises to about 25%.

This table lists all pergens with an unsplit octave up to the fifth-splits. In each column, the pergens are sorted by the size of the generator. The generator is listed followed by a, b and c from its mapping. All pergens with an unsplit octave are perfect.
{| class="wikitable"
|+Pergens of the form (P8, x), showing generator and mapping (a = 1)
!unsplit
!half-splits
!third-splits
!quarter-splits
!fifth-splits
!sixth-splits
|-
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (1 1 1)
|P4/2 (1 2 -2)
|P4/3 (1 2 -3)
|P4/4 (1 2 -4)
|P4/5 (1 2 -5)
|P4/6 (1 2 -6)
|-
|
|P5/2 (1 1 2)
|P5/3 (1 1 3)
|P5/4 (1 1 4)
|P5/5 (1 1 5)
|P5/6 (1 1 6)
|-
|
|
|P11/3 (1 3 -3)
|P11/4 (1 3 -4)
|P11/5 (1 3 -5)
|P11/6 (1 3 -6)
|-
|
|
|
|P12/4 (1 0 4)
|P12/5 (1 0 5)
|P12/6 (1 0 6)
|-
|
|
|
|
|ccP4/5 (1 4 -5)
|ccP4/6 (1 4 -6)
|-
|
|
|
|
|
|ccP5/6 (1 -1 6)
|}
Of all the half-octave pergens, half of every other column (i.e. 25%) are imperfect. Imperfect pergens occur whenever b is not a multiple of a.
{| class="wikitable"
|+Pergens of the form (P8/2, x), showing generator and mapping (a = 2)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (2 2 1)
|'''M2/4 (2 3 2)'''
|P4/3 (2 4 -3)
|'''M2/8 (2 3 4)'''
|P4/5 (2 4 -5)
|'''M2/12 (2 3 6)'''
|-
|
|P4/2 (2 4 -2)
|P5/3 (2 2 3)
|P4/4 (2 4 -4)
|P5/5 (2 2 5)
|P4/6 (2 4 -6)
|-
|
|
|P11/3 (2 6 -3)
|P5/4 (2 2 4)
|P11/5 (2 6 -5)
|P5/6 (2 2 6)
|-
|
|
|
|'''cm7/8 (2 5 -4)'''
|P12/5 (2 0 5)
|P11/6 (2 6 -6)
|-
|
|
|
|
|ccP4/5 (2 8 -5)
|'''cm7/12 (2 5 -6)'''
|-
|
|
|
|
|
|'''cM9/12 (2 1 6)'''
|}
Note that some of these pergens, when put in mingen form, become imperfect. For example, (P8/2, P11/3) becomes (P8/2, M2/6). Also note that for many of these pergens, the generators are comma-sized, and MOS scales will either be very "hard" (L/s very large) or else will contain very many notes per octave. For example, to bring the L/s ratio down to about 5, (P8/2, M2/4) needs a 16 note scale, and (P8/2, P11/3) needs a 28 note scale!

Of all the third-octave pergens, two-thirds of every third column (2/9 or 22%) are imperfect:
{| class="wikitable"
|+Pergens of the form (P8/3, x), showing generator and mapping (a = 3)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (3 3 1)
|P4/2 (3 6 -2)
|'''m3/9 (3 5 -3)'''
|P4/4 (3 6 -4)
|P4/5 (3 6 -5)
|'''m3/18 (3 5 -6)'''
|-
|
|P5/2 (3 3 2)
|'''M6/9 (3 4 3)'''
|P5/4 (3 3 4)
|P5/5 (3 3 5)
|'''M6/18 (3 4 6)'''
|-
|
|
|P4/3 (3 6 -3)
|P11/4 (3 9 -4)
|P11/5 (3 9 -5)
|P4/6 (3 6 -6)
|-
|
|
|
|P12/4 (3 0 4)
|P12/5 (5 0 5)
|P5/6 (3 3 6)
|-
|
|
|
|
|ccP4/5 (3 12 -5)
|'''ccm3/18 (3 7 -6)'''
|-
|
|
|
|
|
|'''ccM6/18 (3 2 6)'''
|}
Of all the quarter-octave pergens, imperfection occurs in half of every 4th column and 3/4 of every 4th column (5/16 or 31.25%).
{| class="wikitable"
|+Pergens of the form (P8/4, x), showing generator and mapping (a = 4)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
!c = ±7
!c = ±8
|-
|P5 (4 4 1)
|'''m6/8 (4 7 -2)'''
|P4/3 (4 8 -3)
|'''M2/8 (4 6 4)'''
|P4/5 (4 8 -5)
|P4/6 (4 8 -6)
|P4/7 (4 8 -7)
|'''M2/16 (4 6 8)'''
|-
|
|P4/2 (4 8 -2)
|P5/3 (4 4 3)
|'''m6/16 (4 7 -4)'''
|P5/5 (4 4 5)
|P5/6 (4 4 6)
|P5/7 (4 4 7)
|'''m6/32 (4 7 -8)'''
|-
|
|
|P11/3 (4 12 -3)
|'''M10/16 (4 5 4)'''
|P11/5 (4 12 -5)
|P11/6 (4 12 -6)
|P11/7 (4 12 -7)
|'''M10/32 (4 5 8)'''
|-
|
|
|
|P4/4 (4 8 -4)
|P12/5 (4 0 5)
|'''m6/24 (4 7 -6)'''
|P12/7 (4 0 7)
|P4/8 (4 8 -8)
|-
|
|
|
|
|ccP4/5 (4 16 -5)
|'''M10/24 (4 5 6)'''
|ccP4/7 (4 16 -7)
|P5/8 (4 4 8)
|-
|
|
|
|
|
|'''ccm6/24 (4 9 -6)'''
|ccP5/7 (4 -4 7)
|'''ccm6/32 (4 9 -8)'''
|-
|
|
|
|
|
|
|c3P4/7 (4 20 -7)
|'''cm7/16 (4 10 -8)'''
|-
|
|
|
|
|
|
|
|'''c3M3/32 (4 3 8)'''
|}
Percentage of imperfect pergens in each category:
{| class="wikitable"
|+
!(P8, x)
!(P8/2, x)
!(P8/3, x)
!(P8/4, x)
!(P8/5, x)
!(P8/6, x)
!(P8/7, x)
|-
|none
|1/4
|2/9
|5/16
|4/25
|5/12
|6/49
|-
|0%
|25%
|22.22%
|31.25%
|16%
|41.67%
|12.24%
|}

== Addenda (Spring 2026) ==

=== Initial commas ===
As one stacks generators and octave-reduces, at some point one overshoots or undershoots the octave by an interval of about a quartertone or less. This small interval is the pergen's initial comma. For example, (P8, P5)'s initial comma is the pythagorean comma, its next comma is Mercator's comma, etc. Each comma has a certain genspan, here 12 and 53. The genspan of the initial comma limits the size of a scale one can construct, assuming one wants to avoid overly-small steps. Thus one can have a pythagorean scale of up to 12 notes, but a 13-note scale will have a very small step. Note that the genspan gives the maximum notes per period, not per octave. Assuming one also wants to avoid extreme L/s step ratios also limits the maximum notes per octave.

The table below lists the initial comma of various pergens. "±" indicates a tippy pergen. "c" is the difference between the fifth and 7\12. "abs(6c)" means the absolute value of 6c. The dim 2nd is a pythagorean comma.
{| class="wikitable sortable"
|+Initial comma of each pergen
!#
!pergen
!interval
!cents
!genspan
!notes per octave
|-
!1
!(P8, P5)
|±d2
|abs(12c)
|±12G
|12
|-
!2
!(P8/2, P5)
|±d2/2
|abs(6c)
|±6G
|12
|-
!3
!(P8, P4/2)
|m2/2
|50¢ - 2.5c
|5G
|5
|-
!4
!(P8, P5/2)
|A1/2
|50¢ + 3.5c
|7G
|7
|-
!5
!(P8/2, P4/2)
| colspan="2" | ''same as #3 (P8, P4/2)''
|5G
|10
|-
!6
!(P8/3, P5)
|±d2/3
|abs(4c)
|±4G
|12
|-
!7
!(P8, P4/3)
|A1/3
|33.3¢ + 2.33c
| -7G
|7
|-
!8
!(P8, P5/3)
|m2/3
|33.3¢ - 1.67c
| -5G
|5
|-
!9
!(P8, P11/3)
|M2/3
|66.7¢ + 0.67c
|2G
|2 (or >= 14)
|-
!10
!(P8/3, P4/2)
|A2/6
|50¢ + 1.5c
|3G
|9
|-
!11
!(P8/3, P5/2)
|m3/6
|50¢ - 0.5c
|1G
|3
|-
!12
!(P8/2, P4/3)
| colspan="2" | ''same as #7 (P8, P4/3)''
| -7G
|14
|-
!13
!(P8/2, P5/3)
| colspan="2" | ''same as #8 (P8, P5/3)''
| -5G
|10
|-
!14
!(P8/2, P11/3)
|M2/6
|33.3¢ + 0.33c
|1G
|2
|-
!15
!(P8/3, P4/3)
| colspan="2" | ''same as #7 (P8, P4/3)''
| -7G
|21
|-
!16
!(P8/4, P5)
|±d2/4
|abs(3c)
|±3G
|12
|-
!17
!(P8, P4/4)
| colspan="2" | ''same as #3 (P8, P4/2)''
|10G
|10
|-
!18
!(P8, P5/4)
|A1/4
|25¢ + 1.75c
|7G
|7
|-
!19
!(P8, P11/4)
|dd3/4
|25¢ - 4.25c
| -17G
|17
|-
!20
!(P8, P12/4)
|m2/4
|25¢ - 1.25c
| -5G
|5
|-
!21
!(P8/4, P4/2)
|M2/4
|50¢ + c/2
|G
|4
|-
!22
!(P8/2, M2/4)
|M2/4
|50¢ + c/2
|G
|2
|-
!23
!(P8/2, P4/4)
|m2/4
|25¢ - 1.25c
|5G
|10
|-
!24
!(P8/2, P5/4)
| colspan="2" |''same as #18 (P8, P5/4)''
|7G
|14
|-
!25
!(P8/4, P4/3)
|d4/12
|33.3¢ - 0.67c
|2G
|8
|}
The initial comma of #9 (P8, P11/3) is about 67¢, which is not too small to be a scale step. But if there are more than 2 notes per 8ve, the L/s ratio becomes enormous. The ratio only becomes reasonable (roughly 3) when there are at least 14 notes per octave.

Note the initial comma is often equivalent to the uninflected EU divided by the height. For example, (P8, P4/2) has comma m2/2 and EU vvm2. In other words, the up-arrow is often the initial comma.

This suggests a new algorithm for finding a good EU for a pergen. Search the cents table (in the Notation Guide For rank-2 Pergens pdf, the first table of each pergen) for a small step. The search can be easily done by computer. Then derive the EU from that small step.

For example, pergen #25 (P8/4, P4/3) has a 33¢ step at 2G - P. Thus ^1 = 2G - P. Multiplying 2G by 3 gives us unsplit 4ths, and multiplying P by 4 gives us unsplit octaves. Thus we must multiply the up-arrow by 12 to get an unsplit 3-limit interval. 12 ups = 24G - 12P = 8P4 - 3P8 = dim 4th. Thus the EU is v12d4, and ^12C = Fb. But the pergenLister program lists the single-pair notation for this pergen as having an EU of ^12d94. Thus the pergenLister algorithm missed a much simpler EU, and hence a much simpler notation.

Not all initial commas imply a valid notation. For pergen #66 (P8, P5/6), the initial comma is P - 10G = m2/3 = 33.3¢ - 1.67c. The implied EU is v3m2. But this notation is incomplete because it only notates 3 of the 6 fifthchains. There must be 6 arrows in the EU, not 3. Or else there must be a second EU with 2 arrows. Fortunately, the next comma on the genchain is 21G - 2P = A1/2 = 50¢ - 3.5c, which implies a double-pair notation with EU' = \\A1. This is the notation found by pergenLister.

True doubles require double-pair notation and thus require finding two commas.

[[Category:Regular temperament theory]]
[[Category:Notation]]
[[Category:Pages with proofs]]

User:Eufalesio/Ultimate

2026-05-14T23:22:46Z

TallKite: suggested edits, feel free to undo any you don't like

is 41&53&217, with mapping [⟨1 0 0 25 -33 -13], ⟨0 1 0 -14 23 12], ⟨0 0 1 0 0 -1]]. It's otherwise known by in the wiki as ''[[cassaschismic]]'' (technical info inside), also [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here]]; but I will simply call it '''Ultimate'''. My reasoning of this will become clear. Or at least, I expect you to understand why it's clear in my mind.

== Quick definition ==
Ultimate can be easily defined in the 13-limit as nullifying the [[2080/2079|sinaisma]], [[minisma]], and [[eufalesma]]. This indirectly means that the [[Symbiotic comma|salozo]], [[Nexus comma|tribilo]], [[Wilschisma|sathoyo]], [[Olympia|salururu]], [[Garischisma|sasaru]], [[Argyria|lolotrizo]] commas, among an infinitude more, are all nullified too.

Ultimate has the following notable equations:

* [[Apotome]] = [[77/72]] (salozoma tempered out)
* [[77/72]] = [[27/26]] * [[36/35]] (eufalesma tempered out)
* [[Pythagorean comma|Poma]] = [[64/63]] (sasaruma tempered out)
* 2 pomas = [[33/32]] (salururuma tempered out)
* [[36/35]] = [[1053/1024]] (minisma tempered out)
* [[32/27]] = [[11/10]] * [[14/13]] (sinaisma tempered out)

et cetera...

The [[pergen]] is (P8, P5, ^1), where ^1 is the "minicomma"; a 3~5c interval that represents 385/384, 352/351, 5120/5103, 513/512, the layoma, etc. 4:5:6:7:9:11:13 is notated as P1 ^d4 P5 dd8 M9 AAA9 vAAA4. Using pomas (pythagorean commas) improves this notation. ''(to do: write out this chord with pomas)''

=== Interval list ===
Here is a quick compressed cheat sheet of octave-reduced intervals. This is a MASSIVE simplification with many (infinitely many) intervals left out for the sake of brevity. For every entry here, ''(to do: complete this sentence)''
{| class="wikitable"
!
! colspan="5" |minicomma-span
|-
!'''Fifth-span'''
!-2
!-1
!0
!+1
!+2
|-
!0
|
|720/361
|1/1
|
|
|-
!1
|
|256/171
|'''3/2'''
|
|
|-
!2
|
|64/57
|'''9/8'''
|
|
|-
!3
|
|'''32/19'''
|27/16
|
|
|-
!4
|
|24/19
|81/64
|
|
|-
!5
|
|36/19
|243/128
|
|
|-
!6
|
|64/45
|729/512
|
|
|-
!7
|
|'''16/15'''
|77/72
|
|
|-
!8
|
|'''8/5'''
|77/48
|'''5/4'''
|
|-
!9
|
|6/5
|77/64
|
|
|-
!10
|
|9/5
|65/36
|
|
|-
!11
|
|27/20
|65/48
|
|
|-
!12
|
|81/80
|64/63
|
|
|-
!13
|
|243/160
|32/21
|
|
|-
!14
|
|729/640
|'''8/7'''
|
|
|-
!15
|
|416/243
|12/7
|
|
|-
!16
|
|104/81
|9/7
|
|
|-
!17
|
|52/27
|27/14
|
|
|-
!18
|
|13/9
|81/56
|
|
|-
!19
|
|13/12
|88/81
|
|
|-
!20
|
|'''13/8'''
|44/27
|
|
|-
!21
|
|39/32
|11/9
|
|
|-
!22
|
|64/35
|11/6
|
|
|-
!23
|
|48/35
|'''11/8'''
|
|
|-
!24
|
|36/35
|33/32
|
|
|}

== Justification ==
The chain of fifths is a very important framework historically. It's been in Western music THE way to think about everything all the way from plainchant to Renaissance meantone temperaments to the modern day; where the 12-pitch-class circle of fifths is taught; 12edo, a massively over-represented tuning. It has a bit of a bad reputation in the xen circles, but the more I researched, the more I realized it is a '''paragon''', and that its position nowadays is very much well earned.

My main aim is to expand tonality with JI, and there is no better way to do so than to also extend the fundamental tuning framework to its logical conclusion.

12edo introduces the [[compton]] framework, which closes the chain of fifths with 12 flat, but proportionally very good fifths. Compton ''sensu stricto'' uses an independent generator to each all the different primes, and is generally a very good temperament. However, if the fifths are tuned sharper to become closer to just, the chain goes on for longer...

41edo introduces the [[cassandra]] framework, which thanks to its incredibly accurate fifths, introduces the poma as an accidental. In cassandra, the poma acts as 81/80~64/63 at around ~29c, and two of them add to 1053/1024~33/32~36/35. 41edo is a bit overtempered but notable as a proportionally coarse but good system. 53edo has practically pure fifths and very good p5 (prime 5) and p13 (prime 13), but p7 and p11 are tuned worse.

94edo is in my opinion the optimal cassandra equal tuning with a poma of 25.5c, tempered just enough for the 13-limit to reach <4 cents of error and very easy to use.

However, if you forgo p5 and p13 for the chain of fifths, you end up with [[gary]]. Gary is a serendipitous temperament, the same as cassandra but optimized for the [[zala]]. 135edo is an essentially perfect tuning, reaching sub-cent levels of error in the subgroup, with a poma of 26.6 c.

Ultimate is not just an extension of the concept, but what I believe to be the '''end''' of that extension. Ultimate adds an independent minicomma generator for p5 and p13, which acts as 385/384, ~352/351, ~5120/5103, layoma, ... etc, being around 4.4c. It doesn't begin to make sense up until 217edo, but 270edo and 311edo are arguably the best tunings. 217edo is alright, but not the best, which would make it a waste of time if it weren't a multiple of 31edo.

'''The key reasons''' on why Ultimate is ultimate, is ultimately due to the fact that 270edo and 311edo are inside the supported equal tunings, because going any further would make the edosteps not discernible, and because no other edo in their vicinity is as good as them. Beyond that, I see no reason to use edos.

270edo and 311edo inherit a chain of fifths that is consistent with cassandra, which itself is an extension of the circle of fifths. The only addition is a single edostep, and respectively, the entire 13-limit is tuned to unfathomable precision, and the 41-limit is fully accessible and very well tuned. However, I prefer sticking to the 13-limit, so 270edo is an optimal equal tuning.

=== Precision levels ===
{{EDOs|12e, 41, 53, 94, 217, 270, 311}} are all part of the same rank-3 tuning, so it allows a piece or a production to be written using the notation, which encodes the same mappings. Of course, using the notation to its fullest extent only makes sense for the finer 217, 270, 311. This necessarily means that there are levels of precision to Ultimate. (The notation ideas are heavily WIP)

=== 12e ===
The coarsest tuning that makes sense. It can be written just with sharps and flats, since the poma and the minicomma are tempered out in all its possible expressions. 12e because patent val tunes 11/8 as a tritone, not fourth. The cassandra mapping is based on 11/8 as a kind of fourth, not tritone. Either way, p11 is NOT there. Consider it an extremely coarse [[yazatha]] tuning.

=== 41 ===
The coarsest cassandra tuning. It can be written with sharps and flats, plus ↑ and '''↓''' for the pomas. In the case of 41edo, there is no need for double pomas, because the apotome can be split in half. Thus, half sharps and half flats can be used instead of two pomas. ONLY in 41edo. Ideal for 11-limit pieces with acoustic instruments, like the well known [[Kite guitar]], albeit, it is not a cassandra layout, but [[Skip fretting system 41 2 13]]. The cassandra layout is [[skip fretting system 41 3 7]].

=== 53 ===
Another good cassandra tuning. Just like 41edo, It can be written with sharps and flats, plus ↑ and '''↓''' for the pomas. The poma can be doubled into ⇑ and '''⇓''' to reach p11 and p13. It is playable and around the extremum possible inside the [[Lumatone mapping for 103edo|Lumatone]], which despite having a p7 and p11 that are not too well tuned; it has good 13-limit capabilities. It can be used in a guitar with the [[skip freting system 53 4 9]].

=== 94 ===
Best cassandra tuning. Just like 53edo, it can be written with sharps and flats, plus ↑ and '''↓''' for the pomas. The poma can be doubled into ⇑ and '''⇓''' to reach p11 and p13. Since the chain takes much longer to close, / and \ may be used to raise or lower by a half-poma. (Note that a half-poma implies a half-octave, and thus an even-numbered edo.) This tuning is optimal and technically usable in the Lumatone, but only as a subset, requiring more than one preset to reach within the [[Standard Lumatone mapping for Pythagorean]]. The cassandra layout can be used in a guitar with the [[skip fretting system 94 7 16]]. However, in a 6-string guitar there will be no other unisons.

=== 217, 270, 311 ===
They all work much the same way. They can be written with sharps and flats, ↑ and '''↓''' for the pomas, ⇑ and '''⇓''' for doubled pomas, and the addition of ^ and v for the minicomma, taken directly from the [[Kite's ups and downs notation|ups-and-downs notation.]] This is completely unfeasible to use with a Lumatone, with any acoustic instrument, isomorphically; though, it can still be used in a DAW without much problem. Because ultimate is rank-3, the layout is 3D and thus it is impossible to play on a flat surface, requiring some sort of eldritch holographic "keyspace".

217edo could ''in theory'' be used with binary valve or key systems in woodwinds, granted they have the intonation precision to reliably hit pitches within a maximum error of 2.76 cents. Which I know won't happen. Best course would be to tune the instrument to 31edo plus a slide to nudge everything into the right place, but that's not Ultimate. That's [[birds]].

=== Ultimate ''sensu stricto'' ===
It is possible to forgo edos altogether and use Ultimate as is, providing the absolute best tuning possible. However, it's a rank-3 system. It has no extra unisons unlike the equal tunings. This is reserved for when 270edo is not just enough, and beating is something to avoid as much as possible.

== The special place of 94edo and 270edo ==
Of all the equal tunings supported by Ultimate, the best ones are 94edo and 270edo. They have the key property of being even, and thus also tempers out the [[kalisma]], allowing the poma to be split in halves. Using them this way is reminiscent of [[Gariwizmic]], a very similar temperament to Ultimate, but with the minicomma found deep in the generator chain, not independent. This is useful for easier navigation within a DAW.

It's possible to use Gariwizmic wholesale, but I wouldn't recommend it. For that, Ultimate is a much better choice overall. Gariwizmic would only provide the structure, not the tuning.

Talk:Kite's thoughts on pergens

2026-05-14T10:26:50Z

TallKite: /* Pergens for no-3 temperaments? */

{{WSArchiveLink}}
Note to self: "Mids never appear in the perchain." Check that expanding the definition of mid intervals to include the 4th and 5th hasn't changed this. [[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 04:46, 30 January 2020 (UTC)

== Cmloegcmluin's clarification questions ==

I came across this page yesterday because Jason suggested it as a better approach to the problem TAMNAMS is trying to solve. But I can't get very far before I'm lost. "Both fractions are always of the form 1/N, thus the octave and/or the 3-limit interval is split into N parts. The interval which is split into multiple generators is the multigen. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc." What is N? And what are these conventional names P5, M6, m7? --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 16:57, 15 April 2021 (UTC)

: In (P8, P5/3), N is 3. In (P8/4, P5), N is 4. In (P8/2, M2/4), well, I didn't state that very well, I guess there are two N's, which may or may not be equal. Later I adopt a notation (P8/m, M/n), where M stands for multigen. So for (P8/2, M2/4) we have m=2, n=4.

: P stands for perfect, M for major (or multigen if not followed by a number) and m for minor. It's a 3-limit interval, so M2 = major 2nd = 9/8, A4 = aug 4th = 729/512, etc.

:: Okay, thanks. If there's a place where that terminology is explained, that might be great to link to (people like me who never formally studied music and are unfamiliar with those abbreviations might benefit.

:: And re: N, what I meant was more like: what does it mean? Does the letter N stand for something? Or does that value represent something I would be familiar with? Or is it just some abstract number to work with? It's probably obvious to some readers (esp. given the way you matter-of-factly present it) but I have no idea myself. Thanks for indulging my questions. Hopefully I can at least help improve the accessibility of the ideas for others. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 23:04, 15 April 2021 (UTC)

::: Explanation of M2, m7, etc.: https://en.wikipedia.org/wiki/Interval_(music)#Main_intervals. Knowing these terms really helps with chord names like CM6 and Dm7.
::: N doesn't stand for anything. It's like x or y in algebra. But N isn't really important. The important thing is the pergen. I only wrote "Both fractions are always of the form 1/N" because I wanted to make clear that there is no "two-thirds-of-a-fifth" pergen. IOW the fraction always has a numerator of 1. If N still confuses you, I suggest you just ignore it. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 05:39, 16 April 2021 (UTC)

== Kite's thoughts on canonical generators, which imply canonical mappings ==

The canonical generator of a rank-2 pergen is called the best-multigen generator. This is found by:
# minimizing the absolute value of the 2nd number in a multigen's monzo (e.g. P5/2 over M2/4 because P5 = (-1 1) and M2 = (-3 2)). I suspect this also minimizes the multigen's fraction.
# next, choosing the least-cents voicing of the multigen (e.g. P4/2 over P12/2 and P11/3 over P12/3)
# and finally, if the generator is P4, using P5 instead, to follow historical practice.

But a very popular canonical generator is the one that has the least cents. (x31eq.com's mappings always imply this generator.) If the least-cents generator differs from the best-multigen generator, it's always of the form G - nP or nP - G, where P = period, G = best-multigen generator, and n = round ((cents of G)/(cents of P)). A pergen that uses this alternate generator is called a least-cents pergen.

The best-multigen generator works better harmonically, and the least-cents generator works better melodically. For example with (P8/2, P5), MOS scales can be thought of as two familiar chains of 5ths, offset by a half-octave. Whereas with the corresponding least-cents pergen (P8/2, M2/2), MOS scales can be thought of as each half of the octave being filled in by a stack of semitones. But the best-multigen generator has the advantage that it simplifies the process of finding an ideal notation. This is why the best-multigen pergen is the canonical pergen.

The least-cents generator differs from the best-multigen generator in two cases. In the trivial case, the unsplit pergen becomes (P8, P4). The other case occurs sometimes but not always when the period is not an octave. The least-cents multigen in this case is always imperfect, because step #1 is skipped. PergenLister displays the least-cents generator's cents in the 3rd column. It usually displays this generator as a fraction of an imperfect multigen in the "Unreduced Pergen" column. However unreduced pergens #16 and #27 are not least-cents. This is because the purpose of this column is to find an alternate notation, not to find the least-cents generator.

'''Thus for many temperaments there are two canonical mappings''', one for each type of canonical generator. Except for the trivial case, both are useful and IMO the xenwiki should list both. For example, Sagugu aka Srutal is (P8/2, P5), which implies the mapping [(2 2 7) (0 1 -2)]. But the least-cents pergen (P8/2, M2/2) implies the mapping [(2 3 5) (0 1 -2)]. Another example: Trigu aka Augmented is (P8/3, P5) which implies [(3 3 4) (0 1 0)]. But the least-cents pergen (P8/3, m3/3) implies [(3 5 0) (0 -1 0)]. However any temperament in which the least-cents generator is the same as the best-multigen generator has only one canonical mapping, e.g. Zozo aka Semaphore which is (P8, P4/2).

This table shows the least-cents generator for pergens 1-32, if it differs from the best-multigen generator (asterisk indicates a true double):
{| class="wikitable" style="text-align:center;"
|+
! colspan="3" |best-multigen generator
! colspan="3" |least-cents generator
|-
! colspan="3" |unsplit pergen
! colspan="2" |generator
!cents
|-
!1
|(P8, P5)
|700 + c
|P - G
|P4
|500 - c
|-
! colspan="3" |half-split pergens
! colspan="3" |
|-
!2
|(P8/2, P5)
|700 + c
|G - P
|M2/2
|100 + c
|-
!3
|(P8, P4/2)
|250 - c/2
|
|
|
|-
!4
|(P8, P5/2)
|350 + c/2
|
|
|
|-
!5
|(P8/2, P4/2) *
|250 - c/2
|
|
|
|-
! colspan="3" |third-split pergens
! colspan="3" |
|-
!6
|(P8/3, P5)
|700 + c
|2P - G
|m3/3
|100 - c
|-
!7
|(P8, P4/3)
|167 - c/3
|
|
|
|-
!8
|(P8, P5/3)
|233 + c/3
|
|
|
|-
!9
|(P8, P11/3)
|567 - c/3
|
|
|
|-
!10
|(P8/3, P4/2)
|250 - c/2
|P - G
|M6/6
|150 + c/2
|-
!11
|(P8/3, P5/2)
|350 + c/2
|P - G
|m3/6
|50 - c/2
|-
!12
|(P8/2, P4/3)
|167 - c/3
|
|
|
|-
!13
|(P8/2, P5/3)
|233 + c/3
|
|
|
|-
!14
|(P8/2, P11/3)
|567 - c/3
|P - G
|M2/6
|33 + c/6
|-
!15
|(P8/3, P4/3) *
|167 - c/3
|
|
|
|-
! colspan="3" |quarter-split pergens
! colspan="3" |
|-
!16
|(P8/4, P5)
|700 + c
|G - 2P
|M3/4
|100 + c
|-
!17
|(P8, P4/4)
|125 - c/4
|
|
|
|-
!18
|(P8, P5/4)
|175 + c/4
|
|
|
|-
!19
|(P8, P11/4)
|425 - c/4
|
|
|
|-
!20
|(P8, P12/4)
|475 + c/4
|
|
|
|-
!21
|(P8/4, P4/2) *
|250 - c/2
|P - G
|M2/4
|50 + c/2
|-
!22
|(P8/2, M2/4)
|50 + c/2
|
|
|
|-
!23
|(P8/2, P4/4) *
|125 - c/4
|
|
|
|-
!24
|(P8/2, P5/4) *
|175 + c/4
|
|
|
|-
!25
|(P8/4, P4/3)
|167 - c/3
|P - G
|M10/12
|133 + c/3
|-
!26
|(P8/4, P5/3)
|233 + c/3
|P - G
|m6/12
|67 - c/3
|-
!27
|(P8/4, P11/3)
|567 - c/3
|2P - G
|M2/6
|33 + c/3
|-
!28
|(P8/3, P4/4)
|125 - c/4
|
|
|
|-
!29
|(P8/3, P5/4)
|175 + c/4
|
|
|
|-
!30
|(P8/3, P11/4)
|425 - c/4
|G - P
|m3/12
|25 - c/4
|-
!31
|(P8/3, P12/4)
|475 + c/4
|G - P
|M6/12
|75 + c/4
|-
!32
|(P8/4, P4/4) *
|125 - c/4
|
|
|
|}

== Updated terminology ==
Ups and downs are collectively called arrows. Lifts and drops are collectively called slants. Arrows and slants are collectively called inflections. Thus a better term for "bare" enharmonic is uninflected enharmonic. The article has been edited to correct this. Also, the term "E" for enharmonic interval has been replaced with EI. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 02:19, 17 December 2023 (UTC)

==Pergens for no-3 temperaments?==
Hey TallKite, I was just wondering if you could explain how to use pergens to describe no-3 temperaments. Also, how do I determine tunings such as 3/7-comma mintaka? Thanks! [[User:MisterShafXen|MisterShafXen]] ([[User talk:MisterShafXen|talk]]) 11:57, 10 May 2026 (UTC)
: Glad to help! For no-3 temps, see https://en.xen.wiki/w/Kite%27s_thoughts_on_pergens#Notating_non-8ve_and_no-5ths_pergens. Analogous to imperfect 2.3 pergens like (P8/2, M2/4), there are imperfect 2.5 pergens that would use A5, defined as 25/16. And likewise d4, A7, etc.
: For mintaka, the period is P12 = 3/1 and the generator is 11/7. The 3.7.11 mapping is [(1 3 3)(0 -3 -2)]. Thus prime 7 equals 3 periods minus 3 generators. Thus the multigen is 27/7. The pergen is #9 in the 3.7 column (P12, cM7/3), where cM7 means P12 = 3/1 plus a M3 = 9/7.
: 3/7-comma mintaka means the multigen 27/7 is sharpened or flattened by 3/7 of a comma, 4.473¢. The mintaka comma has 7 in the denominator, so tempering it out implies sharpening prime 7. Thus prime 7 is sharpened by 4.473¢, and the multigen 27/7 is flattened by 4.473¢. (We're assuming a just prime 3.)
: If you're into temps that don't use 2 and/or 3, you might find pergen squares useful, since they apply to all JI subgroups.
: --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 10:26, 14 May 2026 (UTC)

Kite's ups and downs notation

2026-05-14T08:19:28Z

TallKite: /* Definition */

== Definition ==
Ups and Downs (or ^v) is a notation system that can notate almost every [[EDO|edo]]. The up symbol "^" and the down symbol "v" indicate raising/lowering a note (or widening/narrowing an interval) by one EDOstep. The mid symbol, "~" is for intervals exactly midway between major and minor, e.g. 3\24 is a mid 2nd. The mid 4th (~4) is midway between perfect and augmented, i.e. halfway-augmented, and the mid 5th (~5) is a halfway-diminished 5th.

Ups and downs can also notate any [[Tour of Regular Temperaments|rank-2 temperament]], although some temperaments require an additional pair of accidentals, lifts and drops (/ and \). In this context, an up or a lift represents sharpening by a [[comma]] that has been tempered, but not tempered out. For example, in [[Porcupine|Triyo aka Porcupine]], an up/down represents raising/lowering by a tempered 81/80, and lifts/drops aren't used. In practice, the two uses of the notation often coincide perfectly. Triyo is supported by both 15edo and 22edo, and both edos map 81/80 to one EDOstep. Thus if Triyo is tuned to 15edo, an up simultaneously means both a tempered 81/80 and 1\15. Likewise, if tuned to 22edo, the up means both 81/80 and 1\22. If not tuned to an edo at all, then the up only means 81/80. Thus a piece written in Triyo can be converted to a piece written in 22edo by simply writing "22edo" on the top of the page.

Ups and downs can also be used to notate rank-3 just intonation subgroups such as 2.3.5 or 2.3.7 or 2.3.11. See [[Kite's thoughts on ups and downs notation for rank-3 JI|Ups and downs notation for Rank-3 JI]].

'''This page only discusses notation of edos.''' However, the notation of chords and chord progressions applies to all situations. For notation of rank-2 and rank-3 temperaments, see the [[pergen|pergens]] article.

For more on edo notation, see the [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf '''Notation guide for edos 5-72'''], which also covers chord names, slash chords, staff notation, key signatures, and scale trees.

== Explanation (a 22edo example) ==
To understand the ups and downs notation, let's start with an edo that doesn't need it. 19edo is easy to notate because 7 fifths reduced by 4 octaves adds up to one EDOstep. C♯ is right next to C, and the keyboard runs {{nowrap|C, C♯, D♭, D, D♯, E♭, E}} etc. Conventional notation works perfectly with 19edo as long as you remember that C♯ and D♭ are different notes.

In contrast, 22edo is hard to notate because 7 fifths reduces to ''three'' EDOsteps, and the usual chain of fifths {{dash|E♭, B♭, F, C, G, D, A, E, B, F♯, C♯}} etc. creates the scale {{dash|C, D♭, B♯, C♯, D, E♭, F♭, D♯, E, F}}. That's very confusing because B♯–D♭ looks ascending on the page but sounds descending, and a 4:5:6 major chord is written {{dash|C, D♯, G}}, and the 5/4, usually a major third, becomes an augmented second. Some people forgo the chain of fifths for a maximally even scale like {{dash|C, D, E, F, G, A, B, C}}. But that's confusing because G–D and A–E are diminished 5ths. And if your piece is in G or A, that's really confusing. A notation system should work in every key!

The solution is to use the sharp symbol to mean "raised by 7 fifths", and to use the up-arrow symbol to mean "sharpened by one EDOstep". 22edo can be written {{dash|C, Db, ^Db, vD, D, Eb, ^Eb, vE, E, F}} etc. The notes are pronounced up-D-flat, down-D, etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp. The up or down comes before the note name to make naming chords easy.

The names change depending on the key, just like in conventional notation where F# in D major becomes Gb in Db major. So the B scale is {{dash|B, C, ^C, vC#, C#, D, ^D, vD#, D#, E}} etc.

The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible.

=== Relative notation and interval arithmetic ===
Ups and downs can be used not only for absolute notation (note names) but also for relative notation (intervals, chords and scales). Relative notation for 22edo intervals: {{dash|P1, m2, ^m2, vM2, M2, m3, ^m3, vM3, M3, P4, ^4/d5, vA4/^d5, A4/v5, P5}} etc. That's pronounced upminor 2nd, downmajor 3rd, etc. You can apply this pattern to any 22edo key. The '''plain''' notes (those without ups or downs) always form a chain of fifths.

A core principle of ups and downs notation is that '''interval arithmetic is always preserved'''. Ups and downs are simply added in:

{| class="wikitable" style="text-align: center;"
|-
!
! Interval between two notes
! Note plus an interval
! Sum of two intervals
|-
! conventional
| C to E = M3
| C + M3 = E
| M2 + M2 = M3
|-
! rowspan="2" | with ups and downs
| ^C to E = vM3
| ^C + M3 = ^E
| ^M2 + M2 = ^M3
|-
| C to ^E = ^M3
| C + ^M3 = ^E
| M2 + vM2 = vM3
|-
! (cancelling)
| ^C to ^E = M3
| ^C + vM3 = E
| ^M2 + vM2 = M3
|-
! (combining)
| ^C to vE = vvM3
| ^C + ^M3 = ^^E
| vM2 + vM2 = vvM3
|}

The same logic holds for a note minus an interval (C - vm3 = ^A) or one interval minus another interval (M3 - vM2 = ^M2).

=== "Arrow" as a term for EDOstep ===
Up and down are short for up-arrow and down-arrow, and arrow refers to both. Sometimes the name of a notation symbol comes to mean that which the symbol indicates. Just as "bar" (the vertical line that separates measures) has come to mean "measure", "[[arrow]]" has also come to mean "EDOstep".

=== Enharmonic unisons ===
Conventionally, in C you use D# instead of Eb when you have a Gaug chord. You have the freedom to spell your notes how you like, to make your chords look right. Likewise, in 22edo, D♭ can be spelled ^C or vB♯ or even ^^B (double-up B, or '''dup''' B for short, rhymes with "cup"). Respelling is done by adding or subtracting an [[Enharmonic unisons in ups and downs notation|enharmonic unison]], '''EU''' for short.

From the [[Pergen|pergens]] article: "Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. Enharmonic unisons are like vanishing commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of EUs needed always equals the difference between the notation's rank and the tuning's rank."

Since 22edo is rank-1, and conventional notation plus ups and downs is rank-3, two EUs are needed to define the notation: v3A1 and vm2. Either EU can be added to or subtracted from any note to respell the note. For example, ^C + vm2 = Db and ^^Eb + v3A1 = vE. Any combination of these two EUs is also an EU, for example their sum v4M2. Thus ^^F = ^^F + v4M2 = vvG (double-down G, or '''dud''' G for short, rhymes with "cud").

=== Larger EDOs ===
In larger edos, triple-arrows, quadruple-arrows, etc. can occur. Up, dup, trup and quup all rhyme, as do dud, trud and quud.

{| class="wikitable" style="text-align:center;"
|+ style="font-size: 105%;" | Symbols and words for multiple arrows
|-
! Written
! Spoken
! Etymology
!
! Written
! Spoken
! Etymology
|-
| ^
| up
|
! 1 arrow
| v
| down
|
|-
| ^^
| dup
| '''d'''ouble-'''up'''
! 2 arrows
| vv
| dud
| rowspan="4" | "-d" for down replaces "-p" for up
|-
| ^^^
| trup
| '''tr'''iple-'''up'''
! 3 arrows
| vvv
| trud
|-
| v>
| quup "kwup"
| '''qu'''adruple-'''up'''
! 4 arrows
| ^<
| quud "kwud"
|-
| >
| quip
| '''qui'''ntuple-u'''p'''
! 5 arrows
| <
| quid
|}

(In addition to dup, trup, etc. there is dub, trip, quad and quin, used for multiple sharps/flats and multiple lifts/drops, e.g. dubsharp or triplift.)

Very large edos can go well beyond 5 arrows. The sequence of names resembles tally counting I, II, III, IIII, <s>||||</s>. But the sequence of ''symbols'' resembles roman numerals I, II, III, IV, V. Thus 4 ups is spoken quup but written v>.

{| class="wikitable" style="text-align: center;"
|-
| up ^
| dup ^^
| trup ^^^
| quup v>
| quip >
| upquip ^>
| dupquip ^^>
| trupquip ^^^>
| quupquip v>>
| quipquip >>
| upquipquip ^>>
| dupquipquip ^^>>
| trupquipquip ^^^>>
| quupquipquip v>>>
| triplequip >>>
|-
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
! 14
! 15
|-
| down v
| dud vv
| trud vvv
| quud ^<
| quid <
| downquid v<
| dudquid vv<
| trudquid vvv<
| quudquid ^<<
| quidquid <<
| downquidquid v<<
| dudquidquid vv<<
| trudquidquid vvv<<
| quudquidquid ^<<<
| triplequid <<<
|}

Lifts and drops (/ and \) can be used for microinflections of less than an edostep, since they look like part of an arrow.
{| class="wikitable"
|+
|^
|up
| rowspan="2" |arrow
| rowspan="4" |inflection
| rowspan="6" |alteration
|-
|v
|down
|-
|/
|lift
| rowspan="2" |slash
|-
|\
|drop
|-
|#
|sharp
| colspan="2" rowspan="2" |accidental
|-
|b
|flat
|}
For very large edos in which commas like 81/80 and 64/63 are many edosteps, the color notation accidental pairs yo/gu and zo/ru can be "edoized" to stand for a certain number of edosteps. For example, in [[311edo]], 81/80 is 6 edosteps. Thus g means ^> and y means v<. The colors can be combined with arrows as in upyo or dudgu (^y or vvg). Likewise, 64/63 is 7 edosteps, thus r means ^^> and z means vv<.

===Staff Notation===
For staff notation, put an arrow to the left of the note and any sharp or flat it might have. Like sharps and flats, an arrow applies to any similar note that follows in the measure. If C is upped, any other C in the same octave inherits the up. If an up-C is followed by a down-C, the down-arrow replaces the up-arrow.

But what happens when accidentals are mixed with arrows? What if the key signature makes that upped C be sharp? Or what if there is a C with a sharp just before the upped C? Does the up-arrow override or "cancel" the sharp? And what if an upped C is followed by a sharpened C?

There are several possible ways to handle this issue. The default is the simplest way, to explicitly specify both arrows and accidentals every time. Thus any accidental or arrow cancels any previous ones. An arrow by itself implies a natural sign.

{| class="wikitable" style="text-align:center;"
|-
! rowspan="2" | Start with this
! colspan="6" | Turn it into this
|-
! C
! ^C
! ^^C
! C#
! ^C#
! ^^C#
|-
! C
|        
| ^
| ^^
| #
| ^#
| ^^#
|-
! ^C
| <big>♮</big>
|        
| ^^
| #
| ^#
| ^^#
|-
! ^^C
| <big>♮</big>
| ^
|        
| #
| ^#
| ^^#
|-
! C#
| <big>♮</big>
| ^
| ^^
|        
| ^#
| ^^#
|-
! ^C#
| <big>♮</big>
| ^
| ^^
| #
|        
| ^^#
|-
! ^^C#
| <big>♮</big>
| ^
| ^^
| #
| ^#
|        
|}

See [[Kite Guitar originals#Cancelling rules]] for another way.

For more on staff notation, see the [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf Notation Guide for EDOs 5-72].

=== Key signatures ===
Key signatures follow the conventional practice, expanded to allow for double-sharps and double flats in some edos. For example, 19edo has the key of Bbb with a key signature of B𝄫 E𝄫 A♭ D♭ G♭ C♭ F♭. Some edos have upped/downed tonics, e.g. 24edo has the key of vD with a key signature of F♯ C♯ (v). The (v) is a "global down" that downs all 7 notes of the vD scale. See also [[Kite Guitar originals#Scales and key signatures]] for the use of '''arrow stacks'''.

=== Placement of the arrow ===
It might seem more natural to place the arrow after the note, for example B^ or Bb^. But the arrow must come first, to make chord names unambiguous. Otherwise B^m could mean either a minor chord rooted on B^ or an upminor chord rooted on B. (Chord names are explained fully below.)

The issue arises because while English normally places the adjective before the noun, it doesn't do so with sharps and flats. A flattened B should logically be called "flat B" not "B flat", and be written bB not Bb. If it were, then it would seem very natural to have the up come first, as in ^bB. This would be the typical English adjective-adjective-noun construction. Instead we must use ^Bb, an unnatural adjective-noun-adjective construction. This issue fortunately arises only for note names. On the staff, the flat comes before the note, so naturally the up comes before the flat. In relative notation, the quality comes before the interval, as in minor 3rd and augmented 4th, or in jazz terms flat 3rd and sharp 4th. So terms like upminor 3rd and downsharp 4th have a natural adjective-adjective-noun construction.

=== Further notes ===
Edo intervals are often written as 7\22. This can also be written as vM3\22. This is useful when comparing edos, e.g. vM3\22 vs. vM3\15.

== Examples: edos 12-24 ==
Sharp-1, flat-2, etc. refer to the [[sharpness]], the number of arrows made by seven 5ths minus four 8ves. All sharp-1 and flat-1 edos can be notated without ups and downs, because the up is exactly equivalent to a sharp or flat.

A ring is a circle of 5ths. In multi-ring (aka ringy) edos like 14, 15 and 24, a single ring doesn't contain all the edo's notes. In contrast, edos like 12, 19 and 22 are single-ring. It's possible to notate any single-ring edo with conventional notation if notes are permitted to be out of order (e.g. 22edo could have C Db B# C# D). But multi-ring edos absolutely require ups and downs.

13edo and 18edo aren't compatible with heptatonic notation, because the minor 2nd is descending. Thus the minor 3rd is flatter than the major 2nd, the 4th is flatter than the major 3rd, etc. These edos are best notated using the 2nd best fifth, as 13b and 18b.

There are four flat-N edos on this list. 16edo and 23edo are flat-1, 18b is flat-2 and 13b is flat-3. There are two ways to notate such edos: with sharp lowering the pitch, and major/aug narrower than minor/dim, or with sharp raising the pitch, and major/aug wider than minor/dim. Both notations are shown. In the 2nd notation, note that a fifth above B is Fb, not F#.

12edo is sharp-1, thus doesn't need ups and downs. Enharmonic unison: d2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[12-edo|12edo]] {{normal|sharp-1}}
| '''D'''
| D#/Eb
| '''E'''
| '''F'''
| F#/Gb
| '''G'''
| G#/Ab
| '''A'''
| A#/Bb
| '''B'''
| '''C'''
| C#/Db
| '''D'''
|-
| P1
| A1/m2
| M2
| m3
| M3
| P4
| A4/d5
| P5
| m6
| M6
| m7
| M7
| P8
|}

There are two ways to notate 13b-edo. The enharmonic unisons for the 1st notation are ^3A1 and vM2. For the 2nd they are v3A1 and vm2.

{| class="wikitable" style="text-align:center;"
|-
! rowspan="4" | [[13-edo|13b-edo]] {{normal|flat-3}}
! rowspan="2" | Sharp lowers the pitch, major narrower than minor
| '''D'''
| '''E'''
| ^E/F#
| vEb/^F#
| Eb/vF
| '''F'''
| '''G'''
| '''A'''
| '''B'''
| ^B/C#
| vBb/^C#
| Bb/vC
| '''C'''
| '''D'''
|-
| P1
| M2
| ^M2/M3
| vm2/^M3
| m2/vm3
| m3
| P4
| P5
| M6
| ^M6/M7
| vm6/^M7
| m6/vm7
| m7
| P8
|-
! rowspan="2" | Sharp raises the pitch, major wider than minor
| '''D'''
| '''E'''
| ^E/Fb
| vE#/^Fb
| E#/vF
| '''F'''
| '''G'''
| '''A'''
| '''B'''
| ^B/Cb
| vB#/^Cb
| B#/vC
| '''C'''
| '''D'''
|-
| P1
| m2
| ^m2/m3
| vM2/^m3
| M2/vM3
| M3
| P4
| P5
| m6
| ^m6/m7
| vM6/^m7
| M6/vM7
| M7
| P8
|}

Because every 14edo interval is perfect, the quality can be omitted. Sharps and flats can also be omitted. 14edo contains 2 rings of 7edo: an up/down-ring and a plain-ring. Enharmonic unisons: A1 and vvm2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[14-edo|14edo]] {{normal|sharp-0}}
| '''D'''
| ^D/vE
| '''E'''
| ^E/vF
| '''F'''
| ^F/vG
| '''G'''
| ^G/vA
| '''A'''
| ^A/vB
| '''B'''
| ^B/vC
| '''C'''
| ^C/vD
| '''D'''
|-
| 1
| ^1/v2
| 2
| ^2/v3
| 3
| ^3/v4
| 4
| ^4/v5
| 5
| ^5/v6
| 6
| ^6/v7
| 7
| ^7/v8
| 8
|}

15edo contains 3 rings of 5edo: an up-ring, a down-ring, and a plain-ring. Enharmonic unisons: v3A1 and m2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[15-edo|15edo]] {{normal|sharp-3}}
| '''D'''
| ^D
| vE
| '''E/F'''
| ^F
| vG
| '''G'''
| ^G
| vA
| '''A'''
| ^A
| vB
| '''B/C'''
| ^C
| vD
| '''D'''
|-
| P1
| ^m2
| vM2
| M2/m3
| ^m3
| vM3
| M3/P4
| ^4
| v5
| P5
| ^m6
| vM6
| M6/m7
| ^m7
| vM7
| P8
|}

16edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic unison: either AA2 or dd2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="4" | [[16-edo|16edo]] {{normal|flat-1}}
! rowspan="2" | Sharp lowers the pitch, major narrower than minor
| '''D'''
| Db/E#
| '''E'''
| Eb
| F#
| '''F'''
| Fb/G#
| '''G'''
| Gb/A#
| '''A'''
| Ab/B#
| '''B'''
| Bb
| C#
| '''C'''
| Cb/D#
| '''D'''
|-
| P1
| A2
| M2
| m2/A3
| M3
| m3
| d3/A4
| P4
| d4/A5
| P5
| d5/A6
| M6
| m6/A7
| M7
| m7
| d7
| P8
|-
! rowspan="2" | Sharp raises the pitch, major wider than minor
| '''D'''
| D#/Eb
| '''E'''
| E#
| Fb
| '''F'''
| F#/Gb
| '''G'''
| G#/Ab
| '''A'''
| A#/Bb
| '''B'''
| B#
| Cb
| '''C'''
| C#/Db
| '''D'''
|-
| P1
| d2
| m2
| M2
| m3
| M3
| A3
| P4
| A4/d5
| P5
| d6
| m6
| M6/d7
| m7
| M7
| A7
| P8
|}

17edo is sharp-2 and thus has mid intervals. Enharmonic unisons: vvA1 and vm2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[17edo]] {{normal|sharp-2}}
| '''D'''
| ^D/Eb
| D#/vE
| '''E'''
| '''F'''
| ^F/Gb
| F#/vG
| '''G'''
| ^G/Ab
| G#/vA
| '''A'''
| ^A/Bb
| A#/vB
| '''B'''
| '''C'''
| ^C/Db
| C#/vD
| '''D'''
|-
| P1
| ^1/m2
| A1/~2
| M2
| m3
| ~3
| M3
| P4
| ^4/~4/d5
| A4/v5/~5
| P5
| m6
| ~6
| M6
| m7
| ~7
| M7
| P8
|}

18b-edo contains 2 rings of 9edo: an up/down-ring and a plain-ring. There are two ways to notate it. Enharmonic unisons: either ^^A1 and vvM2, or vvA1 and vvm2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="4" | '''[[18-edo|18b-edo]]''' flat-2
! rowspan="2" | Sharp lowers, major is narrower
| '''D'''
| ^D/vE
| '''E'''
| ^E
| Eb/F#
| vF
| '''F'''
| ^F/vG
| '''G'''
| ^G/vA
| '''A'''
| ^A/vB
| '''B'''
| ^B
| Bb/C#
| vC
| '''C'''
| ^C/vD
| '''D'''
|-
| P1
| ^1/vM2
| M2
| ~2
| m2/M3
| ~3
| m3
| ^m3/v4
| P4
| ^4/v5
| P5
| ^5/vM6
| M6
| ~6
| m6/M7
| ~7
| m7
| ^m2/d8
| P8
|-
! rowspan="2" | Sharp raises, major is wider
| '''D'''
| ^D/vE
| '''E'''
| ^E
| E#/Fb
| vF
| '''F'''
| ^F/vG
| '''G'''
| ^G/vA
| '''A'''
| ^A/vB
| '''B'''
| ^B
| B#/Cb
| vC
| '''C'''
| ^C/vD
| '''D'''
|-
| P1
| ^1/vm2
| m2
| ~2
| M2/m3
| ~3
| M3
| ^M3/v4
| P4
| ^4/v5
| P5
| ^5/vm6
| m6
| ~6
| M6/m7
| ~7
| M7
| ^M7/d8
| P8
|}

19edo is sharp-1, thus doesn't need ups and downs. Enharmonic unison: dd2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[19-edo|19edo]] {{normal|sharp-1}}
| '''D'''
| D#
| Eb
| '''E'''
| E#/Fb
| '''F'''
| F#
| Gb
| '''G'''
| G#
| Ab
| '''A'''
| A#
| Bb
| '''B'''
| B#/Cb
| '''C'''
| C#
| Db
| '''D'''
|-
| P1
| d2
| m2
| M2
| d3
| m3
| M3
| A3
| P4
| A4
| d5
| P5
| A5
| m6
| M6
| d7
| m7
| M7
| A7
| P8
|}

20edo contains 4 rings of 5edo: an up-ring, a down-ring, a dup/dud-ring, and a plain-ring. Enharmonic unisons: v4A1 and m2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[20-edo|20edo]] {{normal|sharp-4}}
| '''D'''
| ^D
| ^^D/vvE
| vE
| '''E/F'''
| ^F
| ^^F/vvG
| vG
| '''G'''
| ^G
| ^^G/vvA
| vA
| '''A'''
| ^A
| ^^A/vvB
| vB
| '''B/C'''
| ^C
| ^^C/vvD
| vD
| '''D'''
|-
| P1/m2
| ^m2
| ~2
| vM2
| M2/m3
| ^m3
| ~3
| vM3
| M3/P4
| ^4
| ~4/~5
| v5
| P5/m6
| ^m6
| ~6
| vM6
| M6/m7
| ^m7
| ~7
| vM7
| P8
|}

Because every 21edo interval is perfect, the quality can be omitted. 21edo contains 3 rings of 7edo: an up-ring, a down-ring and a plain-ring. Enharmonic unisons: A1 and v3m2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[21-edo|21edo]] {{normal|sharp-0}}
| '''D'''
| ^D
| vE
| '''E'''
| ^E
| vF
| '''F'''
| ^F
| vG
| '''G'''
| ^G
| vA
| '''A'''
| ^A
| vB
| '''B'''
| ^B
| vC
| '''C'''
| ^C
| vD
| '''D'''
|-
| 1
| ^1
| v2
| 2
| ^2
| v3
| 3
| ^3
| v4
| 4
| ^4
| v5
| 5
| ^5
| v6
| 6
| ^6
| v7
| 7
| ^7
| v8
| 8
|}

22edo is sharp-3. Enharmonic unisons: v3A1 and vm2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[22-edo|22edo]] {{normal|sharp-3}}
| '''D'''
| ^D/Eb
| vD#/^Eb
| D#/vE
| '''E'''
| '''F'''
| ^F/Gb
| vF#/^Gb
| F#/vG
| '''G'''
| ^G/Ab
| vG#/^Ab
| G#/vA
| '''A'''
| etc.
|-
| P1
| ^1/m2
| vA1/^m2
| vM2
| M2
| m3
| ^m3
| vM3
| M3
| P4
| ^4/d5
| vA4/^d5
| A4/v5
| P5
| etc.
|}

23edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic unison: either A32 or d32.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="4" | [[23-edo|23edo]] {{normal|flat-1}}
! rowspan="2" | Sharp lowers, major is narrower
| '''D'''
| Db
| E#
| '''E'''
| Eb
| Ebb/Fx
| F#
| '''F'''
| Fb
| G#
| '''G'''
| Gb
| A#
| '''A'''
| Ab
| B#
| '''B'''
| Bb
| Bbb/Cx
| C#
| '''C'''
| Cb
| D#
| '''D'''
|-
| P1
| d1
| A2
| M2
| m2
| d2/A3
| M3
| m3
| d3
| A4
| P4
| d4
| A5
| P5
| d5
| A6
| M6
| m6
| d6/A7
| M7
| m7
| d7
| A8
| P8
|-
! rowspan="2" | Sharp raises, major is wider
| '''D'''
| D#
| Eb
| '''E'''
| E#
| Ex/Fbb
| Fb
| '''F'''
| F#
| Gb
| '''G'''
| G#
| Ab
| '''A'''
| A#
| Bb
| '''B'''
| B#
| Bx/Cbb
| Cb
| '''C'''
| C#
| Db
| '''D'''
|-
| P1
| A1
| d2
| m2
| M2
| A2/d3
| m3
| M3
| A3
| d4
| P4
| A4
| d5
| P5
| A5
| d6
| m6
| M6
| A6/d7
| m7
| M7
| A7
| d8
| P8
|}

24edo contains 2 rings of 12edo: an up/down-ring and a plain-ring. Enharmonic unisons: vvA1 and d2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[24-edo|24edo]] {{normal|sharp-2}}
| '''D'''
| ^D/vEb
| D#/Eb
| ^D#/vE
| '''E'''
| ^E/vF
| '''F'''
| ^F
| F#/Gb
| vG
| '''G'''
| ^G/vAb
| G#/Ab
| ^G#/vA
| '''A'''
| etc.
|-
| P1
| ^1/vm2
| A1/m2
| ~2
| M2
| ^M2/vm3
| m3
| ~3
| M3
| ^M3/v4
| P4
| ^4/~4
| A4/d5
| v5/~5
| P5
| etc.
|}

== Chords and chord progressions==
Chord names are based on jazz chord names. See Jim Aiken's book ''A Player's Guide to Chords & Harmony''. Alterations are enclosed in parentheses, additions never are. Alterations always come last in the chord name. Examples:

* [[19edo chords]]
* [[22edo chords]]
* [[24edo chord names]]
* [[31edo chord names]]
* [[41edo chord names]]
* [[Kite Guitar chord shapes (downmajor tuning)]]

In [[Sharpness|sharp-0]] edos aka perfect edos (7, 14, 21, 28 and 35), every interval is perfect, and there is no major or minor. In the following lists of chord names, omit major, minor, dim and aug. Substitute up for upmajor and upminor, and down for downmajor and downminor. The C-E-G chord is called "C perfect" or simply "C".

Chord progressions use ups/downs notation to name the roots, e.g. Cv - Gv - vA^m - F or Iv - Vv - vVI^m - IVv. In relative notation, '''never use lower case roman numerals''' for minor chords, because both vIIm and VIIm would be written vii.

=== Triads ===
The major chord and various alterations of it:
*C E G = C = "C" or "C major" (in perfect edos, "C" or "C perfect")
*C ^E G = C^ = "C up" or "C upmajor"
*C vE G = Cv = "C down" or "C downmajor" (in sharp-2 edos, C~ = "C mid")
* C vvE G = Cvv = "C dud" or "C dudmajor" (in sharp-4 edos, C~ = "C mid", in sharp-6 edos, C^~ = "C upmid")
This table shows how altering the 3rd or the 5th affects the name of the triad. The conventional abbreviations for aug and dim are + and o. These are rather cryptic, and can be replaced with the more obvious and intuitive a and d. Likewise the symbols Δ and − can be replaced with M and m.

{| class="wikitable" style="text-align:center;"
|-
!
! Major
! Minor
! sus4
! sus2
! colspan="2" | Augmented
! colspan="2" | Diminished
|-
! what's downed
! C E G
! C Eb G
! C F G
! C D G
! colspan="2" | C E G#
! colspan="2" | C Eb Gb
|-
! nothing
| C
| Cm
| C4
| C2
| Ca
| C+
| Cd
| Co
|-
! 3rd
| Cv
| Cvm
| Cv4
| Cv2
| Cva
| Cv+
| Cvd
| Cvo
|-
! 5th
| C(v5)
| Cm(v5)
| C4(v5)
| C2(v5)
| Ca(v5)
| C+(v5)
| Cd(v5)
| Co(v5)
|-
! 3rd, 5th
| Cv(v5)
| Cvm(v5)
| Cv4(v5)
| Cv2(v5)
| Cva(v5)
| Cv+(v5)
| Cvd(v5)
| Cvo(v5)
|}

Note that the dim chord is a triad, not a tetrad. A dim tetrad should always be written Co7, never Co. In jazz, the 7 is omitted because dim triads are so much rarer than dim tetrads. But ups and downs notation is meant to work for all genres, not just jazz. So the dim triad and the dim tetrad need different names.

Many edos have notes between the major 3rd and the perfect 4th, creating triads impossible in 12edo, such as:
*C Fb G = C(d4) or C(b4) = "C dim-four" or "C sus-flat-four"
*C E# G = C(a3) or C(#3) = "C aug-three" or "C sus-sharp-three"
*C Ebb G = C(d3) or C(bb3) = "C dim-three" or "C sus-double-flat-three"
*C D# G = C(a2) or C(#2) = "C aug-two" or "C sus-sharp-two"
The "sus" is needed so that C(#2) doesn't sound like C#2, which is C# D# G#.

=== Global arrows ===
A global arrow occurs between the chord root and the conventional chord type (e.g. C^m7). It raises or lowers the 3rd, and also the 6th, 7th or 11th, if present. Thus C down-nine is the usual C9 chord with the 3rd and 7th downed: Cv9 = C vE G vBb D. A global-mid chord has a mid 3rd, 6th, 7th, and/or 11th. Mnemonic: every other note of a stacked-3rds chord is affected: '''6th''' - root - '''3rd''' - 5th - '''7th''' - 9th - '''11th''' - 13th. Note that the 6th is affected, but the 13th is not.

The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd. Furthermore, in larger edos, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the 5th. By the same logic, if the 7th is upped or downed, the 11th would be too.

A 2nd or 4th in a sus chord is also affected: C4 = C F G but Cv4 = C vF G = "C down-four" or "C sus-down-four". But Cv7(4) = C F G vBb

Every conventional chord can accept a global arrow, with one exception: it's pointless for a C5 chord, because there is no 3rd, 6th or 7th to alter. Thus Cv5 is invalid. But C(v5) is valid, and if someone says "C down five", it means C(v5) = C E vG.

=== Sixth and seventh chords ===
If the 7th is not a perfect 5th or a dim 5th above the 3rd, the chord is named as a triad with an added 7th. An added 7th is usually preceded by a comma (the actual punctuation mark, not an interval), which is spoken as "add":
*C E G Bb = C7 = "C seven" (conventional chord)
*C vE G Bb = Cv,7 = "C down add-seven"
*C E G vBb = C,v7 = "C add down-seven"
*C vE G vBb = Cv7 = "C down seven" (global down)
All 7th chords follow this same pattern. Likewise, if the 6th is not a perfect 4th or aug 4th above the 3rd, it's an add-6 chord. Permitting add-7 chords has the added benefit that the wordy "minor-7 flat-5" and the illogical "half-dim" can both be replaced with "dim add-7", written Cd,7.

In the table below, if a chord is '''bolded''', the comma punctuation is not spoken as "add".
{| class="wikitable" style="text-align:center;"
|-
!
! maj7
! dom7
! min7
! colspan="3" | dim-add-7 or min7(b5) or half-dim
! colspan="2" | dim7
! maj6
! min6
|-
! what's downed
! C E G B
! C E G Bb
! C Eb G Bb
! colspan="3" | C Eb Gb Bb
! colspan="2" | C Eb Gb Bbb
! C E G A
! C Eb G A
|-
! nothing
| CM7
| C7
| Cm7
| Cd,7
| Cm7(b5)
| Cø
| Cd7
| Co7
| C6
| Cm6
|-
! 3rd
| Cv,M7
| Cv,7
| Cvm,7
| Cvd,7
| Cvm,7(b5)
| Cø(v3)
| '''Cvd,d7'''
| Cvod7
| Cv,6
| Cvm,6
|-
! 5th
| CM7(v5)
| C7(v5)
| Cm7(v5)
| Cd,7(v5)
| Cm7(vb5)
| Cø(v5)
| Cd7(v5)
| Co7(v5)
| C6(v5)
| Cm6(v5)
|-
! 6th/7th
| C,vM7
| C,v7
| Cmv7
| Cdv7
| Cmv7(b5)
| Cø(v7)
| Cdvd7
| Covd7
| C,v6
| Cmv6
|-
! 3rd, 5th
| Cv,M7(v5)
| Cv,7(v5)
| Cvm,7(v5)
| Cvd,7(v5)
| Cvm,7(vb5)
| Cø(v3v5)
| '''Cvd,d7(v5)'''
| Cvod7(v5)
| Cv,6(v5)
| Cvm,6(v5)
|-
! 3rd, 6th/7th
| CvM7
| Cv7
| Cvm7
| Cvdv7
| Cvm7(b5)
| Cvø
| Cvd7
| Cvo7
| Cv6
| Cvm6
|-
! 5th, 6th/7th
| C,vM7(v5)
| C,v7(v5)
| Cmv7(v5)
| Cdv7(v5)
| Cmv7(vb5)
| Cø(v5v7)
| Cdvd7(v5)
| Covd7(v5)
| C,v6(v5)
| Cm,v6(v5)
|-
! 3rd, 5th, 6th/7th
| CvM7(v5)
| Cv7(v5)
| Cvm7(v5)
| Cvdv7(v5)
| Cvm7(vb5)
| Cvø(v5)
| Cvd7(v5)
| Cvo7(v5)
| Cv6(v5)
| Cvm6(v5)
|}

Various unusual tetrads:
* {{nowrap|C vE G ^Bb {{=}} Cv^7 {{=}} "C down up-seven"}} (in sharp-2 edos 17, 24, 31, etc. {{nowrap|C~7 {{=}} "C mid-seven"}})
* {{nowrap|C E G A# {{=}} C,#6}} or {{nowrap|C,A6 {{=}} "C add sharp-six"}} or "C add aug-six"
* {{nowrap|C E G Ab {{=}} C,b6}} or {{nowrap|C,m6 {{=}} "C add flat-six"}} or "C add minor-six"
* {{nowrap|C E G Bbb {{=}} C,bb7}} or {{nowrap|C,d7 {{=}} "C add double-flat-seven"}} or "C add dim-seven" (19edo's 4:5:6:7 chord)
* {{nowrap|C E G B# {{=}} C,#7}} or {{nowrap|C,A7 {{=}} "C add sharp-seven"}} or "C add aug-seven"
* {{nowrap|C E G Cb {{=}} C,b8}} or {{nowrap|C,d8 {{=}} "C add flat-eight"}} or "C add dim-eight"

=== Ninth chords ===
As in conventional chord naming, a sharp-9 or flat-9 chord is always named as a 7th chord with an added 9th. Thus {{nowrap|B D# F# A C}} is named B7b9 (not Bb9 which would be {{nowrap|Bb D F A C}}). Likewise C#7b9 not C#b9, even thought the latter is clearly the same flat-9 chord as the former. Likewise Cm7b9 not Cmb9, etc.

Double alterations need only a single pair of parentheses, e.g. C E vG vB D is named CM9(v5v7). Double additions mostly need only a single comma, e.g. C E G vBb vD is named C,v7v9. But certain 6/9 chords require two commas. In '''bolded''' 6/9 chords, the comma between the 6 and the 9 is not spoken as "add". However any comma before "6" is, e.g. Cv,6,9 is "C down add six nine".

{| class="wikitable" style="text-align:center;"
|-
!
! add9
! maj9
! dom9
! min9
! dom7b9
! maj6/9
! min6/9
|-
! what's downed
! C E G D
! C E G B D
! C E G Bb D
! C Eb G Bb D
! C E G Bb Db
! C E G A D
! C Eb G A D
|-
! nothing
| C,9
| CM9
| C9
| Cm9
| C7b9
| '''C6,9'''
| '''Cm6,9'''
|-
! 3rd
| Cv,9
| CM9(v3)
| C9(v3)
| Cm9(v3)
| Cv,7b9
| '''Cv,6,9'''
| '''Cvm,6,9'''
|-
! 5th
| C,9(v5)
| CM9(v5)
| C9(v5)
| Cm9(v5)
| C7b9(v5)
| '''C6,9(v5)'''
| '''Cm6,9(v5)'''
|-
! 6th/7th
| ------
| CM9(v7)
| C9(v7)
| Cm9(v7)
| C,v7b9
| '''C,v6,9'''
| '''Cmv6,9'''
|-
! 9th
| C,v9
| CM7v9
| C7v9
| Cm7v9
| C7vb9
| C6v9
| Cm6v9
|-
! 3rd, 5th
| Cv,9(v5)
| CM9(v3v5)
| C9(v3v5)
| Cm9(v3v5)
| Cv,7b9(v5)
| '''Cv,6,9(v5)'''
| '''Cvm,6,9(v5)'''
|-
! 3rd, 6th/7th
| ------
| CvM9
| Cv9
| Cvm9
| Cv7b9
| '''Cv6,9'''
| '''Cvm6,9'''
|-
! 3rd, 9th
| Cv,v9
| Cv,M7v9 or CM7v9(v3)
| Cv,7v9 or C7v9(v3)
| Cvm,7v9 or Cm7v9(v3)
| Cv,7vb9 or C7vb9(v3)
| Cv,6v9 or C6v9(v3)
| Cvm,6v9 or Cm6v9(v3)
|-
! 5th, 6th/7th
| ------
| CM9(v5v7)
| C9(v5v7)
| Cm9(v5v7)
| C,v7b9(v5)
| '''C,v6,9(v5)'''
| '''Cm,v6,9(v5)'''
|-
! 5th, 9th
| C,v9(v5)
| CM7v9(v5)
| C7v9(v5)
| Cm7v9(v5)
| C7vb9(v5)
| C6v9(v5)
| Cm6v9(v5)
|-
! 6th/7th, 9th
| ------
| C,vM7v9
| C,v7v9
| Cmv7v9
| C,v7vb9
| C,v6v9
| Cmv6v9
|-
! 3rd, 5th, 6th/7th
| ------
| CvM9(v5)
| Cv9(v5)
| Cvm9(v5)
| Cv7b9(v5)
| '''Cv6,9(v5)'''
| '''Cvm6,9(v5)'''
|-
! 3rd, 5th, 9th
| Cv,v9(v5)
| Cv,M7v9(v5) or CM7v9(v3v5)
| Cv,7v9(v5) or C7v9(v3v5)
| Cvm,7v9(v5) or Cm7v9(v3v5)
| Cv,7vb9(v5) or C7vb9(v3v5)
| Cv,6v9(v5) or C6v9(v3v5)
| Cvm,6v9(v5) or Cm6v9(v3v5)
|-
! 3rd, 6th/7th, 9th
| ------
| CvM7v9
| Cv7v9
| Cvm7v9
| Cv7vb9
| Cv6v9
| Cvm6v9
|-
! 5th, 6th/7th, 9th
| ------
| C,vM7v9(v5)
| C,v7v9(v5)
| Cmv7v9(v5)
| C,v7vb9(v5)
| C,v6v9(v5)
| Cmv6v9(v5)
|-
! 3rd, 5th, 6th/7th, 9th
| ------
| CvM7v9(v5)
| Cv7v9(v5)
| Cvm7v9(v5)
| Cv7vb9(v5)
| Cv6v9(v5)
| Cvm6v9(v5)
|}

=== Rules for punctuation usage ===
Tetrads, pentads, etc. often require a comma (the actual punctuation mark) to ensure correct parsing of the chord name. Only use a comma when needed, to reduce clutter and standardize chord names. A comma is needed in {{nowrap|Cv,7 {{=}} C vE G Bb}} because omitting it makes {{nowrap|Cv7 {{=}} C vE G vBb}}, a different chord. But C7,v9 is incorrect because C7v9 is the same chord.

The rule is, omit the comma unless doing so changes the chord. This simple rule suffices in most situations. What follows is a detailed analysis, designed to aid in writing computer code that automates chord naming.

A comma separates an added note and prevents it from merging with what comes before it. The comma is unneeded in C7v9 because the 7 can't merge with the down to make a 7v. But Cm,7 is incorrect even though the m and the 7 can merge, because Cm7 is the same chord.

The various components of a chord name are either numbers (for the 6th, 7th, 9th, etc.) or adjectives (up, down, mid, sharp, flat, major, minor, aug and dim). These adjectives usually modify the following number, but they sometimes modify the preceding root, e.g. Caug or C#7. Up, down and mid can't modify the preceding root.

A comma is always needed to separate a number from a number (Cv6,9). It's usually needed to separate an adjective from a number (Cv,7). The only exception is for certain conventional chords like Cm7 where separation is unneeded. A comma is always needed to separate the root of a plain major chord from an adjective (D,v7) or a number (Eb,9). It's never needed to separate a number from an adjective (C7^9). It's needed to separate an adjective from an adjective only if the two adjectives could apply to a single noun. There are six types of such adjective pairs.

* up followed by any adjective except down (C^,^9 or C^,~7 or C^,#9 or C^,b9 or C^,M7 or C^,m6 or C^,a7 or C^,d7)
* down followed by any adjective except up
* sharp followed by sharp (C#,#9)
* flat followed by flat (Bb,b9)
* aug followed by aug (Ca,a7)
* dim followed by dim (Cd,d9)

No other adjective pair can apply to a single noun, thus the comma is omitted:

* {{nowrap|Cv^9 {{=}} C vE G ^D}} (an interval can't be both upped and downed)
* {{nowrap|CmM7 {{=}} C Eb G B}} (an interval can't be both minor and major) *
* {{nowrap|Cma7 {{=}} C Eb G B#}} (an interval can't be both minor and aug) **
* {{nowrap|Cm#11 {{=}} C Eb G F#}} (an interval can't be both minor and sharp)
* {{nowrap|Cvmm6 {{=}} C vEb G Ab}} (an interval can't be doubly minor)
* {{nowrap|Cmv7 {{=}} C Eb G vBb}} (an interval can be downminor, but it can't be minordown)
* {{nowrap|C~v7 {{=}} C vvE G vBb}} in a sharp-4 edo (an interval can be downmid, but it can't be middown)
* {{nowrap|C~~9 = C vvE G vvD}} in a sharp-4 edo (an interval can't be doubly mid)
<nowiki/>* But beware of the minor-major chord. CvmM7 means C vEb G vB and Cvm,M7 means C vEb G B.

<nowiki/>**But since Cma7 can be read as an alternate spelling of Cmaj7, adding a comma is wise: Cm,a7.

In the spoken name, a comma is almost always pronounced as "add". The only exceptions are:

* a comma separating two numbers: C6,9 is spoken as "C six nine"
* a comma separating two ups or two downs: Cv,v9 is spoken as "C-down down-nine", since Cvv9 would be "C dud-nine"
* a comma separating two sharps or two flats: C#,#9 is "C sharp sharp-nine" since C##9 would be "C double-sharp nine"
* a comma separating two augs or two dims: Cvd,d7 is "C down-dim dim-seven", since Cvdd7 would be "C down-double-dim-seven"

Of course, there's no great harm in saying "add" when it isn't strictly needed.
{| class="wikitable"
|+ style="font-size: 105%;" | When to write a comma or say "add"
|-
! colspan="2" rowspan="2" |
! colspan="2" | Component after the possible comma
|-
! adjective
! number
|-
! rowspan="3" | Component before the possible comma
! root
| comma always "add" always
| comma always "add" always
|-
! adjective
| comma sometimes "add" sometimes if comma, never if no comma
| comma usually "add" always if comma, never if no comma
|-
! number
| comma never "add" never
| comma always "add" never
|}

More examples, in which the comma is almost always spoken as "add":

*B9 = B D# F# AvC#
*B,9 = B D# F# C#
*Bb9 = Bb D F Ab C
* Bb,9 = Bb D F C
*B,b9 = B D# F# C
*B7b9 = B D# F# A C
* Bbb9 = Bbb Db Fb Abb Cb
*Bbb,9 = Bbb Db Fb Cb
*Bb,b9 = Bb D F Cb (no "add", "B flat flat-nine")
* B,bb9 = B D# F# Cbb

== Cross-edo considerations ==
In 22edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19edo, it's 0-6-11 = 0¢-379¢-695¢. The two chords sound quite different, because "major 3rd" is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound ru (7-under) and 19edo major chords sound yo (5-over).

A chord quality like "major" refers not to the sound but to the function of the chord. If you want to play a {{nowrap|{{dash|I, VIm, IIm, V, I}}}} progression without pitch shifts or tonic drift, you can do that in any edo, as long as you use only major and minor chords. The notation tells you what kind of chord can be used to play that progression. In 22edo, the chord that you need sounds like a ru chord.

In other words, {{nowrap|{{dash|I, VIm, IIm, V, I}}}} in just intonation implies {{nowrap|{{dash|Iy, VIg, IIg, Vy, Iy}}}}, but this implication only holds in those edos in which major sounds yo. Because 22edo's yo chord {{nowrap|{{dash|0, 7, 13}} {{=}} {{dash|0{{c}}, 382{{c}}, 709{{c}}}}}} is downmajor, it doesn't work in that progression.

Another example: {{nowrap|{{dash|I7, bVII7, IV7, I7}}}}. To play this progression without shifts or drifts, the 7th in the I7 chord must be a minor 7th. in 22edo, that 7th sounds zo (7-over, thus 7/4). In 19edo, it sounds gu (5-under, thus 9/5).

== Ups and downs solfege ==
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down. See [[Uniform solfege|Uniform Solfege]].

== See also ==
* [[Enharmonic unisons in ups and downs notation]]
* [[Lambda ups and downs notation]]

Ups and downs notation was invented by [[Kite Giedraitis]] in early 2014.

{{Navbox notation}}
{{Todo|intro}}

[[Category:Ups and downs notation| ]] 
[[Category:Notation]]

22/7

2026-05-10T22:39:21Z

TallKite: add color name

{{Infobox Interval|Name=undecimal minor thirteenth, first pi approximant|Color name= c1or5, Coloru 5th|Sound=}}22/7 is a [[11-limit]] interval one octave above [[11/7]].

== Approximation to π ==
It is the first non trivial convergent to the continued fraction of [[Acoustic pi|acoustic π]]. The next in the series is 333/106, a much more complex 53-limit interval. The difference between π and 22/7 is of only 0.697 cents.

Kite's thoughts on pergens

2026-05-10T08:42:53Z

TallKite: /* Addenda (Spring 2026) */ completed

A '''pergen''' (pronounced "peer-jen", from '''per'''iod and '''gen'''erator) is a way of classifying a [[regular temperament]] solely by its [[periods and generators|period and generator(s)]]. For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. Every rank-2, rank-3, rank-4, etc. temperament has a pergen. Assuming the prime [[subgroup]] includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a fifth or some fraction of some 3-limit interval. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every [[strong extension]] of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but do not uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.

Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka Triyoti]] has the 5th sharpened by one-fifth of [[250/243]] ({{monzo| 1 -5 3 }}). Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Triguti tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name. (Note these uses of comma fractions are not convention universally; temperament pages tend to use comma fractions to indicate the damage to the generator rather than the fifth. This is analogous to indicating the amount of stretching of an edo by the damage to the edostep rather than to the octave. But the latter approach is much more common, because the damage to the octave is much more audible and thus much more musically relevant than the damage to an edostep, which often doesn't correspond to a simple ratio. Likewise for pergens: the damage to Triyoti/Porcupine's generator, which is both 10/9 and 11/10, is less relevant than the damage to Triyoti's 4th or 5th.)

Overwhelmed? See [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf ''Notation guide for rank-2 pergens''] for practical notation examples.

{{See also| Rank-2 temperaments by mapping of 3 }}

= Definition =
If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. Both fractions are always of the form 1/N, thus the octave and/or the 3-limit interval is '''split''' into N parts. The interval which is split into multiple generators is the '''multigen'''. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.

For example, the Srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. Dicot aka Yoyoti (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine aka Triyoti is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore aka Zozoti, a pun on "semi-fourth", is of course half-fourth.

Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both Srutal aka Saguguti and Injera aka Gu & Biruyoti sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using '''ups and downs''' (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and downs notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.

The largest category contains all single-comma rank-2 temperaments with a comma of the form 2x 3y P or 2x 3y P-1, where P is a prime > 3 (a '''higher prime'''), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called '''unsplit'''.

Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multigen, and as a tie-breaker, to minimize the size in cents of the multigen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.

For example, Srutal could be (P8/2, M2/2), but P5 is preferred because it is unsplit. Or it could be (P8/2, P12), but P5 is preferred because it is smaller. Or it could be (P8/2, P4), but P5 is always preferred over P4. Note that P5/2 is not preferred over P4/2. For example, Decimal is (P8/2, P4/2), not (P8/2, P5/2).

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! colspan="4" | example temperaments
|-
! | written
! | spoken
! | comma(s)
! | name
! colspan="2" | [[Color notation|color name]]
|-
| | (P8, P5)
| | unsplit
| | 81/80
| | Meantone
| | Guti
| | gT
|-
| | "
| | "
| | 64/63
| | Archy
| | Ruti
| | rT
|-
| | "
| | "
| | (-14,8,1)
| | Schismic
| | Layoti
| | LyT
|-
| | (P8/2, P5)
| | half-8ve
| | (11, -4, -2)
| | Srutal
| | Saguguti
| | sggT
|-
| | "
| | "
| | 81/80, 50/49
| | Injera
| | Gu & Biruyoti
| | g&rryyT
|-
| | (P8, P5/2)
| | half-5th
| | 25/24
| | Dicot
| | Yoyoti
| | yyT
|-
| | "
| | "
| | (-1,5,0,0,-2)
| | Mohajira
| | Luluti
| | 1uuT
|-
| | (P8, P4/2)
| | half-4th
| | 49/48
| | Semaphore
| | Zozoti
| | zzT
|-
| | (P8/2, P4/2)
| | half-everything
| | 25/24, 49/48
| | Decimal
| | Yoyo & Zozoti
| | yy&zzT
|-
| | (P8, P4/3)
| | third-4th
| | 250/243
| | Porcupine
| | Triyoti
| | y3T
|-
| | (P8, P11/3)
| | third-11th
| | (12,-1,0,0,-3)
| | Satrilu
| | Satriluti
| | s1u3T
|-
| | (P8/4, P5)
| | quarter-8ve
| | (3,4,-4)
| | Diminished
| | Quadguti
| | g4T
|-
| | (P8/2, M2/4)
| | half-8ve quarter-tone
| | (-17,2,0,0,4)
| | Laquadlo
| | Laquadloti
| | L1o4T
|-
| | (P8, P12/5)
| | fifth-12th
| | (-10,-1,5)
| | Magic
| | Laquinyoti
| | Ly5T
|}

(P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum/difference of octaves and fifths, every single 3-limit interval is also split in half.

The multigen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: bi- splits something into two parts, tri- into three parts, etc. For a comma with monzo (a,b,c,d...), the '''color depth''' is GCD (c,d...).

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[Kite's_color_notation|Color notation]] (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yo, (a,b,-1) = gu, (a,b,0,1) = zo, (a,b,0,-1) = ru, (a,b,0,0,1) = lo/ilo, (a,b,0,0,-1) = lu, and (a,b) = wa. Examples: 5/4 = y3 = yo 3rd, 7/5 = zg5 = zogu 5th, etc.

For example, Marvel aka Ruyoyoti (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). But colors can be replaced with ups and downs, to be higher-prime-agnostic. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.

More examples: Trizoguti (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Biruyoti (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, Biruyoti Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.

A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with microtonal accidentals: (P8, P5, ^1, /1,...).

=Derivation=

For any comma, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents, where GCD (0,x) = x. The comma will split the octave into m parts, and if n > m, it will split some 3-limit interval into n parts.

In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is '''dependent''' on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two '''independent''' primes.

For example, consider Sawa & Ruyoyoti (2.3.5.7 with commas 256/243 and 225/224). The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Notating multi-EDO pergens|multi-EDO pergens]] below).

To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [http://x31eq.com/temper/uv.html x31eq.com/temper/uv.html] that will find such a matrix, it's the reduced mapping. Next make a '''square mapping''' by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.

For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.

2/1 = P8 = x·P, thus P = P8/x

3/1 = P12 = y·P + z·G, thus G = [P12 - y·(P8/x)] / z = [-y·P8 + x·P12] / xz = (-y, x) / xz

M's 3-limit monzo is (-y, x), or (y, -x) if z is negative. To get alternate generators, add i periods to G, with i ranging from -x (subtracting a full octave) to +x (adding a full octave).

G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz

'''The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| <= x'''

A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen.

For example, Porcupine aka Triyoti (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 <= i <= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).

Rank-3 pergens are trickier to find. For example, Breedsmic aka Bizozoguti is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7 x31.com] gives us this matrix:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 5/1
! | 7/1
|-
! | period
| | 1
| | 1
| | 1
| | 2
|-
! | gen1
| | 0
| | 2
| | 1
| | 1
|-
! | gen2
| | 0
| | 0
| | 2
| | 1
|}

Thus 2/1 = P, 3/1 = P + 2·G1, 5/1 = P + G1 + 2·G2, and 7/1 = 2·P + G1 + G2. Discard the last column to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 5/1
|-
! | period
| | 1
| | 1
| | 1
|-
! | gen1
| | 0
| | 2
| | 1
|-
! | gen2
| | 0
| | 0
| | 2
|}

Use an [http://wims.unice.fr/wims/wims.cgi?session=GF84B8C7BF.1&lang=en&cmd=reply&module=tool%2Flinear%2Fmatmult.en&matA=1+1+1%0D%0A0+2+1%0D%0A0+0+2&matB=&show=A%5E-1 online tool] to invert it. Here "/4" means that each entry is to be divided by the determinant of the last matrix, which is 4.

{| class="wikitable" style="text-align:center;"
|-
! |
! | period
! | gen1
! | gen2
! |
|-
! | 2/1
| | 4
| | -2
| | -1
| |
|-
! | 3/1
| | 0
| | 2
| | -1
| |
|-
! | 5/1
| | 0
| | 0
| | 2
| | /4
|}

Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25:6)/4.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a double octave to the multigen. The alternate gens are P11/2 and P19/2, both of which are much larger, so the best gen1 is P5/2.

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96:25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128:75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675:512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.

Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 7/1
|-
! | period
| | 1
| | 1
| | 2
|-
! | gen1
| | 0
| | 2
| | 1
|-
! | gen2
| | 0
| | 0
| | 1
|}

This inverts to this matrix:

{| class="wikitable" style="text-align:center;"
|-
! |
! | period
! | gen1
! | gen2
! |
|-
! | 2/1
| | 2
| | -1
| | -3
| |
|-
! | 3/1
| | 0
| | 1
| | -1
| |
|-
! | 7/1
| | 0
| | 0
| | 2
| | /2
|}

Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, /1) with /1 = 64/63, rank-3 half-5th. This is far better than (P8, P5/2, (96:25)/4).

The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is > 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible. Using 7 instead of 5 in the pergen is very common for rank-3. See [[User:TallKite/Catalog of eleven-limit rank three temperaments with Color names]] for more examples.

=Applications=

Pergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources.

Another obvious application is to categorize regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), Semihemi is (P8/2, P4/2), Triforce is (P8/3, P4/2), both Tetracot and Semihemififths are (P8, P5/4), Fourfives is (P8/4, P5/5), Pental is (P8/5, P5), and Fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, either because it contains more than 2 primes, or because the split multigen isn't actually a generator. For example, Sensei, or Semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.

Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example Porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first Porcupine is Triyoti, and the second one is Triyo & Ruti. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.

Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See [[pergen#Further Discussion-Chord names and scale names|Chord names and scale names]] below.

The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.

All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and downs notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Luluti and Dicot aka Yoyoti are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.

One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, which is octave-equivalent, fifth-generated and heptatonic. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and all notation used here is backwards compatible.

Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, Schismic aka Layoti is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C vE G. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below.

Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multigen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a [[pergen#Further Discussion-Supplemental materials-Notaion guide PDF|notation guide]]. It even allows every pergen to be numbered.

The '''[[enharmonic unison]]''', or more briefly the '''EU''', can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's EU. The pergen and the EU together define the notation. (''Edited to add: not quite accurate, see the Addenda.'')

The '''genchain''' (chain of generators) in the table is only a short section of the full genchain.

C - G implies ...Eb Bb F C G D A E B F# C#...C - ^Eb=vE - G implies ...F -- ^Ab=vA -- C -- ^Eb=vE -- G -- ^Bb=vB -- D...

If the octave is split, the table has a '''perchain''' ("peer-chain", chain of periods) that shows the octave: C -- vF#=^Gb -- C. Genchains have a theoretically infinite length, but perchains have a finite length. The full rank-2 lattice has genchains running horizontally and perchains running vertically.

{| class="wikitable" style="text-align:center;"
|-
! |
! | pergen
! | enharmonic
unison(s)
! | equivalence(s)
! | split
interval(s)
! | perchain(s) and/or
genchains(s)
! | examples
|-
| | 1
| | (P8, P5)
unsplit
| | none
| | none
| | none
| | C - G
| | Pythagorean, Meantone, Dominant,
Schismic, Mavila, Archy, etc.
|-
! |
! | half-splits
! |
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5)
half-8ve
| | ^^d2 (if 5th
> 700¢
| | ^^C = B#
| | P8/2 = vA4 = ^d5
| | C - vF#=^Gb - C
| | Srutal aka Saguguti
^1 = 81/80
|-
| |
| | "
| | vvd2 (if 5th

< 700¢)
| | ^^C = Db
| | P8/2 = ^A4 = vd5
| | C - ^F#=vGb - C
| | Injera aka Gu & Biruyoti

^1 = 64/63
|-
| |
| | "
| | vvM2
| | ^^C = D
| | P8/2 = ^4 = v5
| | C - ^F=vG - C
| | Thothoti, if 13/8 = M6

^1 = 27/26
|-
| | 3
| | (P8, P4/2)

half-4th
| | vvm2
| | ^^C = Db
| | P4/2 = ^M2 = vm3
| | C - ^D=vEb - F
| | Semaphore aka Zozoti

^1 = 64/63
|-
| |
| | "
| | ^^dd2
| | ^^C = B##
| | P4/2 = vA2 = ^d3
| | C - vD#=^Ebb - F
| | Lala-yoyoti

^1 = 81/80
|-
| | 4
| | (P8, P5/2)

half-5th
| | vvA1
| | ^^C = C#
| | P5/2 = ^m3 = vM3
| | C - ^Eb=vE - G
| | Mohajira aka Luluti

^1 = 33/32
|-
| | 5
| | (P8/2, P4/2)

half-

everything
| | \\m2,

vvA1,

^^\\d2,

vv\\M2
| | //C = Db

^^C = C#

^^//C = D
| | P4/2 = /M2 = \m3

P5/2 = ^m3 = vM3

P8/2 = v/A4 = ^\d5

= ^/4 = v\5
| | C - /D=\Eb - F,

C - ^Eb=vE - G,

C - v/F#=^\Gb - C,

C - ^/F=v\G - C
| | Zozo & Luluti

^1 = 33/32

/1 = 64/63
|-
| |
| | "
| | ^^d2,

\\m2,

vv\\A1
| | ^^ C= B#

//C = Db

^^//C = C#
| | P8/2 = vA4 = ^d5

P4/2 = /M2 = \m3

P5/2 = ^/m3 = v\M3
| | C - vF#=^Gb - C,

C - /D=\Eb - F,

C - ^/Eb=v\E - G
| | Sagugu & Zozoti

^1 = 81/80

/1 = 64/63
|-
| |
| | "
| | ^^d2,

\\A1,

^^\\m2
| | ^^C = B#

//C = C#

^^\\C = B
| | P8/2 = vA4 = ^d5

P5/2 = /m3 = \M3

P4/2 =v/M2 = ^\m3
| | C - vF#=^Gb - C,

C - /Eb=\E - G,

C - v/D=^\Eb - F
| | Sagugu & Luluti

^1 = 81/80

/1 = 33/32
|-
! |
! | third-splits
! |
! |
! |
! |
! |
|-
| | 6
| | (P8/3, P5)

third-8ve
| | ^3d2
| | ^3C = B#
| | P8/3 = vM3 = ^^d4
| | C - vE - ^Ab - C
| | Augmented aka Triguti

^1 = 81/80
|-
| | 7
| | (P8, P4/3)

third-4th
| | v3A1
| | ^3C = C#
| | P4/3 = vM2 = ^^m2
| | C - vD - ^Eb - F
| | Porcupine aka Triyoti

^1 = 81/80
|-
| | 8
| | (P8, P5/3)

third-5th
| | v3m2
| | ^3C = Db
| | P5/3 = ^M2 = vvm3
| | C - ^D - vF - G
| | Slendric aka Latrizoti

^1 = 64/63
|-
| | 9
| | (P8, P11/3)

third-11th
| | ^3dd2
| | ^3C = B##
| | P11/3 = vA4 = ^^dd5
| | C - vF# - ^Cb - F
| | Satriluti, if 11/8 = A4

^1 = 729/704
|-
| |
| | "
| | v3M2
| | ^3C = D
| | P11/3 = ^4 = vv5
| | C - ^F - vC - F
| | Satriluti, if 11/8 = P4

^1 = 33/32
|-
| | 10
| | (P8/3, P4/2)

third-8ve, half-4th
| | v6A2
| | ^6C = D#
| | P8/3 = ^^m3 = v4A4

P4/2 = ^3m2 = v3M3
| | C - ^^Eb - vvA - C

C - ^3Db=v3E - F
| | Tribiloti, if 11/8 = P4

^1 = 33/32
|-
| |
| | "
| | ^3d2,

\\m2
| | ^3C = B#

//C = Db
| | P8/3 = vM3 = ^^d4

P4/2 = /M2 = \m3
| | C - vE - ^Ab - C

C - /D=\Eb - F
| | Triforce aka Trigu & Zozoti

^1 = 81/80, /1 = 64/63
|-
| | 11
| | (P8/3, P5/2)

third-8ve, half-5th
| | ^3d2

\\A1
| | ^3C = B#

//C = C#
| | P8/3 = vM3 = ^^d4

P5/2 = /m3 = \M3
| | C - vE - ^Ab - C

C - /Eb=\E - G
| | Satribizoti

^1 = 49/48, /1 = 343/324
|-
| | 12
| | (P8/2, P4/3)

half-8ve, third-4th
| | ^^d2

\3A1
| | ^^C = Dbb

/3C = C#
| | P8/2 = vA4 = ^d5

P4/3 = \M2 = //m2
| | C - vF#=^Gb - C

C - \D - /Eb - F
| | Latribiruti

^1 = 1029/1024, /1 = 49/48
|-
| | 13
| | (P8/2, P5/3)

half-8ve, third-5th
| | ^6d32
| | ^6C = B#3
| | P8/2 = v3AA4 = ^3dd5

P5/3 = vvA2 = ^4dd3
| | C - v3Fx=^3Gbb C

C - vvD# - ^^Fb - G
| | Latribiyoti

^1 = 81/80
|-
| |
| | "
| | ^^d2,

\3m2
| | ^^C = B#

/3C = Db
| | P8/2 = vA4 = ^d5

P5/3 = /M2 = \\m3
| | C - vF#=^Gb - C

C - /D - \F - G
| | Lemba aka Latrizo & Biruyoti

^1 = (10,-6,1,-1), /1 = 64/63
|-
| | 14
| | (P8/2, P11/3)

half-8ve, third-11th
| | v6M2
| | ^6C = D
| | P8/2 = ^34 = v35

P11/3 = ^^4 = v45
| | C - ^3F=v3G - C

C - ^^F - vvC - F
| | Latribiloti, if 11/8 = P4

^1 = 33/32
|-
| | 15
| | (P8/3, P4/3)

third-

everything
| | v3d2,

\3A1
| | ^3C = Dbb

/3C = C#
| | P8/3 = ^M3 = vvd4

P4/3 = \M2 = //m2

P5/3 = v/M2
| | C - ^E - vAb - C

C - \D - /Eb - F

C - v/D - ^\F - G
| | Triyo & Triguti

^1 = 64/63

/1 = 81/80
|-
| |
| | "
| | ^3d2,

\3m2
| | ^3C = B#

/3C = Db
| | P8/3 = vM3 = ^^d4

P5/3 = /M2 = \\m3

P4/3 = v\M2
| | C - vE - ^Ab - C

C - /D - \F - G

C - v\D - ^/Eb - F
| | Trigu & Latrizoti

^1 = 81/80

/1 = 64/63
|-
| |
| | "
| | v3A1,

\3m2
| | ^3C = C#

/3C = Db
| | P4/3 = vM2 = ^^m2

P5/3 = /M2 = \\m3

P8/3 = v/M3
| | C - vD - ^Eb - F

C - /D - \F - G

C - v/E - ^\Ab - C
| | Triyo & Latrizoti

^1 = 81/80

/1 = 64/63
|-
! |
! | quarter-splits
! |
! |
! |
! |
! |
|-
| | 16
| | (P8/4, P5)
| | ^4d2
| | ^4C = B#
| | P8/4 = vm3 = ^3A2
| | C vEb vvGb=^^F# ^A C
| | Diminished aka Quadguti
|-
| | 17
| | (P8, P4/4)
| | ^4dd2
| | ^4C = B##
| | P4/4 = ^m2 = v3AA1
| | C ^Db ^^Ebb=vvD# vE F
| | Negri aka Laquadyoti
|-
| | 18
| | (P8, P5/4)
| | v4A1
| | ^4C = C#
| | P5/4 = vM2 = ^3m2
| | C vD vvE=^^Eb ^F G
| | Tetracot aka Saquadyoti
|-
| | 19
| | (P8, P11/4)
| | v4dd3
| | ^4C = Eb3
| | P11/4 = ^M3 = v3dd5
| | C ^E ^^G# vDb F
| | Squares aka Laquadruti
|-
| | 20
| | (P8, P12/4)
| | v4m2
| | ^4C = Db
| | P12/4 = v4 = ^3M3
| | C vF vvBb=^^A ^D G
| | Vulture aka Sasa-quadyoti
|-
| |
| | etc.
| |
| |
| |
| |
| |
|}
The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably it isn't particularly complex.

Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢).

==Tipping points==

Removing the ups and downs from an EU makes an '''uninflected''' EU, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s EU is ^^d2, the uninflected EU is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the EU vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore '''up may need to be swapped with down, depending on the size of the 5th''' in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' EUs are upped or downed as if the 5th were just.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every uninflected EU, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the uninflected EU.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | uninflected EU
! | 3-exponent
! | tipping

point edo
! | edo's 5th
! | upping range
! | downing range
! | if the 5th is just
|-
| | M2
| | C - D
| | 2
| | 2-edo
| | 600¢
| | none
| | all
| | downed
|-
| | m3
| | C - Eb
| | -3
| | 3-edo
| | 800¢
| | none
| | all
| | downed
|-
| | m2
| | C - Db
| | -5
| | 5-edo
| | 720¢
| | none
| | all
| | downed
|-
| | A1
| | C - C#
| | 7
| | 7-edo
| | ~686¢
| | 600-686¢
| | 686¢-720¢
| | downed
|-
| | d2
| | C - Dbb
| | -12
| | 12-edo
| | 700¢
| | 700-720¢
| | 600-700¢
| | upped
|-
| | dd3
| | C - Eb3
| | -17
| | 17-edo
| | ~706¢
| | 706-720¢
| | 600-706¢
| | downed
|-
| | dd2
| | C - Db3
| | -19
| | 19-edo
| | ~695¢
| | 695-720¢
| | 600-695¢
| | upped
|-
| | d34
| | C - Fb3
| | -22
| | 22-edo
| | ~709¢
| | 709-720¢
| | 600-709¢
| | downed
|-
| | d32
| | C - Db4
| | -26
| | 26-edo
| | ~692¢
| | 692-720¢
| | 600-692¢
| | upped
|-
| | d44
| | C - Fb4
| | -29
| | 29-edo
| | ~703¢
| | 703-720¢
| | 600-703¢
| | downed
|-
| | d43
| | C - Eb5
| | -31
| | 31-edo
| | ~697¢
| | 697-720¢
| | 600-697¢
| | upped
|-
| | etc.
| |
| |
| |
| |
| |
| |
| |
|}

=Further Discussion=

==Naming very large intervals==

So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, adding an 8ve is indicated by "c" for '''compound''' (a conventional music theory term). Thus 10/3 = cM6 = compound major 6th, 9/2 = ccM2 or cM9, etc. For a pergen with an unsplit octave, the multigen is always some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen M is less than 3 octaves, and can be P4, P5, P11, P12, ccP4 or ccP5. The last one can be spoken as "coco-fifth". Tripe compound can be spoken as "trico".

==Secondary splits==

Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. Porcupine aka Triyoti) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with:

P4/3: C - vD - ^Eb - F

A4:/3 C - D - E - F# (the lack of ups and downs indicates that this interval was already split into 3 parts)

m7/3: C - ^Eb - vG - Bb (because m7 is already split into halves, we also have m7/6: C - vD - ^Eb - F - vG - ^Ab - Bb)

M7/3: C - vE - ^G - B

m10/3: C - F - Bb - Eb (m10 is already split into 3 parts, thus m10/9 also occurs)

M10/3: C - ^F - vB - E

Of course, an equal-step melody can span other intervals besides 3-limit ones. Simply extend or cut short the earlier examples to find other melodies:

^m3/2: C - vD - ^Eb (^m3 = 6/5)

^m6/5: C - vD - ^Eb - F - vG - ^Ab (^m6 = 8/5)

vm9/4: C - ^Eb - vG - Bb - ^Db (vm9 = 32/15)

vM7/2: C - ^F - vB (vM7 = 15/8, probably more harmonious than M7 = 243/128)

More remote intervals include A1, d4, d7 and d10. These unfortunately require a very long genchain. The most interesting melodically is A1: C - ^C - vC# - C#. From C to C# is seven 5ths, which equals 21 generators, so the genchain would have to contain 22 notes if it had no gaps. Note that A1 is the uninflected EU of third-4th. The uninflected EU is always a secondary split.

For a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occurring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.

Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof [[pergen#Further Discussion-Various proofs|below]]). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If only the 8ve is split, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the EU is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. "Every" means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.

The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occurring splits are listed too, under "all pergens". (P8/3, P4/2) has all the secondary splits that (P8/3, P5) and (P8, P4/2) have, plus additional ones.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! | secondary splits of a 12th or less
|-
| colspan="2" | all pergens
| | M3/2, A4/3, d5/2, A5/4, m7/2, M9/2, m10/3, A11/2
|-
! colspan="2" | half-splits
! |
|-
| | (P8/2, P5)
| | half-8ve
| | M2/2, M3/4, A4/6, A5/8, m6/2, M10/2, d12/2, A12/2
|-
| | (P8, P4/2)
| | half-4th
| | m2/2, M6/2, m7/4, A8/2, m10/6, A11/4, P12/2
|-
| | (P8, P5/2)
| | half-5th
| | A1/2, m3/2, M7/2, m9/2, P11/2
|-
| | (P8/2, P4/2)
| | half-everything
| | every 3-limit interval is split twice as much as before
|-
! colspan="2" | third-splits
! |
|-
| | (P8/3, P5)
| | third-8ve
| | m3/3, M6/3, d5/6, A11/3, d12/3
|-
| | (P8, P4/3)
| | third-4th
| | A1/3, m7/6, M7/3, m10/9, M10/3
|-
| | (P8, P5/3)
| | third-5th
| | m2/3, m6/3, M9/6, A8/3, A12/3
|-
| | (P8, P11/3)
| | third-11th
| | M2/3, M3/6, A4/9, A5/12, m9/3, P12/3
|-
| | (P8/3, P4/2)
| | third-8ve half-4th
| | third-8ve splits, half-4th splits, M6/6, m10/6, A11/12
|-
| | (P8/3, P5/2)
| | third-8ve half-5th
| | third-8ve splits, half-5th splits, m3/6, d5/12
|-
| | (P8/2, P4/3)
| | half-8ve third-4th
| | half-8ve splits, third-4th splits, A4/6, M10/6
|-
| | (P8/2, P5/3)
| | half-8ve third-5th
| | half-8ve splits, third-5th splits, m6/6, M9/6, A12/6
|-
| | (P8/2, P11/3)
| | half-8ve third-11th
| | half-8ve splits, third-11th splits, M2/6, M3/12, A4/18, A5/24
|-
| | (P8/3, P4/3)
| | third-everything
| | every 3-limit interval is split three times as much as before
|}

==Singles and doubles==

If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a '''single-split''' pergen. If it has two fractions, it's a '''double-split''' pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called '''single-pair''' notation because it adds only a single pair of accidentals to conventional notation. '''Double-pair''' notation uses both ups/downs and lifts/drops. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the EU for single-pair notation is a 3rd or larger, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs.

In this article, for double-pair notation, the period uses ups and downs, and the generator uses lifts and drops. But the choice of which pair of accidentals is used for what is arbitrary, and ups/downs could be exchanged with lifts/drops.

Every double-split pergen is either a '''true double''' or a '''false double'''. A true double, like third-everything (P8/3, P4/3) or half-8ve quarter-4th (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-8ve quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.

A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.

==Finding an example temperament==

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4⋅P and P8. If P is 6/5, the comma is 4⋅P - P8 = (6/5)4 ÷ (2/1) = 648/625, making the Diminished temperament aka Quadguti. If P is 7/6, the comma is P8 - 4⋅P = (2/1) · (7/6)-4, making the Quadruti temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.

Another method: if the generator's cents are known, look on the genchain for an interval that approximates a ratio of color depth ±1. Let the interval be I, and the genspan of this interval be x. Then n⋅x gens = n⋅I = x⋅M, where M is the multigen and M/n is the generator. The comma can be found from this equation, if n and x are coprime. For example, suppose (P8, P5/5) has G = 140¢. The genchain is all multiples of 140¢. Looking at the cents, 280¢ is about 7/6. Thus 2G = 7/6, and 10G = 5⋅(7/6) = 2⋅P5. Thus 2⋅P5 - 5⋅(7/6) = 0G = 0¢, and the comma is (3, 7, 0, -5). If the period is split and the generator isn't, use the perchain instead of the genchain. For example, (P8/7, P5) has a period of 141¢. 2 gens = 343¢, about 11/9. Thus 7⋅(11/9) = 2⋅P8, and the comma is (-2, -14, 0, 0, 7), making Saseploti.

If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.

Finding the comma(s) for a double-split pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is '''explicitly false'''. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4⋅G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).

If the pergen is not explicitly false, put the pergen in its '''unreduced''' form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n⋅P8 - m⋅M)/nm) = (P8/m, M'/n'). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P < G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2⋅P8 - 3⋅P5) / (3⋅2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This is explicitly false, thus the comma can be found from m3/6 alone. G' is about 50¢, and the comma is 6⋅G' - m3. The comma splits both the octave and the fifth.

This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit. To summarize:

<ul><li>'''A'''<nowiki/> '''double-split pergen is explicitly false if m = |b|, and not explicitly false if m > |b|.'''</li><li>'''A double-split pergen is a true double if and only if neither it nor its unreduced form is explicitly false'''''<nowiki/>'''''<nowiki/>.'''</li><li>'''A double-split pergen is a true double if''' '''GCD (m, n) > |b|,''' '''and a false double if GCD (m, n) = |b|.'''</li></ul>

A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.

Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an '''alternate''' generator. A generator or period plus or minus any number of EUs makes an '''equivalent''' generator or period. An equivalent generator is always the same size in cents, since the EU is always 0¢. An equivalent generator is the same interval, merely notated differently. For example: (P8, P5/2) has generator ^m3 and equivalent generator vM3. Another example, half-8ve (P8/2, P5) has period vA4 and equivalent period ^d5. It has generator P5 and alternate generators P4 and vA1. vA1 is equivalent to ^m2.

Of the two equivalent generators, the preferred generator is always the smaller one (smaller degree, or if the degrees are the same, more diminished). This is because the equivalent generator can be more easily found by adding the EU, rather than subtracting it. For example, ^m3 + vvA1 = vM3 is an easier calculation than vM3 - vvA1 = ^m3. This is particularly true with complex EUs like ^6dd2.

There are also alternate EUs, see below. For double-pair notation, there are also equivalent EUs.

==Ratio and cents of the accidentals==

The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2x · 3y · P±1, where P is a higher prime. They are called mapping commas because they equate or map the ratio P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for prime 5 include 81/80, 135/128, and the schisma aka Layoma = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 (besides 1/1 of course) are the currently employed commas and combinations of them.

If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in Lemba aka Latrizo & Biruyoti, where ^1 equals 64/63 minus 81/80.

Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also [[Pergen#Notating_multi-EDO_pergens|multi-EDO pergens]] pergens below.

Cents can be assigned to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + 7c. Since the EU = 0¢, we can derive the cents of the up symbol. If the EU is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its EU.

In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the EU is an A1.

This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, basic information for playing the score. Examples:
* 15-edo: # = 240¢, ^ = 80¢ (^ = third-sharp)
* 16-edo: # = -75¢
* 17-edo: # = 141¢, ^ = 71¢ (^ = half-sharp)
* 18b-edo: # = -133¢, ^ = 67¢ (^ = half-sharp)
* 19-edo: # = 63¢
* 21-edo: ^ = 57¢ (if used, # = 0¢)
* 22-edo: # = 164¢, ^ = 55¢ (^ = third-sharp)
* quarter-comma Meantone: # = 76¢
* fifth-comma Meantone: # = 84¢
* third-comma Archy aka Ruti: # = 177¢
* eighth-comma Porcupine aka Triyoti: # = 157¢, ^ = 52¢ (^ = third-sharp)
* seventh-comma Srutal aka Sagugu & Zoquadyoti: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #, seventh-comma means 1/7 of 2048/2025)
* third-comma Injera aka Gu & Biruyoti: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)
* eighth-comma Hedgehog aka Triyo & Biruyoti: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)
The last five examples are a generalization of the practice of naming meantone tunings as a fraction of a comma, e.g. quarter-comma. The 5th is sharpened or flattened by some fraction of the first comma in the color name. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both Srutal and Injera are half-8ve, but their optimal tunings are very different.

==Finding a notation for a pergen==

There are multiple notations for a given pergen, depending on the EU(s). Preferably, the EU's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:

<ul><li>For (P8/m, M/n), P8 = m⋅P + x⋅EU and M = n⋅G + y⋅EU', with 0 < |x| <= m/2 and 0 < |y| <= n/2</li><li>x is the count for EU, with EU occurring x times in one octave, and x⋅EU is the octave's '''multi-EU'''</li><li>y is the count for EU', with EU' occurring y times in one multigen, and y⋅EU' is the multigen's multi-EU</li><li>For false doubles using single-pair notation, EU = EU', but x and y are usually different, making different multi-EUs</li><li>The unreduced pergen is (P8/m, M'/n'), with a new enharmonic unison EU" and new counts, P8 = m⋅P + x'⋅EU", and M' = n'⋅G' + y'⋅EU"</li></ul>

The '''keyspan''' of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The '''[[stepspan]]''' of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.

Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a '''gedra''', analogous to a monzo, but written in brackets not parentheses: 3/2 = (-1,1) is a 7-semitone 5th, thus (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval with a monzo (a,b), there is a unique gedra [k,s], and vice versa:

<ul><li>k = 12a + 19b</li><li>s = 7a + 11b</li></ul>The matrix [(12,19) (7,11)] is unimodular, and can be inverted, and (a,b) can be derived from [k,s]:

<ul><li>a = -11k + 19s</li><li>b = 7k - 12s</li></ul>Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].

Gedras greatly facilitate finding a pergen's period, generator and EU(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an EU with the smallest possible keyspan and stepspan, which is the best EU for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up.

For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = m3. The EU can also be found using gedras: x⋅EU = M - n⋅G = P5 - 2⋅m3 = [7,4] - 2⋅[3,2] = [7,4] - [6,4] = [1,0] = A1.

Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. x⋅EU = P8 - m⋅P = P8 - 5⋅M2 = [12,7] - 5⋅[2,1] = [2,2] = 2⋅[1,1] = 2⋅m2. Because x = 2, EU will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2⋅m2 = d3). The EU's '''count''' is 2. The uninflected EU is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus EU = v5m2. Since P8 = 5⋅P + 2⋅EU, the period must be ^^M2, to make the ups and downs come out even. The number of the period's (or generator's) ups or downs always equals the count. Equipped with the period and the EU, the perchain is easily found:

P1 -- ^^M2=v3m3 -- v4 -- ^5 -- ^3M6=vvm7 -- P8
C -- ^^D=v3Eb -- vF -- ^G -- ^3A=vvBb -- C

Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has an uninflected generator [5,3]/5 = [1,1] = m2. The uninflected EU is P4 - 5⋅m2 = [5,3] - 5⋅[1,1] = [5,3] - [5,5] = [0,-2] = -2⋅[0,1] = two descending d2's. The d2 must be upped, and EU = ^5d2. Since P4 = 5⋅G - 2⋅EU, G must be ^^m2. The genchain is:

P1 -- ^^m2=v3A1 -- vM2 -- ^m3 -- ^3d4=vvM3 -- P4C -- ^^Db -- vD -- ^Eb -- vvE -- F

To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and EU from the fractional multigen as before. Then deduce the period from the EU. If the multigen was changed by unreducing, find the original generator from the period and the alternate generator.

For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The uninflected alternate generator G' is [1,1]/10 = [0,0] = P1. The uninflected EU is m2 - 10⋅P1 = m2. It must be downed, thus EU = v10m2. Since m2 = 10⋅G' + EU, G' is ^1. The octave plus or minus some number of EUs must equal 5 periods, thus (P8 + x⋅m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5⋅P + 2⋅EU, and P = ^4M2. Next, find the original half-4th generator. P = P8/5 = ~240¢, and G = P4/2 = ~250¢. Because P < G, G' is not P - G but G - P, and G is not P - G' but P + G', which equals ^4M2 + ^1 = ^5M2. While the alternate multigen is more complex than the original multigen, the alternate generator is usually simpler than the original generator.

P1- - ^4M2=v6m3 -- vv4 -- ^^5 -- ^6M6=v4m7 -- P8
C -- ^4D=v6Eb -- vvF -- ^^G -- ^6A=v4Bb -- C
P1 -- ^5M2=v5m3 -- P4
C -- ^5D=v5Eb -- F

To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. Each pair has its own enharmonic unison. For (P8/2, P4/2), the split octave implies P = vA4 and EU = ^^d2, and the split 4th implies G = /M2 and EU' = \\m2.

A false-double pergen can optionally use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair EU is not a unison or a 2nd, as with (P8/2, P4/3).

Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an EU that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So EU = v4dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The uninflected enharmonic unison is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic unison, we use the second pair of accidentals: EU' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic unisons is also an enharmonic unison: EU + EU' = v4\\d3 = 2·vv\m2, and EU - EU' = v4//ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent EUs, and v\4 and v/d4 are equivalent generators. Here is the genchain:

P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11
C -- ^E=v\F -- /Ab=\A -- ^/C=vDb -- F

One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and EU = v12m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the uninflected enharmonic unison: m3 - 3·m2 = [0,-1] = descending d2. Thus EU' = /3d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonic unisons are found from EU + EU' and EU - 2·EU'. Equivalent periods and generators are found from the many enharmonic unisons, which also allow much freedom in chord spelling. Enharmonic unison = v12m3 = /3d2 = v4/m2 = v4\\A1. Period = \M3 = v44 = //d4. Generator = ^\M3 = v34 = ^//d4.

P1 — \M3 — \\A5=/m6 — P8C — \E — /Ab — CP1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^38=v/m9 — P11
C — ^\E — ^^/Ab=vv\A — v/Db — F

It's not yet known if every pergen can avoid large EUs (those of a 3rd or more) with double-pair notation. One situation in which very large EUs occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the EU's stepspan to equal the multigen's stepspan.

Sixth-4th with single-pair notation has an awkward ^6d64 EU. This pergen might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore commas, aka Triyo & Zozoti), and its double-pair notation can also combine both. Third-4th has EU = v3A1 and G'= vM2 = ^^m2. Half-4th has EU' = \\m2 and G' = /M2 = \m3. G' - G = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G' - G = /M2 - vM2 = ^/1. Equivalent EUs are v3\\M2 and ^3\\d2.

P1 — ^/1=^\m2 — ^^m2=vM2 — /M2=\m3 — ^m3=vvM3 — v/M3=v\4 — P4
C — ^/C=^\Db — ^^Db=vD — /D=\Eb — ^Eb=vvE — v/E=v\F — F

When ups and downs are used to notate edos, a third symbol is used, a '''mid''' , written ~. The mid is only for relative notation (intervals), never for absolute notation (notes). For imperfect intervals, the mid interval is the interval exactly midway between major and minor. In addition, the mid-4th is midway between perfect and augmented, i.e. half-augmented, and the mid-5th is half-diminished. For example, 17edo's 2nds and 3rds run minor-mid-major. This avoids having to constantly choose between the equivalent terms upminor and downmajor. 72edo's 2nds and 3rds run minor, upminor, downmid, mid, upmid, downmajor, major. This makes the terms more concise. Mids can be used in pergen notation too. For single-pair, EU must be an A1, with an even number of downs. Using mids with double-pair notation is trickier. Mids never appear in the perchain. If one accidental pair appears only in the perchain and the other pair only in the genchain, then the mids appear only in the genchain, if at all.

==Alternate enharmonic unisons==

Sometimes the EU found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, ccM6/12), a false double. The uninflected alternate generator is ccM6/12 = [33,19]/12 = [3,2] = m3. The uninflected EU is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The EU becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v12A2, which is an improvement but still awkward. The period is ^4m3 and the generator is v3M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.

P1 -- ^4m3 -- v4M6 -- P8
C -- ^4Eb -- v4A -- C
P1 -- v3M2 -- v6M3=^6m2 -- ^3m3 -- P4
C -- v3D -- v6E=^6Db -- ^3Eb -- F

Because G is a M2 and EU is an A2, the equivalent generator G - EU is a descending A1. Ascending intervals that look descending can be confusing, one has to take into account the eighth-tone ups to see that v3D -- ^6Db is ascending. Double-pair notation may be preferable. This makes P = vM3, EU = ^3d2, G = /m2, and EU' = /4dd2.

P1 -- vM3 -- ^m6 -- P8
C -- vE -- ^Ab -- C
P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4
C -- /Db -- //Ebb=\\D# -- \E -- F

Because the EU found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans, like 5edo or 19edo. Heptatonic stepspans are best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the EU.

To search for an alternate EU, convert the EU to a gedra, then multiply it by the count to get the multi-EU. The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and EU is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-EU. If the stepspan becomes negative, or if it's zero and the keyspan becomes negative, invert the gedra. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new EU. Choose between ups and downs according to whether the 5th falls in the EU's upping or downing range, see [[pergen#Applications-Tipping points|tipping points]] above. Add n'''·'''count ups or downs to the new multi-EU. Add or subtract the new multi-EU from the multigen (or the octave) to get an interval which splits cleanly into n (or m) parts. Each part is the new generator (or period).

For example, (P8, P5/3) has n = 3, G = ^M2, and EU = v3m2 = [1,1]. G is upped only once, so the count is 1, and the multi-EU is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-EU [-2,1]. This can't be simplified, so the new EU is also [-2,1] = d32. Assuming a reasonably just 5th, EU needs to be upped, so EU = ^3d32. Add the multi-EU ^3[-2,1] to the multigen P5 = [7,4] to get ^3[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^3[-2,1] = v3[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·EU to check: 3·vA2 + 1·^3d32 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d32 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- vD# -- ^Fb -- G. Because d32 = -200¢ - 26·c, ^ = (-d32) / 3 = 67¢ + 8.67·c, about a third-tone.

Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one EU (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain EU. Specifically, the comma tempered out should map to the EU, or the multi-EU if the count is > 1. For example, consider Semaphore aka Zozoti (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 EU makes sense. G = ^M2 and the genchain is C -- ^D=vEb -- F. But consider Lala-yoyoti, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus EU = ^^dd2, G = vA2, and the genchain is C -- vD#=^Ebb -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.

Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 EU is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of EU.

For example, Paralimmal aka Satriluti tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the EU is v3M2, but if 11/8 is notated as a vA4, the EU is ^3dd2.

Sometimes the temperament implies an EU that isn't even a 2nd. For example, Liese aka Gu & Trizoguti (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and EU = 3·vd5 - P11 = v3dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- vGb -- ^B -- F.

This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an EU, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different EUs.

==Chord names and staff notation==

Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[Ups and downs notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.

In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G ^A and C E G vBb are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.

Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara aka Ru & Biruyoti (2.3.5.7 with 64/63 and 50/49) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and EU = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C vE G Bb = Cv,7.

A different temperament may result in the same pergen with the same EU, but may still produce a different name for the same chord. For example, Injera aka Gu & Biruyoti (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 EU is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and EU = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G vBb = C,v7.

Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes. This is in fact a requirement for MOS scales.

Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. Iv -- vIIIv -- vVI^m -- Iv. A Porcupine aka Triyoti (third-4th) comma pump can be written out like so: Cv -- vA^m -- vDv -- [vvB=^Bb]^m -- ^Ebv -- G^m -- Gv -- Cv. Brackets are used to show that vvB and ^Bb are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. vvB is written first to show that this root is a vM6 above the previous root, vD. ^Bb is second to show the P4 relationship to the next root, ^Eb. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either vvB^m or ^Bb^m, or possibly ^BbvvM = ^Bb vD ^F.

Lifts and drops, like ups and downs, precede the note head and any sharps or flats. Scores for melody instruments could instead have them above or below the staff. This score uses ups and downs, and has chord names. It's for the third-4th pergen, with EU = v3A1. As noted above, it can be played in EDOs 15, 22 or 29 as is, because ^1 maps to 1 edostep. But to be played in EDOs 30, 37 or 44, ^1 must be interpreted as two edosteps.

Mizarian Porcupine Overture by Herman Miller (P8, P4/3) ^3C = C#

[[File:Mizarian_Porcupine_Overture.png|alt=Mizarian Porcupine Overture.png|800x692px|Mizarian Porcupine Overture.png]]

==Tipping points and sweet spots==

The tipping point for half-octave with a d2 EU is 700¢, 12-edo's 5th. As noted above, the 5th of Pajara (half-8ve) tends to be sharp, thus it has EU = ^^d2. But Injera, also half-8ve, has a flat 5th, and thus EU = vvd2. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But if a single temperament "tips over", either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.

The tipping point depends on the choice of EU. It's not the temperament that tips, it's the notation. Half-8ve could be notated with an EU of vvM2. The tipping point becomes 600¢, a very unlikely 5th, and tipping is impossible. For single-comma temperaments, the EU usually equals the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of EU is a given. However, the mapping of primes 11 and 13 is not agreed on.

The notation's tipping point is determined by the uninflected EU, which is implied by the vanishing comma. For example, Porcupine aka Triyoti's 250/243 comma is an A1 = (-11,7), which implies an uninflected EU of A1, which implies 7-edo, and a 685.7¢ tipping point. Dicot aka Yoyoti's 25/24 comma is also an A1, and has the same tipping point. Semaphore aka Zozoti's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. See [[pergen#Applications-Tipping points|tipping points]] above for a more complete list.

Double-pair notation has two EUs, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two EUs can be any two of A1, m2 or d2. One can choose to use whichever two EUs best avoid tipping.

An example of a temperament that tips easily is Negri aka Laquadyoti, 2.3.5 and (-14,3,4). Because Negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with Negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and the EU could be either ^4dd2 or v4dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an EU of ^4dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma Negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.

Another "tippy" temperament is found by adding the mapping comma 81/80 to the Negri comma and getting the Latriyoma comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th.

==Notating unsplit pergens==

An unsplit pergen doesn't require ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a mapping comma, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (i.e. the monzo has a final exponent of ±1).

The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate. The up symbol's ratio is always the mapping comma, or its inverse.

{| class="wikitable" style="text-align:center;"
|-
! | 5-limit temperament
! | comma
! | sweet spot
! colspan="2" | no ups or downs
! colspan="3" | with ups and downs
! colspan="2" | up symbol
|-
! | (pergen is unsplit)
! |
! | (5th = 700¢ + c)
! | 5/4 is
! | 4:5:6 chord
! | 5/4 is
! | 4:5:6 chord
! | EU
! | ratio
! | cents
|-
| | Meantone aka Guti
| | 81/80 = P1
| | c = -3¢ to -5¢
| | M3
| | C E G
| | ---
| | ---
| | ---
| | ---
| | ---
|-
| | Mavila aka Layobiti
| | 135/128 = A1
| | c = -21¢ to -22¢
| | m3
| | C Eb G
| | ^M3
| | C ^E G
| | ^A1
| | 80/81 = d1
| | -100¢ - 7c = 47¢-54¢
|-
| | Laguti
| | (-15,11,-1) = A1
| | c = -10¢ to -12¢
| | A3
| | C E# G
| | ^M3
| | C ^E G
| | vA1
| | 80/81 = A1
| | 100¢ + 7c = 26¢-30¢
|-
| | Schismic aka Layoti
| | (-15,8,1) = -d2
| | c = 1.7¢ to 2.0¢
| | d4
| | C Fb G
| | vM3
| | C vE G
| | ^d2
| | 81/80 = -d2
| | 12c = 20¢-24¢
|-
| | Lalaguti
| | (-23,16,-1) = -d2
| | c = -0.9¢ to -1.2¢
| | AA2
| | C D## G
| | vM3
| | C vE G
| | vd2
| | 81/80 = d2
| | -12c = 10¢-15¢
|-
| | Father aka Gubiti
| | 16/15 = m2
| | c = 56¢ to 58¢
| | P4
| | C F G
| | vM3
| | C vE G
| | ^m2
| | 81/80 = -m2
| | -100¢ + 5c = 180-190¢
|-
| | Superpyth aka Sasayoti
| | (12,-9,1) = m2
| | c = 9¢ to 10¢
| | A2
| | C D# G
| | vM3
| | C vE G
| | vm2
| | 81/80 = m2
| | 100¢ - 5c = 50-55¢
|}
The Schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch. The Mavila temperament could perhaps be notated without ups and downs, because 5/4 is still a 3rd, and the 4:5:6 triad still looks like a triad.

For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the Schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-limit comma is the pythagorean comma (-19,12).

A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for Archy aka Ruti (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit Schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C vE G vBb.

==Notating rank-3 pergens==

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. Enharmonic unisons aka EUs are like vanishing commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of EUs needed always equals the difference between the notation's rank and the tuning's rank. Examples:

{| class="wikitable" style="text-align:center;"
|-
! | tuning
! | pergen
! | tuning's rank
! | notation
! | notation's rank without any EUs
! | # of EUs needed
! | EUs
|-
| | 12-edo
| | (P8/12)
| | rank-1
| | conventional
| | rank-2
| | 1
| | EU = d2
|-
| | 19-edo
| | (P8/19)
| | rank-1
| | conventional
| | rank-2
| | 1
| | EU = dd2
|-
| | 15-edo
| | (P8/15)
| | rank-1
| | single-pair
| | rank-3
| | 2
| | EU = m2, EU' = v3A1 = v3M2
|-
| | 24-edo
| | (P8/24)
| | rank-1
| | single-pair
| | rank-3
| | 2
| | EU = d2, EU' = vvA1 = vvm2
|-
| | 3-limit JI aka pythagorean
| | (P8, P5)
| | rank-2
| | conventional
| | rank-2
| | 0
| | ---
|-
| | Meantone aka Guti
| | (P8, P5)
| | rank-2
| | conventional
| | rank-2
| | 0
| | ---
|-
| | Diaschismic aka Saguguti
| | (P8/2, P5)
| | rank-2
| | single-pair
| | rank-3
| | 1
| | EU = ^^d2
|-
| | Semaphore aka Zozoti
| | (P8, P4/2)
| | rank-2
| | single-pair
| | rank-3
| | 1
| | EU = vvm2
|-
| | Decimal aka Yoyo & Zozoti
| | (P8/2, P4/2)
| | rank-2
| | double-pair
| | rank-4
| | 2
| | EU = vvd2, EU' = \\m2 = ^^\\A1
|-
| | 5-limit JI
| | (P8, P5, ^1)
| | rank-3
| | single-pair
| | rank-3
| | 0
| | ---
|-
| | Marvel aka Ruyoyoti
| | (P8, P5, ^1)
| | rank-3
| | single-pair
| | rank-3
| | 0
| | ---
|-
| | Breedsmic aka Bizozoguti
| | (P8, P5/2, ^1)
| | rank-3
| | double-pair
| | rank-4
| | 1
| | EU = \\dd3
|-
| | 7-limit JI
| | (P8, P5, ^1, /1)
| | rank-4
| | double-pair
| | rank-4
| | 0
| | ---
|}
When there is more than one EU, the first one can be added to or subtracted from the 2nd one, to make an equivalent 2nd EU.

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas. There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.

Even the unsplit rank-3 pergen requires single-pair notation, for the 2nd generator. Single-splits and false double-splits require double-pair, true doubles and false triples require triple-pair, and true triples require quadruple-pair. Some false triples and some true doubles may use quadruple-pair as well, to avoid awkward EUs of a 3rd or more. Rather than devising a third or fourth pair of symbols, and a third or fourth pair of adjectives to describe them, one might simply use colors.

A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.

Some examples of 7-limit rank-3 temperaments:

{| class="wikitable" style="text-align:center;"
|-
! | 7-limit temperament
! | comma
! | pergen
! | spoken pergen
! | notation
! | period
! | gen1
! | gen2
! | EU
|-
| | Marvel aka Ruyoyoti
| | 225/224
| | (P8, P5, ^1)
| | rank-3 unsplit
| | single-pair
| | P8
| | P5
| | ^1 = 81/80
| | ---
|-
| | "
| | "
| | "
| | "
| | double-pair
| | "
| | "
| | "
| | ^^\d2
|-
| | Biruyoti
| | 50/49
| | (P8/2, P5, ^1)
| | rank-3 half-8ve
| | double-pair
| | v/A4 = 10/7
| | P5
| | ^1 = 81/80
| | ^^\\d2
|-
| | Trizoguti
| | 1029/1000
| | (P8, P11/3, ^1)
| | rank-3 third-11th
| | double-pair
| | P8
| | ^\d5 = 7/5
| | ^1 = 81/80
| | ^^^\\\dd3
|-
| | Breedsmic aka Bizozoguti
| | 2401/2400
| | (P8, P5/2, ^1)
| | rank-3 half-5th
| | double-pair
| | P8
| | v//A2 = 60/49
| | /1 = 64/63
| | ^^\4dd3
|-
| | Demeter aka Trizo-aguguti
| | 686/675
| | (P8, P5, \m3/2)
| | half-downminor-3rd
| | double-pair
| | P8
| | P5
| | v/A1 = 15/14
| | ^^\\\dd3
|}
If using single-pair notation, Marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - vE - G - vvA#. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - vE - G - \Bb.

There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, Biruyoti is half-8ve, with a d2 comma, like Srutal. Using the same notation as Srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and EU = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as C vE G v/Bb. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C vE G \Bb.

With this standardization, the EU can be derived directly from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)-2 · (64/63)2 · (-19,12). This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the EU never is, therefore the EU = ^^\\d2. The period is found by adding/subtracting the EU from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both EU and P become more complex, but the ratio for /1 becomes simpler.

This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For Biruyoti, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore Biruyoti doesn't tip. Single-pair rank-3 notation has no EU, and thus no tipping point. Double-pair rank-3 notation has 1 EU, but two mapping commas. Rank-3 notations rarely tip.

Unlike the previous examples, Demeter aka Trizo-aguguti's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no EU. But the 4:5:6:7 chord would be spelled C -- ^^^Fbbb -- G -- ^^Bbb, very awkward! Standard double-pair notation is better. Gen2 = v/A1, EU = ^^\\\dd3, and ^^\\\C = A##. Genchain2 is C -- v/C# -- \Eb -- vE -- \\Gb -- v\G -- vvG#=\\\Bbb -- v\\Bb... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own EU, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own EU, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops only for the other kind of accidentals.

There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For Demeter, any combination of \m3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because there will always be a 3-limit comma small enough to be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the '''DOL''' ([[Odd limit|double odd limit]]) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred. And as a tie-breaker, in case of two ratios with the same DOL (such as 5/3 and 6/5), to minimize the size in cents of the multigen2. Since 5/3 = 884.4¢ and 6/5 = 315.6¢, 6/5 is preferred.

If ^1 = 81/80, possible half-split gen2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.

All possible rank-3 pergens can be listed, but the table is much longer than for rank-2 pergens. Here are all the half-split pergens:

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! colspan="4" | prime subgroup used by the pergen
|-
! | unsplit
! colspan="2" | 2.3.5 (^1 = 81/80)
! colspan="2" | 2.3.7 (^1 = 64/63)
|-
| | 1
| | (P8, P5, ^1)
| | rank-3 unsplit
| | same
| | same
|-
! | half-splits
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5, ^1)
| | rank-3 half-8ve
| | same
| | same
|-
| | 3
| | (P8, P4/2, ^1)
| | rank-3 half-4th
| | same
| | same
|-
| | 4
| | (P8, P5/2, ^1)
| | rank-3 half-5th
| | same
| | same
|-
| | 5
| | (P8/2, P4/2, ^1)
| | rank-3 half-everything
| | same
| | same
|-
| | 6
| | (P8, P5, ^m3/2)
| | half-upminor-3rd
| | (P8, P5, ^M2/2)
| | half-upmajor-2nd
|-
| | 7
| | (P8, P5, vM3/2)
| | half-downmajor-3rd
| | (P8, P5, vm3/2)
| | half-downminor-3rd
|-
| | 8
| | (P8, P5, ^m6/2)
| | half-upminor-6th
| | (P8, P5, ^M6/2)
| | half-upmajor-6th
|-
| | 9
| | (P8, P5, vM6/2)
| | half-downmajor-6th
| | (P8, P5, vm7/2)
| | half-downminor-7th
|-
| | 10
| | (P8/2, P5, ^m3/2)
| | half-8ve half-upminor-3rd
| | (P8/2, P5, ^M2/2)
| | half-8ve half-upmajor-2nd
|-
| | 11
| | (P8/2, P5, vM3/2)
| | half-8ve half-downmajor-3rd
| | (P8/2, P5, vm3/2)
| | half-8ve half-downminor-3rd
|-
| | 12
| | (P8, P4/2, vM3/2)
| | half-4th half-downmajor-3rd
| | (P8, P4/2, ^M2/2)
| | half-4th half-upmajor-2nd
|-
| | 13
| | (P8, P4/2, ^m6/2)
| | half-4th half-upminor-6th
| | (P8, P4/2, vm7/2)
| | half-4th half-downminor-7th
|-
| | 14
| | (P8, P5/2, vM3/2)
| | half-5th half-downmajor-3rd
| | (P8, P5/2, ^M2/2)
| | half-5th half-upmajor-2nd
|-
| | 15
| | (P8, P5/2, ^m6/2)
| | half-5th half-upminor-6th
| | (P8, P5/2, vm7/2)
| | half-5th half-downminor-7th
|-
| | 16
| | (P8/2, P4/2, vM3/2)
| | half-everything half-downmajor-3rd
| | (P8/2, P4/2, ^M2/2)
| | half-everything half-upmajor-2nd
|}
There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc.

==Notating multi-EDO pergens==

A multi-EDO pergen is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave, thus it doesn't appear in the pergen. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12. Pergens which imply an edo which doesn't have a decent 5th, e.g. P8/3, P8/4, P8/6, etc., are covered in the next section.

A multi-EDO pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the octave and removes the middle term from the pergen. For example, Blackwood aka Sawati plus Ya is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1). Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:

{| class="wikitable" style="text-align:center;"
|-
! | temperament
! | pergen
! | spoken name
! | enharmonic unisons
! | perchain
! | genchain
! | ^1 ratio
! | /1 ratio
|-
| | Blackwood aka Sawati+ya
| | (P8/5, ^1)
| | rank-2 5-edo
| | EU = m2
| | D E=F G A B=C D
| | D vF#=vG vvB...
| | 81/80 = 16/15
| | ---
|-
| | Whitewood aka Lawati+ya
| | (P8/7, ^1)
| | rank-2 7-edo
| | EU = A1
| | D E F G A B C D
| | D ^F ^^A...
| | 80/81 = 135/128
| | ---
|-
| | 10edo+ya
| | (P8/10, /1)
| | rank-2 10-edo
| | EU = m2, EU' = vvA1 = vvM2
| | D ^D=vE E=F ^F=vG G...
| | D \F#=\G \\B...
| | (see below)
| | 81/80
|-
| | 12edo+la
| | (P8/12, ^1)
| | rank-2 12-edo
| | EU = d2
| | D D#=Eb E F F#=Gb...
| | D ^G ^^C
| | 33/32
| | ---
|-
| | "
| | "
| | "
| | "
| | "
| | D vG#=vAb vvD...
| | 729/704
| | ---
|-
| | 17edo+ya
| | (P8/17, /1)
| | rank-2 17-edo
| | EU = dd3, EU' = vm2 = vvA1
| | D ^D=Eb D#=vE E F...
| | D \F# \\A#=v\\B...
| | 256/243
| | 81/80
|}
If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 10, 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15, or 12/11, or 13/12.

The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.

All multi-EDO pergens are of the form (P8/m, ^1) or (P8/m, /1), and they can be identified solely by their splitting fraction. Multi-EDO pergens are a small minority of rank-2 pergens.

It's possible to have a fifth-8ve pergen with an independent 5th, but there will be very small intervals of about 20¢. Here are two such:

{| class="wikitable" style="text-align:center;"
|-
! | temperament
! | subgroup
! | comma
! | pergen
! | spoken name
! | EU
! | perchain
! | genchain
! | ^1 ratio
|-
| | Laquinzoti
| | 2.3.7
| | (-14,0,0,5)
| | (P8/5, P5)
| | fifth-8ve
| | v5m2
| | D ^^E vG ^A vvC D
| | C G D A E...
| | 49/48
|-
| | Saquinruti
| | 2.3.7
| | (22,-5,0,-5)
| | "
| | "
| | "
| | "
| | "
| | 64/63
|}
Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C ^^D vF ^G vvBb C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section). A multi-EDO pergen should never have an uninflected genchain.

==Notating non-8ve and no-5ths pergens==

In multi-EDO pergens, the 5th is present but not independent. In no-5ths pergens, the 5th is not present, and the prime subgroup doesn't contain 3.

In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any EUs, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.

But in non-8ve and no-5ths pergens, not every name has a note. For example, Biruyoti Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - vF# - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a huge number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk.

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! colspan="6" | prime subgroup used by the pergen
|-
! | unsplit
! | 2.3
! | 2.5 (M3 = 5/4)
! | 2.7 (M2 = 8/7)
! | 3.5 (M6 = 5/3)
! | 3.7 (M3 = 9/7)
! | 5.7 (ccM3 = 5/1, d5 = 7/5)
|-
| | 1
| | (P8, P5)
| | (P8, M3)
| | (P8, M2)
| | (P12, M6)
| | (P12, M3)
| | (ccM3, d5)
|-
! | half-splits
! |
! |
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5)
| | (P8/2, M3)
| | (P8/2, M2)
| | (P12/2, M6)
| | (P12/2, M3)
| | (M9, d5)*
|-
| | 3
| | (P8, P4/2)
| | (P8, M2)*
| | (P8, M2/2)
| | (P12, M6/2)
| | (P12, M2)*
| | (ccM3, m3)*
|-
| | 4
| | (P8, P5/2)
| | (P8, m6/2)
| | (P8, P5)*
| | (P12, P4)*
| | (P12, m10/2)
| | (ccM3, M7)*
|-
| | 5
| | (P8/2, P4/2)
| | (P8/2, M2)*
| | (P8/2, M2/2)
| | (P12/2, M6/2)
| | (P12/2, M3/2)
| | (M9, m3)*
|-
! | third-splits
! |
! |
! |
! |
! |
! |
|-
| | 6
| | (P8/3, P5)
| | (P8/3, M3)
| | (P8/3, M2)
| | (P12/3, M6)
| | (P12/3, M3)
| | (ccM3/3, d5)
|-
| | 7
| | (P8, P4/3)
| | (P8, M3/3)
| | (P8, M2/3)
| | (P12, M6/3)
| | (P12, M3/3)
| | (ccM3, d5/3)
|-
| | 8
| | (P8, P5/3)
| | (P8, m6/3)
| | (P8, m7/3)
| | (P12, m7/3)
| | (P12, P4)*
| | (ccM3, cA6/3)
|-
| | 9
| | (P8, P11/3)
| | (P8, M10/3)
| | (P8, M9/3)
| | (P12, ccM3/3)
| | (P12, cM7/3)
| | (ccM3, ccm7/3)
|-
| | 10
| | (P8/3, P4/2)
| | (P8/3, M2)*
| | (P8/3, M2/2)
| | (P12/3, M6/2)
| | (P12/3, M2)*
| | (ccM3/3, m3)*
|-
| | 11
| | (P8/3, P5/2)
| | (P8/3. m6/2)
| | (P8/3, P5)*
| | (P12/3, P4)*
| | (P12/3, m10/2)
| | (ccM3/3, M7)*
|-
| | 12
| | (P8/2, P4/3)
| | (P8/2, M3/3)
| | (P8/2, M2/3)
| | (P12/2, M6/3)
| | (P12/2, M3/3)
| | (M9, d5/3)*
|-
| | 13
| | (P8/2, P5/3)
| | (P8/2, m6/3)
| | (P8/2, m7/3)
| | (P12/2, m7/3)
| | (P12/2, P4)*
| | (M9, cA6/3)*
|-
| | 14
| | (P8/2, P11/3)
| | (P8/2, M10/3)
| | (P8/2, M9/3)
| | (P12/2, ccM3/3)
| | (P12/2, cM7/3)
| | (M9, ccm7/3)*
|-
| | 15
| | (P8/3, P4/3)
| | (P8/3, M3/3)
| | (P8/3, M2/3)
| | (P12/3, M6/3)
| | (P12/3, P4)*
| | (ccM3/3, d5/3)
|}
For prime subgroup p.q, the unsplit pergen has period p/1. The unsplit pergen's generator is found by dividing q by p until it's less than p/1, and period-inverting if it's more than half of p/1.

Multi-EDO pergens also appear in this table. For example, in row #33, (P8/5, ^1) replaces both 2.5 (P8/5, M3) and 2.7 (P8/5, M2). This has the advantage of avoiding missing notes. Since one fifth of 5/1 is only 25¢ from 7/5, the 5.7 (ccM3/5, d5) can optionally be replaced too.

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! | 2.3
! | 2.5
! | 2.7
! | 3.5
! | 3.7
! | 5.7
|-
| | 33
| | (P8/5, P5)
| | (P8/5, ^1)
| | (P8/5, ^1)
| | (P12/5, M6)
| | (P12/5, M3)
| | (ccM3/5, ^1)
|}

Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the first 3.5 pergen, (P12, M6). The note one M6 above C1 is A1, and the note three P12's above C1 is A5. (There is no A2, A3 or A4.) Referring to a note as simply A is ambiguous.

Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table could be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2), with ^M2 = 8/7 and ^1 = √256/245 = √(81/80 * 64/63 * 64/63) = about 38¢. The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3), with ^M3 = 9/7 and ^1 = √6561/6125 = √[(81/80)3 * (64/63)2] = about 60¢.

==Pergen squares==

Pergen squares, which were discovered by Praveen Venkataramana, are a way to visualize pergens squares in a way that isn't specific to any primes at all. To understand them, let's assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.

For (P8, P5), the pergen square has 4 notes, shown here with octave numbers (ignore the periods).

C2 -- G2 
| . . . . | 
C1 -- G1

Splitting the 8ve or the multigen adds notes to the square. For (P8/2, P5), there are 6 notes. The new notes fall halfway between each 8ve:

C2 --- G2 
vF#1 vC#2 
C1 --- G1

A pergen square shows all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and vC#2 bisects it. vG#2 bisects the G2-A2 M2. Every "downed" note bisects an 8ve, a M2, a cm7 (e.g. D1 to C3), a cM9 (e.g. C1 to D3), and many other intervals.

C2 --- G2 --- D3 --- A3 
vF#1 vC#2 vG#2 vD#3 
C1 --- G1 --- D2 --- A2

Splitting the 5th adds notes to the horizontal edges of the square:

C2 vE2 G2 
| . . . . . . | 
C1 vE1 G1

Each downed note also bisects a P11 (e.g. G1-C3), as well as many other intervals.

C3 vE3 G3 
| . . . . . . | 
C2 vE2 G2 
| . . . . . . | 
C1 vE1 G1

From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square.

C2 ---- G2 
| . ^A1 . | 
C1 ---- G1

^A1 also bisects the P12 from C1 to G2.

Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Pierce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.

[[File:pergen_squares.png|alt=pergen squares.png|pergen squares.png]]

A similar chart could be made for all rank-3 pergens, using pergen cubes.

==Notating tunings with an arbitrary generator==

Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.

The next table lists all the ranges for all multigens up to seventh-splits. One can look up one's generator in the first column and find a possible multigen. Use the octave inverse if G > 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | primary choice
! colspan="4" | secondary choices
|-
! | generator
! | possible multigen
! | generator
! | multigen
! | generator
! | multigen
|-
| | 23-60¢
| | M2/4 (requires P8/2)
| |
| |
| |
| |
|-
| | 69-79¢
| | P4/7
| |
| |
| |
| |
|-
| | 80-92¢
| | P4/6
| |
| |
| |
| |
|-
| | 92-103¢
| | P5/7
| |
| |
| |
| |
|-
| | 96-111¢
| | P4/5
| |
| |
| |
| |
|-
| | 108-120¢
| | P5/6
| |
| |
| |
| |
|-
| | 120-138¢
| | P4/4
| |
| |
| |
| |
|-
| | 129-144¢
| | P5/5
| |
| |
| |
| |
|-
| | 160-185¢
| | P4/3
| | 162-180¢
| | P5/4
| |
| |
|-
| | 215-240¢
| | P5/3
| |
| |
| |
| |
|-
| | 240-277¢
| | P4/2
| | 240-251¢
| | P11/7
| | 264-274¢
| | P12/7
|-
| | 280-292¢
| | P11/6
| |
| |
| |
| |
|-
| | 308-320¢
| | P12/6
| |
| |
| |
| |
|-
| | 323-360¢
| | P5/2
| | 336-351¢
| | P11/5
| |
| |
|-
| | 369-384¢
| | P12/5
| |
| |
| |
| |
|-
| | 411-422¢
| | ccP4/7
| |
| |
| |
| |
|-
| | 420-438¢
| | P11/4
| |
| |
| |
| |
|-
| | 435-446¢
| | ccP5/7
| |
| |
| |
| |
|-
| | 462-480¢
| | P12/4
| | 462-480¢
| | M9/3
| |
| |
|-
| | 480-554¢
| | P4 = P5
| | 480-492¢
| | ccP4/6
| | 508-520¢
| | ccP5/6
|-
| | 560-585¢
| | P11/3
| |
| |
| |
| |
|-
| | 576-591¢
| | ccP4/5
| | 583-593¢
| | cccP4/7
| |
| |
|}
There are gaps in the table, especially near 150¢, 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation. But the total range of possible generators is mostly well covered, providing convenient notation options. In particular, if the tuning's generator is just over 720¢, to avoid descending 2nds, instead of calling the generator a 5th, one can call its inverse a quarter-12th. The generator is notated as an up-5th, and four of them make a ccP4.

The table assumes unsplit octaves. Splitting the octave creates alternate generators. For example, if P = P8/3 = 400¢ and G = 300¢, alternate generators are 100¢ and 500¢. Any of these can be used to find a convenient multigen.

See also the [[Map_of_rank-2_temperaments|map of rank-2 temperaments]].

==Pergens and MOS scales==

Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! colspan="3" | MOS scales of 5-12 notes
! |
! |
! |
|-
| | (P8, P5)
| | unsplit
| | 5 = 2L 3s
| | 7 = 5L 2s
| colspan="2" | 12 = 7L 5s (or 5L 7s)
| |
| |
|-
! colspan="2" | halves
! |
! |
! |
! |
! |
! |
|-
| | (P8/2, P5)
| | half-8ve
| | 6 = 2L 4s
| | 8 = 2L 6s
| | 10 = 2L 8s
| colspan="3" | 12 = 2L 10s (or 10L 2s)
|-
| | (P8, P4/2)
| | half-4th
| | 5 = 4L 1s
| | 9 = 5L 4s
| |
| |
| |
| |
|-
| | (P8, P5/2)
| | half-5th
| | 7 = 3L 4s
| | 10 = 7L 3s
| |
| |
| |
| |
|-
| | (P8/2, P4/2)
| | half-everything
| | 6 = 4L 2s
| | 10 = 4L 6s
| |
| |
| |
| |
|-
! colspan="2" | thirds
! |
! |
! |
! |
! |
! |
|-
| | (P8/3, P5)
| | third-8ve
| | 6 = 3L 3s
| | 9 = 3L 6s
| colspan="2" | 12 = 3L 9s (or 9L 3s)
| |
| |
|-
| | (P8, P4/3)
| | third-4th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 1L 6s
| | 8 = 7L 1s
| |
| |
|-
| | (P8, P5/3)
| | third-5th
| | 5 = 1L 4s
| | 6 = 5L 1s
| | 11 = 5L 6s
| |
| |
| |
|-
| | (P8, P11/3)
| | third-11th
| | 5 = 2L 3s
| | 7 = 2L 5s
| | 9 = 2L 7s
| | 11 = 2L 9s
| |
| |
|-
| | (P8/3, P4/2)
| | third-8ve, half-4th
| | 6 = 3L 3s *
| | 9 = 6L 3s
| |
| |
| |
| |
|-
| | (P8/3, P5/2)
| | third-8ve, half-5th
| | 6 = 3L 3s *
| | 9 = 3L 6s
| | 12 = 3L 9s
| |
| |
| |
|-
| | (P8/2, P4/3)
| | half-8ve, third-4th
| | 6 = 2L 4s *
| | 8 = 6L 2s
| |
| |
| |
| |
|-
| | (P8/2, P5/3)
| | half-8ve, third-5th
| | 6 = 4L 2s *
| | 10 = 6L 4s
| |
| |
| |
| |
|-
| | (P8/2, P11/3)
| | half-8ve, third-11th
| | 6 = 2L 4s *
| | 8 = 2L 6s
| | 10 = 2L 8s
| | 12 = 2L 10s
| |
| |
|-
| | (P8/3, P4/3)
| | third-everything
| | 6 = 3L 3s *
| | 9 = 6L 3s *
| |
| |
| |
| |
|-
! colspan="2" | quarters
! |
! |
! |
! |
! |
! |
|-
| | (P8/4, P5)
| | quarter-8ve
| | 8 = 4L 4s
| colspan="2" | 12 = 4L 8s (or 8L 4s)
| |
| |
| |
|-
| | (P8, P4/4)
| | quarter-4th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 1L 6s
| | 8 = 1L 7s
| | 9 = 1L 8s
| | 10 = 9L 1s
|-
| | (P8, P5/4)
| | quarter-5th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 6L 1s
| |
| |
| |
|-
| | (P8, P11/4)
| | quarter-11th
| | 5 = 3L 2s
| | 8 = 3L 5s
| | 11 = 3L 8s
| |
| |
| |
|-
| | (P8, P12/4)
| | quarter-12th
| | 5 = 3L 2s
| | 8 = 5L 3s
| |
| |
| |
| |
|-
| | (P8/4, P4/2)
| | quarter-8ve half-4th
| | 8 = 4L 4s *
| | 12 = 4L 8s
| |
| |
| |
| |
|-
| | (P8/2, M2/4)
| | half-8ve quarter-tone
| | 6 = 2L 4s *
| | 8 = 2L 6s *
| | 10 = 2L 8s
| | 12 = 2L 10s
| |
| |
|-
| | (P8/2, P4/4)
| | half-8ve quarter-4th
| | 6 = 2L 4s *
| | 8 = 2L 6s *
| | 10 = 8L 2s
| |
| |
| |
|-
| | (P8/2, P5/4)
| | half-8ve quarter-5th
| | 6 = 2L 4s *
| | 8 = 6L 2s *
| |
| |
| |
| |
|-
| | (P8/4, P4/3)
| | quarter-8ve third-4th
| | 8 = 4L 4s *
| | 12 = 8L 4s *
| |
| |
| |
| |
|-
| | (P8/4, P5/3)
| | quarter-8ve third-5th
| | 8 = 4L 4s *
| | 12 = 4L 8s *
| |
| |
| |
| |
|-
| | (P8/4, P11/3)
| | quarter-8ve third-11th
| | 8 = 4L 4s *
| | 12 = 4L 8s *
| |
| |
| |
| |
|-
| | (P8/3, P4/4)
| | third-8ve quarter-4th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 9L 3s *
| |
| |
| |
|-
| | (P8/3, P5/4)
| | third-8ve quarter-5th
| | 6 = 3L 3s *
| | 9 = 6L 3s *
| |
| |
| |
| |
|-
| | (P8/3, P11/4)
| | third-8ve quarter-11th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 3L 9s *
| |
| |
| |
|-
| | (P8/3, P12/4)
| | third-8ve quarter-12th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 3L 9s *
| |
| |
| |
|-
| | (P8/4, P4/4)
| | quarter-everything
| | 8 = 4L 4s *
| | 12 = 8L 4s *
| |
| |
| |
| |
|}

A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the simplest, and also one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided.

{| class="wikitable" style="text-align:center;"
|-
! | MOS scale
! colspan="2" | primary example
! | secondary examples
|-
! | Pentatonic
! |
! |
! |
|-
| | 1L 4s
| | (P8, P5/3) [5]
| | third-5th pentatonic
| | third-4th, quarter-4th, quarter-5th
|-
| | 2L 3s
| | (P8, P5) [5]
| | unsplit pentatonic
| | third-11th
|-
| | 3L 2s
| | (P8, P12/4) [5]
| | quarter-12th pentatonic
| | quarter-11th
|-
| | 4L 1s
| | (P8, P4/2) [5]
| | half-4th pentatonic
| |
|-
! | Hexatonic
! |
! |
! |
|-
| | 1L 5s
| | (P8, P4/3) [6]
| | third-4th hexatonic
| | quarter-4th, quarter-5th, fifth-4th, fifth-5th
|-
| | 2L 4s
| | (P8/2, P5) [6]
| | half-8ve hexatonic
| |
|-
| | 3L 3s
| | (P8/3, P5) [6]
| | third-8ve hexatonic
| |
|-
| | 4L 2s
| | (P8/2, P4/2) [6]
| | half-everything hexatonic
| |
|-
| | 5L 1s
| | (P8, P5/3) [6]
| | third-5th hexatonic
| |
|-
! | Heptatonic
! |
! |
! |
|-
| | 1L 6s
| | (P8, P4/3) [7]
| | third-4th heptatonic
| | quarter-4th, fifth-4th, fifth-5th, sixth-4th, sixth-5th
|-
| | 2L 5s
| | (P8, P11/3) [7]
| | third-11th heptatonic
| | fifth-double-compound-4th, sixth-double-compound-5th
|-
| | 3L 4s
| | (P8, P5/2) [7]
| | half-5th heptatonic
| | fifth-12th
|-
| | 4L 3s
| | (P8, P11/5) [7]
| | fifth-11th heptatonic
| | sixth-12th
|-
| | 5L 2s
| | (P8, P5) [7]
| | unsplit heptatonic
| | sixth-double-compound-4th
|-
| | 6L 1s
| | (P8, P5/4) [7]
| | quarter-5th heptatonic
| |
|-
! | Octotonic
! |
! |
! |
|-
| | 1L 7s
| | (P8, P4/4) [8]
| | quarter-4th octotonic
| | fifth-4th, fifth-5th, sixth-4th, sixth-5th, seventh-4th, seventh-5th
|-
| | 2L 6s
| | (P8/2, P5) [8]
| | half-8ve octotonic
| |
|-
| | 3L 5s
| | (P8, P11/4) [8]
| | quarter-11th octotonic
| | seventh-cc4th, seventh-cc5th
|-
| | 4L 4s
| | (P8/4, P5) [8]
| | quarter-8ve octotonic
| |
|-
| | 5L 3s
| | (P8, P12/4) [8]
| | quarter-12th octotonic
| | (very lopsided, unless 5th is quite flat)
|-
| | 6L 2s
| | (P8/2, P4/3) [8]
| | half-8ve third-4th octotonic
| |
|-
| | 7L 1s
| | (P8, P4/3) [8]
| | third-4th octotonic
| |
|-
! | Nonatonic
! |
! |
! |
|-
| | 1L 8s
| | (P8, P4/4) [9]
| | quarter-4th nonatonic
| | fifth-4th, sixth-4th, sixth-5th, seventh-4th/5th, eighth-4th/5th
|-
| | 2L 7s
| | (P8, c3P5/8) [9]
| | eighth-c35th nonatonic
| | third-11th, fifth-cc4th
|-
| | 3L 6s
| | (P8/3, P5) [9]
| | third-8ve nonatonic
| | third-8ve half-5th
|-
| | 4L 5s
| | (P8, P12/7) [9]
| | seventh-12th nonatonic
| | sixth-11th
|-
| | 5L 4s
| | (P8, P4/2) [9]
| | half-4th nonatonic
| | (lopsided unless 4th is sharp), seventh-11th
|-
| | 6L 3s
| | (P8/3, P4/2) [9]
| | third-8ve half-4th nonatonic
| |
|-
| | 7L 2s
| | (P8, ccP5/6)[9]
| | sixth-cc5th nonatonic
| | (lopsided unless 5th is sharp)
|-
| | 8L 1s
| | (P8, P5/5) [9]
| | fifth-5th nonatonic
| |
|-
! | Decatonic
! |
! |
! |
|-
| | 1L 9s
| | (P8, P5/6) [10]
| | sixth-5th decatonic
| | fifth-4th, sixth-4th, seventh-4th/5th, eighth-4th/5th, ninth-4th/5th
|-
| | 2L 8s
| | (P8/2, P5) [10]
| | half-8ve decatonic
| | half-8ve quartertone, half-8ve third-11th
|-
| | 3L 7s
| | (P8, P12/5) [10]
| | fifth-12th decatonic
| | eighth-cc4th, eighth-cc5th
|-
| | 4L 6s
| | (P8/2, P4/2) [10]
| | half-everything decatonic
| |
|-
| | 5L 5s
| | (P8/5, P5) [10]
| | fifth-8ve decatonic
| | (lopsided unless 5th is quite flat)
|-
| | 6L 4s
| | (P8/2, P5/3) [10]
| | half-8ve third-5th decatonic
| |
|-
| | 7L 3s
| | (P8, P5/2) [10]
| | half-5th decatonic
| | ninth-cc5th
|-
| | 8L 2s
| | (P8/2, P4/4) [10]
| | half-8ve quarter-4th decatonic
| | half-8ve quarter-12th
|-
| | 9L 1s
| | (P8, P4/2) [10]
| | quarter-4th decatonic
| |
|}

The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the Sensei aka Sepgu & Ruyoyoti generator would be (P8, ccP5/7) [5]. The rationale would be that two Sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be Roulette [7] aka Zozoquinguti Nowa [7], with a 2.5.7 pergen (P8, (5/4)/2) = (P8, M3/2) = (P8, M2), where 5·G = A6 = 7/4.

==Pergens and EDOs==

Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored.

Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset.

How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinitely many possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, ccP5/31),... (P8, (i-1,1)/n), where n = 12i+7.

How many edos support a given pergen? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be a multiple of m, and k must be divisible by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.

Given an edo, a period, and a generator, what is the pergen? There is usually more than one right answer. For 10edo with P = 5\10 and G = 2\10, it could be either (P8/2, P4/2) or (P8/2, P5/3). Every coprime period/generator pair results in a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.

'''EDOs Supporting A Pergen'''

This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Both of these edos are incompatible with heptatonic notation, and 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. 6-edo, 11-edo and 23-edo could also be considered ambiguous.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! | supporting edos (12-31 only)
|-
| | (P8, P5)
| | unsplit
| | 12, 13, 13b, 14*, 15*, 16, 17, 18, 18b*, 19, 20*,

21*, 22, 23, 24*, 25*, 26, 27, 28*, 29, 30*, 31
|-
! | halves
! |
! |
|-
| | (P8/2, P5)
| | half-8ve
| | 12, 14, 16, 18, 18b, 20*, 22, 24*, 26, 28*, 30*
|-
| | (P8, P4/2)
| | half-4th
| | 13b, 14, 15*, 18b*, 19, 20*, 23, 24, 25*, 28*, 29, 30*
|-
| | (P8, P5/2)
| | half-5th
| | 13, 14*, 17, 18b, 20*, 21*, 24, 27, 28*, 30*, 31
|-
| | (P8/2, P4/2)
| | half-everything
| | 14, 18b, 20*, 24, 28*, 30*
|-
! | thirds
! |
! |
|-
| | (P8/3, P5)
| | third-8ve
| | 12, 15, 18, 18b*, 21, 24*, 27, 30*
|-
| | (P8, P4/3)
| | third-4th
| | 13b, 14*, 15, 21*, 22, 28*, 29, 30*
|-
| | (P8, P5/3)
| | third-5th
| | 15*, 16, 20*, 21, 25*, 26, 30*, 31
|-
| | (P8, P11/3)
| | third-11th
| | 13, 15, 17, 21, 23, 30*
|-
| | (P8/3, P4/2)
| | third-8ve, half-4th
| | 15, 18b*, 24, 30*
|-
| | (P8/3, P5/2)
| | third-8ve, half-5th
| | 18b, 21, 24, 27, 30
|-
| | (P8/2, P4/3)
| | half-8ve, third-4th
| | 14, 22, 28*, 30
|-
| | (P8/2, P5/3)
| | half-8ve, third-5th
| | 16, 20*, 26, 30*
|-
| | (P8/2, P11/3)
| | half-8ve, third-11th
| | 19, 30
|-
| | (P8/3, P4/3)
| | third-everything
| | 15, 21, 30*
|-
! | quarters
! |
! |
|-
| | (P8/4, P5)
| | quarter-8ve
| | 12, 16, 20, 24*, 28
|-
| | (P8, P4/4)
| | quarter-4th
| | 18b*, 19, 20*, 28, 29, 30*
|-
| | (P8, P5/4)
| | quarter-5th
| | 13, 14*, 20, 21*, 27, 28*
|-
| | (P8, P11/4)
| | quarter-11th
| | 14, 17, 20, 28*, 31
|-
| | (P8, P12/4)
| | quarter-12th
| | 13b, 15*, 18b, 20*, 23, 25*, 28, 30*
|-
| | (P8/4, P4/2)
| | quarter-8ve, half-4th
| | 20, 24, 28
|-
| | (P8/2, M2/4)
| | half-8ve, quarter-tone
| | 18, 20, 22, 24, 26, 28
|-
| | (P8/2, P4/4)
| | half-8ve, quarter-4th
| | 18b, 20*, 28, 30*
|-
| | (P8/2, P5/4)
| | half-8ve, quarter-5th
| | 14, 20, 28*
|-
| | (P8/4, P4/3)
| | quarter-8ve, third-4th
| | 28
|-
| | (P8/4, P5/3)
| | quarter-8ve, third-5th
| | 16, 20
|-
| | (P8/4, P11/3)
| | quarter-8ve, third-11th
| |
|-
| | (P8/3, P4/4)
| | third-8ve, quarter-4th
| | 18b*, 30
|-
| | (P8/3, P5/4)
| | third-8ve, quarter-5th
| | 21, 27
|-
| | (P8/3, P11/4)
| | third-8ve, quarter-11th
| |
|-
| | (P8/3, P12/4)
| | third-8ve, quarter-12th
| | 15, 18b, 30*
|-
| | (P8/4, P4/4)
| | quarter-everything
| | 20, 28
|}
The edos that support the fewest pergens are prime-number edos like 11edo or 13edo. The most "pergen-friendly" edos tend to be ones in which the circle of 5ths doesn't reach every edostep. For example, 24edo supports all half-split pergens, since both P8 and P5 map to an even number of edosteps. 72edo supports all half-splits and all third-splits. 15, 21 and 36 edo support many but not all third-splits (not those with m = 2 or n = 2).

'''Notating a pergen tuned to an EDO'''

If both the pergen and the EDO are notated with ups and downs, what is the relationship between the two kinds of up? If the edo supports the pergen, fully or partially, then the pergen's up equals some multiple of the EDO's up, i.e. some number of edosteps. For third-4th in 22edo or 29edo, the pergen's up = 1 edostep. But in 37edo or 44edo, ^1 = 2 edosteps. For half-8ve in 12edo, ^1 = 0 edosteps, and the ups and downs in the score can simply be ignored. In fact, it seems every pergen in 5edo, 7edo and 12edo has ^1 = 0 edosteps. It's not yet known why.

When notating a piece in a specific pergen meant to be played in a specific EDO, either the pergen notation or the EDO notation can be used. For small or mid-sized EDOs, they're usually identical. If one has to choose, the pergen notation is generally preferred. It's less cluttered. Also, it's easier to mentally double every up and down in larger EDOs than it is to halve them in smaller EDOs.

Half-8ve in 22edo has P = vA4. But in 16edo, P = A4 + 2 edosteps, and ^1 = -2 edosteps. Negative edosteps means up is down, and should be avoided. The notation has tipped, and the period should be notated as ^A4, making ^1 = 2 edostep. Even better would be P = ^4, making ^1 = 1 edostep.

Half-5th has EU = vvA1, hence any EDO in which A1 = 4 edosteps will have ^1 = 2 edosteps. These "doubled EDOs" are 20, 27, 34, 41, 48, 55, etc. The "tripled EDOs" with A1 = 6 edosteps and ^1 = 3 edosteps are every 7th EDO from 30 to 72.

Half-4th has EU = vvm2. Doubled EDOs have m2 = 4 edosteps. These are 23, 28, 33, 38, 43, 48, 53, etc. Tripled EDOs are every 5th one from 47 to 72.

Third-4th has EU = v3A1. Doubled EDOs are the same ones as half-5th's tripled EDOs. Third-5th has EU = v3m2. Doubled EDOs are the same as half-4th's tripled EDOs.

The relationship between a pergen's up and an EDO's up when the pergen uses double-pair notation is complex. Consider half-everything, notated with ups/downs in the perchain and lifts/drops in the genchain. 10edo has ^1 = 1 edostep and /1 = 0 edosteps, thus one simply ignores lifts and drops. But for 24edo, ^1 = 0 edosteps and /1 = 1 edostep. One must ignore ups and downs and convert lifts/drops to edosteps.

'''Pergens Within An EDO'''

See the screenshots in the Supplemental Materials section for a complete list of which pergens are supported by 12, 15 and 19 edo. The list includes multiple pergens per period/generator combination. The next table shows only the simplest pergen for each combination. Combinations are excluded if the period and generator are not coprime, because the scale is contained in a smaller edo. For example, 15edo has no unsplit pergen, because those scales are contained in 5edo. Periods of 1 or 2 edosteps are excluded as too trivial, because the scale is the entire edo. Every step of the edo appears in the genchains, even if the chains are only one step long.

To find the pergen, find the edo, then find the row that corresponds to the period, then find the column that corresponds to the generator. It appears, but has yet to be formally proven, that the number of P/G combinations that N-edo supports is (N-1)/2 rounded down. If we include the trivial combination where P = 1 edostep, the number of combinations would be N/2 rounded up.

{| class="wikitable" style="text-align:center;"
|-
! | EDO
! | Period
! colspan="11" | Generator in edosteps
|-
! |
! | in edosteps
! | 1
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
|-
! | 5
! | 5 = P8
| | P4/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 6
! | 6 = P8
| | P4/2
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 7
! | 7 = P8
| | P4/3
| | P5/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 8
! | 8 = P8
| | P4/3
| | -
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/2
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 9
! | 9 = P8
| | P4/4
| | P4/2
| | -
| | P5
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/3
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 10
! | 10 = P8
| | P4/4
| | -
| | P5/2
| | -
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 5 = P8/2
| | P5
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 11
! | 11 = P8
| | P4/5
| | P5/3
| | P5/2
| | P11/4
| | P5
| |
| |
| |
| |
| |
| |
|-
! | 12
! | 12 = P8
| | P4/5
| | -
| | -
| | -
| | P5
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/2
| | P5
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/3
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/4
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 13b
! | 13 = P8
| | P4/6
| | P4/3
| | P4/2
| | P12/5
| | P12/4
| | P5
| |
| |
| |
| |
| |
|-
! | 14
! | 14 = P8
| | P4/6
| | -
| | P4/2
| | -
| | P11/4
| | -
| |
| |
| |
| |
| |
|-
! | "
! | 7 = P8/2
| | P5
| | P4/3
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 15
! | 15 = P8
| | P4/6
| | P4/3
| | -
| | P12/6
| | -
| | -
| | P11/3
| |
| |
| |
| |
|-
! | "
! | 5 = P8/3
| | P5
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/5
| | P4/3
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 16
! | 16 = P8
| | P4/7
| | -
| | P5/3
| | -
| | P12/5
| | -
| | P5
| |
| |
| |
| |
|-
! | "
! | 8 = P8/2
| | P5
| | -
| | P5/3
| | -
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/4
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 17
! | 17 = P8
| | P4/7
| | P5/5
| | P11/8
| | P11/6
| | P5/2
| | P11/4
| | P5
| | P11/3
| |
| |
| |
|-
! | 18b
! | 18 = P8
| | P4/8
| | -
| | -
| | -
| | P5/2
| | -
| | P12/4
| | -
| |
| |
| |
|-
! | "
! | 9 = P8/2
| | P5
| | P4/4
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/3
| | P5/2
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/6
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 19
! | 19 = P8
| | P4/8
| | P4/4
| | P11/9
| | P4/2
| | P12/6
| | P12/5
| | ccP5/7
| | P5
| | P11/3
| |
| |
|-
! | 20
! | 20 = P8
| | P4/8
| | -
| | P5/4
| | -
| | -
| | -
| | P11/4
| | -
| | c3P5/8
| |
| |
|-
! | "
! | 10 = P8/2
| | M2/4
| | -
| | P5/4
| | -
| | -
| |
| |
| |
| |
| |
| |
|-
! | "
! | 5 = P8/4
| | P4/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/5
| | P5/4
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 21
! | 21 = P8
| | P4/9
| | P5/6
| | -
| | P5/3
| | P11/6
| | -
| | -
| | c3P4/9
| | -
| | P11/3
| |
|-
! | "
! | 7 = P8/3
| | P5/2
| | P5
| | P4/3
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/7
| | P5/3
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 22
! | 22 = P8
| | P4/9
| | -
| | P4/3
| | -
| | P12/7
| | -
| | P12/5
| | -
| | P5
| | -
| |
|-
! | "
! | 11 = P8/2
| | M2/4
| | P5
| | P4/3
| | P12/5
| | P12/7
| |
| |
| |
| |
| |
| |
|-
! | 23
! | 23 = P8
| | P4/10
| | P4/5
| | P11/11
| | P12/9
| | P4/2
| | P12/6
| | ccP4/8
| | ccP4/7
| | P12/4
| | P5
| | P11/3
|-
! | 24
! | 24 = P8
| | P4/10
| | -
| | -
| | -
| | P4/2
| | -
| | P5/2
| | -
| | -
| | -
| | c4P5/10
|-
! | "
! | 12 = P8/2
| | M2/4
| | -
| | -
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
|-
! | "
! | 8 = P8/3
| | P5/2
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/4
| | P4/2
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/6
| | P4/2
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/8
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! |
! |
! | 1
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
|}
Larger edos can have very awkward pergens. For example, 31edo with G = 14/31 is (P8, c5P4/12). It's much simpler to think of the generator as the downfifth = 17\31, and the pergen as the pseudopergen (P8, v5).

'''EDO-pair names'''

Just as a pair of edos and a prime subgroup can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo & N'-edo, m = GCD (N,N'). The period P equals both (N/m)\N and (N'/m)\N'. For example, for 12edo and 16edo, m = 4, and the period is both 3\12 and 4\16.

For each edo, find the nearest '''edomapping''' (patent val) for the 2.3 subgroup. If the edo has a "b" wart, e.g. 13b-edo, use the second-nearest edomapping. Form a 2x2 matrix from these edomappings. Let d be the determinant of this matrix. If d = m, -m or 0, the generator is the 5th, and the pergen is simply (P8/m, P5).

For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.

If |d| ≠ m or 0, P5 is not the generator, and we must find the multigen. Take the ratio of the two edos N/N' and reduce it by m. In the scale tree ([http://tallkite.com/misc_files/Scale-Tree-Complete.pdf pdf] or [http://tallkite.com/misc_files/Scale-Tree-Complete.jpg jpeg]), let g/g' be the smallest ancestor of this ratio. The generator G maps to both g\N and g'\N'. The two edos come closest to coinciding here.

For example, for 7-edo and 17-edo, m = 1 but d = 2, so G ≠ P5. The smallest ancestor of 7/17 is 2/5. G maps to both 2\7 and 5\17, which are both neutral 3rds. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths. 2\7 = 343¢ and 5\17 = 353¢, and their difference is only 1\N", where N" = LCM (N, N'). This 10¢ difference is the least difference between any 7edo note and any 17edo note, except that the two neutral 6ths differ by the same 10¢, and of course some note pairs coincide exactly, such as 0\7 and 0\17.

Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found from the scale tree, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this new vector with either edomapping is zero. Treating this new vector as a monzo, since the dot product is zero, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for. If the comma is (a, b, c), then a·P8 + b·P5 + c·G = 0, and G = (b-a, -b) / c.

For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2).

To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 (Dicot aka Yoyo). 11/9 also works, it yields 243/242 (Mohajira aka Lulu).

If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a multi-EDO pergen.

If the 1st edo is 7edo, the pergen indicates the natural heptatonic notation for the 2nd edo, e.g. 12edo = unsplit, 15edo = third-4th and 17edo = half-5th.

The closer two edos are in the scale tree, the smaller the splitting fractions in the pergen they make. Examples:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 12-edo
! | 13b-edo
! | 14-edo
! | 15-edo
! | 16-edo
! | 17-edo
! | 18b-edo
! | 19-edo
! | 20-edo
|-
! | 13b-edo
| | (P8, P5/7)
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 14-edo
| | (P8/2, P5)
| | (P8, P4/6)
| |
| |
| |
| |
| |
| |
| |
|-
! | 15-edo
| | (P8/3, P5)
| | (P8, c5P5/12)
| | (P8, P4/6)
| |
| |
| |
| |
| |
| |
|-
! | 16-edo
| | (P8/4, P5)
| | (P8, P12/5)
| | (P8/2, P5)
| | (P8, P5/9)
| |
| |
| |
| |
| |
|-
! | 17-edo
| | (P8, P5)
| | (P8, ccP5/11)
| | (P8, P11/4)
| | (P8, P11/3)
| | (P8/2, P4/7)
| |
| |
| |
| |
|-
! | 18b-edo
| | (P8/6, P5)
| | (P8, P12/4)
| | (P8/2, P4/2)
| | (P8/3, P12/4)
| | (P8/2, P5)
| | (P8, P5/10)
| |
| |
| |
|-
! | 19-edo
| | (P8, P5)
| | (P8, P12/10)
| | (P8, P4/2)
| | (P8, P12/6)
| | (P8, P12/5)
| | (P8, P11/3)
| | (P8, P4/8)
| |
| |
|-
! | 20-edo
| | (P8/4, P5)
| | (P8, ccP4/16)
| | (P8/2, P5/4)
| | (P8/5, ^1)
| | (P8/4, P5/3)
| | (P8, P11/4)
| | (P8/2, P4/8)
| | (P8, P4/8)
| |
|-
! | 21-edo
| | (P8/3, P5)
| | (P8, c3P4/9)
| | (P8/7, ^1)
| | (P8/3, P4/3)
| | (P8, P5/3)
| | (P8, P11/6)
| | (P8/3, P5/2)
| | (P8, P11/3)
| | (P8, P5/12)
|-
! | 22-edo
| | (P8/2, P5)
| | (P8, c3P4/15)
| | (P8/2, P4/3)
| | (P8, P4/3)
| | (P8/2, P12/5)
| | (P8, P5)
| | (P8/2, P12/7)
| | (P8, P12/5)
| | (P8/2, M2/4)
|-
! | 23-edo
| | (P8, P4/5)
| | (P8, ccP4/8)
| | (P8, P4/2)
| | (P8, P12/12)
| | (P8, P5)
| | (P8, P12/9)
| | (P8, P12/4)
| | (P8, P12/6)
| | (P8, c5P5/16)
|-
! | 24-edo
| | (P8/12, ^1)
| | (P8, c6P4/14)
| | (P8/2, P4/2)
| | (P8/3, P4/2)
| | (P8/8, P5)
| | (P8, P5/2)
| | (P8/6, P4/2)
| | (P8, P4/2)
| | (P8/4, P4/2)
|}

A specific pergen can be converted to an edo pair by finding the range of its generator cents in the [[pergen#Further Discussion-Notating tunings with an arbitrary generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of edos 7, 10 and 17 defines (P8, P5/2).

==Array Keyboards (unfinished)==

Array keyboards have a 2-dimensional layout of keys, and are very appropriate for rank-2 tunings. A good layout can be found from the tuning's pergen. First find an edo N-edo that is compatible with the pergen, then arrange the keys in N columns to the 8ve, with each row usually containing the multigen interval. The unsplit pergen in 7 columns:

{| class="wikitable" style="text-align:center;"
|-
| | D#
| |
| |
| |
| |
| |
| |
| |
|-
| | D
| | E
| | F#
| | G#
| | A#
| |
| |
| |
|-
| | Db
| | Eb
| | F
| | G
| | A
| | B
| | C#
| | D#
|-
| |
| |
| |
| | Gb
| | Ab
| | Bb
| | C
| | D
|-
| |
| |
| |
| |
| |
| |
| |
| | Db
|}
Higher notes are at the top of each column. The rows would actually be angled so that the two D's are at the same horizontal level. The vertical steps are A1 and the horizontal steps are M2, and the keyboard is defined as 7(A1, M2).

The third-4th keyboard is 7(A1/3 = ^1 = vvA1, P4/3 = vM2 = ^^m2).

{| class="wikitable" style="text-align:center;"
|-
| | vD#
| | ^E
| | F#
| | vG#
| | ^A
| | B
| | vC#
| | ^D
|-
| | ^D
| | E
| | vF#
| | ^G
| | A
| | vB
| | ^C
| | D
|-
| | D
| | vE
| | ^F
| | G
| | vA
| | ^B
| | C
| | vD
|-
| | vD
| | ^Eb
| | F
| | vG
| | ^Ab
| | Bb
| | vC
| | ^Db
|}

''Hypothesis: Let the 5th's keyspan (i.e. column-span) be F. In order for the keyboard to have the pitches in order, the fifth must fall between the two Stern-Brocot ancestors of F\N (simplified if possible). For example, an 8-column keyboard has F = 5, the ancestors of 5\8 are 3\5 and 2\3, and the 5th must be between 720¢ and 800¢. Thus the most musically useful N values are 5, 7, 10, 12 and 14.''

(more to come)

==Supplemental materials==

===Notation guide PDF===

This PDF is a rank-2 notation guide that shows the full lattice for the first 32 pergens, up through the quarter-splits block. It also includes the single-split pergens from the fifth-split, sixth-split and seventh-split blocks. It includes alternate EUs for many pergens.

[http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf '''<big>TallKite.com/misc_files/notation guide for rank-2 pergens.pdf</big>''']

{| class="wikitable" style="text-align:center;"
|+Table of contents for the N'''otation Guide for Rank-2 Pergens''' (* indicates a true double)
|-
! colspan="3" |unsplit
! colspan="3" |quarter-splits
! colspan="3" |single-split fifth-splits
! colspan="3" |single-split seventh-splits
|-
!1
|(P8, P5)
|unsplit
!16
|(P8/4, P5)
|quarter-8ve
!33
|(P8/5, P5)
|fifth-8ve
!96
|(P8/7, P5)
|seventh-8ve
|-
! colspan="3" |half-splits
!17
|(P8, P4/4)
|quarter-4th
!34
|(P8, P4/5)
|fifth-4th
!97
|(P8, P4/7)
|seventh-4th
|-
!2
|(P8/2, P5)
|half-8ve
!18
|(P8, P5/4)
|quarter-5th
!35
|(P8, P5/5)
|fifth-5th
!98
|(P8, P5/7)
|seventh-5th
|-
!3
|(P8, P4/2)
|half-4th
!19
|(P8, P11/4)
|quarter-11th
!36
|(P8, P11/5)
|fifth-11th
!99
|(P8, P11/7)
|seventh-11th
|-
!4
|(P8, P5/2)
|half-5th
!20
|(P8, P12/4)
|quarter-12th
!37
|(P8, P12/5)
|fifth-12th
!100
|(P8, P12/7)
|seventh-12th
|-
!5
|(P8/2, P4/2) *
|half-everything *
!21
|(P8/4, P4/2) *
|quarter-8ve, half-4th *
!38
|(P8, ccP4/5)
|fifth-coco-4th
!101
|(P8, ccP4/7)
|seventh-coco-4th
|-
! colspan="3" |third-splits
!22
|(P8/2, M2/4)
|half-8ve, quarter-tone
! colspan="3" |single-split sixth-splits
!102
|(P8, ccP5/7)
|seventh-coco-5th
|-
!6
|(P8/3, P5)
|third-8ve
!23
|(P8/2, P4/4) *
|half-8ve, quarter-4th *
!64
|(P8/6, P5)
|sixth-8ve
!103
|(P8, c3P4/7)
|seventh-trico-4th
|-
!7
|(P8, P4/3)
|third-4th
!24
|(P8/2, P5/4) *
|half-8ve, quarter-5th *
!65
|(P8, P4/6)
|sixth-4th
| colspan="3" rowspan="9" |
|-
!8
|(P8, P5/3)
|third-5th
!25
|(P8/4, P4/3)
|quarter-8ve, third-4th
!66
|(P8, P5/6)
|sixth-5th
|-
!9
|(P8, P11/3)
|third-11th
!26
|(P8/4, P5/3)
|quarter-8ve, third-5th
!67
|(P8, P11/6)
|sixth-11th
|-
!10
|(P8/3, P4/2)
|third-8ve, half-4th
!27
|(P8/4, P11/3)
|quarter-8ve, third-11th
!68
|(P8, P12/6)
|sixth-12th
|-
!11
|(P8/3, P5/2)
|third-8ve, half-5th
!28
|(P8/3, P4/4)
|third-8ve, quarter-4th
!69
|(P8, ccP4/6)
|sixth-coco-4th
|-
!12
|(P8/2, P4/3)
|half-8ve, third-4th
!29
|(P8/3, P5/4)
|third-8ve, quarter-5th
!70
|(P8, ccP5/6)
|sixth-coco-5th
|-
!13
|(P8/2, P5/3)
|half-8ve, third-5th
!30
|(P8/3, P11/4)
|third-8ve, quarter-11th
| colspan="3" rowspan="3" |
|-
!14
|(P8/2, P11/3)
|half-8ve, third-11th
!31
|(P8/3, P12/4)
|third-8ve, quarter-12th
|-
!15
|(P8/3, P4/3) *
|third-everything *
!32
|(P8/4, P4/4) *
|quarter-everything *
|}Screenshots of the first 2 pages:

[[File:pergens_1.png|alt=pergens 1.png|704x948px|pergens 1.png]]

[[File:pergens_2.png|alt=pergens 2.png|704x760px|pergens 2.png]]

===PergenLister===

PergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonic unisons for each one. Alternate EUs are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair.

http://www.tallkite.com/apps/pergenLister.html (written in Jesusonic, runs inside Reaper or ReaJS)

The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. EUs of a 3rd or more are in red.

Screenshots of the first 69 pergens:

[[File:alt-pergenLister_1.png|alt=alt-pergenLister 1.png|800x427px|alt-pergenLister 1.png]]

[[File:alt-pergenLister_2.png|alt=alt-pergenLister 2.png|800x455px|alt-pergenLister 2.png]]

The first 29 pergens supported by 12edo:

[[File:alt-pergenLister_12edo.png|alt=alt-pergenLister 12edo.png|800x449px|alt-pergenLister 12edo.png]]

Some of the pergens supported by 15edo. A red asterisk means partial support.

[[File:alt-pergenLister_15edo.png|alt=alt-pergenLister 15edo.png|800x493px|alt-pergenLister 15edo.png]]

Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.

[[File:alt-pergenLister_19edo.png|alt=alt-pergenLister 19edo.png|800x459px|alt-pergenLister 19edo.png]]

Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:

If z > 0, then y is at least ceiling (x·(3/2 - z/2)) and at most floor (x·5/3)

If z < 0, then y is at least ceiling (x·3/2) and at most floor (x·(5/3 - z/2))

Next we loop through all combinations of x and z in such a way that larger values of x and z come last:

i = 1; loop (maxFraction,

<ul><li>j = 1; loop (i - 1,<ul><li>makeMapping (i, j); makeMapping (i, -j);</li><li>makeMapping (j, i); makeMapping (j, -i);</li><li>j += 1;</li></ul></li><li>);</li><li>makeMapping (i, i); makeMapping (i, -i);</li><li>i += 1;</li></ul>);

The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.

==Various proofs (unfinished)==

Although not yet rigorously proven, the two false-double tests have been empirically verified by pergenLister.

The interval P8/2 has a "ratio" of the square root of 2, which equals 21/2, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the '''pergen matrix''' [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.

Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise a, b, and n could all be reduced by GCD (a,b), and the multigen could be simplified. Since GCD (a, b) = 1 and -am/b is an integer, it follows that m must be a multiple of |b| as well.

Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|) = |b| · r. If r = 1, then GCD (m, n) = |b|, and vice versa, which is the proposed test for a false double.

If m = |b|, is the pergen explicitly false? Does splitting (a,b) into n generators also split P8 into m periods?

(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5

Because n is a multiple of b, n/b is an integer

M/b = (n/b)·M/n = (n/b)·G 
(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)

Let c and d be the bezout pair of a+b and b, with c·(a+b) + d·b = 1

Since the pergen is a double-split, m > 1, therefore |b| > 1, therefore c ≠ 0

c·(a+b)·P8 = c·b·((n/b)·G - P5) 
(1 - d·b)·P8 = c·b·((n/b)·G - P5) 
P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5) 
P8/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)

Therefore P8 is split into m periods 
Therefore if m = |b|, the pergen is explicitly false

Assume the pergen is a false double, and there's a comma C that splits both P8 and (a,b) appropriately. Can we prove r = 1? Let Q = the higher prime that C uses. Express P, G and C as monzos of the prime subgroup 2.3.Q, by expanding the 2x2 pergen matrix to a 3x3 matrix A:

P = (1/m, 0, 0) 
G = (a/n, b/n, 0) 
C = (u, v, w)

Here u, v and w are integers. If GCD (u, v, w) > 1, simplify C so that it = 1. The inverse of A expresses 2, 3 and Q in terms of P, G and C. If C is tempered out, the C column can be discarded, making the usual 3x2 period-generator mapping. However, if C is not tempered out, the inverse of A is a 3x3 period-generator-comma mapping, which is simply a change of basis. For example, 5-limit JI can be generated by 2/1, 3/2 and 81/80. Here is the inverse of A:

2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) 
3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C) 
Q = Q/1 = ((av-bu)m/wb, -vn/wb, 1/w) · (P, G, C)

Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. Every pergen except the unsplit one requires |w| > 1, so the last column almost always has a fraction. To avoid fractions in the first two columns, A must be unimodular '''''[I think, not sure]''''', and we have wb/mn = ±1, and w = ±mn/b. Substituting for w, we have:

2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) 
3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C) 
Q = Q/1 = (±(av-bu)/n, ±(-v)/m, ±b/mn) · (P, G, C)

For v/m to be an integer, v must equal i·m for some integer i. Likewise, av-bu must equal j·n for some integer j. Thus bu = av - jn = iam - jn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb. Substituting, we get bu = iaprb - jqrb, and u = r(iap - jq). Furthermore, v = im = iprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r > 1, this contradicts the requirement that GCD (u, v, w) = 1, therefore r must be 1, and GCD (m, n) = |b|, and all false doubles pass the false-double test.

Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q?

Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b.

a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)

Thus a·P8 splits into b parts, and since m = |b|, a·P8 splits into m parts. Proceed as before with a bezout pair to find the monzo for P8/m.

Next, assume the pergen isn't explicitly false. The unreduced form is (P8/m, (n - am, -bm) / mn). Substituting in m = pb and n = qb, with p and q coprime, we get (P8/m, (q - ap, -pb) / pqb).

Assume the pergen is a true double, and r > 1. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.

Given a pergen (P8/m, (a,b)/n), how many parts is an arbitrary interval (a',b') split into?

(a',b') = (a'·b, b'·b) / b = (a'·b - a·b', 0) / b + (a·b', b'·b) / b = (a'·b - a·b')·P8 / b + b'·(a,b) / b = (a'·b - a·b')·(m/b)·P + b'·(n/b)·G

Therefore (a',b') is split into GCD (a'·b - a·b')·(m/b), b'·(n/b)) parts.

If m = 1, then b = ±1, and we have GCD (a' ± a·b', b'·n)

If n = 1, then a = -1 and b = 1, and we have GCD (a'·m + b'·m, b') = GCD (a'·m, b')

If m = 1 and n = 1, we have GCD (a', b') = the naturally occurring split.

If m = n (nth-everything), we have n · GCD (a', b')

The multigen and the arbitrary interval can be expressed as gedras:

(a,b) = [k,s] = (-11k+19s, 7k-12s)

(a',b') = [k',s'] = (-11k'+19s', 7k'-12s')

a'·b - a·b' works out to be k·s' - k'·s, and we have GCD ((k·s' - k'·s)·m/b, b'·n/b)

If s is a multiple of n (happens when EU is an A1) and s' is a multiple of n, let s = x·n and s' = y·n

GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')

Thus every such interval is split, e.g. the half-5th pergen splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. This is an easy way to determine if an interval is split or not by the pergen.

To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts

if r > 1, it's a true double

a·P8 = (a,0) = (a,b) - (0,b) = M - b·P12

M = n·G = qrb·G

a·P8 = qrb·G - b·P12 = b·(qr·G - P12)

Let c and d be the bezout pair of a and b, with c·a + d·b = 1

If |b| = 1, let c = 1 and d = ±a, to avoid c = 0

ca·P8 = cb·(qr·G - P12)

(1 - d·b)·P8 = c·b·(qr·G - P12)

P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) = b · (d,-c) + bcqr·G

== Glossary ==
to find the definition, use control-F to search for the first (bolded) occurrence of the word on this page.

pergen 
split 
multigen 
ups and downs (the ^ and v symbols) 
higher prime (any prime > 3) 
color depth 
dependent/independent 
square mapping 
lifts and drops (the / and \ symbols) 
enharmonic unison, EU 
uninflected 
genchain 
perchain 
compound (increased by an octave) 
single-split, double-split 
single-pair, double-pair (number of new accidentals in the notation) 
true double, false double 
explicitly false 
unreduced 
alternate vs. equivalent (generator or period) 
mapping comma 
keyspan 
stepspan 
gedra 
count 
mid 
edomapping 
upspan 
liftspan

chain number 
single-chain 
multi-chain 
arrow comma

==Miscellaneous Notes==

=== Combining pergens ===
Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). If adding a comma to a temperament doesn't change the pergen, it's a strong extension, otherwise it's a weak extension.

General rules for combining pergens:

<ul><li>(P8/m, M/n) + (P8, P5) = (P8/m, M/n)</li>
<li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li>
<li>(P8/m, P5) + (P8/m', P5) = (P8/m", P5), where m" = LCM (m,m')</li>
<li>(P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n')</li></ul>

However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.

If a false double pergen can be broken down into two simpler ones, that may help with finding a double-pair notation with an EU of a 2nd or less. For example, sixth-4th's single pair notation has an EU of a 4th. But since sixth-4th is half-4th plus third-4th, and those two have a good EU, sixth-4th can be notated with one pair from half-4th and another from third-4th.

=== Expanding gedras ===
Gedras can be expanded to 5-limit or higher by including another keyspan that is compatible with 7 and 12, such as 9 or 16. But a more useful approach is for the third number to be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]:

k = 12a + 19b + 28c + 34d 
s = 7a + 11b + 14c + 20d 
g = -c 
r = -d

a = -11k + 19s - 4g + 6r 
b = 7k - 12s + 4g - 2r 
c = -g 
d = -r

Gedras can also be expanded by adding an entry for ups/downs. The entry shows the '''upspan''', which is the number of ups the interval has. Downed intervals have a negative upspan. An entry for '''liftspan''' can also be added. For example, vM3 = [4,2,-1], and ^^\4 = [5,3,2,-1].

=== Height of a pergen ===
The LCM of the pergen's two splitting fractions could be called the '''height''' of the pergen. For example, (P8, P5) has height 1, and (P8/2, M2/4) has height 4. In single-pair notation, the EU's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.

=== Generalizing the pergen ===
See [[User:AthiTrydhen/Abstract pergens]]

=== Credits ===
Pergens were discovered by [[KiteGiedraitis|Kite Giedraitis]] in 2017, and developed with the help of [[Praveen Venkataramana]].

== Addenda (late 2023) ==
=== New terminology===
All temperaments have a '''chain number''', which is the number of fifthchains in the temperament's lattice. Any (P8, P5) temperament has a chain number of 1, and is '''single-chain'''. All other pergens are '''multi-chain'''. For example, Porcupine/Triyoti has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Saguguti has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m * n / f, where M is the multigen and f is the absolute value of M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.

===The EU(s) define the pergen===
The pergen can be derived directly from the EU(s). Thus the EU(s) define both the pergen and the notation. An EU can be thought of as a comma in the 2.3.^ subgroup. One derives a mapping from this comma as one would for any 3-prime comma, and one derives the pergen by inverting the first 2 columns of the mapping.

For example, if the EU is vvd2, the 2.3.^ monzo is [19 -12 -2]. There are many possible mappings, but only one that gives a canonical pergen: [(2 2 7) (0 1 -6)]. Discarding the last column and inverting gives us [(1/2 0) (-1 1)] = (P8/2, P5). Another example: vvA1 = [-11 7 -2]. The mapping [(1 1 -2) (0 2 7)] reduces to [(1 1) (0 2)], which inverts to [(1 0) (-1/2 1/2)] = (P8, P5/2).

One can explore the universe of possible EUs, and thus possible pergen notations, more easily if using the gedra format, expressing the EU as a combination of A1's, d2's and arrows. Thus vvA1 = [1 0 -2], v3m2 = [1 1 -3], etc. Unlike a conventional 2.3.^ monzo, the first two numbers in a A1.d2.^1 monzo are fairly small, and the second number is never negative (since it's an EU). And we can require that the first two numbers be coprime (see the next section). All this facilitates one's search.

===Simplifying a "squared" EU===
Consider an uninflected EU of AA1. AA1 is "squared" in the sense that AA1 = A1 + A1, thus both numbers in its ratio are square numbers. If it had an even upspan (the number of ups), it would obviously be an invalid EU, for if v4AA1 = 0¢, then so does vvA1, and v4AA1 could be replaced with vvA1. So the upspan must be odd.

Consider an EU of v3AA1. The pergen is (P8, P4/3). Here are the [[twin squares]].

<math>
\begin{array} {rrr}
\text{P8} \\
\text{^m2} \\
\text{vvvAA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & {\color {Green}0} \\
8 & -5 & {\color {Green}1} \\
\hline
{\color {Red}-22} & {\color {Red}14} & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & {\color {Red}2} \\
0 & -3 & {\color {Red}-14} \\
\hline
{\color {Green}0} & {\color {Green}-1} & -5 \\
\end{array} \right]
</math>

The two red numbers in the lower left are both even, because AA1 is a squared ratio. As a result, the two red numbers in the upper right must both be even to ensure their rows' dot products with the EU are zero, which is an even number. If we halve all the red numbers and double all the green numbers, all the various row dot products will be unchanged, and the twin squares will remain valid:

<math>
\begin{array} {rrr}
\text{P8} \\
\text{^^m2} \\
\text{vvvA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & {\color {Green}0} \\
8 & -5 & {\color {Green}2} \\
\hline
{\color {Red}-11} & {\color {Red}7} & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & {\color {Red}1} \\
0 & -3 & {\color {Red}-7} \\
\hline
{\color {Green}0} & {\color {Green}-2} & -5 \\
\end{array} \right]
</math>

But while the EU has become simpler, the generator has become more complex in that it is dup not up. This is remedied by adding the EU to it. Changes are in red:

<math>
\begin{array} {rrr}
\text{P8} \\
\text{vM2} \\
\text{vvvA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & 0 \\
{\color {Red}-3} & {\color {Red}2} & {\color {Red}-1} \\
\hline
-11 & 7 & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & 1 \\
0 & -3 & -7 \\
\hline
{\color {Red}0} & {\color {Red}1} & {\color {Red}2} \\
\end{array} \right]
</math>

Following this procedure, it's always possible to simplify a squared (or cubed, etc.) EU.

===Arrow commas===
The '''[[arrow]] comma''' is the ratio that equals the up [[arrow]]. This term overlaps a lot with the term mapping comma, but it isn't quite identical to it. For 5-limit temperaments the arrow comma is almost always 81/80, and for 7-limit it's almost always 64/63. But other commas can occur.

Consider Triyoti/Porcupine which is (P8, P4/3). The vanishing comma or '''VC''' is 250/243, which has a 2.3.5 monzo of [2 -5 3]. The EU is v3A1, which has a 2.3.^ monzo of [-11 7 -3]. The arrow comma or '''AC''' equals an up, therefore it vanishes when downed. The downed AC (or '''vAC''') can be expressed as a 2.3.5.^ monzo. For Triyoti/Porcupine with an EU of v3A1, the vAC is v(81/80) or [-4 4 -1 -1].

===The three commas ===
Thus when we consider a single-comma temperament along with its notation, there are three commas of interest, the VC, the vAC and the EU. In a 5-limit rank-2 temperament, they can all be expressed as 2.3.5.^ monzos.

Any two of these 3 commas determines the third comma. Thus we can approach temperaments and notations from 6 different angles. We can start with any one of these three commas and give it a specific monzo. Then we can choose one of the other two commas, experiment with a variety of specific monzos and examine the results. Let's start with specifying the VC, experimenting with the vAC and seeing what the EU turns out to be.

The EU always equals the VC (possibly inverted) plus or minus some number of vAC's. That number is whatever is needed to eliminate the higher prime. The VC must be inverted if the resulting EU would otherwise have a negative stepspan, or is a diminished unison.

In our Triyoti example, 250/243 plus 3 downed syntonic commas = v3A1. As 2.3.5.^ monzos, we have [1 -5 3 0] + 3·[-4 4 -1 -1] = [-11 7 0 -3]. Note the zeros. The VC always has a zero arrow-count and the EU always has a zero prime-5-count.

Visualizing the three commas in the lattice, one starts at the VC and heads towards the row of 3-limit intervals via the vAC. Wherever one lands is the uninflected EU. One can try other AC's besides 81/80. The AC's prime-5-count must be ±1, so [[135/128|Layobi]] (135/128) or [[Schisma|Layo]] (the schisma) are possibilities. But either of these would lead to a very remote EU (v3AA1 and v3d44 respectively), making a very awkward notation.

Next let's specify the AC, experiment with the VC and see what the EU turns out to be. For 5-limit temperaments, one can require that the AC always be 81/80, and derive the EU (and thus the notation) from the VC. For example, Saguguti/Diaschismic has VC = [11 -4 -2 0]. Subtracting two vAC's makes [19 -12 0 2] = ^^d2. This is in fact the recommended EU for (P8/2, P5).

More examples: Laquinyoti/Magic is (P8, P12/5) and has VC = [-10 -1 5 0]. Adding five vAC's makes [-30 19 0 -5]. This appears to be a AA7 but is actually a negative 2nd. We invert to get [30 -19 0 5] = ^5dd2. Guguti/Bug has VC = 27/25 = [0 3 -2 0]. Subtracting two vAC's makes [8 -5 0 2] = ^^m2. Again, this is the recommended EU.

Let's try a 7-limit temperament with the obvious vAC of 64/63 = [6 -2 -1 -1]. Zozoti/Semaphore has VC = 49/48 = [-4 -1 2 0]. Adding two vAC's makes [8 -5 0 -2] = vvm2.{{todo|complete section|comment=apply to multi-comma temperaments|inline=1}}

== Addenda (late 2024) ==

=== Chord names ===
When naming chords, it's very convenient to have the freedom to rename an aug 4th as a dim 5th, or a minor 10th as an aug ninth. Thus for some pergens, an extra pair of accidentals is used. Some examples:

* [[Chords of meantone]] (P8, P5) (^1 = -d2 = pythagorean comma)
* [[Chords of diaschismic]] (P8/2, P5)
* [[Chords of hemififths]] (P8, P5/2) (/1 = vm2 = ~81/80 = ~64/63)
* [[Chords of porcupine]] (P8, P4/3)
* [[Chords of magic]] (P8, P12/5) (/1 = ^^d2)

=== Frequency of imperfect pergens ===
Imperfect pergens occur when there are multiple genchains (i.e. the octave is split), and the fifth is on a different genchain than the tonic, and also on a different perchain. How often do they occur? In order to answer that, we need to survey all pergens in order. But the question of how to do that depends on how they are sorted. The pergenLister app sorts them by the size of the larger denominator. Using this order, pergenLister finds about 4% of all pergens are imperfect. But they can also be sorted by their canonical mappings [(a b) (0 c)]. If sorted by a (octave fraction), and then by |c| (perfect multigen's fraction), more complex pergens appear sooner, and the percentage rises to about 25%.

This table lists all pergens with an unsplit octave up to the fifth-splits. In each column, the pergens are sorted by the size of the generator. The generator is listed followed by a, b and c from its mapping. All pergens with an unsplit octave are perfect.
{| class="wikitable"
|+Pergens of the form (P8, x), showing generator and mapping (a = 1)
!unsplit
!half-splits
!third-splits
!quarter-splits
!fifth-splits
!sixth-splits
|-
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (1 1 1)
|P4/2 (1 2 -2)
|P4/3 (1 2 -3)
|P4/4 (1 2 -4)
|P4/5 (1 2 -5)
|P4/6 (1 2 -6)
|-
|
|P5/2 (1 1 2)
|P5/3 (1 1 3)
|P5/4 (1 1 4)
|P5/5 (1 1 5)
|P5/6 (1 1 6)
|-
|
|
|P11/3 (1 3 -3)
|P11/4 (1 3 -4)
|P11/5 (1 3 -5)
|P11/6 (1 3 -6)
|-
|
|
|
|P12/4 (1 0 4)
|P12/5 (1 0 5)
|P12/6 (1 0 6)
|-
|
|
|
|
|ccP4/5 (1 4 -5)
|ccP4/6 (1 4 -6)
|-
|
|
|
|
|
|ccP5/6 (1 -1 6)
|}
Of all the half-octave pergens, half of every other column (i.e. 25%) are imperfect. Imperfect pergens occur whenever b is not a multiple of a.
{| class="wikitable"
|+Pergens of the form (P8/2, x), showing generator and mapping (a = 2)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (2 2 1)
|'''M2/4 (2 3 2)'''
|P4/3 (2 4 -3)
|'''M2/8 (2 3 4)'''
|P4/5 (2 4 -5)
|'''M2/12 (2 3 6)'''
|-
|
|P4/2 (2 4 -2)
|P5/3 (2 2 3)
|P4/4 (2 4 -4)
|P5/5 (2 2 5)
|P4/6 (2 4 -6)
|-
|
|
|P11/3 (2 6 -3)
|P5/4 (2 2 4)
|P11/5 (2 6 -5)
|P5/6 (2 2 6)
|-
|
|
|
|'''cm7/8 (2 5 -4)'''
|P12/5 (2 0 5)
|P11/6 (2 6 -6)
|-
|
|
|
|
|ccP4/5 (2 8 -5)
|'''cm7/12 (2 5 -6)'''
|-
|
|
|
|
|
|'''cM9/12 (2 1 6)'''
|}
Note that some of these pergens, when put in mingen form, become imperfect. For example, (P8/2, P11/3) becomes (P8/2, M2/6). Also note that for many of these pergens, the generators are comma-sized, and MOS scales will either be very "hard" (L/s very large) or else will contain very many notes per octave. For example, to bring the L/s ratio down to about 5, (P8/2, M2/4) needs a 16 note scale, and (P8/2, P11/3) needs a 28 note scale!

Of all the third-octave pergens, two-thirds of every third column (2/9 or 22%) are imperfect:
{| class="wikitable"
|+Pergens of the form (P8/3, x), showing generator and mapping (a = 3)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (3 3 1)
|P4/2 (3 6 -2)
|'''m3/9 (3 5 -3)'''
|P4/4 (3 6 -4)
|P4/5 (3 6 -5)
|'''m3/18 (3 5 -6)'''
|-
|
|P5/2 (3 3 2)
|'''M6/9 (3 4 3)'''
|P5/4 (3 3 4)
|P5/5 (3 3 5)
|'''M6/18 (3 4 6)'''
|-
|
|
|P4/3 (3 6 -3)
|P11/4 (3 9 -4)
|P11/5 (3 9 -5)
|P4/6 (3 6 -6)
|-
|
|
|
|P12/4 (3 0 4)
|P12/5 (5 0 5)
|P5/6 (3 3 6)
|-
|
|
|
|
|ccP4/5 (3 12 -5)
|'''ccm3/18 (3 7 -6)'''
|-
|
|
|
|
|
|'''ccM6/18 (3 2 6)'''
|}
Of all the quarter-octave pergens, imperfection occurs in half of every 4th column and 3/4 of every 4th column (5/16 or 31.25%).
{| class="wikitable"
|+Pergens of the form (P8/4, x), showing generator and mapping (a = 4)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
!c = ±7
!c = ±8
|-
|P5 (4 4 1)
|'''m6/8 (4 7 -2)'''
|P4/3 (4 8 -3)
|'''M2/8 (4 6 4)'''
|P4/5 (4 8 -5)
|P4/6 (4 8 -6)
|P4/7 (4 8 -7)
|'''M2/16 (4 6 8)'''
|-
|
|P4/2 (4 8 -2)
|P5/3 (4 4 3)
|'''m6/16 (4 7 -4)'''
|P5/5 (4 4 5)
|P5/6 (4 4 6)
|P5/7 (4 4 7)
|'''m6/32 (4 7 -8)'''
|-
|
|
|P11/3 (4 12 -3)
|'''M10/16 (4 5 4)'''
|P11/5 (4 12 -5)
|P11/6 (4 12 -6)
|P11/7 (4 12 -7)
|'''M10/32 (4 5 8)'''
|-
|
|
|
|P4/4 (4 8 -4)
|P12/5 (4 0 5)
|'''m6/24 (4 7 -6)'''
|P12/7 (4 0 7)
|P4/8 (4 8 -8)
|-
|
|
|
|
|ccP4/5 (4 16 -5)
|'''M10/24 (4 5 6)'''
|ccP4/7 (4 16 -7)
|P5/8 (4 4 8)
|-
|
|
|
|
|
|'''ccm6/24 (4 9 -6)'''
|ccP5/7 (4 -4 7)
|'''ccm6/32 (4 9 -8)'''
|-
|
|
|
|
|
|
|c3P4/7 (4 20 -7)
|'''cm7/16 (4 10 -8)'''
|-
|
|
|
|
|
|
|
|'''c3M3/32 (4 3 8)'''
|}
Percentage of imperfect pergens in each category:
{| class="wikitable"
|+
!(P8, x)
!(P8/2, x)
!(P8/3, x)
!(P8/4, x)
!(P8/5, x)
!(P8/6, x)
!(P8/7, x)
|-
|none
|1/4
|2/9
|5/16
|4/25
|5/12
|6/49
|-
|0%
|25%
|22.22%
|31.25%
|16%
|41.67%
|12.24%
|}

== Addenda (Spring 2026) ==

As one stacks generators and octave-reduces, at some point one overshoots or undershoots the octave by an interval of about a quartertone or less. This small interval is the pergen's initial comma. For example, (P8, P5)'s initial comma is the pythagorean comma, its next comma is Mercator's comma, etc. Each comma has a certain genspan, here 12 and 53. The genspan of the initial comma limits the size of a scale one can construct, assuming one wants to avoid overly-small steps. Thus one can have a pythagorean scale of up to 12 notes, but a 13-note scale will have a very small step. Note that the genspan gives the maximum notes per period, not per octave. Assuming one also wants to avoid extreme L/s step ratios also limits the maximum notes per octave.

The table below lists the initial comma of various pergens. "±" indicates a tippy pergen. "c" is the difference between the fifth and 7\12. "abs(6c)" means the absolute value of 6c. The dim 2nd is a pythagorean comma.
{| class="wikitable sortable"
|+Initial comma of each pergen
!#
!pergen
!interval
!cents
!genspan
!notes per octave
|-
!1
!(P8, P5)
|±d2
|abs(12c)
|±12G
|12
|-
!2
!(P8/2, P5)
|±d2/2
|abs(6c)
|±6G
|12
|-
!3
!(P8, P4/2)
|m2/2
|50¢ - 2.5c
|5G
|5
|-
!4
!(P8, P5/2)
|A1/2
|50¢ + 3.5c
|7G
|7
|-
!5
!(P8/2, P4/2)
| colspan="2" | ''same as #3 (P8, P4/2)''
|5G
|10
|-
!6
!(P8/3, P5)
|±d2/3
|abs(4c)
|±4G
|12
|-
!7
!(P8, P4/3)
|A1/3
|33.3¢ + 2.33c
| -7G
|7
|-
!8
!(P8, P5/3)
|m2/3
|33.3¢ - 1.67c
| -5G
|5
|-
!9
!(P8, P11/3)
|M2/3
|66.7¢ + 0.67c
|2G
|2 (or >= 14)
|-
!10
!(P8/3, P4/2)
|A2/6
|50¢ + 1.5c
|3G
|9
|-
!11
!(P8/3, P5/2)
|m3/6
|50¢ - 0.5c
|1G
|3
|-
!12
!(P8/2, P4/3)
| colspan="2" | ''same as #7 (P8, P4/3)''
| -7G
|14
|-
!13
!(P8/2, P5/3)
| colspan="2" | ''same as #8 (P8, P5/3)''
| -5G
|10
|-
!14
!(P8/2, P11/3)
|M2/6
|33.3¢ + 0.33c
|1G
|2
|-
!15
!(P8/3, P4/3)
| colspan="2" | ''same as #7 (P8, P4/3)''
| -7G
|21
|-
!16
!(P8/4, P5)
|±d2/4
|abs(3c)
|±3G
|12
|-
!17
!(P8, P4/4)
| colspan="2" | ''same as #3 (P8, P4/2)''
|10G
|10
|-
!18
!(P8, P5/4)
|A1/4
|25¢ + 1.75c
|7G
|7
|-
!19
!(P8, P11/4)
|dd3/4
|25¢ - 4.25c
| -17G
|17
|-
!20
!(P8, P12/4)
|m2/4
|25¢ - 1.25c
| -5G
|5
|-
!21
!(P8/4, P4/2)
|M2/4
|50¢ + c/2
|G
|4
|-
!22
!(P8/2, M2/4)
|M2/4
|50¢ + c/2
|G
|2
|-
!23
!(P8/2, P4/4)
|m2/4
|25¢ - 1.25c
|5G
|10
|-
!24
!(P8/2, P5/4)
| colspan="2" |''same as #18 (P8, P5/4)''
|7G
|14
|-
!25
!(P8/4, P4/3)
|d4/12
|33.3¢ - 0.67c
|2G
|8
|}
The initial comma of #9 (P8, P11/3) is about 67¢, which is not too small to be a scale step. But if there are more than 2 notes per 8ve, the L/s ratio becomes enormous. The ratio only becomes reasonable (roughly 3) when there are at least 14 notes per octave.

Note the initial comma is often equivalent to the uninflected EU divided by the height. For example, (P8, P4/2) has comma m2/2 and EU vvm2. In other words, the up-arrow is often the initial comma.

This suggests a new algorithm for finding a good EU for a pergen. Search the cents table (in the Notation Guide For rank-2 Pergens pdf, the first table of each pergen) for a small step. The search can be easily done by computer. Then derive the EU from that small step.

For example, pergen #25 (P8/4, P4/3) has a 33¢ step at 2G - P. Thus ^1 = 2G - P. Multiplying 2G by 3 gives us unsplit 4ths, and multiplying P by 4 gives us unsplit octaves. Thus we must multiply the up-arrow by 12 to get an unsplit 3-limit interval. 12 ups = 24G - 12P = 8P4 - 3P8 = dim 4th. Thus the EU is v12d4, and ^12C = Fb. But the pergenLister program lists the single-pair notation for this pergen as having an EU of ^12d94. Thus the pergenLister algorithm missed a much simpler EU, and hence a much simpler notation.

Not all initial commas imply a valid notation. For pergen #66 (P8, P5/6), the initial comma is P - 10G = m2/3 = 33.3¢ - 1.67c. The implied EU is v3m2. But this notation is incomplete because it only notates 3 of the 6 fifthchains. There must be 6 arrows in the EU, not 3. Or else there must be a second EU with 2 arrows. The next comma on the genchain is 21G - 2P = A1/2 = 50¢ - 3.5c, which implies a double-pair notation with EU' = \\A1. This is the notation found by pergenLister.

True doubles require double-pair notation and thus require finding two commas.

[[Category:Regular temperament theory]]
[[Category:Notation]]
[[Category:Pages with proofs]]

270edo

2026-05-10T00:58:04Z

TallKite: /* Ups and downs notation */ added the spoken terms

{{Infobox ET}}
{{ED intro}} 270edo's step size is called a '''tredek''' when used as an [[interval size unit]].

== Theory ==
270edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] and almost [[Consistency #Consistency to distance d|consistent to distance 2]] in it, missing [[15/13]] and [[26/15]] as they have 25.8% error ([[tempering out]] [[676/675]]). It is the 11th [[zeta gap edo]], the 13th [[zeta integral edo]], the 23rd [[zeta peak edo]], and the 18th [[zeta peak integer edo]], making it a [[strict zeta edo]].

In the [[5-limit]] it tempers out the [[ennealimma]], {{monzo| 1 -27 18 }}, the [[vulture comma]], {{monzo| 24 -21 4 }}, and the [[vishnuzma]], {{monzo| 23 6 -14 }}.

In the [[7-limit]] it tempers out the [[2401/2400|breedsma]] (2401/2400), the [[4375/4374|ragisma]] (4375/4374), and by extension the [[wizma]] (420175/419904), and the [[landscape comma]] (250047/250000) so that it [[support]]s [[ennealimmal]] temperament. It also tempers out the [[quasiorwellisma]] (29360128/29296875) and the [[garischisma]] (33554432/33480783).

In the [[11-limit]], it tempers out the lehmerisma ([[3025/3024]]), the vishdel comma ([[5632/5625]]), the kalisma ([[9801/9800]]), the [[symbiotic comma]] (19712/19683), the [[nexus comma]] (1771561/1769472), and the [[quartisma]] (117440512/117406179). Notably, it is consistent to distance 3 in the [[11-odd-limit]], and almost to distance 4 ((11/10)4 and (20/11)4 are a hair off, 50.4%).

Finally, in the [[13-limit]] it is slightly worse but still excellent. It tempers out [[676/675]], [[1001/1000]], [[1716/1715]], and [[2080/2079]], making it an [[The Archipelago|archipelago]] tuning, and the [[optimal patent val]] for some of the archipelago temperaments such as [[hemiennealimmal]], [[vulture]], [[eagle]], and [[avicenna (temperament)|avicenna]].

The excellent tuning accuracy does not bar it from the utility of [[essentially tempered chord]]s, including [[sinbadmic chords]] in the 13-odd-limit, and [[island chords]] in the 15-odd-limit.

Beyond the 13-limit, the approximated [[17/1|harmonic 17]] is more than 1/3-edostep, but the [[19/1|harmonic 19]] is very accurately tuned. [[17/13]] and its [[octave complement]] [[26/17]] are the only inconsistently approximated [[21-odd-limit]] intervals, each barely missing the mark (50.4% relative error). The [[23/1|harmonic 23]] is more than 1/3-edostep flat, which incurs more inconsistencies in the next odd limits yet makes 270edo viable but tricky for the full [[23-limit]]. It tempers out [[715/714]], [[936/935]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], [[2058/2057]], and [[2431/2430]] in the [[17-limit]]; [[1216/1215]], [[1331/1330]], [[1521/1520]], [[1540/1539]], and [[1729/1728]] in the [[19-limit]]. If the full 23-limit is desired, then [[460/459]], [[529/528]], [[736/735]], [[897/896]], [[1288/1287]], 1311/1309, and 1771/1768 are further tempered out.

The harmonics [[29/1|29]] and [[31/1|31]] are also more than 1/3-edostep sharp, but not as sharp as the 17 to incur inconsistency ([[29/26]] and [[31/26]] are critically sharp but still consistent). This makes 270edo consistent in the no-17/13 no-23 [[35-odd-limit]]. Notably, it tempers out [[784/783]], [[900/899]], and [[1024/1023]], while inflating [[841/840]] and [[961/960]].

On top of this, its step size is small enough as to arguably give a good enough approximation for any relatively simple JI consonance (beyond the 13-limit on which it is spot on), as the maximum error (assuming consistency) is only 2.{{overline|2}}{{c}}, yet having a step size that ''can'' be [[just-noticeable difference|discernible]].

If, however, you want a edo for more rounded, consistent very high-limit use, the obvious alternative choice is [[311edo]], which is in many ways dual to 270edo as it emphasizes consistency and accuracy in very high-prime-limit and high-odd-limit situations at the expense of lower ones, and is a [[prime edo]] as opposed to a very composite one. While 270edo approximates the first 16 harmonics with astounding accuracy, 311edo approximates the first 42 but not as accurately – strongly favouring the approximation of as many harmonics as possible.

=== Prime harmonics ===
{{Harmonics in equal|270|prec=3|intervals=prime|columns=11}}
{{Harmonics in equal|270|prec=3|intervals=prime|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 270edo (continued)}}

=== Subsets and supersets ===
270 is a very composite number. The prime factorization is {{nowrap|270 {{=}} 2 × 33 × 5}}, with divisors {{EDOs| 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, and 135 }}. This means that 270edo can be conceptualised as the superset of, for example, [[10edo]] and [[27edo]], which are both interesting and somewhat peculiar in their own right.

[[540edo]], which divides the edostep in two, and [[810edo]], which divides the edostep in three, provide good correction for harmonics 17, 23, and beyond.

== Intervals ==
As 270edo is a large edo, its intervals can be found on a separate page: [[Table of 270edo intervals]].

== Notation ==
=== Ups and downs notation ===
270edo can be notated using [[Kite's ups and downs notation|ups and downs]] with Stein-Zimmerman quarter-tone accidentals representing half-sharps and half-flats. These can be spoken as ''sha'' and ''fla''. For example, the note 12\270 above C is C downsha, and the note 39\270 above C is C shasharp.
{{Ups and downs sharpness|270|true}}

=== Sagittal notation ===
The [[Sagittal notation]] for 270edo uses alterations of the Promethian set. Since the apotome can be split in two, a SZ half-sharp and a half-flat may be used.
{| class="wikitable center-all" data-darkreader-inline-color=""
! colspan="2" |+ edosteps
|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
|11
|12
|13
|14
|15
|16
|17
|18
|19
|20
|21
|22
|23
|24
|25
|26
|-
! rowspan="3" |Symbol
!SZ
| rowspan="3" |<big>{{sagittal||(}}</big>
| rowspan="3" |<big>{{sagittal|)|(}}</big>
| rowspan="3" |<big>{{sagittal|)~|}}</big>
| rowspan="3" |<big>{{Sagittal|~|(}}</big>
| rowspan="3" |<big>{{Sagittal|/|}}</big>
| rowspan="3" |<big>{{Sagittal||)}}</big>
| rowspan="3" |<big>{{sagittal||\}}</big>
| rowspan="3" |<big>{{sagittal|(|}}</big>
| rowspan="3" |<big>{{sagittal|(|(}}</big>
| rowspan="3" |<big>{{sagittal|//|}}</big>
| rowspan="3" |<big>{{Sagittal|/|)}}</big>
| rowspan="3" |<big>{{Sagittal|/|\}}</big>
|<big>{{Sagittal|t}}</big>
|{{Sagittal||(}}{{sagittal|t}}
|{{Sagittal|)|(}}{{sagittal|t}}
|{{Sagittal|)~|}}{{sagittal|t}}
|{{Sagittal|~|(}}{{sagittal|t}}
|{{Sagittal|/|}}{{sagittal|t}}
|{{Sagittal||)}}{{sagittal|t}}
|{{Sagittal||\}}{{sagittal|t}}
|{{Sagittal|(|}}{{sagittal|t}}
|{{Sagittal|(|(}}{{sagittal|t}}
|{{Sagittal|//|}}{{sagittal|t}}
|{{Sagittal|/|)}}{{sagittal|t}}
|{{Sagittal|/|\}}{{sagittal|t}}
| rowspan="2" |<big>{{Sagittal|#}}</big>
|-
!Evo
| rowspan="2" |<big>{{sagittal|)/|\}}</big>
|{{sagittal|\!/}}{{sagittal|#}}
|{{sagittal|\!)}}{{sagittal|#}}
|{{sagittal|\\!}}{{sagittal|#}}
|{{sagittal|(!(}}{{sagittal|#}}
|{{sagittal|(!}}{{sagittal|#}}
|{{sagittal|!/}}{{sagittal|#}}
|{{sagittal|!)}}{{sagittal|#}}
|{{sagittal|\!}}{{sagittal|#}}
|{{sagittal|~!(}}{{sagittal|#}}
|{{sagittal|)~!}}{{sagittal|#}}
|{{sagittal|)!(}}{{sagittal|#}}
|{{sagittal|!(}}{{sagittal|#}}
|-
!Revo
|<big>{{Sagittal|(|)}}</big>
|<big>{{sagittal|(|\}}</big>
|<big>{{sagittal|)||(}}</big>
|<big>{{sagittal|~||(}}</big>
|<big>{{sagittal|)||~}}</big>
|<big>{{sagittal|/||}}</big>
|<big>{{Sagittal|||)}}</big>
|<big>{{Sagittal|||\}}</big>
|<big>{{sagittal|(||(}}</big>
|<big>{{sagittal|~||\}}</big>
|<big>{{sagittal|//||}}</big>
|<big>{{sagittal|/||)}}</big>
|<big>{{Sagittal|/||\}}</big>
|}
Alternate spellings in the Promethean set (comma tempered out):

* {{sagittal|)|}} = {{sagittal||(}} (2621440/2617839)
* {{sagittal|)|(}} = {{sagittal|~|}} (1949696/1948617)
* {{Sagittal|/|}} = {{sagittal|)|~}} ([[1216/1215]])
* {{Sagittal|~|(}} = {{Sagittal|~~|}} (22528/22491)
* {{sagittal||\}} = {{sagittal|)|)}} ([[1540/1539]])
* {{sagittal|(|}} = {{sagittal|~|)}} ([[19712/19683]])
* {{sagittal|(|(}} = {{sagittal|~|\}} (20493/20480)
* {{Sagittal|/|)}} = {{sagittal|)//|}} = {{sagittal|(|~}} ([[729/728]]) (1540/1539)
* {{Sagittal|/|\}} = {{sagittal|(/|}} ([[131072/130977]]) ''([[3969/3968]])''

See [[Sagittal notation#Revo|apotome complements]] for equivalent accidental pairs.

== Approximation to JI ==
=== 23-odd-limit interval mappings ===
{{15-odd-limit|270|23}}

=== Higher-limit JI ===
270edo's approximation to higher harmonics, starting from 29, demonstrates a somewhat monotonous sharp tendency. This allows it to be considered as a temperament of very high limits – specifically the [[53-limit]]. In fact, 270edo is the first edo to be [[diamond monotone|monotonic]] in the 47- through 51-odd-limit, using the 270i val with the sharp mapping of 23.

For primes 37 and 41, this means the pairs [[37/36]] and [[38/37]], and the pairs [[41/40]] and [[42/41]], are distinct, observing [[1369/1368]] ({{S|37}}) and [[1681/1680]] ({{S|41}}). In fact 38/37, [[39/38]], [[40/39]], and 41/40 are tempered together. The sharp mapping for prime 23 is required here so that [[37/33]] (198.071{{C}} just) is not tuned wider [[46/41]] (199.212{{C}} just). Prime 43 then fits naturally with 42/41, [[43/42]], [[44/43]], and [[45/44]] all tempered together, while 47 may be added such that [[48/47]] is tempered together with [[49/48]], [[50/49]], and [[51/50]]. Again the sharp mapping for prime 23 is required so that [[46/45]] is tempered together with 45/44 and that [[47/46]] is tempered together with 48/47. Prime 53, if desired, is tuned with [[51/50]]~[[53/52]] and [[52/51]]~[[54/53]], so monotonicity is unavoidably lost in the 53-odd-limit.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| {{Monzo| 23 6 -14 }}, {{monzo| 24 -21 4 }}
| {{Mapping| 270 428 627 }}
| −0.1069
| 0.0759
| 1.71
|-
| 2.3.5.7
| 2401/2400, 4375/4374, 29360128/29296875
| {{Mapping| 270 428 627 758 }}
| −0.0858
| 0.0752
| 1.69
|-
| 2.3.5.7.11
| 2401/2400, 3025/3024, 4375/4374, 5632/5625
| {{Mapping| 270 428 627 758 934 }}
| −0.0567
| 0.0889
| 2.00
|-
| 2.3.5.7.11.13
| 676/675, 1001/1000, 1716/1715, 3025/3024, 4096/4095
| {{Mapping| 270 428 627 758 934 999 }}
| −0.0235
| 0.1100
| 2.48
|-
| 2.3.5.7.11.13.19
| 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728
| {{Mapping| 270 428 627 758 934 999 1147 }}
| −0.0290
| 0.1028
| 2.31
|- style="border-top: double;"
| 2.3.5.7.11.13.17
| 676/675, 715/714, 936/935, 1001/1000, 1225/1224, 4096/4095
| {{Mapping| 270 428 627 758 934 999 1104 }}
| −0.0799
| 0.1718
| 3.86
|-
| 2.3.5.7.11.13.17.19
| 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1225/1224, 1331/1330
| {{Mapping| 270 428 627 758 934 999 1104 1147 }}
| −0.0777
| 0.1608
| 3.62
|-
| 2.3.5.7.11.13.17.19.23
| 460/459, 529/528, 676/675, 715/714, 736/735, 936/935, 1001/1000, 1216/1215
| {{Mapping| 270 428 627 758 934 999 1104 1147 1221 }}
| −0.0296
| 0.2037
| 4.58
|}
* 270et has lower [[Tenney-Euclidean temperament measures #TE simple badness|relative errors]] than any previous equal temperaments in the 11-, 13-, 19-, and 23-limit. It is the first to beat [[72edo|72]] in the 11-limit, [[224edo|224]] in the 13-limit, and [[217edo|217]] in the 19- and 23-limit. The next equal temperament that has lower absolute or relative error in the 11-limit is [[342edo|342]], in the 13-limit [[494edo|494]], in the 23-limit [[282edo|282]]; and in the 19-limit, [[311edo|311]] for absolute error and [[581edo|581]] for relative error. It is also a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit, and the edo with the lowest [[TE logflat badness]] in the 11-limit, 13-limit and 19-limit up until [[342edo]], [[96478edo]] and [[3395edo]] respectively.
* 23-limit is not the subgroup it does best, with the no-23 29- and 31-limit approximated even better.
* It is best in the 2.3.5.7.11.13.19 subgroup, having the least absolute error until [[552edo|552]], and the least relative error until [[2190edo|2190]].
* It is also notable in the 17-limit, with lower absolute errors than smaller equal temperaments despite inconsistency in the [[17-odd-limit|corresponding odd limit]].

=== Commas ===
{| class="commatable wikitable center-1 center-2 right-3 center-6"
! [[Harmonic limit|Prime limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Cent]]s
! [[Monzo]]
! colspan="2" | [[Kite's color notation|Color name]]
! Name(s)
|-
| 5
| <abbr title="10485760000/10460353203">[[Vulture comma|(22 digits)]]</abbr>
| 4.20
| {{Monzo| 24 -21 4 }}
| Sasaquadyo
| ssy4
| Vulture comma
|-
| 5
| [[Vishnuzma|(20 digits)]]
| 3.34
| {{Monzo| 23 6 -14 }}
| Sasepbigu
| sg14
| Vishnuzma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| 3.80
| {{Monzo| 25 -14 0 -1 }}
| Sasaru
| ssr
| Garischisma
|-
| 7
| [[2401/2400]]
| 0.72
| {{Monzo| -5 -1 -2 4 }}
| Bizozogu
| z4gg
| Breedsma
|-
| 7
| [[4375/4374]]
| 0.40
| {{Monzo| -1 -7 4 1 }}
| Zoquadyo
| zy4
| Ragisma
|-
| 7
| [[Quasiorwellisma|(16 digits)]]
| 3.73
| {{Monzo| 22 -1 -10 1 }}
| Sazoquinbigu
| szg10
| Quasiorwellisma
|-
| 11
| <abbr title="94489280512/94143178827">[[Pythrabian comma|(22 digits)]]</abbr>
| 6.35
| {{Monzo| 33 -23 0 0 1 }}
| Trisalo
| s1o3
| Pythrabian comma
|-
| 11
| [[5632/5625]]
| 2.15
| {{Monzo| 9 -2 -4 0 1 }}
| Saloquagu
| s1og4
| Vishdel comma
|-
| 11
| [[Nexus comma|(12 digits)]]
| 2.04
| {{Monzo| -16 -3 0 0 6 }}
| Tribilo
| 1o3
| Nexus comma
|-
| 11
| [[3025/3024]]
| 0.57
| {{Monzo| -4 -3 2 -1 2 }}
| Loloruyoyo
| 1ooryy
| Lehmerisma
|-
| 11
| [[9801/9800]]
| 0.18
| {{Monzo| -3 4 -2 -2 2 }}
| Bilorugu
| (1org)2
| Kalisma
|-
| 13
| [[676/675]]
| 2.56
| {{Monzo| 2 -3 -2 0 0 2 }}
| Bithogu
| 3oogg
| Island comma, parizeksma
|-
| 13
| [[1001/1000]]
| 1.73
| {{Monzo| -3 0 -3 1 1 1 }}
| Tholozotrigu
| 3o1ozg3
| Fairytale comma, sinbadma
|-
| 13
| [[2080/2079]]
| 0.83
| {{Monzo| 5 -3 1 -1 -1 1 }}
| Tholuruyo
| 3o1ury
| Ibnsinma, sinaisma
|-
| 13
| [[4096/4095]]
| 0.42
| {{Monzo| 12 -2 -1 -1 0 -1 }}
| Sathurugu
| s3urg
| Minisma
|-
|17
| [[12376/12375]]
| 0.14
| {{Monzo| 3 -2 -3 1 -1 1 1 }}
| Sotholuzotrigu
| 7o3o1uzg3
| Flashma
|-
| 19
| [[1216/1215]]
| 1.42
| 2.3.5.19 {{Monzo| 6 -5 -1 1 }}
| Sanogu
| s9og
| Password, Eratosthenes' comma
|-
|19
|[[11859211/11859210|(16 digits)]]
|0.00
|{{Monzo|-1 -4 -1 1 -4 1 0 4}}
|Quadno-athoquadlu-azogu
|9o43o1u4zg
|Tredekisma
|-
| 23
| [[529/528]]
| 3.24
| 2.3.11.23 {{monzo| -4 -1 -1 2 }}
| Bitwetho-alu
| 23oo1u
| Preziosisma
|-
| 29
| [[784/783]]
| 2.20
| 2.3.7.29 {{monzo| 4 -3 2 -1 }}
| Twenuzozo
| 23uzz
| Biminorisma
|-
| 31
| [[621/620]]
| 2.79
| 2.3.5.23.31 {{monzo| -2 3 -1 1 -1 }}
| Thiwutwethogu
| 31u23og
| Owowhatsthisma
|}
<references group="note"/>

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Associated ratio*
! Temperament
|-
| 1
| 1\270
| 4.{{overline|4}}
| 385/384
| [[Keenanose]]
|-
| 1
| 29\270
| 128.{{overline|8}}
| 14/13
| [[Tertiathirds]]
|-
| 1
| 61\270
| 271.{{overline|1}}
| 90/77
| [[Quasiorwell]]
|-
| 1
| 71\270
| 315.{{overline|5}}
| 6/5
| [[Acrokleismic]] / counteracro
|-
| 1
| 79\270
| 351.{{overline|1}}
| 49/40
| [[Newt]]
|-
| 1
| 97\270
| 431.{{overline|1}}
| 77/60
| [[Lockerbie]]
|-
| 1
| 107\270
| 475.{{overline|5}}
| 25/19
| [[Vulture]]
|-
| 2
| 14\270
| 62.{{overline|2}}
| 28/27
| [[Eagle]]
|-
| 2
| 16\270
| 71.{{overline|1}}
| 25/24
| [[Vishnu]] / ananta / acyuta
|-
| 2
| 112\270 (23\270)
| 497.{{overline|7}} (102.{{overline|2}})
| 4/3 (35/33)
| [[Gariwizmic]]
|-
| 2
| 28\270
| 124.{{overline|4}}
| 275/256
| [[Semivulture]]
|-
| 2
| 47\270
| 208.{{overline|8}}
| 44/39
| [[Abigail]]
|-
| 2
| 52\270
| 231.{{overline|1}}
| 8/7
| [[Orga]]
|-
| 2
| 131\270 (4\270)
| 582.{{overline|2}} (17.{{overline|7}})
| 7/5 (99/98)
| [[Quarvish]]
|-
| 3
| 17\270
| 75.{{overline|5}}
| 24/23
| [[Terture]]
|-
| 3
| 31\270
| 137.{{overline|7}}
| 13/12
| [[Avicenna]]
|-
| 5
| 83\270 (25\270)
| 368.{{overline|8}} (111.{{overline|1}})
| 1024/891 (16/15)
| [[Quintosec]]
|-
| 6
| 112\270 (4\270)
| 497.{{overline|7}} (97.{{overline|7}})
| 4/3 (128/121)
| [[Sextile]]
|-
| 9
| 71\270 (11\270)
| 315.{{overline|5}} (48.{{overline|8}})
| 6/5 (36/35)
| [[Ennealimmal]] / ennealimmia
|-
| 10
| 16\270 (11\270)
| 71.{{overline|1}} (48.{{overline|8}})
| 25/24 (36/35)
| [[Decavish]]
|-
| 10
| 56\270 (2\270)
| 248.{{overline|8}} (8.{{overline|8}})
| 15/13 (176/175)
| [[Decoid]]
|-
| 10
| 71\270 (10\270)
| 315.{{overline|5}} (44.{{overline|4}})
| 6/5 (40/39)
| [[Deca]]
|-
| 18
| 71\270 (4\270)
| 248.{{overline|8}} (17.{{overline|7}})
| 15/13 (99/98)
| [[Hemiennealimmal]]
|-
| 18
| 71\270 (2\270)
| 475.{{overline|5}} (8.{{overline|8}})
| 1053/800 (1287/1280)
| [[Semihemiennealimmal]]
|-
| 27
| 61\270 (1\270)
| 271.{{overline|1}} (4.{{overline|4}})
| 1375/1176 (385/384)
| [[Trinealimmal]]
|-
| 30
| 82\270 (1\270)
| 364.{{overline|4}} (4.{{overline|4}})
| 216/175 (385/384)
| [[Zinc]]
|-
| 45
| 59\270 (1\270)
| 262.{{overline|2}} (4.{{overline|4}})
| 64/55 (385/384)
| [[Rhodium]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Scales ==
=== Mos scales ===
* [[Ennealimmal]][45]: 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2
* [[Vishnu]][34]: 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7

=== Harmonic scales ===
270edo very accurately approximates the mode 16 of [[harmonic series]]. The scale in adjacent steps is 24, 22, 21, 20, 19, 18, 17, 17, 16, 15, 15, 14, 14, 13, 13, 12. Four interval pairs are conflated: 23/22~24/23, 26/25~27/26, 28/27~29/28, and 30/29~31/30.

It further does decently in the mode 24. The scale in adjacent steps is 16, 15, 15, 14, 14, 13, 13, 12, 12, 12, 11, 11, 11, 10, 10, 10, 10, 9, 9, 9, 9, 9, 8, 8.

=== Other scales ===
* [[Maeve Gutierrez #Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]] (''octave reduced: 37 23 93 65 52'')
* [[Maeve Gutierrez #Moonglade scale|Gutierrez Moonglade scale]] (24 tones): 3 17 22 3 20 5 17 25 5 14 22 5 3 16 18 4 19 5 5 12 2 6 17 5

== External links ==
* [http://tonalsoft.com/enc/t/tredek.aspx tredek, 270-edo] on [[Tonalsoft Encyclopedia]]

[[Category:Avicenna (temperament)]]
[[Category:Eagle]]
[[Category:Hemiennealimmal]]
[[Category:Vulture]]

Kite's thoughts on pergens

2026-05-04T00:54:57Z

TallKite: /* Addenda (Spring 2026) */ alternative algorithm for finding EUs -- look for a nearby small scale step and equate that to an up-arrow

A '''pergen''' (pronounced "peer-jen") is a way of classifying a [[regular temperament]] solely by its [[periods and generators|period and generator(s)]]. For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. Every rank-2, rank-3, rank-4, etc. temperament has a pergen. Assuming the prime [[subgroup]] includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a fifth or some fraction of some 3-limit interval. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every [[strong extension]] of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but do not uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.

Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka Triyoti]] has the 5th sharpened by one-fifth of [[250/243]] ({{monzo| 1 -5 3 }}). Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Trigu tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name. (Note these uses of comma fractions are not convention universally; temperament pages tend to use comma fractions to indicate the damage to the generator rather than the fifth. This is analogous to indicating the amount of stretching of an edo by the damage to the edostep rather than to the octave. But the latter approach is much more common, because the damage to the octave is much more audible and thus much more musically relevant than the damage to an edostep, which often doesn't correspond to a simple ratio. Likewise for pergens: the damage to Triyoti/Porcupine's generator, which is both 10/9 and 11/10, is less relevant than the damage to Triyoti's 4th or 5th.)

Overwhelmed? See [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf ''Notation guide for rank-2 pergens''] for practical notation examples.

{{See also| Rank-2 temperaments by mapping of 3 }}

= Definition =
If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. Both fractions are always of the form 1/N, thus the octave and/or the 3-limit interval is '''split''' into N parts. The interval which is split into multiple generators is the '''multigen'''. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.

For example, the Srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. Dicot aka Yoyoti (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine aka Triyoti is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore aka Zozoti, a pun on "semi-fourth", is of course half-fourth.

Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both Srutal aka Saguguti and Injera aka Gu & Biruyoti sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using '''ups and downs''' (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and downs notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.

The largest category contains all single-comma rank-2 temperaments with a comma of the form 2x 3y P or 2x 3y P-1, where P is a prime > 3 (a '''higher prime'''), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called '''unsplit'''.

Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multigen, and as a tie-breaker, to minimize the size in cents of the multigen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.

For example, Srutal could be (P8/2, M2/2), but P5 is preferred because it is unsplit. Or it could be (P8/2, P12), but P5 is preferred because it is smaller. Or it could be (P8/2, P4), but P5 is always preferred over P4. Note that P5/2 is not preferred over P4/2. For example, Decimal is (P8/2, P4/2), not (P8/2, P5/2).

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! colspan="4" | example temperaments
|-
! | written
! | spoken
! | comma(s)
! | name
! colspan="2" | [[Color notation|color name]]
|-
| | (P8, P5)
| | unsplit
| | 81/80
| | Meantone
| | Guti
| | gT
|-
| | "
| | "
| | 64/63
| | Archy
| | Ruti
| | rT
|-
| | "
| | "
| | (-14,8,1)
| | Schismic
| | Layoti
| | LyT
|-
| | (P8/2, P5)
| | half-8ve
| | (11, -4, -2)
| | Srutal
| | Saguguti
| | sggT
|-
| | "
| | "
| | 81/80, 50/49
| | Injera
| | Gu & Biruyoti
| | g&rryyT
|-
| | (P8, P5/2)
| | half-5th
| | 25/24
| | Dicot
| | Yoyoti
| | yyT
|-
| | "
| | "
| | (-1,5,0,0,-2)
| | Mohajira
| | Luluti
| | 1uuT
|-
| | (P8, P4/2)
| | half-4th
| | 49/48
| | Semaphore
| | Zozoti
| | zzT
|-
| | (P8/2, P4/2)
| | half-everything
| | 25/24, 49/48
| | Decimal
| | Yoyo & Zozoti
| | yy&zzT
|-
| | (P8, P4/3)
| | third-4th
| | 250/243
| | Porcupine
| | Triyoti
| | y3T
|-
| | (P8, P11/3)
| | third-11th
| | (12,-1,0,0,-3)
| | Satrilu
| | Satriluti
| | s1u3T
|-
| | (P8/4, P5)
| | quarter-8ve
| | (3,4,-4)
| | Diminished
| | Quadguti
| | g4T
|-
| | (P8/2, M2/4)
| | half-8ve quarter-tone
| | (-17,2,0,0,4)
| | Laquadlo
| | Laquadloti
| | L1o4T
|-
| | (P8, P12/5)
| | fifth-12th
| | (-10,-1,5)
| | Magic
| | Laquinyoti
| | Ly5T
|}

(P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum/difference of octaves and fifths, every single 3-limit interval is also split in half.

The multigen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: bi- splits something into two parts, tri- into three parts, etc. For a comma with monzo (a,b,c,d...), the '''color depth''' is GCD (c,d...).

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[Kite's_color_notation|Color notation]] (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yo, (a,b,-1) = gu, (a,b,0,1) = zo, (a,b,0,-1) = ru, (a,b,0,0,1) = lo/ilo, (a,b,0,0,-1) = lu, and (a,b) = wa. Examples: 5/4 = y3 = yo 3rd, 7/5 = zg5 = zogu 5th, etc.

For example, Marvel aka Ruyoyoti (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). But colors can be replaced with ups and downs, to be higher-prime-agnostic. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.

More examples: Trizoguti (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Biruyoti (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, Biruyoti Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.

A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with microtonal accidentals: (P8, P5, ^1, /1,...).

=Derivation=

For any comma, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents, where GCD (0,x) = x. The comma will split the octave into m parts, and if n > m, it will split some 3-limit interval into n parts.

In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is '''dependent''' on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two '''independent''' primes.

For example, consider Sawa & Ruyoyoti (2.3.5.7 with commas 256/243 and 225/224). The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Notating multi-EDO pergens|multi-EDO pergens]] below).

To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [http://x31eq.com/temper/uv.html x31eq.com/temper/uv.html] that will find such a matrix, it's the reduced mapping. Next make a '''square mapping''' by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.

For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.

2/1 = P8 = x·P, thus P = P8/x

3/1 = P12 = y·P + z·G, thus G = [P12 - y·(P8/x)] / z = [-y·P8 + x·P12] / xz = (-y, x) / xz

M's 3-limit monzo is (-y, x), or (y, -x) if z is negative. To get alternate generators, add i periods to G, with i ranging from -x (subtracting a full octave) to +x (adding a full octave).

G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz

'''The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| <= x'''

A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen.

For example, Porcupine aka Triyoti (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 <= i <= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).

Rank-3 pergens are trickier to find. For example, Breedsmic aka Bizozoguti is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7 x31.com] gives us this matrix:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 5/1
! | 7/1
|-
! | period
| | 1
| | 1
| | 1
| | 2
|-
! | gen1
| | 0
| | 2
| | 1
| | 1
|-
! | gen2
| | 0
| | 0
| | 2
| | 1
|}

Thus 2/1 = P, 3/1 = P + 2·G1, 5/1 = P + G1 + 2·G2, and 7/1 = 2·P + G1 + G2. Discard the last column to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 5/1
|-
! | period
| | 1
| | 1
| | 1
|-
! | gen1
| | 0
| | 2
| | 1
|-
! | gen2
| | 0
| | 0
| | 2
|}

Use an [http://wims.unice.fr/wims/wims.cgi?session=GF84B8C7BF.1&lang=en&cmd=reply&module=tool%2Flinear%2Fmatmult.en&matA=1+1+1%0D%0A0+2+1%0D%0A0+0+2&matB=&show=A%5E-1 online tool] to invert it. Here "/4" means that each entry is to be divided by the determinant of the last matrix, which is 4.

{| class="wikitable" style="text-align:center;"
|-
! |
! | period
! | gen1
! | gen2
! |
|-
! | 2/1
| | 4
| | -2
| | -1
| |
|-
! | 3/1
| | 0
| | 2
| | -1
| |
|-
! | 5/1
| | 0
| | 0
| | 2
| | /4
|}

Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25:6)/4.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a double octave to the multigen. The alternate gens are P11/2 and P19/2, both of which are much larger, so the best gen1 is P5/2.

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96:25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128:75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675:512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.

Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 7/1
|-
! | period
| | 1
| | 1
| | 2
|-
! | gen1
| | 0
| | 2
| | 1
|-
! | gen2
| | 0
| | 0
| | 1
|}

This inverts to this matrix:

{| class="wikitable" style="text-align:center;"
|-
! |
! | period
! | gen1
! | gen2
! |
|-
! | 2/1
| | 2
| | -1
| | -3
| |
|-
! | 3/1
| | 0
| | 1
| | -1
| |
|-
! | 7/1
| | 0
| | 0
| | 2
| | /2
|}

Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, /1) with /1 = 64/63, rank-3 half-5th. This is far better than (P8, P5/2, (96:25)/4).

The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is > 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible. Using 7 instead of 5 in the pergen is very common for rank-3. See [[User:TallKite/Catalog of eleven-limit rank three temperaments with Color names]] for more examples.

=Applications=

Pergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources.

Another obvious application is to categorize regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), Semihemi is (P8/2, P4/2), Triforce is (P8/3, P4/2), both Tetracot and Semihemififths are (P8, P5/4), Fourfives is (P8/4, P5/5), Pental is (P8/5, P5), and Fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, either because it contains more than 2 primes, or because the split multigen isn't actually a generator. For example, Sensei, or Semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.

Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example Porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first Porcupine is Triyoti, and the second one is Triyo & Ruti. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.

Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See [[pergen#Further Discussion-Chord names and scale names|Chord names and scale names]] below.

The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.

All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and downs notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Luluti and Dicot aka Yoyoti are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.

One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, which is octave-equivalent, fifth-generated and heptatonic. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and all notation used here is backwards compatible.

Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, Schismic aka Layoti is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C vE G. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below.

Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multigen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a [[pergen#Further Discussion-Supplemental materials-Notaion guide PDF|notation guide]]. It even allows every pergen to be numbered.

The '''[[enharmonic unison]]''', or more briefly the '''EU''', can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's EU. The pergen and the EU together define the notation. (''Edited to add: not quite accurate, see the Addenda.'')

The '''genchain''' (chain of generators) in the table is only a short section of the full genchain.

C - G implies ...Eb Bb F C G D A E B F# C#...C - ^Eb=vE - G implies ...F -- ^Ab=vA -- C -- ^Eb=vE -- G -- ^Bb=vB -- D...

If the octave is split, the table has a '''perchain''' ("peer-chain", chain of periods) that shows the octave: C -- vF#=^Gb -- C. Genchains have a theoretically infinite length, but perchains have a finite length. The full rank-2 lattice has genchains running horizontally and perchains running vertically.

{| class="wikitable" style="text-align:center;"
|-
! |
! | pergen
! | enharmonic
unison(s)
! | equivalence(s)
! | split
interval(s)
! | perchain(s) and/or
genchains(s)
! | examples
|-
| | 1
| | (P8, P5)
unsplit
| | none
| | none
| | none
| | C - G
| | Pythagorean, Meantone, Dominant,
Schismic, Mavila, Archy, etc.
|-
! |
! | half-splits
! |
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5)
half-8ve
| | ^^d2 (if 5th
> 700¢
| | ^^C = B#
| | P8/2 = vA4 = ^d5
| | C - vF#=^Gb - C
| | Srutal aka Saguguti
^1 = 81/80
|-
| |
| | "
| | vvd2 (if 5th

< 700¢)
| | ^^C = Db
| | P8/2 = ^A4 = vd5
| | C - ^F#=vGb - C
| | Injera aka Gu & Biruyoti

^1 = 64/63
|-
| |
| | "
| | vvM2
| | ^^C = D
| | P8/2 = ^4 = v5
| | C - ^F=vG - C
| | Thothoti, if 13/8 = M6

^1 = 27/26
|-
| | 3
| | (P8, P4/2)

half-4th
| | vvm2
| | ^^C = Db
| | P4/2 = ^M2 = vm3
| | C - ^D=vEb - F
| | Semaphore aka Zozoti

^1 = 64/63
|-
| |
| | "
| | ^^dd2
| | ^^C = B##
| | P4/2 = vA2 = ^d3
| | C - vD#=^Ebb - F
| | Lala-yoyoti

^1 = 81/80
|-
| | 4
| | (P8, P5/2)

half-5th
| | vvA1
| | ^^C = C#
| | P5/2 = ^m3 = vM3
| | C - ^Eb=vE - G
| | Mohajira aka Luluti

^1 = 33/32
|-
| | 5
| | (P8/2, P4/2)

half-

everything
| | \\m2,

vvA1,

^^\\d2,

vv\\M2
| | //C = Db

^^C = C#

^^//C = D
| | P4/2 = /M2 = \m3

P5/2 = ^m3 = vM3

P8/2 = v/A4 = ^\d5

= ^/4 = v\5
| | C - /D=\Eb - F,

C - ^Eb=vE - G,

C - v/F#=^\Gb - C,

C - ^/F=v\G - C
| | Zozo & Luluti

^1 = 33/32

/1 = 64/63
|-
| |
| | "
| | ^^d2,

\\m2,

vv\\A1
| | ^^ C= B#

//C = Db

^^//C = C#
| | P8/2 = vA4 = ^d5

P4/2 = /M2 = \m3

P5/2 = ^/m3 = v\M3
| | C - vF#=^Gb - C,

C - /D=\Eb - F,

C - ^/Eb=v\E - G
| | Sagugu & Zozoti

^1 = 81/80

/1 = 64/63
|-
| |
| | "
| | ^^d2,

\\A1,

^^\\m2
| | ^^C = B#

//C = C#

^^\\C = B
| | P8/2 = vA4 = ^d5

P5/2 = /m3 = \M3

P4/2 =v/M2 = ^\m3
| | C - vF#=^Gb - C,

C - /Eb=\E - G,

C - v/D=^\Eb - F
| | Sagugu & Luluti

^1 = 81/80

/1 = 33/32
|-
! |
! | third-splits
! |
! |
! |
! |
! |
|-
| | 6
| | (P8/3, P5)

third-8ve
| | ^3d2
| | ^3C = B#
| | P8/3 = vM3 = ^^d4
| | C - vE - ^Ab - C
| | Augmented aka Triguti

^1 = 81/80
|-
| | 7
| | (P8, P4/3)

third-4th
| | v3A1
| | ^3C = C#
| | P4/3 = vM2 = ^^m2
| | C - vD - ^Eb - F
| | Porcupine aka Triyoti

^1 = 81/80
|-
| | 8
| | (P8, P5/3)

third-5th
| | v3m2
| | ^3C = Db
| | P5/3 = ^M2 = vvm3
| | C - ^D - vF - G
| | Slendric aka Latrizoti

^1 = 64/63
|-
| | 9
| | (P8, P11/3)

third-11th
| | ^3dd2
| | ^3C = B##
| | P11/3 = vA4 = ^^dd5
| | C - vF# - ^Cb - F
| | Satriluti, if 11/8 = A4

^1 = 729/704
|-
| |
| | "
| | v3M2
| | ^3C = D
| | P11/3 = ^4 = vv5
| | C - ^F - vC - F
| | Satriluti, if 11/8 = P4

^1 = 33/32
|-
| | 10
| | (P8/3, P4/2)

third-8ve, half-4th
| | v6A2
| | ^6C = D#
| | P8/3 = ^^m3 = v4A4

P4/2 = ^3m2 = v3M3
| | C - ^^Eb - vvA - C

C - ^3Db=v3E - F
| | Tribiloti, if 11/8 = P4

^1 = 33/32
|-
| |
| | "
| | ^3d2,

\\m2
| | ^3C = B#

//C = Db
| | P8/3 = vM3 = ^^d4

P4/2 = /M2 = \m3
| | C - vE - ^Ab - C

C - /D=\Eb - F
| | Triforce aka Trigu & Zozoti

^1 = 81/80, /1 = 64/63
|-
| | 11
| | (P8/3, P5/2)

third-8ve, half-5th
| | ^3d2

\\A1
| | ^3C = B#

//C = C#
| | P8/3 = vM3 = ^^d4

P5/2 = /m3 = \M3
| | C - vE - ^Ab - C

C - /Eb=\E - G
| | Satribizoti

^1 = 49/48, /1 = 343/324
|-
| | 12
| | (P8/2, P4/3)

half-8ve, third-4th
| | ^^d2

\3A1
| | ^^C = Dbb

/3C = C#
| | P8/2 = vA4 = ^d5

P4/3 = \M2 = //m2
| | C - vF#=^Gb - C

C - \D - /Eb - F
| | Latribiruti

^1 = 1029/1024, /1 = 49/48
|-
| | 13
| | (P8/2, P5/3)

half-8ve, third-5th
| | ^6d32
| | ^6C = B#3
| | P8/2 = v3AA4 = ^3dd5

P5/3 = vvA2 = ^4dd3
| | C - v3Fx=^3Gbb C

C - vvD# - ^^Fb - G
| | Latribiyoti

^1 = 81/80
|-
| |
| | "
| | ^^d2,

\3m2
| | ^^C = B#

/3C = Db
| | P8/2 = vA4 = ^d5

P5/3 = /M2 = \\m3
| | C - vF#=^Gb - C

C - /D - \F - G
| | Lemba aka Latrizo & Biruyoti

^1 = (10,-6,1,-1), /1 = 64/63
|-
| | 14
| | (P8/2, P11/3)

half-8ve, third-11th
| | v6M2
| | ^6C = D
| | P8/2 = ^34 = v35

P11/3 = ^^4 = v45
| | C - ^3F=v3G - C

C - ^^F - vvC - F
| | Latribiloti, if 11/8 = P4

^1 = 33/32
|-
| | 15
| | (P8/3, P4/3)

third-

everything
| | v3d2,

\3A1
| | ^3C = Dbb

/3C = C#
| | P8/3 = ^M3 = vvd4

P4/3 = \M2 = //m2

P5/3 = v/M2
| | C - ^E - vAb - C

C - \D - /Eb - F

C - v/D - ^\F - G
| | Triyo & Triguti

^1 = 64/63

/1 = 81/80
|-
| |
| | "
| | ^3d2,

\3m2
| | ^3C = B#

/3C = Db
| | P8/3 = vM3 = ^^d4

P5/3 = /M2 = \\m3

P4/3 = v\M2
| | C - vE - ^Ab - C

C - /D - \F - G

C - v\D - ^/Eb - F
| | Trigu & Latrizoti

^1 = 81/80

/1 = 64/63
|-
| |
| | "
| | v3A1,

\3m2
| | ^3C = C#

/3C = Db
| | P4/3 = vM2 = ^^m2

P5/3 = /M2 = \\m3

P8/3 = v/M3
| | C - vD - ^Eb - F

C - /D - \F - G

C - v/E - ^\Ab - C
| | Triyo & Latrizoti

^1 = 81/80

/1 = 64/63
|-
! |
! | quarter-splits
! |
! |
! |
! |
! |
|-
| | 16
| | (P8/4, P5)
| | ^4d2
| | ^4C = B#
| | P8/4 = vm3 = ^3A2
| | C vEb vvGb=^^F# ^A C
| | Diminished aka Quadguti
|-
| | 17
| | (P8, P4/4)
| | ^4dd2
| | ^4C = B##
| | P4/4 = ^m2 = v3AA1
| | C ^Db ^^Ebb=vvD# vE F
| | Negri aka Laquadyoti
|-
| | 18
| | (P8, P5/4)
| | v4A1
| | ^4C = C#
| | P5/4 = vM2 = ^3m2
| | C vD vvE=^^Eb ^F G
| | Tetracot aka Saquadyoti
|-
| | 19
| | (P8, P11/4)
| | v4dd3
| | ^4C = Eb3
| | P11/4 = ^M3 = v3dd5
| | C ^E ^^G# vDb F
| | Squares aka Laquadruti
|-
| | 20
| | (P8, P12/4)
| | v4m2
| | ^4C = Db
| | P12/4 = v4 = ^3M3
| | C vF vvBb=^^A ^D G
| | Vulture aka Sasa-quadyoti
|-
| |
| | etc.
| |
| |
| |
| |
| |
|}
The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably it isn't particularly complex.

Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢).

==Tipping points==

Removing the ups and downs from an EU makes an '''uninflected''' EU, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s EU is ^^d2, the uninflected EU is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the EU vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore '''up may need to be swapped with down, depending on the size of the 5th''' in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' EUs are upped or downed as if the 5th were just.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every uninflected EU, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the uninflected EU.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | uninflected EU
! | 3-exponent
! | tipping

point edo
! | edo's 5th
! | upping range
! | downing range
! | if the 5th is just
|-
| | M2
| | C - D
| | 2
| | 2-edo
| | 600¢
| | none
| | all
| | downed
|-
| | m3
| | C - Eb
| | -3
| | 3-edo
| | 800¢
| | none
| | all
| | downed
|-
| | m2
| | C - Db
| | -5
| | 5-edo
| | 720¢
| | none
| | all
| | downed
|-
| | A1
| | C - C#
| | 7
| | 7-edo
| | ~686¢
| | 600-686¢
| | 686¢-720¢
| | downed
|-
| | d2
| | C - Dbb
| | -12
| | 12-edo
| | 700¢
| | 700-720¢
| | 600-700¢
| | upped
|-
| | dd3
| | C - Eb3
| | -17
| | 17-edo
| | ~706¢
| | 706-720¢
| | 600-706¢
| | downed
|-
| | dd2
| | C - Db3
| | -19
| | 19-edo
| | ~695¢
| | 695-720¢
| | 600-695¢
| | upped
|-
| | d34
| | C - Fb3
| | -22
| | 22-edo
| | ~709¢
| | 709-720¢
| | 600-709¢
| | downed
|-
| | d32
| | C - Db4
| | -26
| | 26-edo
| | ~692¢
| | 692-720¢
| | 600-692¢
| | upped
|-
| | d44
| | C - Fb4
| | -29
| | 29-edo
| | ~703¢
| | 703-720¢
| | 600-703¢
| | downed
|-
| | d43
| | C - Eb5
| | -31
| | 31-edo
| | ~697¢
| | 697-720¢
| | 600-697¢
| | upped
|-
| | etc.
| |
| |
| |
| |
| |
| |
| |
|}

=Further Discussion=

==Naming very large intervals==

So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, adding an 8ve is indicated by "c" for '''compound''' (a conventional music theory term). Thus 10/3 = cM6 = compound major 6th, 9/2 = ccM2 or cM9, etc. For a pergen with an unsplit octave, the multigen is always some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen M is less than 3 octaves, and can be P4, P5, P11, P12, ccP4 or ccP5. The last one can be spoken as "coco-fifth". Tripe compound can be spoken as "trico".

==Secondary splits==

Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. Porcupine aka Triyoti) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with:

P4/3: C - vD - ^Eb - F

A4:/3 C - D - E - F# (the lack of ups and downs indicates that this interval was already split into 3 parts)

m7/3: C - ^Eb - vG - Bb (because m7 is already split into halves, we also have m7/6: C - vD - ^Eb - F - vG - ^Ab - Bb)

M7/3: C - vE - ^G - B

m10/3: C - F - Bb - Eb (m10 is already split into 3 parts, thus m10/9 also occurs)

M10/3: C - ^F - vB - E

Of course, an equal-step melody can span other intervals besides 3-limit ones. Simply extend or cut short the earlier examples to find other melodies:

^m3/2: C - vD - ^Eb (^m3 = 6/5)

^m6/5: C - vD - ^Eb - F - vG - ^Ab (^m6 = 8/5)

vm9/4: C - ^Eb - vG - Bb - ^Db (vm9 = 32/15)

vM7/2: C - ^F - vB (vM7 = 15/8, probably more harmonious than M7 = 243/128)

More remote intervals include A1, d4, d7 and d10. These unfortunately require a very long genchain. The most interesting melodically is A1: C - ^C - vC# - C#. From C to C# is seven 5ths, which equals 21 generators, so the genchain would have to contain 22 notes if it had no gaps. Note that A1 is the uninflected EU of third-4th. The uninflected EU is always a secondary split.

For a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occurring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.

Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof [[pergen#Further Discussion-Various proofs|below]]). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If only the 8ve is split, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the EU is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. "Every" means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.

The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occurring splits are listed too, under "all pergens". (P8/3, P4/2) has all the secondary splits that (P8/3, P5) and (P8, P4/2) have, plus additional ones.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! | secondary splits of a 12th or less
|-
| colspan="2" | all pergens
| | M3/2, A4/3, d5/2, A5/4, m7/2, M9/2, m10/3, A11/2
|-
! colspan="2" | half-splits
! |
|-
| | (P8/2, P5)
| | half-8ve
| | M2/2, M3/4, A4/6, A5/8, m6/2, M10/2, d12/2, A12/2
|-
| | (P8, P4/2)
| | half-4th
| | m2/2, M6/2, m7/4, A8/2, m10/6, A11/4, P12/2
|-
| | (P8, P5/2)
| | half-5th
| | A1/2, m3/2, M7/2, m9/2, P11/2
|-
| | (P8/2, P4/2)
| | half-everything
| | every 3-limit interval is split twice as much as before
|-
! colspan="2" | third-splits
! |
|-
| | (P8/3, P5)
| | third-8ve
| | m3/3, M6/3, d5/6, A11/3, d12/3
|-
| | (P8, P4/3)
| | third-4th
| | A1/3, m7/6, M7/3, m10/9, M10/3
|-
| | (P8, P5/3)
| | third-5th
| | m2/3, m6/3, M9/6, A8/3, A12/3
|-
| | (P8, P11/3)
| | third-11th
| | M2/3, M3/6, A4/9, A5/12, m9/3, P12/3
|-
| | (P8/3, P4/2)
| | third-8ve half-4th
| | third-8ve splits, half-4th splits, M6/6, m10/6, A11/12
|-
| | (P8/3, P5/2)
| | third-8ve half-5th
| | third-8ve splits, half-5th splits, m3/6, d5/12
|-
| | (P8/2, P4/3)
| | half-8ve third-4th
| | half-8ve splits, third-4th splits, A4/6, M10/6
|-
| | (P8/2, P5/3)
| | half-8ve third-5th
| | half-8ve splits, third-5th splits, m6/6, M9/6, A12/6
|-
| | (P8/2, P11/3)
| | half-8ve third-11th
| | half-8ve splits, third-11th splits, M2/6, M3/12, A4/18, A5/24
|-
| | (P8/3, P4/3)
| | third-everything
| | every 3-limit interval is split three times as much as before
|}

==Singles and doubles==

If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a '''single-split''' pergen. If it has two fractions, it's a '''double-split''' pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called '''single-pair''' notation because it adds only a single pair of accidentals to conventional notation. '''Double-pair''' notation uses both ups/downs and lifts/drops. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the EU for single-pair notation is a 3rd or larger, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs.

In this article, for double-pair notation, the period uses ups and downs, and the generator uses lifts and drops. But the choice of which pair of accidentals is used for what is arbitrary, and ups/downs could be exchanged with lifts/drops.

Every double-split pergen is either a '''true double''' or a '''false double'''. A true double, like third-everything (P8/3, P4/3) or half-8ve quarter-4th (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-8ve quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.

A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.

==Finding an example temperament==

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4⋅P and P8. If P is 6/5, the comma is 4⋅P - P8 = (6/5)4 ÷ (2/1) = 648/625, the diminished temperament. If P is 7/6, the comma is P8 - 4⋅P = (2/1) · (7/6)-4, the Quadru temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.

Another method: if the generator's cents are known, look on the genchain for an interval that approximates a ratio of color depth ±1. Let the interval be I, and the genspan of this interval be x. Then n⋅x gens = n⋅I = x⋅M, where M is the multigen and M/n is the generator. The comma can be found from this equation, if n and x are coprime. For example, suppose (P8, P5/5) has G = 140¢. The genchain is all multiples of 140¢. Looking at the cents, 280¢ is about 7/6. Thus 2G = 7/6, and 10G = 5⋅(7/6) = 2⋅P5. Thus 2⋅P5 - 5⋅(7/6) = 0G = 0¢, and the comma is (3, 7, 0, -5). If the period is split and the generator isn't, use the perchain instead of the genchain. For example, (P8/7, P5) has a period of 141¢. 2 gens = 343¢, about 11/9. Thus 7⋅(11/9) = 2⋅P8, and the comma is (-2, -14, 0, 0, 7), Saseplo.

If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.

Finding the comma(s) for a double-split pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is '''explicitly false'''. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4⋅G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).

If the pergen is not explicitly false, put the pergen in its '''unreduced''' form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n⋅P8 - m⋅M)/nm) = (P8/m, M'/n'). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P < G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2⋅P8 - 3⋅P5) / (3⋅2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This is explicitly false, thus the comma can be found from m3/6 alone. G' is about 50¢, and the comma is 6⋅G' - m3. The comma splits both the octave and the fifth.

This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit. To summarize:

<ul><li>'''A'''<nowiki/> '''double-split pergen is explicitly false if m = |b|, and not explicitly false if m > |b|.'''</li><li>'''A double-split pergen is a true double if and only if neither it nor its unreduced form is explicitly false'''''<nowiki/>'''''<nowiki/>.'''</li><li>'''A double-split pergen is a true double if''' '''GCD (m, n) > |b|,''' '''and a false double if GCD (m, n) = |b|.'''</li></ul>

A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.

Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an '''alternate''' generator. A generator or period plus or minus any number of EUs makes an '''equivalent''' generator or period. An equivalent generator is always the same size in cents, since the EU is always 0¢. An equivalent generator is the same interval, merely notated differently. For example: (P8, P5/2) has generator ^m3 and equivalent generator vM3. Another example, half-8ve (P8/2, P5) has period vA4 and equivalent period ^d5. It has generator P5 and alternate generators P4 and vA1. vA1 is equivalent to ^m2.

Of the two equivalent generators, the preferred generator is always the smaller one (smaller degree, or if the degrees are the same, more diminished). This is because the equivalent generator can be more easily found by adding the EU, rather than subtracting it. For example, ^m3 + vvA1 = vM3 is an easier calculation than vM3 - vvA1 = ^m3. This is particularly true with complex EUs like ^6dd2.

There are also alternate EUs, see below. For double-pair notation, there are also equivalent EUs.

==Ratio and cents of the accidentals==

The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2x · 3y · P±1, where P is a higher prime. They are called mapping commas because they equate or map the ratio P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for prime 5 include 81/80, 135/128, and the schisma aka Layoma = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 (besides 1/1 of course) are the currently employed commas and combinations of them.

If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in Lemba aka Latrizo & Biruyoti, where ^1 equals 64/63 minus 81/80.

Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also [[Pergen#Notating_multi-EDO_pergens|multi-EDO pergens]] pergens below.

Cents can be assigned to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + 7c. Since the EU = 0¢, we can derive the cents of the up symbol. If the EU is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its EU.

In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the EU is an A1.

This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, basic information for playing the score. Examples:
* 15-edo: # = 240¢, ^ = 80¢ (^ = third-sharp)
* 16-edo: # = -75¢
* 17-edo: # = 141¢, ^ = 71¢ (^ = half-sharp)
* 18b-edo: # = -133¢, ^ = 67¢ (^ = half-sharp)
* 19-edo: # = 63¢
* 21-edo: ^ = 57¢ (if used, # = 0¢)
* 22-edo: # = 164¢, ^ = 55¢ (^ = third-sharp)
* quarter-comma Meantone: # = 76¢
* fifth-comma Meantone: # = 84¢
* third-comma Archy aka Ruti: # = 177¢
* eighth-comma Porcupine aka Triyoti: # = 157¢, ^ = 52¢ (^ = third-sharp)
* seventh-comma Srutal aka Sagugu & Zoquadyoti: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #, seventh-comma means 1/7 of 2048/2025)
* third-comma Injera aka Gu & Biruyoti: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)
* eighth-comma Hedgehog aka Triyo & Biruyoti: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)
The last five examples are a generalization of the practice of naming meantone tunings as a fraction of a comma, e.g. quarter-comma. The 5th is sharpened or flattened by some fraction of the first comma in the color name. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both Srutal and Injera are half-8ve, but their optimal tunings are very different.

==Finding a notation for a pergen==

There are multiple notations for a given pergen, depending on the EU(s). Preferably, the EU's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:

<ul><li>For (P8/m, M/n), P8 = m⋅P + x⋅EU and M = n⋅G + y⋅EU', with 0 < |x| <= m/2 and 0 < |y| <= n/2</li><li>x is the count for EU, with EU occurring x times in one octave, and x⋅EU is the octave's '''multi-EU'''</li><li>y is the count for EU', with EU' occurring y times in one multigen, and y⋅EU' is the multigen's multi-EU</li><li>For false doubles using single-pair notation, EU = EU', but x and y are usually different, making different multi-EUs</li><li>The unreduced pergen is (P8/m, M'/n'), with a new enharmonic unison EU" and new counts, P8 = m⋅P + x'⋅EU", and M' = n'⋅G' + y'⋅EU"</li></ul>

The '''keyspan''' of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The '''[[stepspan]]''' of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.

Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a '''gedra''', analogous to a monzo, but written in brackets not parentheses: 3/2 = (-1,1) is a 7-semitone 5th, thus (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval with a monzo (a,b), there is a unique gedra [k,s], and vice versa:

<ul><li>k = 12a + 19b</li><li>s = 7a + 11b</li></ul>The matrix [(12,19) (7,11)] is unimodular, and can be inverted, and (a,b) can be derived from [k,s]:

<ul><li>a = -11k + 19s</li><li>b = 7k - 12s</li></ul>Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].

Gedras greatly facilitate finding a pergen's period, generator and EU(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an EU with the smallest possible keyspan and stepspan, which is the best EU for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up.

For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = m3. The EU can also be found using gedras: x⋅EU = M - n⋅G = P5 - 2⋅m3 = [7,4] - 2⋅[3,2] = [7,4] - [6,4] = [1,0] = A1.

Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. x⋅EU = P8 - m⋅P = P8 - 5⋅M2 = [12,7] - 5⋅[2,1] = [2,2] = 2⋅[1,1] = 2⋅m2. Because x = 2, EU will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2⋅m2 = d3). The EU's '''count''' is 2. The uninflected EU is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus EU = v5m2. Since P8 = 5⋅P + 2⋅EU, the period must be ^^M2, to make the ups and downs come out even. The number of the period's (or generator's) ups or downs always equals the count. Equipped with the period and the EU, the perchain is easily found:

P1 -- ^^M2=v3m3 -- v4 -- ^5 -- ^3M6=vvm7 -- P8
C -- ^^D=v3Eb -- vF -- ^G -- ^3A=vvBb -- C

Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has an uninflected generator [5,3]/5 = [1,1] = m2. The uninflected EU is P4 - 5⋅m2 = [5,3] - 5⋅[1,1] = [5,3] - [5,5] = [0,-2] = -2⋅[0,1] = two descending d2's. The d2 must be upped, and EU = ^5d2. Since P4 = 5⋅G - 2⋅EU, G must be ^^m2. The genchain is:

P1 -- ^^m2=v3A1 -- vM2 -- ^m3 -- ^3d4=vvM3 -- P4C -- ^^Db -- vD -- ^Eb -- vvE -- F

To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and EU from the fractional multigen as before. Then deduce the period from the EU. If the multigen was changed by unreducing, find the original generator from the period and the alternate generator.

For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The uninflected alternate generator G' is [1,1]/10 = [0,0] = P1. The uninflected EU is m2 - 10⋅P1 = m2. It must be downed, thus EU = v10m2. Since m2 = 10⋅G' + EU, G' is ^1. The octave plus or minus some number of EUs must equal 5 periods, thus (P8 + x⋅m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5⋅P + 2⋅EU, and P = ^4M2. Next, find the original half-4th generator. P = P8/5 = ~240¢, and G = P4/2 = ~250¢. Because P < G, G' is not P - G but G - P, and G is not P - G' but P + G', which equals ^4M2 + ^1 = ^5M2. While the alternate multigen is more complex than the original multigen, the alternate generator is usually simpler than the original generator.

P1- - ^4M2=v6m3 -- vv4 -- ^^5 -- ^6M6=v4m7 -- P8
C -- ^4D=v6Eb -- vvF -- ^^G -- ^6A=v4Bb -- C
P1 -- ^5M2=v5m3 -- P4
C -- ^5D=v5Eb -- F

To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. Each pair has its own enharmonic unison. For (P8/2, P4/2), the split octave implies P = vA4 and EU = ^^d2, and the split 4th implies G = /M2 and EU' = \\m2.

A false-double pergen can optionally use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair EU is not a unison or a 2nd, as with (P8/2, P4/3).

Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an EU that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So EU = v4dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The uninflected enharmonic unison is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic unison, we use the second pair of accidentals: EU' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic unisons is also an enharmonic unison: EU + EU' = v4\\d3 = 2·vv\m2, and EU - EU' = v4//ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent EUs, and v\4 and v/d4 are equivalent generators. Here is the genchain:

P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11
C -- ^E=v\F -- /Ab=\A -- ^/C=vDb -- F

One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and EU = v12m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the uninflected enharmonic unison: m3 - 3·m2 = [0,-1] = descending d2. Thus EU' = /3d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonic unisons are found from EU + EU' and EU - 2·EU'. Equivalent periods and generators are found from the many enharmonic unisons, which also allow much freedom in chord spelling. Enharmonic unison = v12m3 = /3d2 = v4/m2 = v4\\A1. Period = \M3 = v44 = //d4. Generator = ^\M3 = v34 = ^//d4.

P1 — \M3 — \\A5=/m6 — P8C — \E — /Ab — CP1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^38=v/m9 — P11
C — ^\E — ^^/Ab=vv\A — v/Db — F

It's not yet known if every pergen can avoid large EUs (those of a 3rd or more) with double-pair notation. One situation in which very large EUs occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the EU's stepspan to equal the multigen's stepspan.

Sixth-4th with single-pair notation has an awkward ^6d64 EU. This pergen might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore commas, aka Triyo & Zozoti), and its double-pair notation can also combine both. Third-4th has EU = v3A1 and G'= vM2 = ^^m2. Half-4th has EU' = \\m2 and G' = /M2 = \m3. G' - G = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G' - G = /M2 - vM2 = ^/1. Equivalent EUs are v3\\M2 and ^3\\d2.

P1 — ^/1=^\m2 — ^^m2=vM2 — /M2=\m3 — ^m3=vvM3 — v/M3=v\4 — P4
C — ^/C=^\Db — ^^Db=vD — /D=\Eb — ^Eb=vvE — v/E=v\F — F

When ups and downs are used to notate edos, a third symbol is used, a '''mid''' , written ~. The mid is only for relative notation (intervals), never for absolute notation (notes). For imperfect intervals, the mid interval is the interval exactly midway between major and minor. In addition, the mid-4th is midway between perfect and augmented, i.e. half-augmented, and the mid-5th is half-diminished. For example, 17edo's 2nds and 3rds run minor-mid-major. This avoids having to constantly choose between the equivalent terms upminor and downmajor. 72edo's 2nds and 3rds run minor, upminor, downmid, mid, upmid, downmajor, major. This makes the terms more concise. Mids can be used in pergen notation too. For single-pair, EU must be an A1, with an even number of downs. Using mids with double-pair notation is trickier. Mids never appear in the perchain. If one accidental pair appears only in the perchain and the other pair only in the genchain, then the mids appear only in the genchain, if at all.

==Alternate enharmonic unisons==

Sometimes the EU found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, ccM6/12), a false double. The uninflected alternate generator is ccM6/12 = [33,19]/12 = [3,2] = m3. The uninflected EU is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The EU becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v12A2, which is an improvement but still awkward. The period is ^4m3 and the generator is v3M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.

P1 -- ^4m3 -- v4M6 -- P8
C -- ^4Eb -- v4A -- C
P1 -- v3M2 -- v6M3=^6m2 -- ^3m3 -- P4
C -- v3D -- v6E=^6Db -- ^3Eb -- F

Because G is a M2 and EU is an A2, the equivalent generator G - EU is a descending A1. Ascending intervals that look descending can be confusing, one has to take into account the eighth-tone ups to see that v3D -- ^6Db is ascending. Double-pair notation may be preferable. This makes P = vM3, EU = ^3d2, G = /m2, and EU' = /4dd2.

P1 -- vM3 -- ^m6 -- P8
C -- vE -- ^Ab -- C
P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4
C -- /Db -- //Ebb=\\D# -- \E -- F

Because the EU found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans, like 5edo or 19edo. Heptatonic stepspans are best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the EU.

To search for an alternate EU, convert the EU to a gedra, then multiply it by the count to get the multi-EU. The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and EU is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-EU. If the stepspan becomes negative, or if it's zero and the keyspan becomes negative, invert the gedra. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new EU. Choose between ups and downs according to whether the 5th falls in the EU's upping or downing range, see [[pergen#Applications-Tipping points|tipping points]] above. Add n'''·'''count ups or downs to the new multi-EU. Add or subtract the new multi-EU from the multigen (or the octave) to get an interval which splits cleanly into n (or m) parts. Each part is the new generator (or period).

For example, (P8, P5/3) has n = 3, G = ^M2, and EU = v3m2 = [1,1]. G is upped only once, so the count is 1, and the multi-EU is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-EU [-2,1]. This can't be simplified, so the new EU is also [-2,1] = d32. Assuming a reasonably just 5th, EU needs to be upped, so EU = ^3d32. Add the multi-EU ^3[-2,1] to the multigen P5 = [7,4] to get ^3[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^3[-2,1] = v3[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·EU to check: 3·vA2 + 1·^3d32 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d32 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- vD# -- ^Fb -- G. Because d32 = -200¢ - 26·c, ^ = (-d32) / 3 = 67¢ + 8.67·c, about a third-tone.

Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one EU (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain EU. Specifically, the comma tempered out should map to the EU, or the multi-EU if the count is > 1. For example, consider Semaphore aka Zozoti (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 EU makes sense. G = ^M2 and the genchain is C -- ^D=vEb -- F. But consider Lala-yoyoti, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus EU = ^^dd2, G = vA2, and the genchain is C -- vD#=^Ebb -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.

Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 EU is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of EU.

For example, Paralimmal aka Satriluti tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the EU is v3M2, but if 11/8 is notated as a vA4, the EU is ^3dd2.

Sometimes the temperament implies an EU that isn't even a 2nd. For example, Liese aka Gu & Trizoguti (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and EU = 3·vd5 - P11 = v3dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- vGb -- ^B -- F.

This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an EU, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different EUs.

==Chord names and staff notation==

Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[Ups and downs notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.

In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G ^A and C E G vBb are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.

Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara aka Ru & Biruyoti (2.3.5.7 with 64/63 and 50/49) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and EU = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C vE G Bb = Cv,7.

A different temperament may result in the same pergen with the same EU, but may still produce a different name for the same chord. For example, Injera aka Gu & Biruyoti (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 EU is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and EU = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G vBb = C,v7.

Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes. This is in fact a requirement for MOS scales.

Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. Iv -- vIIIv -- vVI^m -- Iv. A Porcupine aka Triyoti (third-4th) comma pump can be written out like so: Cv -- vA^m -- vDv -- [vvB=^Bb]^m -- ^Ebv -- G^m -- Gv -- Cv. Brackets are used to show that vvB and ^Bb are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. vvB is written first to show that this root is a vM6 above the previous root, vD. ^Bb is second to show the P4 relationship to the next root, ^Eb. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either vvB^m or ^Bb^m, or possibly ^BbvvM = ^Bb vD ^F.

Lifts and drops, like ups and downs, precede the note head and any sharps or flats. Scores for melody instruments could instead have them above or below the staff. This score uses ups and downs, and has chord names. It's for the third-4th pergen, with EU = v3A1. As noted above, it can be played in EDOs 15, 22 or 29 as is, because ^1 maps to 1 edostep. But to be played in EDOs 30, 37 or 44, ^1 must be interpreted as two edosteps.

Mizarian Porcupine Overture by Herman Miller (P8, P4/3) ^3C = C#

[[File:Mizarian_Porcupine_Overture.png|alt=Mizarian Porcupine Overture.png|800x692px|Mizarian Porcupine Overture.png]]

==Tipping points and sweet spots==

The tipping point for half-octave with a d2 EU is 700¢, 12-edo's 5th. As noted above, the 5th of Pajara (half-8ve) tends to be sharp, thus it has EU = ^^d2. But Injera, also half-8ve, has a flat 5th, and thus EU = vvd2. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But if a single temperament "tips over", either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.

The tipping point depends on the choice of EU. It's not the temperament that tips, it's the notation. Half-8ve could be notated with an EU of vvM2. The tipping point becomes 600¢, a very unlikely 5th, and tipping is impossible. For single-comma temperaments, the EU usually equals the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of EU is a given. However, the mapping of primes 11 and 13 is not agreed on.

The notation's tipping point is determined by the uninflected EU, which is implied by the vanishing comma. For example, Porcupine aka Triyoti's 250/243 comma is an A1 = (-11,7), which implies an uninflected EU of A1, which implies 7-edo, and a 685.7¢ tipping point. Dicot aka Yoyoti's 25/24 comma is also an A1, and has the same tipping point. Semaphore aka Zozoti's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. See [[pergen#Applications-Tipping points|tipping points]] above for a more complete list.

Double-pair notation has two EUs, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two EUs can be any two of A1, m2 or d2. One can choose to use whichever two EUs best avoid tipping.

An example of a temperament that tips easily is Negri aka Laquadyoti, 2.3.5 and (-14,3,4). Because Negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with Negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and the EU could be either ^4dd2 or v4dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an EU of ^4dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma Negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.

Another "tippy" temperament is found by adding the mapping comma 81/80 to the Negri comma and getting the Latriyoma comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th.

==Notating unsplit pergens==

An unsplit pergen doesn't require ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a mapping comma, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (i.e. the monzo has a final exponent of ±1).

The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate. The up symbol's ratio is always the mapping comma, or its inverse.

{| class="wikitable" style="text-align:center;"
|-
! | 5-limit temperament
! | comma
! | sweet spot
! colspan="2" | no ups or downs
! colspan="3" | with ups and downs
! colspan="2" | up symbol
|-
! | (pergen is unsplit)
! |
! | (5th = 700¢ + c)
! | 5/4 is
! | 4:5:6 chord
! | 5/4 is
! | 4:5:6 chord
! | EU
! | ratio
! | cents
|-
| | Meantone aka Guti
| | 81/80 = P1
| | c = -3¢ to -5¢
| | M3
| | C E G
| | ---
| | ---
| | ---
| | ---
| | ---
|-
| | Mavila aka Layobiti
| | 135/128 = A1
| | c = -21¢ to -22¢
| | m3
| | C Eb G
| | ^M3
| | C ^E G
| | ^A1
| | 80/81 = d1
| | -100¢ - 7c = 47¢-54¢
|-
| | Laguti
| | (-15,11,-1) = A1
| | c = -10¢ to -12¢
| | A3
| | C E# G
| | ^M3
| | C ^E G
| | vA1
| | 80/81 = A1
| | 100¢ + 7c = 26¢-30¢
|-
| | Schismic aka Layoti
| | (-15,8,1) = -d2
| | c = 1.7¢ to 2.0¢
| | d4
| | C Fb G
| | vM3
| | C vE G
| | ^d2
| | 81/80 = -d2
| | 12c = 20¢-24¢
|-
| | Lalaguti
| | (-23,16,-1) = -d2
| | c = -0.9¢ to -1.2¢
| | AA2
| | C D## G
| | vM3
| | C vE G
| | vd2
| | 81/80 = d2
| | -12c = 10¢-15¢
|-
| | Father aka Gubiti
| | 16/15 = m2
| | c = 56¢ to 58¢
| | P4
| | C F G
| | vM3
| | C vE G
| | ^m2
| | 81/80 = -m2
| | -100¢ + 5c = 180-190¢
|-
| | Superpyth aka Sasayoti
| | (12,-9,1) = m2
| | c = 9¢ to 10¢
| | A2
| | C D# G
| | vM3
| | C vE G
| | vm2
| | 81/80 = m2
| | 100¢ - 5c = 50-55¢
|}
The Schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch. The Mavila temperament could perhaps be notated without ups and downs, because 5/4 is still a 3rd, and the 4:5:6 triad still looks like a triad.

For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the Schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-limit comma is the pythagorean comma (-19,12).

A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for Archy aka Ruti (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit Schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C vE G vBb.

==Notating rank-3 pergens==

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. Enharmonic unisons aka EUs are like vanishing commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of EUs needed always equals the difference between the notation's rank and the tuning's rank. Examples:

{| class="wikitable" style="text-align:center;"
|-
! | tuning
! | pergen
! | tuning's rank
! | notation
! | notation's rank without any EUs
! | # of EUs needed
! | EUs
|-
| | 12-edo
| | (P8/12)
| | rank-1
| | conventional
| | rank-2
| | 1
| | EU = d2
|-
| | 19-edo
| | (P8/19)
| | rank-1
| | conventional
| | rank-2
| | 1
| | EU = dd2
|-
| | 15-edo
| | (P8/15)
| | rank-1
| | single-pair
| | rank-3
| | 2
| | EU = m2, EU' = v3A1 = v3M2
|-
| | 24-edo
| | (P8/24)
| | rank-1
| | single-pair
| | rank-3
| | 2
| | EU = d2, EU' = vvA1 = vvm2
|-
| | 3-limit JI aka pythagorean
| | (P8, P5)
| | rank-2
| | conventional
| | rank-2
| | 0
| | ---
|-
| | Meantone aka Gu
| | (P8, P5)
| | rank-2
| | conventional
| | rank-2
| | 0
| | ---
|-
| | Diaschismic aka Sagugu
| | (P8/2, P5)
| | rank-2
| | single-pair
| | rank-3
| | 1
| | EU = ^^d2
|-
| | Semaphore aka Zozo
| | (P8, P4/2)
| | rank-2
| | single-pair
| | rank-3
| | 1
| | EU = vvm2
|-
| | Decimal aka Yoyo & Zozo
| | (P8/2, P4/2)
| | rank-2
| | double-pair
| | rank-4
| | 2
| | EU = vvd2, EU' = \\m2 = ^^\\A1
|-
| | 5-limit JI
| | (P8, P5, ^1)
| | rank-3
| | single-pair
| | rank-3
| | 0
| | ---
|-
| | Marvel aka Ruyoyo
| | (P8, P5, ^1)
| | rank-3
| | single-pair
| | rank-3
| | 0
| | ---
|-
| | Breedsmic aka Bizozogu
| | (P8, P5/2, ^1)
| | rank-3
| | double-pair
| | rank-4
| | 1
| | EU = \\dd3
|-
| | 7-limit JI
| | (P8, P5, ^1, /1)
| | rank-4
| | double-pair
| | rank-4
| | 0
| | ---
|}
When there is more than one EU, the first one can be added to or subtracted from the 2nd one, to make an equivalent 2nd EU.

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas. There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.

Even the unsplit rank-3 pergen requires single-pair notation, for the 2nd generator. Single-splits and false double-splits require double-pair, true doubles and false triples require triple-pair, and true triples require quadruple-pair. Some false triples and some true doubles may use quadruple-pair as well, to avoid awkward EUs of a 3rd or more. Rather than devising a third or fourth pair of symbols, and a third or fourth pair of adjectives to describe them, one might simply use colors.

A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.

Some examples of 7-limit rank-3 temperaments:

{| class="wikitable" style="text-align:center;"
|-
! | 7-limit temperament
! | comma
! | pergen
! | spoken pergen
! | notation
! | period
! | gen1
! | gen2
! | EU
|-
| | Marvel aka Ruyoyoti
| | 225/224
| | (P8, P5, ^1)
| | rank-3 unsplit
| | single-pair
| | P8
| | P5
| | ^1 = 81/80
| | ---
|-
| | "
| | "
| | "
| | "
| | double-pair
| | "
| | "
| | "
| | ^^\d2
|-
| | Biruyoti
| | 50/49
| | (P8/2, P5, ^1)
| | rank-3 half-8ve
| | double-pair
| | v/A4 = 10/7
| | P5
| | ^1 = 81/80
| | ^^\\d2
|-
| | Trizoguti
| | 1029/1000
| | (P8, P11/3, ^1)
| | rank-3 third-11th
| | double-pair
| | P8
| | ^\d5 = 7/5
| | ^1 = 81/80
| | ^^^\\\dd3
|-
| | Breedsmic aka Bizozoguti
| | 2401/2400
| | (P8, P5/2, ^1)
| | rank-3 half-5th
| | double-pair
| | P8
| | v//A2 = 60/49
| | /1 = 64/63
| | ^^\4dd3
|-
| | Demeter aka Trizo-aguguti
| | 686/675
| | (P8, P5, \m3/2)
| | half-downminor-3rd
| | double-pair
| | P8
| | P5
| | v/A1 = 15/14
| | ^^\\\dd3
|}
If using single-pair notation, Marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - vE - G - vvA#. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - vE - G - \Bb.

There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, Biruyo is half-8ve, with a d2 comma, like Srutal. Using the same notation as Srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and EU = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as C vE G v/Bb. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C vE G \Bb.

With this standardization, the EU can be derived directly from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)-2 · (64/63)2 · (-19,12). This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the EU never is, therefore the EU = ^^\\d2. The period is found by adding/subtracting the EU from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both EU and P become more complex, but the ratio for /1 becomes simpler.

This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For Biruyoti, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore Biruyoti doesn't tip. Single-pair rank-3 notation has no EU, and thus no tipping point. Double-pair rank-3 notation has 1 EU, but two mapping commas. Rank-3 notations rarely tip.

Unlike the previous examples, Demeter aka Trizo-aguguti's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no EU. But the 4:5:6:7 chord would be spelled C -- ^^^Fbbb -- G -- ^^Bbb, very awkward! Standard double-pair notation is better. Gen2 = v/A1, EU = ^^\\\dd3, and ^^\\\C = A##. Genchain2 is C -- v/C# -- \Eb -- vE -- \\Gb -- v\G -- vvG#=\\\Bbb -- v\\Bb... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own EU, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own EU, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops only for the other kind of accidentals.

There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For Demeter, any combination of \m3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because there will always be a 3-limit comma small enough to be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the '''DOL''' ([[Odd limit|double odd limit]]) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred. And as a tie-breaker, in case of two ratios with the same DOL (such as 5/3 and 6/5), to minimize the size in cents of the multigen2. Since 5/3 = 884.4¢ and 6/5 = 315.6¢, 6/5 is preferred.

If ^1 = 81/80, possible half-split gen2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.

All possible rank-3 pergens can be listed, but the table is much longer than for rank-2 pergens. Here are all the half-split pergens:

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! colspan="4" | prime subgroup used by the pergen
|-
! | unsplit
! colspan="2" | 2.3.5 (^1 = 81/80)
! colspan="2" | 2.3.7 (^1 = 64/63)
|-
| | 1
| | (P8, P5, ^1)
| | rank-3 unsplit
| | same
| | same
|-
! | half-splits
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5, ^1)
| | rank-3 half-8ve
| | same
| | same
|-
| | 3
| | (P8, P4/2, ^1)
| | rank-3 half-4th
| | same
| | same
|-
| | 4
| | (P8, P5/2, ^1)
| | rank-3 half-5th
| | same
| | same
|-
| | 5
| | (P8/2, P4/2, ^1)
| | rank-3 half-everything
| | same
| | same
|-
| | 6
| | (P8, P5, ^m3/2)
| | half-upminor-3rd
| | (P8, P5, ^M2/2)
| | half-upmajor-2nd
|-
| | 7
| | (P8, P5, vM3/2)
| | half-downmajor-3rd
| | (P8, P5, vm3/2)
| | half-downminor-3rd
|-
| | 8
| | (P8, P5, ^m6/2)
| | half-upminor-6th
| | (P8, P5, ^M6/2)
| | half-upmajor-6th
|-
| | 9
| | (P8, P5, vM6/2)
| | half-downmajor-6th
| | (P8, P5, vm7/2)
| | half-downminor-7th
|-
| | 10
| | (P8/2, P5, ^m3/2)
| | half-8ve half-upminor-3rd
| | (P8/2, P5, ^M2/2)
| | half-8ve half-upmajor-2nd
|-
| | 11
| | (P8/2, P5, vM3/2)
| | half-8ve half-downmajor-3rd
| | (P8/2, P5, vm3/2)
| | half-8ve half-downminor-3rd
|-
| | 12
| | (P8, P4/2, vM3/2)
| | half-4th half-downmajor-3rd
| | (P8, P4/2, ^M2/2)
| | half-4th half-upmajor-2nd
|-
| | 13
| | (P8, P4/2, ^m6/2)
| | half-4th half-upminor-6th
| | (P8, P4/2, vm7/2)
| | half-4th half-downminor-7th
|-
| | 14
| | (P8, P5/2, vM3/2)
| | half-5th half-downmajor-3rd
| | (P8, P5/2, ^M2/2)
| | half-5th half-upmajor-2nd
|-
| | 15
| | (P8, P5/2, ^m6/2)
| | half-5th half-upminor-6th
| | (P8, P5/2, vm7/2)
| | half-5th half-downminor-7th
|-
| | 16
| | (P8/2, P4/2, vM3/2)
| | half-everything half-downmajor-3rd
| | (P8/2, P4/2, ^M2/2)
| | half-everything half-upmajor-2nd
|}
There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc.

==Notating multi-EDO pergens==

A multi-EDO pergen is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave, thus it doesn't appear in the pergen. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12. Pergens which imply an edo which doesn't have a decent 5th, e.g. P8/3, P8/4, P8/6, etc., are covered in the next section.

A multi-EDO pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the octave and removes the middle term from the pergen. For example, Blackwood aka Sawati plus Ya is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1). Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:

{| class="wikitable" style="text-align:center;"
|-
! | temperament
! | pergen
! | spoken name
! | enharmonic unisons
! | perchain
! | genchain
! | ^1 ratio
! | /1 ratio
|-
| | Blackwood aka Sawati+ya
| | (P8/5, ^1)
| | rank-2 5-edo
| | EU = m2
| | D E=F G A B=C D
| | D vF#=vG vvB...
| | 81/80 = 16/15
| | ---
|-
| | Whitewood aka Lawati+ya
| | (P8/7, ^1)
| | rank-2 7-edo
| | EU = A1
| | D E F G A B C D
| | D ^F ^^A...
| | 80/81 = 135/128
| | ---
|-
| | 10edo+ya
| | (P8/10, /1)
| | rank-2 10-edo
| | EU = m2, EU' = vvA1 = vvM2
| | D ^D=vE E=F ^F=vG G...
| | D \F#=\G \\B...
| | (see below)
| | 81/80
|-
| | 12edo+la
| | (P8/12, ^1)
| | rank-2 12-edo
| | EU = d2
| | D D#=Eb E F F#=Gb...
| | D ^G ^^C
| | 33/32
| | ---
|-
| | "
| | "
| | "
| | "
| | "
| | D vG#=vAb vvD...
| | 729/704
| | ---
|-
| | 17edo+ya
| | (P8/17, /1)
| | rank-2 17-edo
| | EU = dd3, EU' = vm2 = vvA1
| | D ^D=Eb D#=vE E F...
| | D \F# \\A#=v\\B...
| | 256/243
| | 81/80
|}
If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 10, 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15, or 12/11, or 13/12.

The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.

All multi-EDO pergens are of the form (P8/m, ^1) or (P8/m, /1), and they can be identified solely by their splitting fraction. Multi-EDO pergens are a small minority of rank-2 pergens.

It's possible to have a fifth-8ve pergen with an independent 5th, but there will be very small intervals of about 20¢. Here are two such:

{| class="wikitable" style="text-align:center;"
|-
! | temperament
! | subgroup
! | comma
! | pergen
! | spoken name
! | EU
! | perchain
! | genchain
! | ^1 ratio
|-
| | Laquinzoti
| | 2.3.7
| | (-14,0,0,5)
| | (P8/5, P5)
| | fifth-8ve
| | v5m2
| | D ^^E vG ^A vvC D
| | C G D A E...
| | 49/48
|-
| | Saquinruti
| | 2.3.7
| | (22,-5,0,-5)
| | "
| | "
| | "
| | "
| | "
| | 64/63
|}
Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C ^^D vF ^G vvBb C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section). A multi-EDO pergen should never have an uninflected genchain.

==Notating non-8ve and no-5ths pergens==

In multi-EDO pergens, the 5th is present but not independent. In no-5ths pergens, the 5th is not present, and the prime subgroup doesn't contain 3.

In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any EUs, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.

But in non-8ve and no-5ths pergens, not every name has a note. For example, Biruyoti Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - vF# - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a huge number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk.

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! colspan="6" | prime subgroup used by the pergen
|-
! | unsplit
! | 2.3
! | 2.5 (M3 = 5/4)
! | 2.7 (M2 = 8/7)
! | 3.5 (M6 = 5/3)
! | 3.7 (M3 = 9/7)
! | 5.7 (ccM3 = 5/1, d5 = 7/5)
|-
| | 1
| | (P8, P5)
| | (P8, M3)
| | (P8, M2)
| | (P12, M6)
| | (P12, M3)
| | (ccM3, d5)
|-
! | half-splits
! |
! |
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5)
| | (P8/2, M3)
| | (P8/2, M2)
| | (P12/2, M6)
| | (P12/2, M3)
| | (M9, d5)*
|-
| | 3
| | (P8, P4/2)
| | (P8, M2)*
| | (P8, M2/2)
| | (P12, M6/2)
| | (P12, M2)*
| | (ccM3, m3)*
|-
| | 4
| | (P8, P5/2)
| | (P8, m6/2)
| | (P8, P5)*
| | (P12, P4)*
| | (P12, m10/2)
| | (ccM3, M7)*
|-
| | 5
| | (P8/2, P4/2)
| | (P8/2, M2)*
| | (P8/2, M2/2)
| | (P12/2, M6/2)
| | (P12/2, M3/2)
| | (M9, m3)*
|-
! | third-splits
! |
! |
! |
! |
! |
! |
|-
| | 6
| | (P8/3, P5)
| | (P8/3, M3)
| | (P8/3, M2)
| | (P12/3, M6)
| | (P12/3, M3)
| | (ccM3/3, d5)
|-
| | 7
| | (P8, P4/3)
| | (P8, M3/3)
| | (P8, M2/3)
| | (P12, M6/3)
| | (P12, M3/3)
| | (ccM3, d5/3)
|-
| | 8
| | (P8, P5/3)
| | (P8, m6/3)
| | (P8, m7/3)
| | (P12, m7/3)
| | (P12, P4)*
| | (ccM3, cA6/3)
|-
| | 9
| | (P8, P11/3)
| | (P8, M10/3)
| | (P8, M9/3)
| | (P12, ccM3/3)
| | (P12, cM7/3)
| | (ccM3, ccm7/3)
|-
| | 10
| | (P8/3, P4/2)
| | (P8/3, M2)*
| | (P8/3, M2/2)
| | (P12/3, M6/2)
| | (P12/3, M2)*
| | (ccM3/3, m3)*
|-
| | 11
| | (P8/3, P5/2)
| | (P8/3. m6/2)
| | (P8/3, P5)*
| | (P12/3, P4)*
| | (P12/3, m10/2)
| | (ccM3/3, M7)*
|-
| | 12
| | (P8/2, P4/3)
| | (P8/2, M3/3)
| | (P8/2, M2/3)
| | (P12/2, M6/3)
| | (P12/2, M3/3)
| | (M9, d5/3)*
|-
| | 13
| | (P8/2, P5/3)
| | (P8/2, m6/3)
| | (P8/2, m7/3)
| | (P12/2, m7/3)
| | (P12/2, P4)*
| | (M9, cA6/3)*
|-
| | 14
| | (P8/2, P11/3)
| | (P8/2, M10/3)
| | (P8/2, M9/3)
| | (P12/2, ccM3/3)
| | (P12/2, cM7/3)
| | (M9, ccm7/3)*
|-
| | 15
| | (P8/3, P4/3)
| | (P8/3, M3/3)
| | (P8/3, M2/3)
| | (P12/3, M6/3)
| | (P12/3, P4)*
| | (ccM3/3, d5/3)
|}
For prime subgroup p.q, the unsplit pergen has period p/1. The unsplit pergen's generator is found by dividing q by p until it's less than p/1, and period-inverting if it's more than half of p/1.

Multi-EDO pergens also appear in this table. For example, in row #33, (P8/5, ^1) replaces both 2.5 (P8/5, M3) and 2.7 (P8/5, M2). This has the advantage of avoiding missing notes. Since one fifth of 5/1 is only 25¢ from 7/5, the 5.7 (ccM3/5, d5) can optionally be replaced too.

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! | 2.3
! | 2.5
! | 2.7
! | 3.5
! | 3.7
! | 5.7
|-
| | 33
| | (P8/5, P5)
| | (P8/5, ^1)
| | (P8/5, ^1)
| | (P12/5, M6)
| | (P12/5, M3)
| | (ccM3/5, ^1)
|}

Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the first 3.5 pergen, (P12, M6). The note one M6 above C1 is A1, and the note three P12's above C1 is A5. (There is no A2, A3 or A4.) Referring to a note as simply A is ambiguous.

Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table could be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2), with ^M2 = 8/7 and ^1 = √256/245 = √(81/80 * 64/63 * 64/63) = about 38¢. The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3), with ^M3 = 9/7 and ^1 = √6561/6125 = √[(81/80)3 * (64/63)2] = about 60¢.

==Pergen squares==

Pergen squares, which were discovered by Praveen Venkataramana, are a way to visualize pergens squares in a way that isn't specific to any primes at all. To understand them, let's assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.

For (P8, P5), the pergen square has 4 notes, shown here with octave numbers (ignore the periods).

C2 -- G2 
| . . . . | 
C1 -- G1

Splitting the 8ve or the multigen adds notes to the square. For (P8/2, P5), there are 6 notes. The new notes fall halfway between each 8ve:

C2 --- G2 
vF#1 vC#2 
C1 --- G1

A pergen square shows all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and vC#2 bisects it. vG#2 bisects the G2-A2 M2. Every "downed" note bisects an 8ve, a M2, a cm7 (e.g. D1 to C3), a cM9 (e.g. C1 to D3), and many other intervals.

C2 --- G2 --- D3 --- A3 
vF#1 vC#2 vG#2 vD#3 
C1 --- G1 --- D2 --- A2

Splitting the 5th adds notes to the horizontal edges of the square:

C2 vE2 G2 
| . . . . . . | 
C1 vE1 G1

Each downed note also bisects a P11 (e.g. G1-C3), as well as many other intervals.

C3 vE3 G3 
| . . . . . . | 
C2 vE2 G2 
| . . . . . . | 
C1 vE1 G1

From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square.

C2 ---- G2 
| . ^A1 . | 
C1 ---- G1

^A1 also bisects the P12 from C1 to G2.

Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Pierce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.

[[File:pergen_squares.png|alt=pergen squares.png|pergen squares.png]]

A similar chart could be made for all rank-3 pergens, using pergen cubes.

==Notating tunings with an arbitrary generator==

Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.

The next table lists all the ranges for all multigens up to seventh-splits. One can look up one's generator in the first column and find a possible multigen. Use the octave inverse if G > 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | primary choice
! colspan="4" | secondary choices
|-
! | generator
! | possible multigen
! | generator
! | multigen
! | generator
! | multigen
|-
| | 23-60¢
| | M2/4 (requires P8/2)
| |
| |
| |
| |
|-
| | 69-79¢
| | P4/7
| |
| |
| |
| |
|-
| | 80-92¢
| | P4/6
| |
| |
| |
| |
|-
| | 92-103¢
| | P5/7
| |
| |
| |
| |
|-
| | 96-111¢
| | P4/5
| |
| |
| |
| |
|-
| | 108-120¢
| | P5/6
| |
| |
| |
| |
|-
| | 120-138¢
| | P4/4
| |
| |
| |
| |
|-
| | 129-144¢
| | P5/5
| |
| |
| |
| |
|-
| | 160-185¢
| | P4/3
| | 162-180¢
| | P5/4
| |
| |
|-
| | 215-240¢
| | P5/3
| |
| |
| |
| |
|-
| | 240-277¢
| | P4/2
| | 240-251¢
| | P11/7
| | 264-274¢
| | P12/7
|-
| | 280-292¢
| | P11/6
| |
| |
| |
| |
|-
| | 308-320¢
| | P12/6
| |
| |
| |
| |
|-
| | 323-360¢
| | P5/2
| | 336-351¢
| | P11/5
| |
| |
|-
| | 369-384¢
| | P12/5
| |
| |
| |
| |
|-
| | 411-422¢
| | ccP4/7
| |
| |
| |
| |
|-
| | 420-438¢
| | P11/4
| |
| |
| |
| |
|-
| | 435-446¢
| | ccP5/7
| |
| |
| |
| |
|-
| | 462-480¢
| | P12/4
| | 462-480¢
| | M9/3
| |
| |
|-
| | 480-554¢
| | P4 = P5
| | 480-492¢
| | ccP4/6
| | 508-520¢
| | ccP5/6
|-
| | 560-585¢
| | P11/3
| |
| |
| |
| |
|-
| | 576-591¢
| | ccP4/5
| | 583-593¢
| | cccP4/7
| |
| |
|}
There are gaps in the table, especially near 150¢, 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation. But the total range of possible generators is mostly well covered, providing convenient notation options. In particular, if the tuning's generator is just over 720¢, to avoid descending 2nds, instead of calling the generator a 5th, one can call its inverse a quarter-12th. The generator is notated as an up-5th, and four of them make a ccP4.

The table assumes unsplit octaves. Splitting the octave creates alternate generators. For example, if P = P8/3 = 400¢ and G = 300¢, alternate generators are 100¢ and 500¢. Any of these can be used to find a convenient multigen.

See also the [[Map_of_rank-2_temperaments|map of rank-2 temperaments]].

==Pergens and MOS scales==

Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! colspan="3" | MOS scales of 5-12 notes
! |
! |
! |
|-
| | (P8, P5)
| | unsplit
| | 5 = 2L 3s
| | 7 = 5L 2s
| colspan="2" | 12 = 7L 5s (or 5L 7s)
| |
| |
|-
! colspan="2" | halves
! |
! |
! |
! |
! |
! |
|-
| | (P8/2, P5)
| | half-8ve
| | 6 = 2L 4s
| | 8 = 2L 6s
| | 10 = 2L 8s
| colspan="3" | 12 = 2L 10s (or 10L 2s)
|-
| | (P8, P4/2)
| | half-4th
| | 5 = 4L 1s
| | 9 = 5L 4s
| |
| |
| |
| |
|-
| | (P8, P5/2)
| | half-5th
| | 7 = 3L 4s
| | 10 = 7L 3s
| |
| |
| |
| |
|-
| | (P8/2, P4/2)
| | half-everything
| | 6 = 4L 2s
| | 10 = 4L 6s
| |
| |
| |
| |
|-
! colspan="2" | thirds
! |
! |
! |
! |
! |
! |
|-
| | (P8/3, P5)
| | third-8ve
| | 6 = 3L 3s
| | 9 = 3L 6s
| colspan="2" | 12 = 3L 9s (or 9L 3s)
| |
| |
|-
| | (P8, P4/3)
| | third-4th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 1L 6s
| | 8 = 7L 1s
| |
| |
|-
| | (P8, P5/3)
| | third-5th
| | 5 = 1L 4s
| | 6 = 5L 1s
| | 11 = 5L 6s
| |
| |
| |
|-
| | (P8, P11/3)
| | third-11th
| | 5 = 2L 3s
| | 7 = 2L 5s
| | 9 = 2L 7s
| | 11 = 2L 9s
| |
| |
|-
| | (P8/3, P4/2)
| | third-8ve, half-4th
| | 6 = 3L 3s *
| | 9 = 6L 3s
| |
| |
| |
| |
|-
| | (P8/3, P5/2)
| | third-8ve, half-5th
| | 6 = 3L 3s *
| | 9 = 3L 6s
| | 12 = 3L 9s
| |
| |
| |
|-
| | (P8/2, P4/3)
| | half-8ve, third-4th
| | 6 = 2L 4s *
| | 8 = 6L 2s
| |
| |
| |
| |
|-
| | (P8/2, P5/3)
| | half-8ve, third-5th
| | 6 = 4L 2s *
| | 10 = 6L 4s
| |
| |
| |
| |
|-
| | (P8/2, P11/3)
| | half-8ve, third-11th
| | 6 = 2L 4s *
| | 8 = 2L 6s
| | 10 = 2L 8s
| | 12 = 2L 10s
| |
| |
|-
| | (P8/3, P4/3)
| | third-everything
| | 6 = 3L 3s *
| | 9 = 6L 3s *
| |
| |
| |
| |
|-
! colspan="2" | quarters
! |
! |
! |
! |
! |
! |
|-
| | (P8/4, P5)
| | quarter-8ve
| | 8 = 4L 4s
| colspan="2" | 12 = 4L 8s (or 8L 4s)
| |
| |
| |
|-
| | (P8, P4/4)
| | quarter-4th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 1L 6s
| | 8 = 1L 7s
| | 9 = 1L 8s
| | 10 = 9L 1s
|-
| | (P8, P5/4)
| | quarter-5th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 6L 1s
| |
| |
| |
|-
| | (P8, P11/4)
| | quarter-11th
| | 5 = 3L 2s
| | 8 = 3L 5s
| | 11 = 3L 8s
| |
| |
| |
|-
| | (P8, P12/4)
| | quarter-12th
| | 5 = 3L 2s
| | 8 = 5L 3s
| |
| |
| |
| |
|-
| | (P8/4, P4/2)
| | quarter-8ve half-4th
| | 8 = 4L 4s *
| | 12 = 4L 8s
| |
| |
| |
| |
|-
| | (P8/2, M2/4)
| | half-8ve quarter-tone
| | 6 = 2L 4s *
| | 8 = 2L 6s *
| | 10 = 2L 8s
| | 12 = 2L 10s
| |
| |
|-
| | (P8/2, P4/4)
| | half-8ve quarter-4th
| | 6 = 2L 4s *
| | 8 = 2L 6s *
| | 10 = 8L 2s
| |
| |
| |
|-
| | (P8/2, P5/4)
| | half-8ve quarter-5th
| | 6 = 2L 4s *
| | 8 = 6L 2s *
| |
| |
| |
| |
|-
| | (P8/4, P4/3)
| | quarter-8ve third-4th
| | 8 = 4L 4s *
| | 12 = 8L 4s *
| |
| |
| |
| |
|-
| | (P8/4, P5/3)
| | quarter-8ve third-5th
| | 8 = 4L 4s *
| | 12 = 4L 8s *
| |
| |
| |
| |
|-
| | (P8/4, P11/3)
| | quarter-8ve third-11th
| | 8 = 4L 4s *
| | 12 = 4L 8s *
| |
| |
| |
| |
|-
| | (P8/3, P4/4)
| | third-8ve quarter-4th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 9L 3s *
| |
| |
| |
|-
| | (P8/3, P5/4)
| | third-8ve quarter-5th
| | 6 = 3L 3s *
| | 9 = 6L 3s *
| |
| |
| |
| |
|-
| | (P8/3, P11/4)
| | third-8ve quarter-11th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 3L 9s *
| |
| |
| |
|-
| | (P8/3, P12/4)
| | third-8ve quarter-12th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 3L 9s *
| |
| |
| |
|-
| | (P8/4, P4/4)
| | quarter-everything
| | 8 = 4L 4s *
| | 12 = 8L 4s *
| |
| |
| |
| |
|}

A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the simplest, and also one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided.

{| class="wikitable" style="text-align:center;"
|-
! | MOS scale
! colspan="2" | primary example
! | secondary examples
|-
! | Pentatonic
! |
! |
! |
|-
| | 1L 4s
| | (P8, P5/3) [5]
| | third-5th pentatonic
| | third-4th, quarter-4th, quarter-5th
|-
| | 2L 3s
| | (P8, P5) [5]
| | unsplit pentatonic
| | third-11th
|-
| | 3L 2s
| | (P8, P12/4) [5]
| | quarter-12th pentatonic
| | quarter-11th
|-
| | 4L 1s
| | (P8, P4/2) [5]
| | half-4th pentatonic
| |
|-
! | Hexatonic
! |
! |
! |
|-
| | 1L 5s
| | (P8, P4/3) [6]
| | third-4th hexatonic
| | quarter-4th, quarter-5th, fifth-4th, fifth-5th
|-
| | 2L 4s
| | (P8/2, P5) [6]
| | half-8ve hexatonic
| |
|-
| | 3L 3s
| | (P8/3, P5) [6]
| | third-8ve hexatonic
| |
|-
| | 4L 2s
| | (P8/2, P4/2) [6]
| | half-everything hexatonic
| |
|-
| | 5L 1s
| | (P8, P5/3) [6]
| | third-5th hexatonic
| |
|-
! | Heptatonic
! |
! |
! |
|-
| | 1L 6s
| | (P8, P4/3) [7]
| | third-4th heptatonic
| | quarter-4th, fifth-4th, fifth-5th, sixth-4th, sixth-5th
|-
| | 2L 5s
| | (P8, P11/3) [7]
| | third-11th heptatonic
| | fifth-double-compound-4th, sixth-double-compound-5th
|-
| | 3L 4s
| | (P8, P5/2) [7]
| | half-5th heptatonic
| | fifth-12th
|-
| | 4L 3s
| | (P8, P11/5) [7]
| | fifth-11th heptatonic
| | sixth-12th
|-
| | 5L 2s
| | (P8, P5) [7]
| | unsplit heptatonic
| | sixth-double-compound-4th
|-
| | 6L 1s
| | (P8, P5/4) [7]
| | quarter-5th heptatonic
| |
|-
! | Octotonic
! |
! |
! |
|-
| | 1L 7s
| | (P8, P4/4) [8]
| | quarter-4th octotonic
| | fifth-4th, fifth-5th, sixth-4th, sixth-5th, seventh-4th, seventh-5th
|-
| | 2L 6s
| | (P8/2, P5) [8]
| | half-8ve octotonic
| |
|-
| | 3L 5s
| | (P8, P11/4) [8]
| | quarter-11th octotonic
| | seventh-cc4th, seventh-cc5th
|-
| | 4L 4s
| | (P8/4, P5) [8]
| | quarter-8ve octotonic
| |
|-
| | 5L 3s
| | (P8, P12/4) [8]
| | quarter-12th octotonic
| | (very lopsided, unless 5th is quite flat)
|-
| | 6L 2s
| | (P8/2, P4/3) [8]
| | half-8ve third-4th octotonic
| |
|-
| | 7L 1s
| | (P8, P4/3) [8]
| | third-4th octotonic
| |
|-
! | Nonatonic
! |
! |
! |
|-
| | 1L 8s
| | (P8, P4/4) [9]
| | quarter-4th nonatonic
| | fifth-4th, sixth-4th, sixth-5th, seventh-4th/5th, eighth-4th/5th
|-
| | 2L 7s
| | (P8, c3P5/8) [9]
| | eighth-c35th nonatonic
| | third-11th, fifth-cc4th
|-
| | 3L 6s
| | (P8/3, P5) [9]
| | third-8ve nonatonic
| | third-8ve half-5th
|-
| | 4L 5s
| | (P8, P12/7) [9]
| | seventh-12th nonatonic
| | sixth-11th
|-
| | 5L 4s
| | (P8, P4/2) [9]
| | half-4th nonatonic
| | (lopsided unless 4th is sharp), seventh-11th
|-
| | 6L 3s
| | (P8/3, P4/2) [9]
| | third-8ve half-4th nonatonic
| |
|-
| | 7L 2s
| | (P8, ccP5/6)[9]
| | sixth-cc5th nonatonic
| | (lopsided unless 5th is sharp)
|-
| | 8L 1s
| | (P8, P5/5) [9]
| | fifth-5th nonatonic
| |
|-
! | Decatonic
! |
! |
! |
|-
| | 1L 9s
| | (P8, P5/6) [10]
| | sixth-5th decatonic
| | fifth-4th, sixth-4th, seventh-4th/5th, eighth-4th/5th, ninth-4th/5th
|-
| | 2L 8s
| | (P8/2, P5) [10]
| | half-8ve decatonic
| | half-8ve quartertone, half-8ve third-11th
|-
| | 3L 7s
| | (P8, P12/5) [10]
| | fifth-12th decatonic
| | eighth-cc4th, eighth-cc5th
|-
| | 4L 6s
| | (P8/2, P4/2) [10]
| | half-everything decatonic
| |
|-
| | 5L 5s
| | (P8/5, P5) [10]
| | fifth-8ve decatonic
| | (lopsided unless 5th is quite flat)
|-
| | 6L 4s
| | (P8/2, P5/3) [10]
| | half-8ve third-5th decatonic
| |
|-
| | 7L 3s
| | (P8, P5/2) [10]
| | half-5th decatonic
| | ninth-cc5th
|-
| | 8L 2s
| | (P8/2, P4/4) [10]
| | half-8ve quarter-4th decatonic
| | half-8ve quarter-12th
|-
| | 9L 1s
| | (P8, P4/2) [10]
| | quarter-4th decatonic
| |
|}

The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the Sensei aka Sepgu & Ruyoyo generator would be (P8, ccP5/7) [5]. The rationale would be that two Sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be Roulette [7] aka Zozoquingu Nowa [7], with a 2.5.7 pergen (P8, (5/4)/2) = (P8, M3/2) = (P8, M2), where 5·G = A6 = 7/4.

==Pergens and EDOs==

Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored.

Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset.

How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinitely many possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, ccP5/31),... (P8, (i-1,1)/n), where n = 12i+7.

How many edos support a given pergen? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be a multiple of m, and k must be divisible by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.

Given an edo, a period, and a generator, what is the pergen? There is usually more than one right answer. For 10edo with P = 5\10 and G = 2\10, it could be either (P8/2, P4/2) or (P8/2, P5/3). Every coprime period/generator pair results in a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.

'''EDOs Supporting A Pergen'''

This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Both of these edos are incompatible with heptatonic notation, and 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. 6-edo, 11-edo and 23-edo could also be considered ambiguous.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! | supporting edos (12-31 only)
|-
| | (P8, P5)
| | unsplit
| | 12, 13, 13b, 14*, 15*, 16, 17, 18, 18b*, 19, 20*,

21*, 22, 23, 24*, 25*, 26, 27, 28*, 29, 30*, 31
|-
! | halves
! |
! |
|-
| | (P8/2, P5)
| | half-8ve
| | 12, 14, 16, 18, 18b, 20*, 22, 24*, 26, 28*, 30*
|-
| | (P8, P4/2)
| | half-4th
| | 13b, 14, 15*, 18b*, 19, 20*, 23, 24, 25*, 28*, 29, 30*
|-
| | (P8, P5/2)
| | half-5th
| | 13, 14*, 17, 18b, 20*, 21*, 24, 27, 28*, 30*, 31
|-
| | (P8/2, P4/2)
| | half-everything
| | 14, 18b, 20*, 24, 28*, 30*
|-
! | thirds
! |
! |
|-
| | (P8/3, P5)
| | third-8ve
| | 12, 15, 18, 18b*, 21, 24*, 27, 30*
|-
| | (P8, P4/3)
| | third-4th
| | 13b, 14*, 15, 21*, 22, 28*, 29, 30*
|-
| | (P8, P5/3)
| | third-5th
| | 15*, 16, 20*, 21, 25*, 26, 30*, 31
|-
| | (P8, P11/3)
| | third-11th
| | 13, 15, 17, 21, 23, 30*
|-
| | (P8/3, P4/2)
| | third-8ve, half-4th
| | 15, 18b*, 24, 30*
|-
| | (P8/3, P5/2)
| | third-8ve, half-5th
| | 18b, 21, 24, 27, 30
|-
| | (P8/2, P4/3)
| | half-8ve, third-4th
| | 14, 22, 28*, 30
|-
| | (P8/2, P5/3)
| | half-8ve, third-5th
| | 16, 20*, 26, 30*
|-
| | (P8/2, P11/3)
| | half-8ve, third-11th
| | 19, 30
|-
| | (P8/3, P4/3)
| | third-everything
| | 15, 21, 30*
|-
! | quarters
! |
! |
|-
| | (P8/4, P5)
| | quarter-8ve
| | 12, 16, 20, 24*, 28
|-
| | (P8, P4/4)
| | quarter-4th
| | 18b*, 19, 20*, 28, 29, 30*
|-
| | (P8, P5/4)
| | quarter-5th
| | 13, 14*, 20, 21*, 27, 28*
|-
| | (P8, P11/4)
| | quarter-11th
| | 14, 17, 20, 28*, 31
|-
| | (P8, P12/4)
| | quarter-12th
| | 13b, 15*, 18b, 20*, 23, 25*, 28, 30*
|-
| | (P8/4, P4/2)
| | quarter-8ve, half-4th
| | 20, 24, 28
|-
| | (P8/2, M2/4)
| | half-8ve, quarter-tone
| | 18, 20, 22, 24, 26, 28
|-
| | (P8/2, P4/4)
| | half-8ve, quarter-4th
| | 18b, 20*, 28, 30*
|-
| | (P8/2, P5/4)
| | half-8ve, quarter-5th
| | 14, 20, 28*
|-
| | (P8/4, P4/3)
| | quarter-8ve, third-4th
| | 28
|-
| | (P8/4, P5/3)
| | quarter-8ve, third-5th
| | 16, 20
|-
| | (P8/4, P11/3)
| | quarter-8ve, third-11th
| |
|-
| | (P8/3, P4/4)
| | third-8ve, quarter-4th
| | 18b*, 30
|-
| | (P8/3, P5/4)
| | third-8ve, quarter-5th
| | 21, 27
|-
| | (P8/3, P11/4)
| | third-8ve, quarter-11th
| |
|-
| | (P8/3, P12/4)
| | third-8ve, quarter-12th
| | 15, 18b, 30*
|-
| | (P8/4, P4/4)
| | quarter-everything
| | 20, 28
|}
The edos that support the fewest pergens are prime-number edos like 11edo or 13edo. The most "pergen-friendly" edos tend to be ones in which the circle of 5ths doesn't reach every edostep. For example, 24edo supports all half-split pergens, since both P8 and P5 map to an even number of edosteps. 72edo supports all half-splits and all third-splits. 15, 21 and 36 edo support many but not all third-splits (not those with m = 2 or n = 2).

'''Notating a pergen tuned to an EDO'''

If both the pergen and the EDO are notated with ups and downs, what is the relationship between the two kinds of up? If the edo supports the pergen, fully or partially, then the pergen's up equals some multiple of the EDO's up, i.e. some number of edosteps. For third-4th in 22edo or 29edo, the pergen's up = 1 edostep. But in 37edo or 44edo, ^1 = 2 edosteps. For half-8ve in 12edo, ^1 = 0 edosteps, and the ups and downs in the score can simply be ignored. In fact, it seems every pergen in 5edo, 7edo and 12edo has ^1 = 0 edosteps. It's not yet known why.

When notating a piece in a specific pergen meant to be played in a specific EDO, either the pergen notation or the EDO notation can be used. For small or mid-sized EDOs, they're usually identical. If one has to choose, the pergen notation is generally preferred. It's less cluttered. Also, it's easier to mentally double every up and down in larger EDOs than it is to halve them in smaller EDOs.

Half-8ve in 22edo has P = vA4. But in 16edo, P = A4 + 2 edosteps, and ^1 = -2 edosteps. Negative edosteps means up is down, and should be avoided. The notation has tipped, and the period should be notated as ^A4, making ^1 = 2 edostep. Even better would be P = ^4, making ^1 = 1 edostep.

Half-5th has EU = vvA1, hence any EDO in which A1 = 4 edosteps will have ^1 = 2 edosteps. These "doubled EDOs" are 20, 27, 34, 41, 48, 55, etc. The "tripled EDOs" with A1 = 6 edosteps and ^1 = 3 edosteps are every 7th EDO from 30 to 72.

Half-4th has EU = vvm2. Doubled EDOs have m2 = 4 edosteps. These are 23, 28, 33, 38, 43, 48, 53, etc. Tripled EDOs are every 5th one from 47 to 72.

Third-4th has EU = v3A1. Doubled EDOs are the same ones as half-5th's tripled EDOs. Third-5th has EU = v3m2. Doubled EDOs are the same as half-4th's tripled EDOs.

The relationship between a pergen's up and an EDO's up when the pergen uses double-pair notation is complex. Consider half-everything, notated with ups/downs in the perchain and lifts/drops in the genchain. 10edo has ^1 = 1 edostep and /1 = 0 edosteps, thus one simply ignores lifts and drops. But for 24edo, ^1 = 0 edosteps and /1 = 1 edostep. One must ignore ups and downs and convert lifts/drops to edosteps.

'''Pergens Within An EDO'''

See the screenshots in the Supplemental Materials section for a complete list of which pergens are supported by 12, 15 and 19 edo. The list includes multiple pergens per period/generator combination. The next table shows only the simplest pergen for each combination. Combinations are excluded if the period and generator are not coprime, because the scale is contained in a smaller edo. For example, 15edo has no unsplit pergen, because those scales are contained in 5edo. Periods of 1 or 2 edosteps are excluded as too trivial, because the scale is the entire edo. Every step of the edo appears in the genchains, even if the chains are only one step long.

To find the pergen, find the edo, then find the row that corresponds to the period, then find the column that corresponds to the generator. It appears, but has yet to be formally proven, that the number of P/G combinations that N-edo supports is (N-1)/2 rounded down. If we include the trivial combination where P = 1 edostep, the number of combinations would be N/2 rounded up.

{| class="wikitable" style="text-align:center;"
|-
! | EDO
! | Period
! colspan="11" | Generator in edosteps
|-
! |
! | in edosteps
! | 1
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
|-
! | 5
! | 5 = P8
| | P4/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 6
! | 6 = P8
| | P4/2
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 7
! | 7 = P8
| | P4/3
| | P5/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 8
! | 8 = P8
| | P4/3
| | -
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/2
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 9
! | 9 = P8
| | P4/4
| | P4/2
| | -
| | P5
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/3
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 10
! | 10 = P8
| | P4/4
| | -
| | P5/2
| | -
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 5 = P8/2
| | P5
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 11
! | 11 = P8
| | P4/5
| | P5/3
| | P5/2
| | P11/4
| | P5
| |
| |
| |
| |
| |
| |
|-
! | 12
! | 12 = P8
| | P4/5
| | -
| | -
| | -
| | P5
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/2
| | P5
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/3
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/4
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 13b
! | 13 = P8
| | P4/6
| | P4/3
| | P4/2
| | P12/5
| | P12/4
| | P5
| |
| |
| |
| |
| |
|-
! | 14
! | 14 = P8
| | P4/6
| | -
| | P4/2
| | -
| | P11/4
| | -
| |
| |
| |
| |
| |
|-
! | "
! | 7 = P8/2
| | P5
| | P4/3
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 15
! | 15 = P8
| | P4/6
| | P4/3
| | -
| | P12/6
| | -
| | -
| | P11/3
| |
| |
| |
| |
|-
! | "
! | 5 = P8/3
| | P5
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/5
| | P4/3
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 16
! | 16 = P8
| | P4/7
| | -
| | P5/3
| | -
| | P12/5
| | -
| | P5
| |
| |
| |
| |
|-
! | "
! | 8 = P8/2
| | P5
| | -
| | P5/3
| | -
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/4
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 17
! | 17 = P8
| | P4/7
| | P5/5
| | P11/8
| | P11/6
| | P5/2
| | P11/4
| | P5
| | P11/3
| |
| |
| |
|-
! | 18b
! | 18 = P8
| | P4/8
| | -
| | -
| | -
| | P5/2
| | -
| | P12/4
| | -
| |
| |
| |
|-
! | "
! | 9 = P8/2
| | P5
| | P4/4
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/3
| | P5/2
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/6
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 19
! | 19 = P8
| | P4/8
| | P4/4
| | P11/9
| | P4/2
| | P12/6
| | P12/5
| | ccP5/7
| | P5
| | P11/3
| |
| |
|-
! | 20
! | 20 = P8
| | P4/8
| | -
| | P5/4
| | -
| | -
| | -
| | P11/4
| | -
| | c3P5/8
| |
| |
|-
! | "
! | 10 = P8/2
| | M2/4
| | -
| | P5/4
| | -
| | -
| |
| |
| |
| |
| |
| |
|-
! | "
! | 5 = P8/4
| | P4/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/5
| | P5/4
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 21
! | 21 = P8
| | P4/9
| | P5/6
| | -
| | P5/3
| | P11/6
| | -
| | -
| | c3P4/9
| | -
| | P11/3
| |
|-
! | "
! | 7 = P8/3
| | P5/2
| | P5
| | P4/3
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/7
| | P5/3
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 22
! | 22 = P8
| | P4/9
| | -
| | P4/3
| | -
| | P12/7
| | -
| | P12/5
| | -
| | P5
| | -
| |
|-
! | "
! | 11 = P8/2
| | M2/4
| | P5
| | P4/3
| | P12/5
| | P12/7
| |
| |
| |
| |
| |
| |
|-
! | 23
! | 23 = P8
| | P4/10
| | P4/5
| | P11/11
| | P12/9
| | P4/2
| | P12/6
| | ccP4/8
| | ccP4/7
| | P12/4
| | P5
| | P11/3
|-
! | 24
! | 24 = P8
| | P4/10
| | -
| | -
| | -
| | P4/2
| | -
| | P5/2
| | -
| | -
| | -
| | c4P5/10
|-
! | "
! | 12 = P8/2
| | M2/4
| | -
| | -
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
|-
! | "
! | 8 = P8/3
| | P5/2
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/4
| | P4/2
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/6
| | P4/2
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/8
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! |
! |
! | 1
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
|}
Larger edos can have very awkward pergens. For example, 31edo with G = 14/31 is (P8, c5P4/12). It's much simpler to think of the generator as the downfifth = 17\31, and the pergen as the pseudopergen (P8, v5).

'''EDO-pair names'''

Just as a pair of edos and a prime subgroup can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo & N'-edo, m = GCD (N,N'). The period P equals both (N/m)\N and (N'/m)\N'. For example, for 12edo and 16edo, m = 4, and the period is both 3\12 and 4\16.

For each edo, find the nearest '''edomapping''' (patent val) for the 2.3 subgroup. If the edo has a "b" wart, e.g. 13b-edo, use the second-nearest edomapping. Form a 2x2 matrix from these edomappings. Let d be the determinant of this matrix. If d = m, -m or 0, the generator is the 5th, and the pergen is simply (P8/m, P5).

For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.

If |d| ≠ m or 0, P5 is not the generator, and we must find the multigen. Take the ratio of the two edos N/N' and reduce it by m. In the scale tree ([http://tallkite.com/misc_files/Scale-Tree-Complete.pdf pdf] or [http://tallkite.com/misc_files/Scale-Tree-Complete.jpg jpeg]), let g/g' be the smallest ancestor of this ratio. The generator G maps to both g\N and g'\N'. The two edos come closest to coinciding here.

For example, for 7-edo and 17-edo, m = 1 but d = 2, so G ≠ P5. The smallest ancestor of 7/17 is 2/5. G maps to both 2\7 and 5\17, which are both neutral 3rds. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths. 2\7 = 343¢ and 5\17 = 353¢, and their difference is only 1\N", where N" = LCM (N, N'). This 10¢ difference is the least difference between any 7edo note and any 17edo note, except that the two neutral 6ths differ by the same 10¢, and of course some note pairs coincide exactly, such as 0\7 and 0\17.

Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found from the scale tree, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this new vector with either edomapping is zero. Treating this new vector as a monzo, since the dot product is zero, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for. If the comma is (a, b, c), then a·P8 + b·P5 + c·G = 0, and G = (b-a, -b) / c.

For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2).

To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 (Dicot aka Yoyo). 11/9 also works, it yields 243/242 (Mohajira aka Lulu).

If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a multi-EDO pergen.

If the 1st edo is 7edo, the pergen indicates the natural heptatonic notation for the 2nd edo, e.g. 12edo = unsplit, 15edo = third-4th and 17edo = half-5th.

The closer two edos are in the scale tree, the smaller the splitting fractions in the pergen they make. Examples:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 12-edo
! | 13b-edo
! | 14-edo
! | 15-edo
! | 16-edo
! | 17-edo
! | 18b-edo
! | 19-edo
! | 20-edo
|-
! | 13b-edo
| | (P8, P5/7)
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 14-edo
| | (P8/2, P5)
| | (P8, P4/6)
| |
| |
| |
| |
| |
| |
| |
|-
! | 15-edo
| | (P8/3, P5)
| | (P8, c5P5/12)
| | (P8, P4/6)
| |
| |
| |
| |
| |
| |
|-
! | 16-edo
| | (P8/4, P5)
| | (P8, P12/5)
| | (P8/2, P5)
| | (P8, P5/9)
| |
| |
| |
| |
| |
|-
! | 17-edo
| | (P8, P5)
| | (P8, ccP5/11)
| | (P8, P11/4)
| | (P8, P11/3)
| | (P8/2, P4/7)
| |
| |
| |
| |
|-
! | 18b-edo
| | (P8/6, P5)
| | (P8, P12/4)
| | (P8/2, P4/2)
| | (P8/3, P12/4)
| | (P8/2, P5)
| | (P8, P5/10)
| |
| |
| |
|-
! | 19-edo
| | (P8, P5)
| | (P8, P12/10)
| | (P8, P4/2)
| | (P8, P12/6)
| | (P8, P12/5)
| | (P8, P11/3)
| | (P8, P4/8)
| |
| |
|-
! | 20-edo
| | (P8/4, P5)
| | (P8, ccP4/16)
| | (P8/2, P5/4)
| | (P8/5, ^1)
| | (P8/4, P5/3)
| | (P8, P11/4)
| | (P8/2, P4/8)
| | (P8, P4/8)
| |
|-
! | 21-edo
| | (P8/3, P5)
| | (P8, c3P4/9)
| | (P8/7, ^1)
| | (P8/3, P4/3)
| | (P8, P5/3)
| | (P8, P11/6)
| | (P8/3, P5/2)
| | (P8, P11/3)
| | (P8, P5/12)
|-
! | 22-edo
| | (P8/2, P5)
| | (P8, c3P4/15)
| | (P8/2, P4/3)
| | (P8, P4/3)
| | (P8/2, P12/5)
| | (P8, P5)
| | (P8/2, P12/7)
| | (P8, P12/5)
| | (P8/2, M2/4)
|-
! | 23-edo
| | (P8, P4/5)
| | (P8, ccP4/8)
| | (P8, P4/2)
| | (P8, P12/12)
| | (P8, P5)
| | (P8, P12/9)
| | (P8, P12/4)
| | (P8, P12/6)
| | (P8, c5P5/16)
|-
! | 24-edo
| | (P8/12, ^1)
| | (P8, c6P4/14)
| | (P8/2, P4/2)
| | (P8/3, P4/2)
| | (P8/8, P5)
| | (P8, P5/2)
| | (P8/6, P4/2)
| | (P8, P4/2)
| | (P8/4, P4/2)
|}

A specific pergen can be converted to an edo pair by finding the range of its generator cents in the [[pergen#Further Discussion-Notating tunings with an arbitrary generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of edos 7, 10 and 17 defines (P8, P5/2).

==Array Keyboards (unfinished)==

Array keyboards have a 2-dimensional layout of keys, and are very appropriate for rank-2 tunings. A good layout can be found from the tuning's pergen. First find an edo N-edo that is compatible with the pergen, then arrange the keys in N columns to the 8ve, with each row usually containing the multigen interval. The unsplit pergen in 7 columns:

{| class="wikitable" style="text-align:center;"
|-
| | D#
| |
| |
| |
| |
| |
| |
| |
|-
| | D
| | E
| | F#
| | G#
| | A#
| |
| |
| |
|-
| | Db
| | Eb
| | F
| | G
| | A
| | B
| | C#
| | D#
|-
| |
| |
| |
| | Gb
| | Ab
| | Bb
| | C
| | D
|-
| |
| |
| |
| |
| |
| |
| |
| | Db
|}
Higher notes are at the top of each column. The rows would actually be angled so that the two D's are at the same horizontal level. The vertical steps are A1 and the horizontal steps are M2, and the keyboard is defined as 7(A1, M2).

The third-4th keyboard is 7(A1/3 = ^1 = vvA1, P4/3 = vM2 = ^^m2).

{| class="wikitable" style="text-align:center;"
|-
| | vD#
| | ^E
| | F#
| | vG#
| | ^A
| | B
| | vC#
| | ^D
|-
| | ^D
| | E
| | vF#
| | ^G
| | A
| | vB
| | ^C
| | D
|-
| | D
| | vE
| | ^F
| | G
| | vA
| | ^B
| | C
| | vD
|-
| | vD
| | ^Eb
| | F
| | vG
| | ^Ab
| | Bb
| | vC
| | ^Db
|}

''Hypothesis: Let the 5th's keyspan (i.e. column-span) be F. In order for the keyboard to have the pitches in order, the fifth must fall between the two Stern-Brocot ancestors of F\N (simplified if possible). For example, an 8-column keyboard has F = 5, the ancestors of 5\8 are 3\5 and 2\3, and the 5th must be between 720¢ and 800¢. Thus the most musically useful N values are 5, 7, 10, 12 and 14.''

(more to come)

==Supplemental materials==

===Notation guide PDF===

This PDF is a rank-2 notation guide that shows the full lattice for the first 32 pergens, up through the quarter-splits block. It also includes the single-split pergens from the fifth-split, sixth-split and seventh-split blocks. It includes alternate EUs for many pergens.

[http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf '''<big>TallKite.com/misc_files/notation guide for rank-2 pergens.pdf</big>''']

{| class="wikitable" style="text-align:center;"
|+Table of contents for the N'''otation Guide for Rank-2 Pergens''' (* indicates a true double)
|-
! colspan="3" |unsplit
! colspan="3" |quarter-splits
! colspan="3" |single-split fifth-splits
! colspan="3" |single-split seventh-splits
|-
!1
|(P8, P5)
|unsplit
!16
|(P8/4, P5)
|quarter-8ve
!33
|(P8/5, P5)
|fifth-8ve
!96
|(P8/7, P5)
|seventh-8ve
|-
! colspan="3" |half-splits
!17
|(P8, P4/4)
|quarter-4th
!34
|(P8, P4/5)
|fifth-4th
!97
|(P8, P4/7)
|seventh-4th
|-
!2
|(P8/2, P5)
|half-8ve
!18
|(P8, P5/4)
|quarter-5th
!35
|(P8, P5/5)
|fifth-5th
!98
|(P8, P5/7)
|seventh-5th
|-
!3
|(P8, P4/2)
|half-4th
!19
|(P8, P11/4)
|quarter-11th
!36
|(P8, P11/5)
|fifth-11th
!99
|(P8, P11/7)
|seventh-11th
|-
!4
|(P8, P5/2)
|half-5th
!20
|(P8, P12/4)
|quarter-12th
!37
|(P8, P12/5)
|fifth-12th
!100
|(P8, P12/7)
|seventh-12th
|-
!5
|(P8/2, P4/2) *
|half-everything *
!21
|(P8/4, P4/2) *
|quarter-8ve, half-4th *
!38
|(P8, ccP4/5)
|fifth-coco-4th
!101
|(P8, ccP4/7)
|seventh-coco-4th
|-
! colspan="3" |third-splits
!22
|(P8/2, M2/4)
|half-8ve, quarter-tone
! colspan="3" |single-split sixth-splits
!102
|(P8, ccP5/7)
|seventh-coco-5th
|-
!6
|(P8/3, P5)
|third-8ve
!23
|(P8/2, P4/4) *
|half-8ve, quarter-4th *
!64
|(P8/6, P5)
|sixth-8ve
!103
|(P8, c3P4/7)
|seventh-trico-4th
|-
!7
|(P8, P4/3)
|third-4th
!24
|(P8/2, P5/4) *
|half-8ve, quarter-5th *
!65
|(P8, P4/6)
|sixth-4th
| colspan="3" rowspan="9" |
|-
!8
|(P8, P5/3)
|third-5th
!25
|(P8/4, P4/3)
|quarter-8ve, third-4th
!66
|(P8, P5/6)
|sixth-5th
|-
!9
|(P8, P11/3)
|third-11th
!26
|(P8/4, P5/3)
|quarter-8ve, third-5th
!67
|(P8, P11/6)
|sixth-11th
|-
!10
|(P8/3, P4/2)
|third-8ve, half-4th
!27
|(P8/4, P11/3)
|quarter-8ve, third-11th
!68
|(P8, P12/6)
|sixth-12th
|-
!11
|(P8/3, P5/2)
|third-8ve, half-5th
!28
|(P8/3, P4/4)
|third-8ve, quarter-4th
!69
|(P8, ccP4/6)
|sixth-coco-4th
|-
!12
|(P8/2, P4/3)
|half-8ve, third-4th
!29
|(P8/3, P5/4)
|third-8ve, quarter-5th
!70
|(P8, ccP5/6)
|sixth-coco-5th
|-
!13
|(P8/2, P5/3)
|half-8ve, third-5th
!30
|(P8/3, P11/4)
|third-8ve, quarter-11th
| colspan="3" rowspan="3" |
|-
!14
|(P8/2, P11/3)
|half-8ve, third-11th
!31
|(P8/3, P12/4)
|third-8ve, quarter-12th
|-
!15
|(P8/3, P4/3) *
|third-everything *
!32
|(P8/4, P4/4) *
|quarter-everything *
|}Screenshots of the first 2 pages:

[[File:pergens_1.png|alt=pergens 1.png|704x948px|pergens 1.png]]

[[File:pergens_2.png|alt=pergens 2.png|704x760px|pergens 2.png]]

===PergenLister===

PergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonic unisons for each one. Alternate EUs are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair.

http://www.tallkite.com/apps/pergenLister.html (written in Jesusonic, runs inside Reaper or ReaJS)

The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. EUs of a 3rd or more are in red.

Screenshots of the first 69 pergens:

[[File:alt-pergenLister_1.png|alt=alt-pergenLister 1.png|800x427px|alt-pergenLister 1.png]]

[[File:alt-pergenLister_2.png|alt=alt-pergenLister 2.png|800x455px|alt-pergenLister 2.png]]

The first 29 pergens supported by 12edo:

[[File:alt-pergenLister_12edo.png|alt=alt-pergenLister 12edo.png|800x449px|alt-pergenLister 12edo.png]]

Some of the pergens supported by 15edo. A red asterisk means partial support.

[[File:alt-pergenLister_15edo.png|alt=alt-pergenLister 15edo.png|800x493px|alt-pergenLister 15edo.png]]

Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.

[[File:alt-pergenLister_19edo.png|alt=alt-pergenLister 19edo.png|800x459px|alt-pergenLister 19edo.png]]

Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:

If z > 0, then y is at least ceiling (x·(3/2 - z/2)) and at most floor (x·5/3)

If z < 0, then y is at least ceiling (x·3/2) and at most floor (x·(5/3 - z/2))

Next we loop through all combinations of x and z in such a way that larger values of x and z come last:

i = 1; loop (maxFraction,

<ul><li>j = 1; loop (i - 1,<ul><li>makeMapping (i, j); makeMapping (i, -j);</li><li>makeMapping (j, i); makeMapping (j, -i);</li><li>j += 1;</li></ul></li><li>);</li><li>makeMapping (i, i); makeMapping (i, -i);</li><li>i += 1;</li></ul>);

The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.

==Various proofs (unfinished)==

Although not yet rigorously proven, the two false-double tests have been empirically verified by pergenLister.

The interval P8/2 has a "ratio" of the square root of 2, which equals 21/2, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the '''pergen matrix''' [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.

Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise a, b, and n could all be reduced by GCD (a,b), and the multigen could be simplified. Since GCD (a, b) = 1 and -am/b is an integer, it follows that m must be a multiple of |b| as well.

Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|) = |b| · r. If r = 1, then GCD (m, n) = |b|, and vice versa, which is the proposed test for a false double.

If m = |b|, is the pergen explicitly false? Does splitting (a,b) into n generators also split P8 into m periods?

(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5

Because n is a multiple of b, n/b is an integer

M/b = (n/b)·M/n = (n/b)·G 
(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)

Let c and d be the bezout pair of a+b and b, with c·(a+b) + d·b = 1

Since the pergen is a double-split, m > 1, therefore |b| > 1, therefore c ≠ 0

c·(a+b)·P8 = c·b·((n/b)·G - P5) 
(1 - d·b)·P8 = c·b·((n/b)·G - P5) 
P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5) 
P8/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)

Therefore P8 is split into m periods 
Therefore if m = |b|, the pergen is explicitly false

Assume the pergen is a false double, and there's a comma C that splits both P8 and (a,b) appropriately. Can we prove r = 1? Let Q = the higher prime that C uses. Express P, G and C as monzos of the prime subgroup 2.3.Q, by expanding the 2x2 pergen matrix to a 3x3 matrix A:

P = (1/m, 0, 0) 
G = (a/n, b/n, 0) 
C = (u, v, w)

Here u, v and w are integers. If GCD (u, v, w) > 1, simplify C so that it = 1. The inverse of A expresses 2, 3 and Q in terms of P, G and C. If C is tempered out, the C column can be discarded, making the usual 3x2 period-generator mapping. However, if C is not tempered out, the inverse of A is a 3x3 period-generator-comma mapping, which is simply a change of basis. For example, 5-limit JI can be generated by 2/1, 3/2 and 81/80. Here is the inverse of A:

2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) 
3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C) 
Q = Q/1 = ((av-bu)m/wb, -vn/wb, 1/w) · (P, G, C)

Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. Every pergen except the unsplit one requires |w| > 1, so the last column almost always has a fraction. To avoid fractions in the first two columns, A must be unimodular '''''[I think, not sure]''''', and we have wb/mn = ±1, and w = ±mn/b. Substituting for w, we have:

2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) 
3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C) 
Q = Q/1 = (±(av-bu)/n, ±(-v)/m, ±b/mn) · (P, G, C)

For v/m to be an integer, v must equal i·m for some integer i. Likewise, av-bu must equal j·n for some integer j. Thus bu = av - jn = iam - jn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb. Substituting, we get bu = iaprb - jqrb, and u = r(iap - jq). Furthermore, v = im = iprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r > 1, this contradicts the requirement that GCD (u, v, w) = 1, therefore r must be 1, and GCD (m, n) = |b|, and all false doubles pass the false-double test.

Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q?

Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b.

a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)

Thus a·P8 splits into b parts, and since m = |b|, a·P8 splits into m parts. Proceed as before with a bezout pair to find the monzo for P8/m.

Next, assume the pergen isn't explicitly false. The unreduced form is (P8/m, (n - am, -bm) / mn). Substituting in m = pb and n = qb, with p and q coprime, we get (P8/m, (q - ap, -pb) / pqb).

Assume the pergen is a true double, and r > 1. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.

Given a pergen (P8/m, (a,b)/n), how many parts is an arbitrary interval (a',b') split into?

(a',b') = (a'·b, b'·b) / b = (a'·b - a·b', 0) / b + (a·b', b'·b) / b = (a'·b - a·b')·P8 / b + b'·(a,b) / b = (a'·b - a·b')·(m/b)·P + b'·(n/b)·G

Therefore (a',b') is split into GCD (a'·b - a·b')·(m/b), b'·(n/b)) parts.

If m = 1, then b = ±1, and we have GCD (a' ± a·b', b'·n)

If n = 1, then a = -1 and b = 1, and we have GCD (a'·m + b'·m, b') = GCD (a'·m, b')

If m = 1 and n = 1, we have GCD (a', b') = the naturally occurring split.

If m = n (nth-everything), we have n · GCD (a', b')

The multigen and the arbitrary interval can be expressed as gedras:

(a,b) = [k,s] = (-11k+19s, 7k-12s)

(a',b') = [k',s'] = (-11k'+19s', 7k'-12s')

a'·b - a·b' works out to be k·s' - k'·s, and we have GCD ((k·s' - k'·s)·m/b, b'·n/b)

If s is a multiple of n (happens when EU is an A1) and s' is a multiple of n, let s = x·n and s' = y·n

GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')

Thus every such interval is split, e.g. the half-5th pergen splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. This is an easy way to determine if an interval is split or not by the pergen.

To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts

if r > 1, it's a true double

a·P8 = (a,0) = (a,b) - (0,b) = M - b·P12

M = n·G = qrb·G

a·P8 = qrb·G - b·P12 = b·(qr·G - P12)

Let c and d be the bezout pair of a and b, with c·a + d·b = 1

If |b| = 1, let c = 1 and d = ±a, to avoid c = 0

ca·P8 = cb·(qr·G - P12)

(1 - d·b)·P8 = c·b·(qr·G - P12)

P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) = b · (d,-c) + bcqr·G

== Glossary ==
to find the definition, use control-F to search for the first (bolded) occurrence of the word on this page.

pergen 
split 
multigen 
ups and downs (the ^ and v symbols) 
higher prime (any prime > 3) 
color depth 
dependent/independent 
square mapping 
lifts and drops (the / and \ symbols) 
enharmonic unison, EU 
uninflected 
genchain 
perchain 
compound (increased by an octave) 
single-split, double-split 
single-pair, double-pair (number of new accidentals in the notation) 
true double, false double 
explicitly false 
unreduced 
alternate vs. equivalent (generator or period) 
mapping comma 
keyspan 
stepspan 
gedra 
count 
mid 
edomapping 
upspan 
liftspan

chain number 
single-chain 
multi-chain 
arrow comma

==Miscellaneous Notes==

=== Combining pergens ===
Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). If adding a comma to a temperament doesn't change the pergen, it's a strong extension, otherwise it's a weak extension.

General rules for combining pergens:

<ul><li>(P8/m, M/n) + (P8, P5) = (P8/m, M/n)</li>
<li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li>
<li>(P8/m, P5) + (P8/m', P5) = (P8/m", P5), where m" = LCM (m,m')</li>
<li>(P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n')</li></ul>

However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.

If a false double pergen can be broken down into two simpler ones, that may help with finding a double-pair notation with an EU of a 2nd or less. For example, sixth-4th's single pair notation has an EU of a 4th. But since sixth-4th is half-4th plus third-4th, and those two have a good EU, sixth-4th can be notated with one pair from half-4th and another from third-4th.

=== Expanding gedras ===
Gedras can be expanded to 5-limit or higher by including another keyspan that is compatible with 7 and 12, such as 9 or 16. But a more useful approach is for the third number to be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]:

k = 12a + 19b + 28c + 34d 
s = 7a + 11b + 14c + 20d 
g = -c 
r = -d

a = -11k + 19s - 4g + 6r 
b = 7k - 12s + 4g - 2r 
c = -g 
d = -r

Gedras can also be expanded by adding an entry for ups/downs. The entry shows the '''upspan''', which is the number of ups the interval has. Downed intervals have a negative upspan. An entry for '''liftspan''' can also be added. For example, vM3 = [4,2,-1], and ^^\4 = [5,3,2,-1].

=== Height of a pergen ===
The LCM of the pergen's two splitting fractions could be called the '''height''' of the pergen. For example, (P8, P5) has height 1, and (P8/2, M2/4) has height 4. In single-pair notation, the EU's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.

=== Generalizing the pergen ===
See [[User:AthiTrydhen/Abstract pergens]]

=== Credits ===
Pergens were discovered by [[KiteGiedraitis|Kite Giedraitis]] in 2017, and developed with the help of [[Praveen Venkataramana]].

== Addenda (late 2023) ==
=== New terminology===
All temperaments have a '''chain number''', which is the number of fifthchains in the temperament's lattice. Any (P8, P5) temperament has a chain number of 1, and is '''single-chain'''. All other pergens are '''multi-chain'''. For example, Porcupine/Triyoti has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Saguguti has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m * n / f, where M is the multigen and f is the absolute value of M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.

===The EU(s) define the pergen===
The pergen can be derived directly from the EU(s). Thus the EU(s) define both the pergen and the notation. An EU can be thought of as a comma in the 2.3.^ subgroup. One derives a mapping from this comma as one would for any 3-prime comma, and one derives the pergen by inverting the first 2 columns of the mapping.

For example, if the EU is vvd2, the 2.3.^ monzo is [19 -12 -2]. There are many possible mappings, but only one that gives a canonical pergen: [(2 2 7) (0 1 -6)]. Discarding the last column and inverting gives us [(1/2 0) (-1 1)] = (P8/2, P5). Another example: vvA1 = [-11 7 -2]. The mapping [(1 1 -2) (0 2 7)] reduces to [(1 1) (0 2)], which inverts to [(1 0) (-1/2 1/2)] = (P8, P5/2).

One can explore the universe of possible EUs, and thus possible pergen notations, more easily if using the gedra format, expressing the EU as a combination of A1's, d2's and arrows. Thus vvA1 = [1 0 -2], v3m2 = [1 1 -3], etc. Unlike a conventional 2.3.^ monzo, the first two numbers in a A1.d2.^1 monzo are fairly small, and the second number is never negative (since it's an EU). And we can require that the first two numbers be coprime (see the next section). All this facilitates one's search.

===Simplifying a "squared" EU===
Consider an uninflected EU of AA1. AA1 is "squared" in the sense that AA1 = A1 + A1, thus both numbers in its ratio are square numbers. If it had an even upspan (the number of ups), it would obviously be an invalid EU, for if v4AA1 = 0¢, then so does vvA1, and v4AA1 could be replaced with vvA1. So the upspan must be odd.

Consider an EU of v3AA1. The pergen is (P8, P4/3). Here are the [[twin squares]].

<math>
\begin{array} {rrr}
\text{P8} \\
\text{^m2} \\
\text{vvvAA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & {\color {Green}0} \\
8 & -5 & {\color {Green}1} \\
\hline
{\color {Red}-22} & {\color {Red}14} & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & {\color {Red}2} \\
0 & -3 & {\color {Red}-14} \\
\hline
{\color {Green}0} & {\color {Green}-1} & -5 \\
\end{array} \right]
</math>

The two red numbers in the lower left are both even, because AA1 is a squared ratio. As a result, the two red numbers in the upper right must both be even to ensure their rows' dot products with the EU are zero, which is an even number. If we halve all the red numbers and double all the green numbers, all the various row dot products will be unchanged, and the twin squares will remain valid:

<math>
\begin{array} {rrr}
\text{P8} \\
\text{^^m2} \\
\text{vvvA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & {\color {Green}0} \\
8 & -5 & {\color {Green}2} \\
\hline
{\color {Red}-11} & {\color {Red}7} & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & {\color {Red}1} \\
0 & -3 & {\color {Red}-7} \\
\hline
{\color {Green}0} & {\color {Green}-2} & -5 \\
\end{array} \right]
</math>

But while the EU has become simpler, the generator has become more complex in that it is dup not up. This is remedied by adding the EU to it. Changes are in red:

<math>
\begin{array} {rrr}
\text{P8} \\
\text{vM2} \\
\text{vvvA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & 0 \\
{\color {Red}-3} & {\color {Red}2} & {\color {Red}-1} \\
\hline
-11 & 7 & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & 1 \\
0 & -3 & -7 \\
\hline
{\color {Red}0} & {\color {Red}1} & {\color {Red}2} \\
\end{array} \right]
</math>

Following this procedure, it's always possible to simplify a squared (or cubed, etc.) EU.

===Arrow commas===
The '''[[arrow]] comma''' is the ratio that equals the up [[arrow]]. This term overlaps a lot with the term mapping comma, but it isn't quite identical to it. For 5-limit temperaments the arrow comma is almost always 81/80, and for 7-limit it's almost always 64/63. But other commas can occur.

Consider Triyoti/Porcupine which is (P8, P4/3). The vanishing comma or '''VC''' is 250/243, which has a 2.3.5 monzo of [2 -5 3]. The EU is v3A1, which has a 2.3.^ monzo of [-11 7 -3]. The arrow comma or '''AC''' equals an up, therefore it vanishes when downed. The downed AC (or '''vAC''') can be expressed as a 2.3.5.^ monzo. For Triyoti/Porcupine with an EU of v3A1, the vAC is v(81/80) or [-4 4 -1 -1].

===The three commas ===
Thus when we consider a single-comma temperament along with its notation, there are three commas of interest, the VC, the vAC and the EU. In a 5-limit rank-2 temperament, they can all be expressed as 2.3.5.^ monzos.

Any two of these 3 commas determines the third comma. Thus we can approach temperaments and notations from 6 different angles. We can start with any one of these three commas and give it a specific monzo. Then we can choose one of the other two commas, experiment with a variety of specific monzos and examine the results. Let's start with specifying the VC, experimenting with the vAC and seeing what the EU turns out to be.

The EU always equals the VC (possibly inverted) plus or minus some number of vAC's. That number is whatever is needed to eliminate the higher prime. The VC must be inverted if the resulting EU would otherwise have a negative stepspan, or is a diminished unison.

In our Triyoti example, 250/243 plus 3 downed syntonic commas = v3A1. As 2.3.5.^ monzos, we have [1 -5 3 0] + 3·[-4 4 -1 -1] = [-11 7 0 -3]. Note the zeros. The VC always has a zero arrow-count and the EU always has a zero prime-5-count.

Visualizing the three commas in the lattice, one starts at the VC and heads towards the row of 3-limit intervals via the vAC. Wherever one lands is the uninflected EU. One can try other AC's besides 81/80. The AC's prime-5-count must be ±1, so [[135/128|Layobi]] (135/128) or [[Schisma|Layo]] (the schisma) are possibilities. But either of these would lead to a very remote EU (v3AA1 and v3d44 respectively), making a very awkward notation.

Next let's specify the AC, experiment with the VC and see what the EU turns out to be. For 5-limit temperaments, one can require that the AC always be 81/80, and derive the EU (and thus the notation) from the VC. For example, Sagugu/Diaschismic has VC = [11 -4 -2 0]. Subtracting two vAC's makes [19 -12 0 2] = ^^d2. This is in fact the recommended EU for (P8/2, P5).

More examples: Laquinyoti/Magic is (P8, P12/5) and has VC = [-10 -1 5 0]. Adding five vAC's makes [-30 19 0 -5]. This appears to be a AA7 but is actually a negative 2nd. We invert to get [30 -19 0 5] = ^5dd2. Guguti/Bug has VC = 27/25 = [0 3 -2 0]. Subtracting two vAC's makes [8 -5 0 2] = ^^m2. Again, this is the recommended EU.

Let's try a 7-limit temperament with the obvious vAC of 64/63 = [6 -2 -1 -1]. Zozo/Semaphore has VC = 49/48 = [-4 -1 2 0]. Adding two vAC's makes [8 -5 0 -2] = vvm2.{{todo|complete section|comment=apply to multi-comma temperaments|inline=1}}

== Addenda (late 2024) ==

=== Chord names ===
When naming chords, it's very convenient to have the freedom to rename an aug 4th as a dim 5th, or a minor 10th as an aug ninth. Thus for some pergens, an extra pair of accidentals is used. Some examples:

* [[Chords of meantone]] (P8, P5) (^1 = -d2 = pythagorean comma)
* [[Chords of diaschismic]] (P8/2, P5)
* [[Chords of hemififths]] (P8, P5/2) (/1 = vm2 = ~81/80 = ~64/63)
* [[Chords of porcupine]] (P8, P4/3)
* [[Chords of magic]] (P8, P12/5) (/1 = ^^d2)

=== Frequency of imperfect pergens ===
Imperfect pergens occur when there are multiple genchains (i.e. the octave is split), and the fifth is on a different genchain than the tonic, and also on a different perchain. How often do they occur? In order to answer that, we need to survey all pergens in order. But the question of how to do that depends on how they are sorted. The pergenLister app sorts them by the size of the larger denominator. Using this order, pergenLister finds about 4% of all pergens are imperfect. But they can also be sorted by their canonical mappings [(a b) (0 c)]. If sorted by a (octave fraction), and then by |c| (perfect multigen's fraction), more complex pergens appear sooner, and the percentage rises to about 25%.

This table lists all pergens with an unsplit octave up to the fifth-splits. In each column, the pergens are sorted by the size of the generator. The generator is listed followed by a, b and c from its mapping. All pergens with an unsplit octave are perfect.
{| class="wikitable"
|+Pergens of the form (P8, x), showing generator and mapping (a = 1)
!unsplit
!half-splits
!third-splits
!quarter-splits
!fifth-splits
!sixth-splits
|-
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (1 1 1)
|P4/2 (1 2 -2)
|P4/3 (1 2 -3)
|P4/4 (1 2 -4)
|P4/5 (1 2 -5)
|P4/6 (1 2 -6)
|-
|
|P5/2 (1 1 2)
|P5/3 (1 1 3)
|P5/4 (1 1 4)
|P5/5 (1 1 5)
|P5/6 (1 1 6)
|-
|
|
|P11/3 (1 3 -3)
|P11/4 (1 3 -4)
|P11/5 (1 3 -5)
|P11/6 (1 3 -6)
|-
|
|
|
|P12/4 (1 0 4)
|P12/5 (1 0 5)
|P12/6 (1 0 6)
|-
|
|
|
|
|ccP4/5 (1 4 -5)
|ccP4/6 (1 4 -6)
|-
|
|
|
|
|
|ccP5/6 (1 -1 6)
|}
Of all the half-octave pergens, half of every other column (i.e. 25%) are imperfect. Imperfect pergens occur whenever b is not a multiple of a.
{| class="wikitable"
|+Pergens of the form (P8/2, x), showing generator and mapping (a = 2)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (2 2 1)
|'''M2/4 (2 3 2)'''
|P4/3 (2 4 -3)
|'''M2/8 (2 3 4)'''
|P4/5 (2 4 -5)
|'''M2/12 (2 3 6)'''
|-
|
|P4/2 (2 4 -2)
|P5/3 (2 2 3)
|P4/4 (2 4 -4)
|P5/5 (2 2 5)
|P4/6 (2 4 -6)
|-
|
|
|P11/3 (2 6 -3)
|P5/4 (2 2 4)
|P11/5 (2 6 -5)
|P5/6 (2 2 6)
|-
|
|
|
|'''cm7/8 (2 5 -4)'''
|P12/5 (2 0 5)
|P11/6 (2 6 -6)
|-
|
|
|
|
|ccP4/5 (2 8 -5)
|'''cm7/12 (2 5 -6)'''
|-
|
|
|
|
|
|'''cM9/12 (2 1 6)'''
|}
Note that some of these pergens, when put in mingen form, become imperfect. For example, (P8/2, P11/3) becomes (P8/2, M2/6). Also note that for many of these pergens, the generators are comma-sized, and MOS scales will either be very "hard" (L/s very large) or else will contain very many notes per octave. For example, to bring the L/s ratio down to about 5, (P8/2, M2/4) needs a 16 note scale, and (P8/2, P11/3) needs a 28 note scale!

Of all the third-octave pergens, two-thirds of every third column (2/9 or 22%) are imperfect:
{| class="wikitable"
|+Pergens of the form (P8/3, x), showing generator and mapping (a = 3)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (3 3 1)
|P4/2 (3 6 -2)
|'''m3/9 (3 5 -3)'''
|P4/4 (3 6 -4)
|P4/5 (3 6 -5)
|'''m3/18 (3 5 -6)'''
|-
|
|P5/2 (3 3 2)
|'''M6/9 (3 4 3)'''
|P5/4 (3 3 4)
|P5/5 (3 3 5)
|'''M6/18 (3 4 6)'''
|-
|
|
|P4/3 (3 6 -3)
|P11/4 (3 9 -4)
|P11/5 (3 9 -5)
|P4/6 (3 6 -6)
|-
|
|
|
|P12/4 (3 0 4)
|P12/5 (5 0 5)
|P5/6 (3 3 6)
|-
|
|
|
|
|ccP4/5 (3 12 -5)
|'''ccm3/18 (3 7 -6)'''
|-
|
|
|
|
|
|'''ccM6/18 (3 2 6)'''
|}
Of all the quarter-octave pergens, imperfection occurs in half of every 4th column and 3/4 of every 4th column (5/16 or 31.25%).
{| class="wikitable"
|+Pergens of the form (P8/4, x), showing generator and mapping (a = 4)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
!c = ±7
!c = ±8
|-
|P5 (4 4 1)
|'''m6/8 (4 7 -2)'''
|P4/3 (4 8 -3)
|'''M2/8 (4 6 4)'''
|P4/5 (4 8 -5)
|P4/6 (4 8 -6)
|P4/7 (4 8 -7)
|'''M2/16 (4 6 8)'''
|-
|
|P4/2 (4 8 -2)
|P5/3 (4 4 3)
|'''m6/16 (4 7 -4)'''
|P5/5 (4 4 5)
|P5/6 (4 4 6)
|P5/7 (4 4 7)
|'''m6/32 (4 7 -8)'''
|-
|
|
|P11/3 (4 12 -3)
|'''M10/16 (4 5 4)'''
|P11/5 (4 12 -5)
|P11/6 (4 12 -6)
|P11/7 (4 12 -7)
|'''M10/32 (4 5 8)'''
|-
|
|
|
|P4/4 (4 8 -4)
|P12/5 (4 0 5)
|'''m6/24 (4 7 -6)'''
|P12/7 (4 0 7)
|P4/8 (4 8 -8)
|-
|
|
|
|
|ccP4/5 (4 16 -5)
|'''M10/24 (4 5 6)'''
|ccP4/7 (4 16 -7)
|P5/8 (4 4 8)
|-
|
|
|
|
|
|'''ccm6/24 (4 9 -6)'''
|ccP5/7 (4 -4 7)
|'''ccm6/32 (4 9 -8)'''
|-
|
|
|
|
|
|
|c3P4/7 (4 20 -7)
|'''cm7/16 (4 10 -8)'''
|-
|
|
|
|
|
|
|
|'''c3M3/32 (4 3 8)'''
|}
Percentage of imperfect pergens in each category:
{| class="wikitable"
|+
!(P8, x)
!(P8/2, x)
!(P8/3, x)
!(P8/4, x)
!(P8/5, x)
!(P8/6, x)
!(P8/7, x)
|-
|none
|1/4
|2/9
|5/16
|4/25
|5/12
|6/49
|-
|0%
|25%
|22.22%
|31.25%
|16%
|41.67%
|12.24%
|}

== Addenda (Spring 2026) ==

WORK IN PROGRESS

As one stacks generators and octave-reduces, at some point one overshoots or undershoots the octave by an interval of about a quartertone or less. This small interval is the pergen's initial comma. For example, (P8, P5)'s initial comma is the pythagorean comma, its next comma is Mercator's comma, etc. Each comma has a certain genspan, here 12 and 53. The genspan of the initial comma limits the size of a scale one can construct, assuming one wants to avoid overly-small steps. Thus one can have a pythagorean scale of up to 12 notes, but a 13-note scale will have a very small step. Note that the genspan gives the maximum notes per period, not per octave. Assuming one also wants to avoid extreme L/s step ratios also limits the maximum notes per octave.

The table below lists the initial comma of various pergens. "±" indicates a tippy pergen. "c" is the difference between the fifth and 7\12. "abs(6c)" means the absolute value of 6c. The dim 2nd is a pythagorean comma.
{| class="wikitable sortable"
|+Initial comma of each pergen
!#
!pergen
!interval
!cents
!genspan
!notes per octave
|-
!1
!(P8, P5)
|±d2
|abs(12c)
|±12G
|12
|-
!2
!(P8/2, P5)
|±d2/2
|abs(6c)
|±6G
|12
|-
!3
!(P8, P4/2)
|m2/2
|50¢ - 2.5c
|5G
|5
|-
!4
!(P8, P5/2)
|A1/2
|50¢ + 3.5c
|7G
|7
|-
!5
!(P8/2, P4/2)
| colspan="2" | ''same as #3 (P8, P4/2)''
|5G
|10
|-
!6
!(P8/3, P5)
|±d2/3
|abs(4c)
|±4G
|12
|-
!7
!(P8, P4/3)
|A1/3
|33.3¢ + 2.33c
| -7G
|7
|-
!8
!(P8, P5/3)
|m2/3
|33.3¢ - 1.67c
| -5G
|5
|-
!9
!(P8, P11/3)
|M2/3
|66.7¢ + 0.67c
|2G
|2
|-
!10
!(P8/3, P4/2)
|A2/6
|50¢ + 1.5c
|3G
|9
|-
!11
!(P8/3, P5/2)
|m3/6
|50¢ - 0.5c
|1G
|3
|-
!12
!(P8/2, P4/3)
| colspan="2" | ''same as #7 (P8, P4/3)''
| -7G
|14
|-
!13
!(P8/2, P5/3)
| colspan="2" | ''same as #8 (P8, P5/3)''
| -5G
|10
|-
!14
!(P8/2, P11/3)
|M2/6
|33.3¢ + 0.33c
|1G
|2
|-
!15
!(P8/3, P4/3)
| colspan="2" | ''same as #7 (P8, P4/3)''
| -7G
|21
|-
!16
!(P8/4, P5)
|±d2/4
|abs(3c)
|±3G
|12
|-
!17
!(P8, P4/4)
| colspan="2" | ''same as #3 (P8, P4/2)''
|10G
|10
|-
!18
!(P8, P5/4)
|A1/4
|25¢ + 1.75c
|7G
|7
|-
!19
!(P8, P11/4)
|dd3/4
|25¢ - 4.25c
| -17G
|17
|-
!20
!(P8, P12/4)
|m2/4
|25¢ - 1.25c
| -5G
|5
|-
!21
!(P8/4, P4/2)
|M2/4
|50¢ + c/2
|G
|4
|-
!22
!(P8/2, M2/4)
|M2/4
|50¢ + c/2
|G
|2
|-
!23
!(P8/2, P4/4)
|m2/4
|25¢ - 1.25c
|5G
|10
|-
!24
!(P8/2, P5/4)
| colspan="2" |''same as #18 (P8, P5/4)''
|7G
|14
|-
!25
!(P8/4, P4/3)
|d4/12
|33.3¢ - 0.67c
|2G
|8
|}
The initial comma of (P8, P11/3) is a rather large 67¢, but if there are more than 2 notes per 8ve, the L/s ratio becomes enormous!

Note the unusability of certain pergens such as (P8/2, P11/3).

Note the initial comma is often equivalent to the uninflected EU divided by the height. For example, (P8, P4/2) has comma m2/2 and EU vvm2. In other words, the up-arrow is often the initial comma.

This suggests a new algorithm for finding a good EU for a pergen. Search the cents table (in the Notation Guide For rank-2 Pergens pdf, the first table of each pergen) for a small step. The search can be easily done by computer. Then derive the EU from that small step.

For example, the pergen #25 (P8/4, P4/3) has a 33¢ step at 2G - P. Thus ^1 = 2G - P. Multiplying 2G by 3 gives us whole 4ths, and multiplying P by 4 gives us whole octaves. Thus we must multiply the up-arrow by 12 to get a 3-limit interval. Thus 12 ups = 24G - 12P = 8P4 - 3P8 = dim 4th. Thus the EU is v12d4, and ^12C = Fb. But the pergenLister program lists the single-pair notation for this pergen as having an EU of ^12d94. Thus the pergenLister algorithm missed a much simpler EU, and hence a much simpler notation.

True doubles require finding two small steps.

[[Category:Regular temperament theory]]
[[Category:Notation]]
[[Category:Pages with proofs]]

Kite's thoughts on pergens

2026-05-03T21:13:32Z

TallKite: update temperament names with -ti, add an addenda

A '''pergen''' (pronounced "peer-jen") is a way of classifying a [[regular temperament]] solely by its [[periods and generators|period and generator(s)]]. For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. Every rank-2, rank-3, rank-4, etc. temperament has a pergen. Assuming the prime [[subgroup]] includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a fifth or some fraction of some 3-limit interval. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every [[strong extension]] of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but do not uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.

Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka Triyoti]] has the 5th sharpened by one-fifth of [[250/243]] ({{monzo| 1 -5 3 }}). Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Trigu tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name. (Note these uses of comma fractions are not convention universally; temperament pages tend to use comma fractions to indicate the damage to the generator rather than the fifth. This is analogous to indicating the amount of stretching of an edo by the damage to the edostep rather than to the octave. But the latter approach is much more common, because the damage to the octave is much more audible and thus much more musically relevant than the damage to an edostep, which often doesn't correspond to a simple ratio. Likewise for pergens: the damage to Triyoti/Porcupine's generator, which is both 10/9 and 11/10, is less relevant than the damage to Triyoti's 4th or 5th.)

Overwhelmed? See [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf ''Notation guide for rank-2 pergens''] for practical notation examples.

{{See also| Rank-2 temperaments by mapping of 3 }}

= Definition =
If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. Both fractions are always of the form 1/N, thus the octave and/or the 3-limit interval is '''split''' into N parts. The interval which is split into multiple generators is the '''multigen'''. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.

For example, the Srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. Dicot aka Yoyoti (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine aka Triyoti is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore aka Zozoti, a pun on "semi-fourth", is of course half-fourth.

Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both Srutal aka Saguguti and Injera aka Gu & Biruyoti sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using '''ups and downs''' (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and downs notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.

The largest category contains all single-comma rank-2 temperaments with a comma of the form 2x 3y P or 2x 3y P-1, where P is a prime > 3 (a '''higher prime'''), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called '''unsplit'''.

Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multigen, and as a tie-breaker, to minimize the size in cents of the multigen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.

For example, Srutal could be (P8/2, M2/2), but P5 is preferred because it is unsplit. Or it could be (P8/2, P12), but P5 is preferred because it is smaller. Or it could be (P8/2, P4), but P5 is always preferred over P4. Note that P5/2 is not preferred over P4/2. For example, Decimal is (P8/2, P4/2), not (P8/2, P5/2).

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! colspan="4" | example temperaments
|-
! | written
! | spoken
! | comma(s)
! | name
! colspan="2" | [[Color notation|color name]]
|-
| | (P8, P5)
| | unsplit
| | 81/80
| | Meantone
| | Guti
| | gT
|-
| | "
| | "
| | 64/63
| | Archy
| | Ruti
| | rT
|-
| | "
| | "
| | (-14,8,1)
| | Schismic
| | Layoti
| | LyT
|-
| | (P8/2, P5)
| | half-8ve
| | (11, -4, -2)
| | Srutal
| | Saguguti
| | sggT
|-
| | "
| | "
| | 81/80, 50/49
| | Injera
| | Gu & Biruyoti
| | g&rryyT
|-
| | (P8, P5/2)
| | half-5th
| | 25/24
| | Dicot
| | Yoyoti
| | yyT
|-
| | "
| | "
| | (-1,5,0,0,-2)
| | Mohajira
| | Luluti
| | 1uuT
|-
| | (P8, P4/2)
| | half-4th
| | 49/48
| | Semaphore
| | Zozoti
| | zzT
|-
| | (P8/2, P4/2)
| | half-everything
| | 25/24, 49/48
| | Decimal
| | Yoyo & Zozoti
| | yy&zzT
|-
| | (P8, P4/3)
| | third-4th
| | 250/243
| | Porcupine
| | Triyoti
| | y3T
|-
| | (P8, P11/3)
| | third-11th
| | (12,-1,0,0,-3)
| | Satrilu
| | Satriluti
| | s1u3T
|-
| | (P8/4, P5)
| | quarter-8ve
| | (3,4,-4)
| | Diminished
| | Quadguti
| | g4T
|-
| | (P8/2, M2/4)
| | half-8ve quarter-tone
| | (-17,2,0,0,4)
| | Laquadlo
| | Laquadloti
| | L1o4T
|-
| | (P8, P12/5)
| | fifth-12th
| | (-10,-1,5)
| | Magic
| | Laquinyoti
| | Ly5T
|}

(P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum/difference of octaves and fifths, every single 3-limit interval is also split in half.

The multigen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: bi- splits something into two parts, tri- into three parts, etc. For a comma with monzo (a,b,c,d...), the '''color depth''' is GCD (c,d...).

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[Kite's_color_notation|Color notation]] (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yo, (a,b,-1) = gu, (a,b,0,1) = zo, (a,b,0,-1) = ru, (a,b,0,0,1) = lo/ilo, (a,b,0,0,-1) = lu, and (a,b) = wa. Examples: 5/4 = y3 = yo 3rd, 7/5 = zg5 = zogu 5th, etc.

For example, Marvel aka Ruyoyoti (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). But colors can be replaced with ups and downs, to be higher-prime-agnostic. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.

More examples: Trizoguti (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Biruyoti (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, Biruyoti Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.

A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with microtonal accidentals: (P8, P5, ^1, /1,...).

=Derivation=

For any comma, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents, where GCD (0,x) = x. The comma will split the octave into m parts, and if n > m, it will split some 3-limit interval into n parts.

In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is '''dependent''' on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two '''independent''' primes.

For example, consider Sawa & Ruyoyoti (2.3.5.7 with commas 256/243 and 225/224). The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Notating multi-EDO pergens|multi-EDO pergens]] below).

To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [http://x31eq.com/temper/uv.html x31eq.com/temper/uv.html] that will find such a matrix, it's the reduced mapping. Next make a '''square mapping''' by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.

For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.

2/1 = P8 = x·P, thus P = P8/x

3/1 = P12 = y·P + z·G, thus G = [P12 - y·(P8/x)] / z = [-y·P8 + x·P12] / xz = (-y, x) / xz

M's 3-limit monzo is (-y, x), or (y, -x) if z is negative. To get alternate generators, add i periods to G, with i ranging from -x (subtracting a full octave) to +x (adding a full octave).

G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz

'''The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| <= x'''

A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen.

For example, Porcupine aka Triyoti (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 <= i <= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).

Rank-3 pergens are trickier to find. For example, Breedsmic aka Bizozoguti is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7 x31.com] gives us this matrix:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 5/1
! | 7/1
|-
! | period
| | 1
| | 1
| | 1
| | 2
|-
! | gen1
| | 0
| | 2
| | 1
| | 1
|-
! | gen2
| | 0
| | 0
| | 2
| | 1
|}

Thus 2/1 = P, 3/1 = P + 2·G1, 5/1 = P + G1 + 2·G2, and 7/1 = 2·P + G1 + G2. Discard the last column to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 5/1
|-
! | period
| | 1
| | 1
| | 1
|-
! | gen1
| | 0
| | 2
| | 1
|-
! | gen2
| | 0
| | 0
| | 2
|}

Use an [http://wims.unice.fr/wims/wims.cgi?session=GF84B8C7BF.1&lang=en&cmd=reply&module=tool%2Flinear%2Fmatmult.en&matA=1+1+1%0D%0A0+2+1%0D%0A0+0+2&matB=&show=A%5E-1 online tool] to invert it. Here "/4" means that each entry is to be divided by the determinant of the last matrix, which is 4.

{| class="wikitable" style="text-align:center;"
|-
! |
! | period
! | gen1
! | gen2
! |
|-
! | 2/1
| | 4
| | -2
| | -1
| |
|-
! | 3/1
| | 0
| | 2
| | -1
| |
|-
! | 5/1
| | 0
| | 0
| | 2
| | /4
|}

Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25:6)/4.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a double octave to the multigen. The alternate gens are P11/2 and P19/2, both of which are much larger, so the best gen1 is P5/2.

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96:25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128:75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675:512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.

Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 2/1
! | 3/1
! | 7/1
|-
! | period
| | 1
| | 1
| | 2
|-
! | gen1
| | 0
| | 2
| | 1
|-
! | gen2
| | 0
| | 0
| | 1
|}

This inverts to this matrix:

{| class="wikitable" style="text-align:center;"
|-
! |
! | period
! | gen1
! | gen2
! |
|-
! | 2/1
| | 2
| | -1
| | -3
| |
|-
! | 3/1
| | 0
| | 1
| | -1
| |
|-
! | 7/1
| | 0
| | 0
| | 2
| | /2
|}

Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, /1) with /1 = 64/63, rank-3 half-5th. This is far better than (P8, P5/2, (96:25)/4).

The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is > 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible. Using 7 instead of 5 in the pergen is very common for rank-3. See [[User:TallKite/Catalog of eleven-limit rank three temperaments with Color names]] for more examples.

=Applications=

Pergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources.

Another obvious application is to categorize regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), Semihemi is (P8/2, P4/2), Triforce is (P8/3, P4/2), both Tetracot and Semihemififths are (P8, P5/4), Fourfives is (P8/4, P5/5), Pental is (P8/5, P5), and Fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, either because it contains more than 2 primes, or because the split multigen isn't actually a generator. For example, Sensei, or Semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.

Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example Porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first Porcupine is Triyoti, and the second one is Triyo & Ruti. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.

Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See [[pergen#Further Discussion-Chord names and scale names|Chord names and scale names]] below.

The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.

All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and downs notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Luluti and Dicot aka Yoyoti are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.

One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, which is octave-equivalent, fifth-generated and heptatonic. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and all notation used here is backwards compatible.

Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, Schismic aka Layoti is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C vE G. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below.

Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multigen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a [[pergen#Further Discussion-Supplemental materials-Notaion guide PDF|notation guide]]. It even allows every pergen to be numbered.

The '''[[enharmonic unison]]''', or more briefly the '''EU''', can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's EU. The pergen and the EU together define the notation. (''Edited to add: not quite accurate, see the Addenda.'')

The '''genchain''' (chain of generators) in the table is only a short section of the full genchain.

C - G implies ...Eb Bb F C G D A E B F# C#...C - ^Eb=vE - G implies ...F -- ^Ab=vA -- C -- ^Eb=vE -- G -- ^Bb=vB -- D...

If the octave is split, the table has a '''perchain''' ("peer-chain", chain of periods) that shows the octave: C -- vF#=^Gb -- C. Genchains have a theoretically infinite length, but perchains have a finite length. The full rank-2 lattice has genchains running horizontally and perchains running vertically.

{| class="wikitable" style="text-align:center;"
|-
! |
! | pergen
! | enharmonic
unison(s)
! | equivalence(s)
! | split
interval(s)
! | perchain(s) and/or
genchains(s)
! | examples
|-
| | 1
| | (P8, P5)
unsplit
| | none
| | none
| | none
| | C - G
| | Pythagorean, Meantone, Dominant,
Schismic, Mavila, Archy, etc.
|-
! |
! | half-splits
! |
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5)
half-8ve
| | ^^d2 (if 5th
> 700¢
| | ^^C = B#
| | P8/2 = vA4 = ^d5
| | C - vF#=^Gb - C
| | Srutal aka Saguguti
^1 = 81/80
|-
| |
| | "
| | vvd2 (if 5th

< 700¢)
| | ^^C = Db
| | P8/2 = ^A4 = vd5
| | C - ^F#=vGb - C
| | Injera aka Gu & Biruyoti

^1 = 64/63
|-
| |
| | "
| | vvM2
| | ^^C = D
| | P8/2 = ^4 = v5
| | C - ^F=vG - C
| | Thothoti, if 13/8 = M6

^1 = 27/26
|-
| | 3
| | (P8, P4/2)

half-4th
| | vvm2
| | ^^C = Db
| | P4/2 = ^M2 = vm3
| | C - ^D=vEb - F
| | Semaphore aka Zozoti

^1 = 64/63
|-
| |
| | "
| | ^^dd2
| | ^^C = B##
| | P4/2 = vA2 = ^d3
| | C - vD#=^Ebb - F
| | Lala-yoyoti

^1 = 81/80
|-
| | 4
| | (P8, P5/2)

half-5th
| | vvA1
| | ^^C = C#
| | P5/2 = ^m3 = vM3
| | C - ^Eb=vE - G
| | Mohajira aka Luluti

^1 = 33/32
|-
| | 5
| | (P8/2, P4/2)

half-

everything
| | \\m2,

vvA1,

^^\\d2,

vv\\M2
| | //C = Db

^^C = C#

^^//C = D
| | P4/2 = /M2 = \m3

P5/2 = ^m3 = vM3

P8/2 = v/A4 = ^\d5

= ^/4 = v\5
| | C - /D=\Eb - F,

C - ^Eb=vE - G,

C - v/F#=^\Gb - C,

C - ^/F=v\G - C
| | Zozo & Luluti

^1 = 33/32

/1 = 64/63
|-
| |
| | "
| | ^^d2,

\\m2,

vv\\A1
| | ^^ C= B#

//C = Db

^^//C = C#
| | P8/2 = vA4 = ^d5

P4/2 = /M2 = \m3

P5/2 = ^/m3 = v\M3
| | C - vF#=^Gb - C,

C - /D=\Eb - F,

C - ^/Eb=v\E - G
| | Sagugu & Zozoti

^1 = 81/80

/1 = 64/63
|-
| |
| | "
| | ^^d2,

\\A1,

^^\\m2
| | ^^C = B#

//C = C#

^^\\C = B
| | P8/2 = vA4 = ^d5

P5/2 = /m3 = \M3

P4/2 =v/M2 = ^\m3
| | C - vF#=^Gb - C,

C - /Eb=\E - G,

C - v/D=^\Eb - F
| | Sagugu & Luluti

^1 = 81/80

/1 = 33/32
|-
! |
! | third-splits
! |
! |
! |
! |
! |
|-
| | 6
| | (P8/3, P5)

third-8ve
| | ^3d2
| | ^3C = B#
| | P8/3 = vM3 = ^^d4
| | C - vE - ^Ab - C
| | Augmented aka Triguti

^1 = 81/80
|-
| | 7
| | (P8, P4/3)

third-4th
| | v3A1
| | ^3C = C#
| | P4/3 = vM2 = ^^m2
| | C - vD - ^Eb - F
| | Porcupine aka Triyoti

^1 = 81/80
|-
| | 8
| | (P8, P5/3)

third-5th
| | v3m2
| | ^3C = Db
| | P5/3 = ^M2 = vvm3
| | C - ^D - vF - G
| | Slendric aka Latrizoti

^1 = 64/63
|-
| | 9
| | (P8, P11/3)

third-11th
| | ^3dd2
| | ^3C = B##
| | P11/3 = vA4 = ^^dd5
| | C - vF# - ^Cb - F
| | Satriluti, if 11/8 = A4

^1 = 729/704
|-
| |
| | "
| | v3M2
| | ^3C = D
| | P11/3 = ^4 = vv5
| | C - ^F - vC - F
| | Satriluti, if 11/8 = P4

^1 = 33/32
|-
| | 10
| | (P8/3, P4/2)

third-8ve, half-4th
| | v6A2
| | ^6C = D#
| | P8/3 = ^^m3 = v4A4

P4/2 = ^3m2 = v3M3
| | C - ^^Eb - vvA - C

C - ^3Db=v3E - F
| | Tribiloti, if 11/8 = P4

^1 = 33/32
|-
| |
| | "
| | ^3d2,

\\m2
| | ^3C = B#

//C = Db
| | P8/3 = vM3 = ^^d4

P4/2 = /M2 = \m3
| | C - vE - ^Ab - C

C - /D=\Eb - F
| | Triforce aka Trigu & Zozoti

^1 = 81/80, /1 = 64/63
|-
| | 11
| | (P8/3, P5/2)

third-8ve, half-5th
| | ^3d2

\\A1
| | ^3C = B#

//C = C#
| | P8/3 = vM3 = ^^d4

P5/2 = /m3 = \M3
| | C - vE - ^Ab - C

C - /Eb=\E - G
| | Satribizoti

^1 = 49/48, /1 = 343/324
|-
| | 12
| | (P8/2, P4/3)

half-8ve, third-4th
| | ^^d2

\3A1
| | ^^C = Dbb

/3C = C#
| | P8/2 = vA4 = ^d5

P4/3 = \M2 = //m2
| | C - vF#=^Gb - C

C - \D - /Eb - F
| | Latribiruti

^1 = 1029/1024, /1 = 49/48
|-
| | 13
| | (P8/2, P5/3)

half-8ve, third-5th
| | ^6d32
| | ^6C = B#3
| | P8/2 = v3AA4 = ^3dd5

P5/3 = vvA2 = ^4dd3
| | C - v3Fx=^3Gbb C

C - vvD# - ^^Fb - G
| | Latribiyoti

^1 = 81/80
|-
| |
| | "
| | ^^d2,

\3m2
| | ^^C = B#

/3C = Db
| | P8/2 = vA4 = ^d5

P5/3 = /M2 = \\m3
| | C - vF#=^Gb - C

C - /D - \F - G
| | Lemba aka Latrizo & Biruyoti

^1 = (10,-6,1,-1), /1 = 64/63
|-
| | 14
| | (P8/2, P11/3)

half-8ve, third-11th
| | v6M2
| | ^6C = D
| | P8/2 = ^34 = v35

P11/3 = ^^4 = v45
| | C - ^3F=v3G - C

C - ^^F - vvC - F
| | Latribiloti, if 11/8 = P4

^1 = 33/32
|-
| | 15
| | (P8/3, P4/3)

third-

everything
| | v3d2,

\3A1
| | ^3C = Dbb

/3C = C#
| | P8/3 = ^M3 = vvd4

P4/3 = \M2 = //m2

P5/3 = v/M2
| | C - ^E - vAb - C

C - \D - /Eb - F

C - v/D - ^\F - G
| | Triyo & Triguti

^1 = 64/63

/1 = 81/80
|-
| |
| | "
| | ^3d2,

\3m2
| | ^3C = B#

/3C = Db
| | P8/3 = vM3 = ^^d4

P5/3 = /M2 = \\m3

P4/3 = v\M2
| | C - vE - ^Ab - C

C - /D - \F - G

C - v\D - ^/Eb - F
| | Trigu & Latrizoti

^1 = 81/80

/1 = 64/63
|-
| |
| | "
| | v3A1,

\3m2
| | ^3C = C#

/3C = Db
| | P4/3 = vM2 = ^^m2

P5/3 = /M2 = \\m3

P8/3 = v/M3
| | C - vD - ^Eb - F

C - /D - \F - G

C - v/E - ^\Ab - C
| | Triyo & Latrizoti

^1 = 81/80

/1 = 64/63
|-
! |
! | quarter-splits
! |
! |
! |
! |
! |
|-
| | 16
| | (P8/4, P5)
| | ^4d2
| | ^4C = B#
| | P8/4 = vm3 = ^3A2
| | C vEb vvGb=^^F# ^A C
| | Diminished aka Quadguti
|-
| | 17
| | (P8, P4/4)
| | ^4dd2
| | ^4C = B##
| | P4/4 = ^m2 = v3AA1
| | C ^Db ^^Ebb=vvD# vE F
| | Negri aka Laquadyoti
|-
| | 18
| | (P8, P5/4)
| | v4A1
| | ^4C = C#
| | P5/4 = vM2 = ^3m2
| | C vD vvE=^^Eb ^F G
| | Tetracot aka Saquadyoti
|-
| | 19
| | (P8, P11/4)
| | v4dd3
| | ^4C = Eb3
| | P11/4 = ^M3 = v3dd5
| | C ^E ^^G# vDb F
| | Squares aka Laquadruti
|-
| | 20
| | (P8, P12/4)
| | v4m2
| | ^4C = Db
| | P12/4 = v4 = ^3M3
| | C vF vvBb=^^A ^D G
| | Vulture aka Sasa-quadyoti
|-
| |
| | etc.
| |
| |
| |
| |
| |
|}
The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably it isn't particularly complex.

Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢).

==Tipping points==

Removing the ups and downs from an EU makes an '''uninflected''' EU, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s EU is ^^d2, the uninflected EU is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the EU vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore '''up may need to be swapped with down, depending on the size of the 5th''' in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' EUs are upped or downed as if the 5th were just.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every uninflected EU, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the uninflected EU.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | uninflected EU
! | 3-exponent
! | tipping

point edo
! | edo's 5th
! | upping range
! | downing range
! | if the 5th is just
|-
| | M2
| | C - D
| | 2
| | 2-edo
| | 600¢
| | none
| | all
| | downed
|-
| | m3
| | C - Eb
| | -3
| | 3-edo
| | 800¢
| | none
| | all
| | downed
|-
| | m2
| | C - Db
| | -5
| | 5-edo
| | 720¢
| | none
| | all
| | downed
|-
| | A1
| | C - C#
| | 7
| | 7-edo
| | ~686¢
| | 600-686¢
| | 686¢-720¢
| | downed
|-
| | d2
| | C - Dbb
| | -12
| | 12-edo
| | 700¢
| | 700-720¢
| | 600-700¢
| | upped
|-
| | dd3
| | C - Eb3
| | -17
| | 17-edo
| | ~706¢
| | 706-720¢
| | 600-706¢
| | downed
|-
| | dd2
| | C - Db3
| | -19
| | 19-edo
| | ~695¢
| | 695-720¢
| | 600-695¢
| | upped
|-
| | d34
| | C - Fb3
| | -22
| | 22-edo
| | ~709¢
| | 709-720¢
| | 600-709¢
| | downed
|-
| | d32
| | C - Db4
| | -26
| | 26-edo
| | ~692¢
| | 692-720¢
| | 600-692¢
| | upped
|-
| | d44
| | C - Fb4
| | -29
| | 29-edo
| | ~703¢
| | 703-720¢
| | 600-703¢
| | downed
|-
| | d43
| | C - Eb5
| | -31
| | 31-edo
| | ~697¢
| | 697-720¢
| | 600-697¢
| | upped
|-
| | etc.
| |
| |
| |
| |
| |
| |
| |
|}

=Further Discussion=

==Naming very large intervals==

So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, adding an 8ve is indicated by "c" for '''compound''' (a conventional music theory term). Thus 10/3 = cM6 = compound major 6th, 9/2 = ccM2 or cM9, etc. For a pergen with an unsplit octave, the multigen is always some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen M is less than 3 octaves, and can be P4, P5, P11, P12, ccP4 or ccP5. The last one can be spoken as "coco-fifth". Tripe compound can be spoken as "trico".

==Secondary splits==

Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. Porcupine aka Triyoti) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with:

P4/3: C - vD - ^Eb - F

A4:/3 C - D - E - F# (the lack of ups and downs indicates that this interval was already split into 3 parts)

m7/3: C - ^Eb - vG - Bb (because m7 is already split into halves, we also have m7/6: C - vD - ^Eb - F - vG - ^Ab - Bb)

M7/3: C - vE - ^G - B

m10/3: C - F - Bb - Eb (m10 is already split into 3 parts, thus m10/9 also occurs)

M10/3: C - ^F - vB - E

Of course, an equal-step melody can span other intervals besides 3-limit ones. Simply extend or cut short the earlier examples to find other melodies:

^m3/2: C - vD - ^Eb (^m3 = 6/5)

^m6/5: C - vD - ^Eb - F - vG - ^Ab (^m6 = 8/5)

vm9/4: C - ^Eb - vG - Bb - ^Db (vm9 = 32/15)

vM7/2: C - ^F - vB (vM7 = 15/8, probably more harmonious than M7 = 243/128)

More remote intervals include A1, d4, d7 and d10. These unfortunately require a very long genchain. The most interesting melodically is A1: C - ^C - vC# - C#. From C to C# is seven 5ths, which equals 21 generators, so the genchain would have to contain 22 notes if it had no gaps. Note that A1 is the uninflected EU of third-4th. The uninflected EU is always a secondary split.

For a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occurring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.

Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof [[pergen#Further Discussion-Various proofs|below]]). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If only the 8ve is split, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the EU is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. "Every" means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.

The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occurring splits are listed too, under "all pergens". (P8/3, P4/2) has all the secondary splits that (P8/3, P5) and (P8, P4/2) have, plus additional ones.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! | secondary splits of a 12th or less
|-
| colspan="2" | all pergens
| | M3/2, A4/3, d5/2, A5/4, m7/2, M9/2, m10/3, A11/2
|-
! colspan="2" | half-splits
! |
|-
| | (P8/2, P5)
| | half-8ve
| | M2/2, M3/4, A4/6, A5/8, m6/2, M10/2, d12/2, A12/2
|-
| | (P8, P4/2)
| | half-4th
| | m2/2, M6/2, m7/4, A8/2, m10/6, A11/4, P12/2
|-
| | (P8, P5/2)
| | half-5th
| | A1/2, m3/2, M7/2, m9/2, P11/2
|-
| | (P8/2, P4/2)
| | half-everything
| | every 3-limit interval is split twice as much as before
|-
! colspan="2" | third-splits
! |
|-
| | (P8/3, P5)
| | third-8ve
| | m3/3, M6/3, d5/6, A11/3, d12/3
|-
| | (P8, P4/3)
| | third-4th
| | A1/3, m7/6, M7/3, m10/9, M10/3
|-
| | (P8, P5/3)
| | third-5th
| | m2/3, m6/3, M9/6, A8/3, A12/3
|-
| | (P8, P11/3)
| | third-11th
| | M2/3, M3/6, A4/9, A5/12, m9/3, P12/3
|-
| | (P8/3, P4/2)
| | third-8ve half-4th
| | third-8ve splits, half-4th splits, M6/6, m10/6, A11/12
|-
| | (P8/3, P5/2)
| | third-8ve half-5th
| | third-8ve splits, half-5th splits, m3/6, d5/12
|-
| | (P8/2, P4/3)
| | half-8ve third-4th
| | half-8ve splits, third-4th splits, A4/6, M10/6
|-
| | (P8/2, P5/3)
| | half-8ve third-5th
| | half-8ve splits, third-5th splits, m6/6, M9/6, A12/6
|-
| | (P8/2, P11/3)
| | half-8ve third-11th
| | half-8ve splits, third-11th splits, M2/6, M3/12, A4/18, A5/24
|-
| | (P8/3, P4/3)
| | third-everything
| | every 3-limit interval is split three times as much as before
|}

==Singles and doubles==

If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a '''single-split''' pergen. If it has two fractions, it's a '''double-split''' pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called '''single-pair''' notation because it adds only a single pair of accidentals to conventional notation. '''Double-pair''' notation uses both ups/downs and lifts/drops. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the EU for single-pair notation is a 3rd or larger, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs.

In this article, for double-pair notation, the period uses ups and downs, and the generator uses lifts and drops. But the choice of which pair of accidentals is used for what is arbitrary, and ups/downs could be exchanged with lifts/drops.

Every double-split pergen is either a '''true double''' or a '''false double'''. A true double, like third-everything (P8/3, P4/3) or half-8ve quarter-4th (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-8ve quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.

A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.

==Finding an example temperament==

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4⋅P and P8. If P is 6/5, the comma is 4⋅P - P8 = (6/5)4 ÷ (2/1) = 648/625, the diminished temperament. If P is 7/6, the comma is P8 - 4⋅P = (2/1) · (7/6)-4, the Quadru temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.

Another method: if the generator's cents are known, look on the genchain for an interval that approximates a ratio of color depth ±1. Let the interval be I, and the genspan of this interval be x. Then n⋅x gens = n⋅I = x⋅M, where M is the multigen and M/n is the generator. The comma can be found from this equation, if n and x are coprime. For example, suppose (P8, P5/5) has G = 140¢. The genchain is all multiples of 140¢. Looking at the cents, 280¢ is about 7/6. Thus 2G = 7/6, and 10G = 5⋅(7/6) = 2⋅P5. Thus 2⋅P5 - 5⋅(7/6) = 0G = 0¢, and the comma is (3, 7, 0, -5). If the period is split and the generator isn't, use the perchain instead of the genchain. For example, (P8/7, P5) has a period of 141¢. 2 gens = 343¢, about 11/9. Thus 7⋅(11/9) = 2⋅P8, and the comma is (-2, -14, 0, 0, 7), Saseplo.

If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.

Finding the comma(s) for a double-split pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is '''explicitly false'''. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4⋅G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).

If the pergen is not explicitly false, put the pergen in its '''unreduced''' form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n⋅P8 - m⋅M)/nm) = (P8/m, M'/n'). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P < G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2⋅P8 - 3⋅P5) / (3⋅2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This is explicitly false, thus the comma can be found from m3/6 alone. G' is about 50¢, and the comma is 6⋅G' - m3. The comma splits both the octave and the fifth.

This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit. To summarize:

<ul><li>'''A'''<nowiki/> '''double-split pergen is explicitly false if m = |b|, and not explicitly false if m > |b|.'''</li><li>'''A double-split pergen is a true double if and only if neither it nor its unreduced form is explicitly false'''''<nowiki/>'''''<nowiki/>.'''</li><li>'''A double-split pergen is a true double if''' '''GCD (m, n) > |b|,''' '''and a false double if GCD (m, n) = |b|.'''</li></ul>

A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.

Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an '''alternate''' generator. A generator or period plus or minus any number of EUs makes an '''equivalent''' generator or period. An equivalent generator is always the same size in cents, since the EU is always 0¢. An equivalent generator is the same interval, merely notated differently. For example: (P8, P5/2) has generator ^m3 and equivalent generator vM3. Another example, half-8ve (P8/2, P5) has period vA4 and equivalent period ^d5. It has generator P5 and alternate generators P4 and vA1. vA1 is equivalent to ^m2.

Of the two equivalent generators, the preferred generator is always the smaller one (smaller degree, or if the degrees are the same, more diminished). This is because the equivalent generator can be more easily found by adding the EU, rather than subtracting it. For example, ^m3 + vvA1 = vM3 is an easier calculation than vM3 - vvA1 = ^m3. This is particularly true with complex EUs like ^6dd2.

There are also alternate EUs, see below. For double-pair notation, there are also equivalent EUs.

==Ratio and cents of the accidentals==

The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2x · 3y · P±1, where P is a higher prime. They are called mapping commas because they equate or map the ratio P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for prime 5 include 81/80, 135/128, and the schisma aka Layoma = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 (besides 1/1 of course) are the currently employed commas and combinations of them.

If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in Lemba aka Latrizo & Biruyoti, where ^1 equals 64/63 minus 81/80.

Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also [[Pergen#Notating_multi-EDO_pergens|multi-EDO pergens]] pergens below.

Cents can be assigned to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + 7c. Since the EU = 0¢, we can derive the cents of the up symbol. If the EU is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its EU.

In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the EU is an A1.

This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, basic information for playing the score. Examples:
* 15-edo: # = 240¢, ^ = 80¢ (^ = third-sharp)
* 16-edo: # = -75¢
* 17-edo: # = 141¢, ^ = 71¢ (^ = half-sharp)
* 18b-edo: # = -133¢, ^ = 67¢ (^ = half-sharp)
* 19-edo: # = 63¢
* 21-edo: ^ = 57¢ (if used, # = 0¢)
* 22-edo: # = 164¢, ^ = 55¢ (^ = third-sharp)
* quarter-comma Meantone: # = 76¢
* fifth-comma Meantone: # = 84¢
* third-comma Archy aka Ruti: # = 177¢
* eighth-comma Porcupine aka Triyoti: # = 157¢, ^ = 52¢ (^ = third-sharp)
* seventh-comma Srutal aka Sagugu & Zoquadyoti: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #, seventh-comma means 1/7 of 2048/2025)
* third-comma Injera aka Gu & Biruyoti: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)
* eighth-comma Hedgehog aka Triyo & Biruyoti: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)
The last five examples are a generalization of the practice of naming meantone tunings as a fraction of a comma, e.g. quarter-comma. The 5th is sharpened or flattened by some fraction of the first comma in the color name. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both Srutal and Injera are half-8ve, but their optimal tunings are very different.

==Finding a notation for a pergen==

There are multiple notations for a given pergen, depending on the EU(s). Preferably, the EU's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:

<ul><li>For (P8/m, M/n), P8 = m⋅P + x⋅EU and M = n⋅G + y⋅EU', with 0 < |x| <= m/2 and 0 < |y| <= n/2</li><li>x is the count for EU, with EU occurring x times in one octave, and x⋅EU is the octave's '''multi-EU'''</li><li>y is the count for EU', with EU' occurring y times in one multigen, and y⋅EU' is the multigen's multi-EU</li><li>For false doubles using single-pair notation, EU = EU', but x and y are usually different, making different multi-EUs</li><li>The unreduced pergen is (P8/m, M'/n'), with a new enharmonic unison EU" and new counts, P8 = m⋅P + x'⋅EU", and M' = n'⋅G' + y'⋅EU"</li></ul>

The '''keyspan''' of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The '''[[stepspan]]''' of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.

Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a '''gedra''', analogous to a monzo, but written in brackets not parentheses: 3/2 = (-1,1) is a 7-semitone 5th, thus (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval with a monzo (a,b), there is a unique gedra [k,s], and vice versa:

<ul><li>k = 12a + 19b</li><li>s = 7a + 11b</li></ul>The matrix [(12,19) (7,11)] is unimodular, and can be inverted, and (a,b) can be derived from [k,s]:

<ul><li>a = -11k + 19s</li><li>b = 7k - 12s</li></ul>Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].

Gedras greatly facilitate finding a pergen's period, generator and EU(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an EU with the smallest possible keyspan and stepspan, which is the best EU for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up.

For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = m3. The EU can also be found using gedras: x⋅EU = M - n⋅G = P5 - 2⋅m3 = [7,4] - 2⋅[3,2] = [7,4] - [6,4] = [1,0] = A1.

Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. x⋅EU = P8 - m⋅P = P8 - 5⋅M2 = [12,7] - 5⋅[2,1] = [2,2] = 2⋅[1,1] = 2⋅m2. Because x = 2, EU will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2⋅m2 = d3). The EU's '''count''' is 2. The uninflected EU is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus EU = v5m2. Since P8 = 5⋅P + 2⋅EU, the period must be ^^M2, to make the ups and downs come out even. The number of the period's (or generator's) ups or downs always equals the count. Equipped with the period and the EU, the perchain is easily found:

P1 -- ^^M2=v3m3 -- v4 -- ^5 -- ^3M6=vvm7 -- P8
C -- ^^D=v3Eb -- vF -- ^G -- ^3A=vvBb -- C

Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has an uninflected generator [5,3]/5 = [1,1] = m2. The uninflected EU is P4 - 5⋅m2 = [5,3] - 5⋅[1,1] = [5,3] - [5,5] = [0,-2] = -2⋅[0,1] = two descending d2's. The d2 must be upped, and EU = ^5d2. Since P4 = 5⋅G - 2⋅EU, G must be ^^m2. The genchain is:

P1 -- ^^m2=v3A1 -- vM2 -- ^m3 -- ^3d4=vvM3 -- P4C -- ^^Db -- vD -- ^Eb -- vvE -- F

To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and EU from the fractional multigen as before. Then deduce the period from the EU. If the multigen was changed by unreducing, find the original generator from the period and the alternate generator.

For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The uninflected alternate generator G' is [1,1]/10 = [0,0] = P1. The uninflected EU is m2 - 10⋅P1 = m2. It must be downed, thus EU = v10m2. Since m2 = 10⋅G' + EU, G' is ^1. The octave plus or minus some number of EUs must equal 5 periods, thus (P8 + x⋅m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5⋅P + 2⋅EU, and P = ^4M2. Next, find the original half-4th generator. P = P8/5 = ~240¢, and G = P4/2 = ~250¢. Because P < G, G' is not P - G but G - P, and G is not P - G' but P + G', which equals ^4M2 + ^1 = ^5M2. While the alternate multigen is more complex than the original multigen, the alternate generator is usually simpler than the original generator.

P1- - ^4M2=v6m3 -- vv4 -- ^^5 -- ^6M6=v4m7 -- P8
C -- ^4D=v6Eb -- vvF -- ^^G -- ^6A=v4Bb -- C
P1 -- ^5M2=v5m3 -- P4
C -- ^5D=v5Eb -- F

To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. Each pair has its own enharmonic unison. For (P8/2, P4/2), the split octave implies P = vA4 and EU = ^^d2, and the split 4th implies G = /M2 and EU' = \\m2.

A false-double pergen can optionally use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair EU is not a unison or a 2nd, as with (P8/2, P4/3).

Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an EU that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So EU = v4dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The uninflected enharmonic unison is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic unison, we use the second pair of accidentals: EU' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic unisons is also an enharmonic unison: EU + EU' = v4\\d3 = 2·vv\m2, and EU - EU' = v4//ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent EUs, and v\4 and v/d4 are equivalent generators. Here is the genchain:

P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11
C -- ^E=v\F -- /Ab=\A -- ^/C=vDb -- F

One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and EU = v12m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the uninflected enharmonic unison: m3 - 3·m2 = [0,-1] = descending d2. Thus EU' = /3d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonic unisons are found from EU + EU' and EU - 2·EU'. Equivalent periods and generators are found from the many enharmonic unisons, which also allow much freedom in chord spelling. Enharmonic unison = v12m3 = /3d2 = v4/m2 = v4\\A1. Period = \M3 = v44 = //d4. Generator = ^\M3 = v34 = ^//d4.

P1 — \M3 — \\A5=/m6 — P8C — \E — /Ab — CP1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^38=v/m9 — P11
C — ^\E — ^^/Ab=vv\A — v/Db — F

It's not yet known if every pergen can avoid large EUs (those of a 3rd or more) with double-pair notation. One situation in which very large EUs occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the EU's stepspan to equal the multigen's stepspan.

Sixth-4th with single-pair notation has an awkward ^6d64 EU. This pergen might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore commas, aka Triyo & Zozoti), and its double-pair notation can also combine both. Third-4th has EU = v3A1 and G'= vM2 = ^^m2. Half-4th has EU' = \\m2 and G' = /M2 = \m3. G' - G = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G' - G = /M2 - vM2 = ^/1. Equivalent EUs are v3\\M2 and ^3\\d2.

P1 — ^/1=^\m2 — ^^m2=vM2 — /M2=\m3 — ^m3=vvM3 — v/M3=v\4 — P4
C — ^/C=^\Db — ^^Db=vD — /D=\Eb — ^Eb=vvE — v/E=v\F — F

When ups and downs are used to notate edos, a third symbol is used, a '''mid''' , written ~. The mid is only for relative notation (intervals), never for absolute notation (notes). For imperfect intervals, the mid interval is the interval exactly midway between major and minor. In addition, the mid-4th is midway between perfect and augmented, i.e. half-augmented, and the mid-5th is half-diminished. For example, 17edo's 2nds and 3rds run minor-mid-major. This avoids having to constantly choose between the equivalent terms upminor and downmajor. 72edo's 2nds and 3rds run minor, upminor, downmid, mid, upmid, downmajor, major. This makes the terms more concise. Mids can be used in pergen notation too. For single-pair, EU must be an A1, with an even number of downs. Using mids with double-pair notation is trickier. Mids never appear in the perchain. If one accidental pair appears only in the perchain and the other pair only in the genchain, then the mids appear only in the genchain, if at all.

==Alternate enharmonic unisons==

Sometimes the EU found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, ccM6/12), a false double. The uninflected alternate generator is ccM6/12 = [33,19]/12 = [3,2] = m3. The uninflected EU is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The EU becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v12A2, which is an improvement but still awkward. The period is ^4m3 and the generator is v3M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.

P1 -- ^4m3 -- v4M6 -- P8
C -- ^4Eb -- v4A -- C
P1 -- v3M2 -- v6M3=^6m2 -- ^3m3 -- P4
C -- v3D -- v6E=^6Db -- ^3Eb -- F

Because G is a M2 and EU is an A2, the equivalent generator G - EU is a descending A1. Ascending intervals that look descending can be confusing, one has to take into account the eighth-tone ups to see that v3D -- ^6Db is ascending. Double-pair notation may be preferable. This makes P = vM3, EU = ^3d2, G = /m2, and EU' = /4dd2.

P1 -- vM3 -- ^m6 -- P8
C -- vE -- ^Ab -- C
P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4
C -- /Db -- //Ebb=\\D# -- \E -- F

Because the EU found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans, like 5edo or 19edo. Heptatonic stepspans are best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the EU.

To search for an alternate EU, convert the EU to a gedra, then multiply it by the count to get the multi-EU. The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and EU is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-EU. If the stepspan becomes negative, or if it's zero and the keyspan becomes negative, invert the gedra. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new EU. Choose between ups and downs according to whether the 5th falls in the EU's upping or downing range, see [[pergen#Applications-Tipping points|tipping points]] above. Add n'''·'''count ups or downs to the new multi-EU. Add or subtract the new multi-EU from the multigen (or the octave) to get an interval which splits cleanly into n (or m) parts. Each part is the new generator (or period).

For example, (P8, P5/3) has n = 3, G = ^M2, and EU = v3m2 = [1,1]. G is upped only once, so the count is 1, and the multi-EU is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-EU [-2,1]. This can't be simplified, so the new EU is also [-2,1] = d32. Assuming a reasonably just 5th, EU needs to be upped, so EU = ^3d32. Add the multi-EU ^3[-2,1] to the multigen P5 = [7,4] to get ^3[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^3[-2,1] = v3[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·EU to check: 3·vA2 + 1·^3d32 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d32 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- vD# -- ^Fb -- G. Because d32 = -200¢ - 26·c, ^ = (-d32) / 3 = 67¢ + 8.67·c, about a third-tone.

Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one EU (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain EU. Specifically, the comma tempered out should map to the EU, or the multi-EU if the count is > 1. For example, consider Semaphore aka Zozoti (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 EU makes sense. G = ^M2 and the genchain is C -- ^D=vEb -- F. But consider Lala-yoyoti, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus EU = ^^dd2, G = vA2, and the genchain is C -- vD#=^Ebb -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.

Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 EU is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of EU.

For example, Paralimmal aka Satriluti tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the EU is v3M2, but if 11/8 is notated as a vA4, the EU is ^3dd2.

Sometimes the temperament implies an EU that isn't even a 2nd. For example, Liese aka Gu & Trizoguti (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and EU = 3·vd5 - P11 = v3dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- vGb -- ^B -- F.

This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an EU, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different EUs.

==Chord names and staff notation==

Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[Ups and downs notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.

In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G ^A and C E G vBb are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.

Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara aka Ru & Biruyoti (2.3.5.7 with 64/63 and 50/49) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and EU = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C vE G Bb = Cv,7.

A different temperament may result in the same pergen with the same EU, but may still produce a different name for the same chord. For example, Injera aka Gu & Biruyoti (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 EU is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and EU = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G vBb = C,v7.

Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes. This is in fact a requirement for MOS scales.

Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. Iv -- vIIIv -- vVI^m -- Iv. A Porcupine aka Triyoti (third-4th) comma pump can be written out like so: Cv -- vA^m -- vDv -- [vvB=^Bb]^m -- ^Ebv -- G^m -- Gv -- Cv. Brackets are used to show that vvB and ^Bb are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. vvB is written first to show that this root is a vM6 above the previous root, vD. ^Bb is second to show the P4 relationship to the next root, ^Eb. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either vvB^m or ^Bb^m, or possibly ^BbvvM = ^Bb vD ^F.

Lifts and drops, like ups and downs, precede the note head and any sharps or flats. Scores for melody instruments could instead have them above or below the staff. This score uses ups and downs, and has chord names. It's for the third-4th pergen, with EU = v3A1. As noted above, it can be played in EDOs 15, 22 or 29 as is, because ^1 maps to 1 edostep. But to be played in EDOs 30, 37 or 44, ^1 must be interpreted as two edosteps.

Mizarian Porcupine Overture by Herman Miller (P8, P4/3) ^3C = C#

[[File:Mizarian_Porcupine_Overture.png|alt=Mizarian Porcupine Overture.png|800x692px|Mizarian Porcupine Overture.png]]

==Tipping points and sweet spots==

The tipping point for half-octave with a d2 EU is 700¢, 12-edo's 5th. As noted above, the 5th of Pajara (half-8ve) tends to be sharp, thus it has EU = ^^d2. But Injera, also half-8ve, has a flat 5th, and thus EU = vvd2. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But if a single temperament "tips over", either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.

The tipping point depends on the choice of EU. It's not the temperament that tips, it's the notation. Half-8ve could be notated with an EU of vvM2. The tipping point becomes 600¢, a very unlikely 5th, and tipping is impossible. For single-comma temperaments, the EU usually equals the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of EU is a given. However, the mapping of primes 11 and 13 is not agreed on.

The notation's tipping point is determined by the uninflected EU, which is implied by the vanishing comma. For example, Porcupine aka Triyoti's 250/243 comma is an A1 = (-11,7), which implies an uninflected EU of A1, which implies 7-edo, and a 685.7¢ tipping point. Dicot aka Yoyoti's 25/24 comma is also an A1, and has the same tipping point. Semaphore aka Zozoti's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. See [[pergen#Applications-Tipping points|tipping points]] above for a more complete list.

Double-pair notation has two EUs, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two EUs can be any two of A1, m2 or d2. One can choose to use whichever two EUs best avoid tipping.

An example of a temperament that tips easily is Negri aka Laquadyoti, 2.3.5 and (-14,3,4). Because Negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with Negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and the EU could be either ^4dd2 or v4dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an EU of ^4dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma Negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.

Another "tippy" temperament is found by adding the mapping comma 81/80 to the Negri comma and getting the Latriyoma comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th.

==Notating unsplit pergens==

An unsplit pergen doesn't require ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a mapping comma, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (i.e. the monzo has a final exponent of ±1).

The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate. The up symbol's ratio is always the mapping comma, or its inverse.

{| class="wikitable" style="text-align:center;"
|-
! | 5-limit temperament
! | comma
! | sweet spot
! colspan="2" | no ups or downs
! colspan="3" | with ups and downs
! colspan="2" | up symbol
|-
! | (pergen is unsplit)
! |
! | (5th = 700¢ + c)
! | 5/4 is
! | 4:5:6 chord
! | 5/4 is
! | 4:5:6 chord
! | EU
! | ratio
! | cents
|-
| | Meantone aka Guti
| | 81/80 = P1
| | c = -3¢ to -5¢
| | M3
| | C E G
| | ---
| | ---
| | ---
| | ---
| | ---
|-
| | Mavila aka Layobiti
| | 135/128 = A1
| | c = -21¢ to -22¢
| | m3
| | C Eb G
| | ^M3
| | C ^E G
| | ^A1
| | 80/81 = d1
| | -100¢ - 7c = 47¢-54¢
|-
| | Laguti
| | (-15,11,-1) = A1
| | c = -10¢ to -12¢
| | A3
| | C E# G
| | ^M3
| | C ^E G
| | vA1
| | 80/81 = A1
| | 100¢ + 7c = 26¢-30¢
|-
| | Schismic aka Layoti
| | (-15,8,1) = -d2
| | c = 1.7¢ to 2.0¢
| | d4
| | C Fb G
| | vM3
| | C vE G
| | ^d2
| | 81/80 = -d2
| | 12c = 20¢-24¢
|-
| | Lalaguti
| | (-23,16,-1) = -d2
| | c = -0.9¢ to -1.2¢
| | AA2
| | C D## G
| | vM3
| | C vE G
| | vd2
| | 81/80 = d2
| | -12c = 10¢-15¢
|-
| | Father aka Gubiti
| | 16/15 = m2
| | c = 56¢ to 58¢
| | P4
| | C F G
| | vM3
| | C vE G
| | ^m2
| | 81/80 = -m2
| | -100¢ + 5c = 180-190¢
|-
| | Superpyth aka Sasayoti
| | (12,-9,1) = m2
| | c = 9¢ to 10¢
| | A2
| | C D# G
| | vM3
| | C vE G
| | vm2
| | 81/80 = m2
| | 100¢ - 5c = 50-55¢
|}
The Schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch. The Mavila temperament could perhaps be notated without ups and downs, because 5/4 is still a 3rd, and the 4:5:6 triad still looks like a triad.

For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the Schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-limit comma is the pythagorean comma (-19,12).

A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for Archy aka Ruti (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit Schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C vE G vBb.

==Notating rank-3 pergens==

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. Enharmonic unisons aka EUs are like vanishing commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of EUs needed always equals the difference between the notation's rank and the tuning's rank. Examples:

{| class="wikitable" style="text-align:center;"
|-
! | tuning
! | pergen
! | tuning's rank
! | notation
! | notation's rank without any EUs
! | # of EUs needed
! | EUs
|-
| | 12-edo
| | (P8/12)
| | rank-1
| | conventional
| | rank-2
| | 1
| | EU = d2
|-
| | 19-edo
| | (P8/19)
| | rank-1
| | conventional
| | rank-2
| | 1
| | EU = dd2
|-
| | 15-edo
| | (P8/15)
| | rank-1
| | single-pair
| | rank-3
| | 2
| | EU = m2, EU' = v3A1 = v3M2
|-
| | 24-edo
| | (P8/24)
| | rank-1
| | single-pair
| | rank-3
| | 2
| | EU = d2, EU' = vvA1 = vvm2
|-
| | 3-limit JI aka pythagorean
| | (P8, P5)
| | rank-2
| | conventional
| | rank-2
| | 0
| | ---
|-
| | Meantone aka Gu
| | (P8, P5)
| | rank-2
| | conventional
| | rank-2
| | 0
| | ---
|-
| | Diaschismic aka Sagugu
| | (P8/2, P5)
| | rank-2
| | single-pair
| | rank-3
| | 1
| | EU = ^^d2
|-
| | Semaphore aka Zozo
| | (P8, P4/2)
| | rank-2
| | single-pair
| | rank-3
| | 1
| | EU = vvm2
|-
| | Decimal aka Yoyo & Zozo
| | (P8/2, P4/2)
| | rank-2
| | double-pair
| | rank-4
| | 2
| | EU = vvd2, EU' = \\m2 = ^^\\A1
|-
| | 5-limit JI
| | (P8, P5, ^1)
| | rank-3
| | single-pair
| | rank-3
| | 0
| | ---
|-
| | Marvel aka Ruyoyo
| | (P8, P5, ^1)
| | rank-3
| | single-pair
| | rank-3
| | 0
| | ---
|-
| | Breedsmic aka Bizozogu
| | (P8, P5/2, ^1)
| | rank-3
| | double-pair
| | rank-4
| | 1
| | EU = \\dd3
|-
| | 7-limit JI
| | (P8, P5, ^1, /1)
| | rank-4
| | double-pair
| | rank-4
| | 0
| | ---
|}
When there is more than one EU, the first one can be added to or subtracted from the 2nd one, to make an equivalent 2nd EU.

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas. There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.

Even the unsplit rank-3 pergen requires single-pair notation, for the 2nd generator. Single-splits and false double-splits require double-pair, true doubles and false triples require triple-pair, and true triples require quadruple-pair. Some false triples and some true doubles may use quadruple-pair as well, to avoid awkward EUs of a 3rd or more. Rather than devising a third or fourth pair of symbols, and a third or fourth pair of adjectives to describe them, one might simply use colors.

A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.

Some examples of 7-limit rank-3 temperaments:

{| class="wikitable" style="text-align:center;"
|-
! | 7-limit temperament
! | comma
! | pergen
! | spoken pergen
! | notation
! | period
! | gen1
! | gen2
! | EU
|-
| | Marvel aka Ruyoyoti
| | 225/224
| | (P8, P5, ^1)
| | rank-3 unsplit
| | single-pair
| | P8
| | P5
| | ^1 = 81/80
| | ---
|-
| | "
| | "
| | "
| | "
| | double-pair
| | "
| | "
| | "
| | ^^\d2
|-
| | Biruyoti
| | 50/49
| | (P8/2, P5, ^1)
| | rank-3 half-8ve
| | double-pair
| | v/A4 = 10/7
| | P5
| | ^1 = 81/80
| | ^^\\d2
|-
| | Trizoguti
| | 1029/1000
| | (P8, P11/3, ^1)
| | rank-3 third-11th
| | double-pair
| | P8
| | ^\d5 = 7/5
| | ^1 = 81/80
| | ^^^\\\dd3
|-
| | Breedsmic aka Bizozoguti
| | 2401/2400
| | (P8, P5/2, ^1)
| | rank-3 half-5th
| | double-pair
| | P8
| | v//A2 = 60/49
| | /1 = 64/63
| | ^^\4dd3
|-
| | Demeter aka Trizo-aguguti
| | 686/675
| | (P8, P5, \m3/2)
| | half-downminor-3rd
| | double-pair
| | P8
| | P5
| | v/A1 = 15/14
| | ^^\\\dd3
|}
If using single-pair notation, Marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - vE - G - vvA#. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - vE - G - \Bb.

There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, Biruyo is half-8ve, with a d2 comma, like Srutal. Using the same notation as Srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and EU = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as C vE G v/Bb. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C vE G \Bb.

With this standardization, the EU can be derived directly from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)-2 · (64/63)2 · (-19,12). This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the EU never is, therefore the EU = ^^\\d2. The period is found by adding/subtracting the EU from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both EU and P become more complex, but the ratio for /1 becomes simpler.

This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For Biruyoti, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore Biruyoti doesn't tip. Single-pair rank-3 notation has no EU, and thus no tipping point. Double-pair rank-3 notation has 1 EU, but two mapping commas. Rank-3 notations rarely tip.

Unlike the previous examples, Demeter aka Trizo-aguguti's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no EU. But the 4:5:6:7 chord would be spelled C -- ^^^Fbbb -- G -- ^^Bbb, very awkward! Standard double-pair notation is better. Gen2 = v/A1, EU = ^^\\\dd3, and ^^\\\C = A##. Genchain2 is C -- v/C# -- \Eb -- vE -- \\Gb -- v\G -- vvG#=\\\Bbb -- v\\Bb... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own EU, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own EU, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops only for the other kind of accidentals.

There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For Demeter, any combination of \m3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because there will always be a 3-limit comma small enough to be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the '''DOL''' ([[Odd limit|double odd limit]]) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred. And as a tie-breaker, in case of two ratios with the same DOL (such as 5/3 and 6/5), to minimize the size in cents of the multigen2. Since 5/3 = 884.4¢ and 6/5 = 315.6¢, 6/5 is preferred.

If ^1 = 81/80, possible half-split gen2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.

All possible rank-3 pergens can be listed, but the table is much longer than for rank-2 pergens. Here are all the half-split pergens:

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! colspan="4" | prime subgroup used by the pergen
|-
! | unsplit
! colspan="2" | 2.3.5 (^1 = 81/80)
! colspan="2" | 2.3.7 (^1 = 64/63)
|-
| | 1
| | (P8, P5, ^1)
| | rank-3 unsplit
| | same
| | same
|-
! | half-splits
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5, ^1)
| | rank-3 half-8ve
| | same
| | same
|-
| | 3
| | (P8, P4/2, ^1)
| | rank-3 half-4th
| | same
| | same
|-
| | 4
| | (P8, P5/2, ^1)
| | rank-3 half-5th
| | same
| | same
|-
| | 5
| | (P8/2, P4/2, ^1)
| | rank-3 half-everything
| | same
| | same
|-
| | 6
| | (P8, P5, ^m3/2)
| | half-upminor-3rd
| | (P8, P5, ^M2/2)
| | half-upmajor-2nd
|-
| | 7
| | (P8, P5, vM3/2)
| | half-downmajor-3rd
| | (P8, P5, vm3/2)
| | half-downminor-3rd
|-
| | 8
| | (P8, P5, ^m6/2)
| | half-upminor-6th
| | (P8, P5, ^M6/2)
| | half-upmajor-6th
|-
| | 9
| | (P8, P5, vM6/2)
| | half-downmajor-6th
| | (P8, P5, vm7/2)
| | half-downminor-7th
|-
| | 10
| | (P8/2, P5, ^m3/2)
| | half-8ve half-upminor-3rd
| | (P8/2, P5, ^M2/2)
| | half-8ve half-upmajor-2nd
|-
| | 11
| | (P8/2, P5, vM3/2)
| | half-8ve half-downmajor-3rd
| | (P8/2, P5, vm3/2)
| | half-8ve half-downminor-3rd
|-
| | 12
| | (P8, P4/2, vM3/2)
| | half-4th half-downmajor-3rd
| | (P8, P4/2, ^M2/2)
| | half-4th half-upmajor-2nd
|-
| | 13
| | (P8, P4/2, ^m6/2)
| | half-4th half-upminor-6th
| | (P8, P4/2, vm7/2)
| | half-4th half-downminor-7th
|-
| | 14
| | (P8, P5/2, vM3/2)
| | half-5th half-downmajor-3rd
| | (P8, P5/2, ^M2/2)
| | half-5th half-upmajor-2nd
|-
| | 15
| | (P8, P5/2, ^m6/2)
| | half-5th half-upminor-6th
| | (P8, P5/2, vm7/2)
| | half-5th half-downminor-7th
|-
| | 16
| | (P8/2, P4/2, vM3/2)
| | half-everything half-downmajor-3rd
| | (P8/2, P4/2, ^M2/2)
| | half-everything half-upmajor-2nd
|}
There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc.

==Notating multi-EDO pergens==

A multi-EDO pergen is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave, thus it doesn't appear in the pergen. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12. Pergens which imply an edo which doesn't have a decent 5th, e.g. P8/3, P8/4, P8/6, etc., are covered in the next section.

A multi-EDO pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the octave and removes the middle term from the pergen. For example, Blackwood aka Sawati plus Ya is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1). Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:

{| class="wikitable" style="text-align:center;"
|-
! | temperament
! | pergen
! | spoken name
! | enharmonic unisons
! | perchain
! | genchain
! | ^1 ratio
! | /1 ratio
|-
| | Blackwood aka Sawati+ya
| | (P8/5, ^1)
| | rank-2 5-edo
| | EU = m2
| | D E=F G A B=C D
| | D vF#=vG vvB...
| | 81/80 = 16/15
| | ---
|-
| | Whitewood aka Lawati+ya
| | (P8/7, ^1)
| | rank-2 7-edo
| | EU = A1
| | D E F G A B C D
| | D ^F ^^A...
| | 80/81 = 135/128
| | ---
|-
| | 10edo+ya
| | (P8/10, /1)
| | rank-2 10-edo
| | EU = m2, EU' = vvA1 = vvM2
| | D ^D=vE E=F ^F=vG G...
| | D \F#=\G \\B...
| | (see below)
| | 81/80
|-
| | 12edo+la
| | (P8/12, ^1)
| | rank-2 12-edo
| | EU = d2
| | D D#=Eb E F F#=Gb...
| | D ^G ^^C
| | 33/32
| | ---
|-
| | "
| | "
| | "
| | "
| | "
| | D vG#=vAb vvD...
| | 729/704
| | ---
|-
| | 17edo+ya
| | (P8/17, /1)
| | rank-2 17-edo
| | EU = dd3, EU' = vm2 = vvA1
| | D ^D=Eb D#=vE E F...
| | D \F# \\A#=v\\B...
| | 256/243
| | 81/80
|}
If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 10, 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15, or 12/11, or 13/12.

The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.

All multi-EDO pergens are of the form (P8/m, ^1) or (P8/m, /1), and they can be identified solely by their splitting fraction. Multi-EDO pergens are a small minority of rank-2 pergens.

It's possible to have a fifth-8ve pergen with an independent 5th, but there will be very small intervals of about 20¢. Here are two such:

{| class="wikitable" style="text-align:center;"
|-
! | temperament
! | subgroup
! | comma
! | pergen
! | spoken name
! | EU
! | perchain
! | genchain
! | ^1 ratio
|-
| | Laquinzoti
| | 2.3.7
| | (-14,0,0,5)
| | (P8/5, P5)
| | fifth-8ve
| | v5m2
| | D ^^E vG ^A vvC D
| | C G D A E...
| | 49/48
|-
| | Saquinruti
| | 2.3.7
| | (22,-5,0,-5)
| | "
| | "
| | "
| | "
| | "
| | 64/63
|}
Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C ^^D vF ^G vvBb C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section). A multi-EDO pergen should never have an uninflected genchain.

==Notating non-8ve and no-5ths pergens==

In multi-EDO pergens, the 5th is present but not independent. In no-5ths pergens, the 5th is not present, and the prime subgroup doesn't contain 3.

In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any EUs, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.

But in non-8ve and no-5ths pergens, not every name has a note. For example, Biruyoti Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - vF# - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a huge number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk.

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! colspan="6" | prime subgroup used by the pergen
|-
! | unsplit
! | 2.3
! | 2.5 (M3 = 5/4)
! | 2.7 (M2 = 8/7)
! | 3.5 (M6 = 5/3)
! | 3.7 (M3 = 9/7)
! | 5.7 (ccM3 = 5/1, d5 = 7/5)
|-
| | 1
| | (P8, P5)
| | (P8, M3)
| | (P8, M2)
| | (P12, M6)
| | (P12, M3)
| | (ccM3, d5)
|-
! | half-splits
! |
! |
! |
! |
! |
! |
|-
| | 2
| | (P8/2, P5)
| | (P8/2, M3)
| | (P8/2, M2)
| | (P12/2, M6)
| | (P12/2, M3)
| | (M9, d5)*
|-
| | 3
| | (P8, P4/2)
| | (P8, M2)*
| | (P8, M2/2)
| | (P12, M6/2)
| | (P12, M2)*
| | (ccM3, m3)*
|-
| | 4
| | (P8, P5/2)
| | (P8, m6/2)
| | (P8, P5)*
| | (P12, P4)*
| | (P12, m10/2)
| | (ccM3, M7)*
|-
| | 5
| | (P8/2, P4/2)
| | (P8/2, M2)*
| | (P8/2, M2/2)
| | (P12/2, M6/2)
| | (P12/2, M3/2)
| | (M9, m3)*
|-
! | third-splits
! |
! |
! |
! |
! |
! |
|-
| | 6
| | (P8/3, P5)
| | (P8/3, M3)
| | (P8/3, M2)
| | (P12/3, M6)
| | (P12/3, M3)
| | (ccM3/3, d5)
|-
| | 7
| | (P8, P4/3)
| | (P8, M3/3)
| | (P8, M2/3)
| | (P12, M6/3)
| | (P12, M3/3)
| | (ccM3, d5/3)
|-
| | 8
| | (P8, P5/3)
| | (P8, m6/3)
| | (P8, m7/3)
| | (P12, m7/3)
| | (P12, P4)*
| | (ccM3, cA6/3)
|-
| | 9
| | (P8, P11/3)
| | (P8, M10/3)
| | (P8, M9/3)
| | (P12, ccM3/3)
| | (P12, cM7/3)
| | (ccM3, ccm7/3)
|-
| | 10
| | (P8/3, P4/2)
| | (P8/3, M2)*
| | (P8/3, M2/2)
| | (P12/3, M6/2)
| | (P12/3, M2)*
| | (ccM3/3, m3)*
|-
| | 11
| | (P8/3, P5/2)
| | (P8/3. m6/2)
| | (P8/3, P5)*
| | (P12/3, P4)*
| | (P12/3, m10/2)
| | (ccM3/3, M7)*
|-
| | 12
| | (P8/2, P4/3)
| | (P8/2, M3/3)
| | (P8/2, M2/3)
| | (P12/2, M6/3)
| | (P12/2, M3/3)
| | (M9, d5/3)*
|-
| | 13
| | (P8/2, P5/3)
| | (P8/2, m6/3)
| | (P8/2, m7/3)
| | (P12/2, m7/3)
| | (P12/2, P4)*
| | (M9, cA6/3)*
|-
| | 14
| | (P8/2, P11/3)
| | (P8/2, M10/3)
| | (P8/2, M9/3)
| | (P12/2, ccM3/3)
| | (P12/2, cM7/3)
| | (M9, ccm7/3)*
|-
| | 15
| | (P8/3, P4/3)
| | (P8/3, M3/3)
| | (P8/3, M2/3)
| | (P12/3, M6/3)
| | (P12/3, P4)*
| | (ccM3/3, d5/3)
|}
For prime subgroup p.q, the unsplit pergen has period p/1. The unsplit pergen's generator is found by dividing q by p until it's less than p/1, and period-inverting if it's more than half of p/1.

Multi-EDO pergens also appear in this table. For example, in row #33, (P8/5, ^1) replaces both 2.5 (P8/5, M3) and 2.7 (P8/5, M2). This has the advantage of avoiding missing notes. Since one fifth of 5/1 is only 25¢ from 7/5, the 5.7 (ccM3/5, d5) can optionally be replaced too.

{| class="wikitable" style="text-align:center;"
|-
! | pergen number
! | 2.3
! | 2.5
! | 2.7
! | 3.5
! | 3.7
! | 5.7
|-
| | 33
| | (P8/5, P5)
| | (P8/5, ^1)
| | (P8/5, ^1)
| | (P12/5, M6)
| | (P12/5, M3)
| | (ccM3/5, ^1)
|}

Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the first 3.5 pergen, (P12, M6). The note one M6 above C1 is A1, and the note three P12's above C1 is A5. (There is no A2, A3 or A4.) Referring to a note as simply A is ambiguous.

Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table could be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2), with ^M2 = 8/7 and ^1 = √256/245 = √(81/80 * 64/63 * 64/63) = about 38¢. The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3), with ^M3 = 9/7 and ^1 = √6561/6125 = √[(81/80)3 * (64/63)2] = about 60¢.

==Pergen squares==

Pergen squares, which were discovered by Praveen Venkataramana, are a way to visualize pergens squares in a way that isn't specific to any primes at all. To understand them, let's assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.

For (P8, P5), the pergen square has 4 notes, shown here with octave numbers (ignore the periods).

C2 -- G2 
| . . . . | 
C1 -- G1

Splitting the 8ve or the multigen adds notes to the square. For (P8/2, P5), there are 6 notes. The new notes fall halfway between each 8ve:

C2 --- G2 
vF#1 vC#2 
C1 --- G1

A pergen square shows all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and vC#2 bisects it. vG#2 bisects the G2-A2 M2. Every "downed" note bisects an 8ve, a M2, a cm7 (e.g. D1 to C3), a cM9 (e.g. C1 to D3), and many other intervals.

C2 --- G2 --- D3 --- A3 
vF#1 vC#2 vG#2 vD#3 
C1 --- G1 --- D2 --- A2

Splitting the 5th adds notes to the horizontal edges of the square:

C2 vE2 G2 
| . . . . . . | 
C1 vE1 G1

Each downed note also bisects a P11 (e.g. G1-C3), as well as many other intervals.

C3 vE3 G3 
| . . . . . . | 
C2 vE2 G2 
| . . . . . . | 
C1 vE1 G1

From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square.

C2 ---- G2 
| . ^A1 . | 
C1 ---- G1

^A1 also bisects the P12 from C1 to G2.

Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Pierce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.

[[File:pergen_squares.png|alt=pergen squares.png|pergen squares.png]]

A similar chart could be made for all rank-3 pergens, using pergen cubes.

==Notating tunings with an arbitrary generator==

Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.

The next table lists all the ranges for all multigens up to seventh-splits. One can look up one's generator in the first column and find a possible multigen. Use the octave inverse if G > 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | primary choice
! colspan="4" | secondary choices
|-
! | generator
! | possible multigen
! | generator
! | multigen
! | generator
! | multigen
|-
| | 23-60¢
| | M2/4 (requires P8/2)
| |
| |
| |
| |
|-
| | 69-79¢
| | P4/7
| |
| |
| |
| |
|-
| | 80-92¢
| | P4/6
| |
| |
| |
| |
|-
| | 92-103¢
| | P5/7
| |
| |
| |
| |
|-
| | 96-111¢
| | P4/5
| |
| |
| |
| |
|-
| | 108-120¢
| | P5/6
| |
| |
| |
| |
|-
| | 120-138¢
| | P4/4
| |
| |
| |
| |
|-
| | 129-144¢
| | P5/5
| |
| |
| |
| |
|-
| | 160-185¢
| | P4/3
| | 162-180¢
| | P5/4
| |
| |
|-
| | 215-240¢
| | P5/3
| |
| |
| |
| |
|-
| | 240-277¢
| | P4/2
| | 240-251¢
| | P11/7
| | 264-274¢
| | P12/7
|-
| | 280-292¢
| | P11/6
| |
| |
| |
| |
|-
| | 308-320¢
| | P12/6
| |
| |
| |
| |
|-
| | 323-360¢
| | P5/2
| | 336-351¢
| | P11/5
| |
| |
|-
| | 369-384¢
| | P12/5
| |
| |
| |
| |
|-
| | 411-422¢
| | ccP4/7
| |
| |
| |
| |
|-
| | 420-438¢
| | P11/4
| |
| |
| |
| |
|-
| | 435-446¢
| | ccP5/7
| |
| |
| |
| |
|-
| | 462-480¢
| | P12/4
| | 462-480¢
| | M9/3
| |
| |
|-
| | 480-554¢
| | P4 = P5
| | 480-492¢
| | ccP4/6
| | 508-520¢
| | ccP5/6
|-
| | 560-585¢
| | P11/3
| |
| |
| |
| |
|-
| | 576-591¢
| | ccP4/5
| | 583-593¢
| | cccP4/7
| |
| |
|}
There are gaps in the table, especially near 150¢, 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation. But the total range of possible generators is mostly well covered, providing convenient notation options. In particular, if the tuning's generator is just over 720¢, to avoid descending 2nds, instead of calling the generator a 5th, one can call its inverse a quarter-12th. The generator is notated as an up-5th, and four of them make a ccP4.

The table assumes unsplit octaves. Splitting the octave creates alternate generators. For example, if P = P8/3 = 400¢ and G = 300¢, alternate generators are 100¢ and 500¢. Any of these can be used to find a convenient multigen.

See also the [[Map_of_rank-2_temperaments|map of rank-2 temperaments]].

==Pergens and MOS scales==

Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! colspan="3" | MOS scales of 5-12 notes
! |
! |
! |
|-
| | (P8, P5)
| | unsplit
| | 5 = 2L 3s
| | 7 = 5L 2s
| colspan="2" | 12 = 7L 5s (or 5L 7s)
| |
| |
|-
! colspan="2" | halves
! |
! |
! |
! |
! |
! |
|-
| | (P8/2, P5)
| | half-8ve
| | 6 = 2L 4s
| | 8 = 2L 6s
| | 10 = 2L 8s
| colspan="3" | 12 = 2L 10s (or 10L 2s)
|-
| | (P8, P4/2)
| | half-4th
| | 5 = 4L 1s
| | 9 = 5L 4s
| |
| |
| |
| |
|-
| | (P8, P5/2)
| | half-5th
| | 7 = 3L 4s
| | 10 = 7L 3s
| |
| |
| |
| |
|-
| | (P8/2, P4/2)
| | half-everything
| | 6 = 4L 2s
| | 10 = 4L 6s
| |
| |
| |
| |
|-
! colspan="2" | thirds
! |
! |
! |
! |
! |
! |
|-
| | (P8/3, P5)
| | third-8ve
| | 6 = 3L 3s
| | 9 = 3L 6s
| colspan="2" | 12 = 3L 9s (or 9L 3s)
| |
| |
|-
| | (P8, P4/3)
| | third-4th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 1L 6s
| | 8 = 7L 1s
| |
| |
|-
| | (P8, P5/3)
| | third-5th
| | 5 = 1L 4s
| | 6 = 5L 1s
| | 11 = 5L 6s
| |
| |
| |
|-
| | (P8, P11/3)
| | third-11th
| | 5 = 2L 3s
| | 7 = 2L 5s
| | 9 = 2L 7s
| | 11 = 2L 9s
| |
| |
|-
| | (P8/3, P4/2)
| | third-8ve, half-4th
| | 6 = 3L 3s *
| | 9 = 6L 3s
| |
| |
| |
| |
|-
| | (P8/3, P5/2)
| | third-8ve, half-5th
| | 6 = 3L 3s *
| | 9 = 3L 6s
| | 12 = 3L 9s
| |
| |
| |
|-
| | (P8/2, P4/3)
| | half-8ve, third-4th
| | 6 = 2L 4s *
| | 8 = 6L 2s
| |
| |
| |
| |
|-
| | (P8/2, P5/3)
| | half-8ve, third-5th
| | 6 = 4L 2s *
| | 10 = 6L 4s
| |
| |
| |
| |
|-
| | (P8/2, P11/3)
| | half-8ve, third-11th
| | 6 = 2L 4s *
| | 8 = 2L 6s
| | 10 = 2L 8s
| | 12 = 2L 10s
| |
| |
|-
| | (P8/3, P4/3)
| | third-everything
| | 6 = 3L 3s *
| | 9 = 6L 3s *
| |
| |
| |
| |
|-
! colspan="2" | quarters
! |
! |
! |
! |
! |
! |
|-
| | (P8/4, P5)
| | quarter-8ve
| | 8 = 4L 4s
| colspan="2" | 12 = 4L 8s (or 8L 4s)
| |
| |
| |
|-
| | (P8, P4/4)
| | quarter-4th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 1L 6s
| | 8 = 1L 7s
| | 9 = 1L 8s
| | 10 = 9L 1s
|-
| | (P8, P5/4)
| | quarter-5th
| | 5 = 1L 4s
| | 6 = 1L 5s
| | 7 = 6L 1s
| |
| |
| |
|-
| | (P8, P11/4)
| | quarter-11th
| | 5 = 3L 2s
| | 8 = 3L 5s
| | 11 = 3L 8s
| |
| |
| |
|-
| | (P8, P12/4)
| | quarter-12th
| | 5 = 3L 2s
| | 8 = 5L 3s
| |
| |
| |
| |
|-
| | (P8/4, P4/2)
| | quarter-8ve half-4th
| | 8 = 4L 4s *
| | 12 = 4L 8s
| |
| |
| |
| |
|-
| | (P8/2, M2/4)
| | half-8ve quarter-tone
| | 6 = 2L 4s *
| | 8 = 2L 6s *
| | 10 = 2L 8s
| | 12 = 2L 10s
| |
| |
|-
| | (P8/2, P4/4)
| | half-8ve quarter-4th
| | 6 = 2L 4s *
| | 8 = 2L 6s *
| | 10 = 8L 2s
| |
| |
| |
|-
| | (P8/2, P5/4)
| | half-8ve quarter-5th
| | 6 = 2L 4s *
| | 8 = 6L 2s *
| |
| |
| |
| |
|-
| | (P8/4, P4/3)
| | quarter-8ve third-4th
| | 8 = 4L 4s *
| | 12 = 8L 4s *
| |
| |
| |
| |
|-
| | (P8/4, P5/3)
| | quarter-8ve third-5th
| | 8 = 4L 4s *
| | 12 = 4L 8s *
| |
| |
| |
| |
|-
| | (P8/4, P11/3)
| | quarter-8ve third-11th
| | 8 = 4L 4s *
| | 12 = 4L 8s *
| |
| |
| |
| |
|-
| | (P8/3, P4/4)
| | third-8ve quarter-4th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 9L 3s *
| |
| |
| |
|-
| | (P8/3, P5/4)
| | third-8ve quarter-5th
| | 6 = 3L 3s *
| | 9 = 6L 3s *
| |
| |
| |
| |
|-
| | (P8/3, P11/4)
| | third-8ve quarter-11th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 3L 9s *
| |
| |
| |
|-
| | (P8/3, P12/4)
| | third-8ve quarter-12th
| | 6 = 3L 3s *
| | 9 = 3L 6s *
| | 12 = 3L 9s *
| |
| |
| |
|-
| | (P8/4, P4/4)
| | quarter-everything
| | 8 = 4L 4s *
| | 12 = 8L 4s *
| |
| |
| |
| |
|}

A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the simplest, and also one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided.

{| class="wikitable" style="text-align:center;"
|-
! | MOS scale
! colspan="2" | primary example
! | secondary examples
|-
! | Pentatonic
! |
! |
! |
|-
| | 1L 4s
| | (P8, P5/3) [5]
| | third-5th pentatonic
| | third-4th, quarter-4th, quarter-5th
|-
| | 2L 3s
| | (P8, P5) [5]
| | unsplit pentatonic
| | third-11th
|-
| | 3L 2s
| | (P8, P12/4) [5]
| | quarter-12th pentatonic
| | quarter-11th
|-
| | 4L 1s
| | (P8, P4/2) [5]
| | half-4th pentatonic
| |
|-
! | Hexatonic
! |
! |
! |
|-
| | 1L 5s
| | (P8, P4/3) [6]
| | third-4th hexatonic
| | quarter-4th, quarter-5th, fifth-4th, fifth-5th
|-
| | 2L 4s
| | (P8/2, P5) [6]
| | half-8ve hexatonic
| |
|-
| | 3L 3s
| | (P8/3, P5) [6]
| | third-8ve hexatonic
| |
|-
| | 4L 2s
| | (P8/2, P4/2) [6]
| | half-everything hexatonic
| |
|-
| | 5L 1s
| | (P8, P5/3) [6]
| | third-5th hexatonic
| |
|-
! | Heptatonic
! |
! |
! |
|-
| | 1L 6s
| | (P8, P4/3) [7]
| | third-4th heptatonic
| | quarter-4th, fifth-4th, fifth-5th, sixth-4th, sixth-5th
|-
| | 2L 5s
| | (P8, P11/3) [7]
| | third-11th heptatonic
| | fifth-double-compound-4th, sixth-double-compound-5th
|-
| | 3L 4s
| | (P8, P5/2) [7]
| | half-5th heptatonic
| | fifth-12th
|-
| | 4L 3s
| | (P8, P11/5) [7]
| | fifth-11th heptatonic
| | sixth-12th
|-
| | 5L 2s
| | (P8, P5) [7]
| | unsplit heptatonic
| | sixth-double-compound-4th
|-
| | 6L 1s
| | (P8, P5/4) [7]
| | quarter-5th heptatonic
| |
|-
! | Octotonic
! |
! |
! |
|-
| | 1L 7s
| | (P8, P4/4) [8]
| | quarter-4th octotonic
| | fifth-4th, fifth-5th, sixth-4th, sixth-5th, seventh-4th, seventh-5th
|-
| | 2L 6s
| | (P8/2, P5) [8]
| | half-8ve octotonic
| |
|-
| | 3L 5s
| | (P8, P11/4) [8]
| | quarter-11th octotonic
| | seventh-cc4th, seventh-cc5th
|-
| | 4L 4s
| | (P8/4, P5) [8]
| | quarter-8ve octotonic
| |
|-
| | 5L 3s
| | (P8, P12/4) [8]
| | quarter-12th octotonic
| | (very lopsided, unless 5th is quite flat)
|-
| | 6L 2s
| | (P8/2, P4/3) [8]
| | half-8ve third-4th octotonic
| |
|-
| | 7L 1s
| | (P8, P4/3) [8]
| | third-4th octotonic
| |
|-
! | Nonatonic
! |
! |
! |
|-
| | 1L 8s
| | (P8, P4/4) [9]
| | quarter-4th nonatonic
| | fifth-4th, sixth-4th, sixth-5th, seventh-4th/5th, eighth-4th/5th
|-
| | 2L 7s
| | (P8, c3P5/8) [9]
| | eighth-c35th nonatonic
| | third-11th, fifth-cc4th
|-
| | 3L 6s
| | (P8/3, P5) [9]
| | third-8ve nonatonic
| | third-8ve half-5th
|-
| | 4L 5s
| | (P8, P12/7) [9]
| | seventh-12th nonatonic
| | sixth-11th
|-
| | 5L 4s
| | (P8, P4/2) [9]
| | half-4th nonatonic
| | (lopsided unless 4th is sharp), seventh-11th
|-
| | 6L 3s
| | (P8/3, P4/2) [9]
| | third-8ve half-4th nonatonic
| |
|-
| | 7L 2s
| | (P8, ccP5/6)[9]
| | sixth-cc5th nonatonic
| | (lopsided unless 5th is sharp)
|-
| | 8L 1s
| | (P8, P5/5) [9]
| | fifth-5th nonatonic
| |
|-
! | Decatonic
! |
! |
! |
|-
| | 1L 9s
| | (P8, P5/6) [10]
| | sixth-5th decatonic
| | fifth-4th, sixth-4th, seventh-4th/5th, eighth-4th/5th, ninth-4th/5th
|-
| | 2L 8s
| | (P8/2, P5) [10]
| | half-8ve decatonic
| | half-8ve quartertone, half-8ve third-11th
|-
| | 3L 7s
| | (P8, P12/5) [10]
| | fifth-12th decatonic
| | eighth-cc4th, eighth-cc5th
|-
| | 4L 6s
| | (P8/2, P4/2) [10]
| | half-everything decatonic
| |
|-
| | 5L 5s
| | (P8/5, P5) [10]
| | fifth-8ve decatonic
| | (lopsided unless 5th is quite flat)
|-
| | 6L 4s
| | (P8/2, P5/3) [10]
| | half-8ve third-5th decatonic
| |
|-
| | 7L 3s
| | (P8, P5/2) [10]
| | half-5th decatonic
| | ninth-cc5th
|-
| | 8L 2s
| | (P8/2, P4/4) [10]
| | half-8ve quarter-4th decatonic
| | half-8ve quarter-12th
|-
| | 9L 1s
| | (P8, P4/2) [10]
| | quarter-4th decatonic
| |
|}

The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the Sensei aka Sepgu & Ruyoyo generator would be (P8, ccP5/7) [5]. The rationale would be that two Sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be Roulette [7] aka Zozoquingu Nowa [7], with a 2.5.7 pergen (P8, (5/4)/2) = (P8, M3/2) = (P8, M2), where 5·G = A6 = 7/4.

==Pergens and EDOs==

Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored.

Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset.

How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinitely many possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, ccP5/31),... (P8, (i-1,1)/n), where n = 12i+7.

How many edos support a given pergen? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be a multiple of m, and k must be divisible by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.

Given an edo, a period, and a generator, what is the pergen? There is usually more than one right answer. For 10edo with P = 5\10 and G = 2\10, it could be either (P8/2, P4/2) or (P8/2, P5/3). Every coprime period/generator pair results in a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.

'''EDOs Supporting A Pergen'''

This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Both of these edos are incompatible with heptatonic notation, and 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. 6-edo, 11-edo and 23-edo could also be considered ambiguous.

{| class="wikitable" style="text-align:center;"
|-
! colspan="2" | pergen
! | supporting edos (12-31 only)
|-
| | (P8, P5)
| | unsplit
| | 12, 13, 13b, 14*, 15*, 16, 17, 18, 18b*, 19, 20*,

21*, 22, 23, 24*, 25*, 26, 27, 28*, 29, 30*, 31
|-
! | halves
! |
! |
|-
| | (P8/2, P5)
| | half-8ve
| | 12, 14, 16, 18, 18b, 20*, 22, 24*, 26, 28*, 30*
|-
| | (P8, P4/2)
| | half-4th
| | 13b, 14, 15*, 18b*, 19, 20*, 23, 24, 25*, 28*, 29, 30*
|-
| | (P8, P5/2)
| | half-5th
| | 13, 14*, 17, 18b, 20*, 21*, 24, 27, 28*, 30*, 31
|-
| | (P8/2, P4/2)
| | half-everything
| | 14, 18b, 20*, 24, 28*, 30*
|-
! | thirds
! |
! |
|-
| | (P8/3, P5)
| | third-8ve
| | 12, 15, 18, 18b*, 21, 24*, 27, 30*
|-
| | (P8, P4/3)
| | third-4th
| | 13b, 14*, 15, 21*, 22, 28*, 29, 30*
|-
| | (P8, P5/3)
| | third-5th
| | 15*, 16, 20*, 21, 25*, 26, 30*, 31
|-
| | (P8, P11/3)
| | third-11th
| | 13, 15, 17, 21, 23, 30*
|-
| | (P8/3, P4/2)
| | third-8ve, half-4th
| | 15, 18b*, 24, 30*
|-
| | (P8/3, P5/2)
| | third-8ve, half-5th
| | 18b, 21, 24, 27, 30
|-
| | (P8/2, P4/3)
| | half-8ve, third-4th
| | 14, 22, 28*, 30
|-
| | (P8/2, P5/3)
| | half-8ve, third-5th
| | 16, 20*, 26, 30*
|-
| | (P8/2, P11/3)
| | half-8ve, third-11th
| | 19, 30
|-
| | (P8/3, P4/3)
| | third-everything
| | 15, 21, 30*
|-
! | quarters
! |
! |
|-
| | (P8/4, P5)
| | quarter-8ve
| | 12, 16, 20, 24*, 28
|-
| | (P8, P4/4)
| | quarter-4th
| | 18b*, 19, 20*, 28, 29, 30*
|-
| | (P8, P5/4)
| | quarter-5th
| | 13, 14*, 20, 21*, 27, 28*
|-
| | (P8, P11/4)
| | quarter-11th
| | 14, 17, 20, 28*, 31
|-
| | (P8, P12/4)
| | quarter-12th
| | 13b, 15*, 18b, 20*, 23, 25*, 28, 30*
|-
| | (P8/4, P4/2)
| | quarter-8ve, half-4th
| | 20, 24, 28
|-
| | (P8/2, M2/4)
| | half-8ve, quarter-tone
| | 18, 20, 22, 24, 26, 28
|-
| | (P8/2, P4/4)
| | half-8ve, quarter-4th
| | 18b, 20*, 28, 30*
|-
| | (P8/2, P5/4)
| | half-8ve, quarter-5th
| | 14, 20, 28*
|-
| | (P8/4, P4/3)
| | quarter-8ve, third-4th
| | 28
|-
| | (P8/4, P5/3)
| | quarter-8ve, third-5th
| | 16, 20
|-
| | (P8/4, P11/3)
| | quarter-8ve, third-11th
| |
|-
| | (P8/3, P4/4)
| | third-8ve, quarter-4th
| | 18b*, 30
|-
| | (P8/3, P5/4)
| | third-8ve, quarter-5th
| | 21, 27
|-
| | (P8/3, P11/4)
| | third-8ve, quarter-11th
| |
|-
| | (P8/3, P12/4)
| | third-8ve, quarter-12th
| | 15, 18b, 30*
|-
| | (P8/4, P4/4)
| | quarter-everything
| | 20, 28
|}
The edos that support the fewest pergens are prime-number edos like 11edo or 13edo. The most "pergen-friendly" edos tend to be ones in which the circle of 5ths doesn't reach every edostep. For example, 24edo supports all half-split pergens, since both P8 and P5 map to an even number of edosteps. 72edo supports all half-splits and all third-splits. 15, 21 and 36 edo support many but not all third-splits (not those with m = 2 or n = 2).

'''Notating a pergen tuned to an EDO'''

If both the pergen and the EDO are notated with ups and downs, what is the relationship between the two kinds of up? If the edo supports the pergen, fully or partially, then the pergen's up equals some multiple of the EDO's up, i.e. some number of edosteps. For third-4th in 22edo or 29edo, the pergen's up = 1 edostep. But in 37edo or 44edo, ^1 = 2 edosteps. For half-8ve in 12edo, ^1 = 0 edosteps, and the ups and downs in the score can simply be ignored. In fact, it seems every pergen in 5edo, 7edo and 12edo has ^1 = 0 edosteps. It's not yet known why.

When notating a piece in a specific pergen meant to be played in a specific EDO, either the pergen notation or the EDO notation can be used. For small or mid-sized EDOs, they're usually identical. If one has to choose, the pergen notation is generally preferred. It's less cluttered. Also, it's easier to mentally double every up and down in larger EDOs than it is to halve them in smaller EDOs.

Half-8ve in 22edo has P = vA4. But in 16edo, P = A4 + 2 edosteps, and ^1 = -2 edosteps. Negative edosteps means up is down, and should be avoided. The notation has tipped, and the period should be notated as ^A4, making ^1 = 2 edostep. Even better would be P = ^4, making ^1 = 1 edostep.

Half-5th has EU = vvA1, hence any EDO in which A1 = 4 edosteps will have ^1 = 2 edosteps. These "doubled EDOs" are 20, 27, 34, 41, 48, 55, etc. The "tripled EDOs" with A1 = 6 edosteps and ^1 = 3 edosteps are every 7th EDO from 30 to 72.

Half-4th has EU = vvm2. Doubled EDOs have m2 = 4 edosteps. These are 23, 28, 33, 38, 43, 48, 53, etc. Tripled EDOs are every 5th one from 47 to 72.

Third-4th has EU = v3A1. Doubled EDOs are the same ones as half-5th's tripled EDOs. Third-5th has EU = v3m2. Doubled EDOs are the same as half-4th's tripled EDOs.

The relationship between a pergen's up and an EDO's up when the pergen uses double-pair notation is complex. Consider half-everything, notated with ups/downs in the perchain and lifts/drops in the genchain. 10edo has ^1 = 1 edostep and /1 = 0 edosteps, thus one simply ignores lifts and drops. But for 24edo, ^1 = 0 edosteps and /1 = 1 edostep. One must ignore ups and downs and convert lifts/drops to edosteps.

'''Pergens Within An EDO'''

See the screenshots in the Supplemental Materials section for a complete list of which pergens are supported by 12, 15 and 19 edo. The list includes multiple pergens per period/generator combination. The next table shows only the simplest pergen for each combination. Combinations are excluded if the period and generator are not coprime, because the scale is contained in a smaller edo. For example, 15edo has no unsplit pergen, because those scales are contained in 5edo. Periods of 1 or 2 edosteps are excluded as too trivial, because the scale is the entire edo. Every step of the edo appears in the genchains, even if the chains are only one step long.

To find the pergen, find the edo, then find the row that corresponds to the period, then find the column that corresponds to the generator. It appears, but has yet to be formally proven, that the number of P/G combinations that N-edo supports is (N-1)/2 rounded down. If we include the trivial combination where P = 1 edostep, the number of combinations would be N/2 rounded up.

{| class="wikitable" style="text-align:center;"
|-
! | EDO
! | Period
! colspan="11" | Generator in edosteps
|-
! |
! | in edosteps
! | 1
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
|-
! | 5
! | 5 = P8
| | P4/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 6
! | 6 = P8
| | P4/2
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 7
! | 7 = P8
| | P4/3
| | P5/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 8
! | 8 = P8
| | P4/3
| | -
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/2
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 9
! | 9 = P8
| | P4/4
| | P4/2
| | -
| | P5
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/3
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 10
! | 10 = P8
| | P4/4
| | -
| | P5/2
| | -
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 5 = P8/2
| | P5
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 11
! | 11 = P8
| | P4/5
| | P5/3
| | P5/2
| | P11/4
| | P5
| |
| |
| |
| |
| |
| |
|-
! | 12
! | 12 = P8
| | P4/5
| | -
| | -
| | -
| | P5
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/2
| | P5
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/3
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/4
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 13b
! | 13 = P8
| | P4/6
| | P4/3
| | P4/2
| | P12/5
| | P12/4
| | P5
| |
| |
| |
| |
| |
|-
! | 14
! | 14 = P8
| | P4/6
| | -
| | P4/2
| | -
| | P11/4
| | -
| |
| |
| |
| |
| |
|-
! | "
! | 7 = P8/2
| | P5
| | P4/3
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 15
! | 15 = P8
| | P4/6
| | P4/3
| | -
| | P12/6
| | -
| | -
| | P11/3
| |
| |
| |
| |
|-
! | "
! | 5 = P8/3
| | P5
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/5
| | P4/3
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 16
! | 16 = P8
| | P4/7
| | -
| | P5/3
| | -
| | P12/5
| | -
| | P5
| |
| |
| |
| |
|-
! | "
! | 8 = P8/2
| | P5
| | -
| | P5/3
| | -
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/4
| | P5
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 17
! | 17 = P8
| | P4/7
| | P5/5
| | P11/8
| | P11/6
| | P5/2
| | P11/4
| | P5
| | P11/3
| |
| |
| |
|-
! | 18b
! | 18 = P8
| | P4/8
| | -
| | -
| | -
| | P5/2
| | -
| | P12/4
| | -
| |
| |
| |
|-
! | "
! | 9 = P8/2
| | P5
| | P4/4
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/3
| | P5/2
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/6
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 19
! | 19 = P8
| | P4/8
| | P4/4
| | P11/9
| | P4/2
| | P12/6
| | P12/5
| | ccP5/7
| | P5
| | P11/3
| |
| |
|-
! | 20
! | 20 = P8
| | P4/8
| | -
| | P5/4
| | -
| | -
| | -
| | P11/4
| | -
| | c3P5/8
| |
| |
|-
! | "
! | 10 = P8/2
| | M2/4
| | -
| | P5/4
| | -
| | -
| |
| |
| |
| |
| |
| |
|-
! | "
! | 5 = P8/4
| | P4/2
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/5
| | P5/4
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 21
! | 21 = P8
| | P4/9
| | P5/6
| | -
| | P5/3
| | P11/6
| | -
| | -
| | c3P4/9
| | -
| | P11/3
| |
|-
! | "
! | 7 = P8/3
| | P5/2
| | P5
| | P4/3
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/7
| | P5/3
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 22
! | 22 = P8
| | P4/9
| | -
| | P4/3
| | -
| | P12/7
| | -
| | P12/5
| | -
| | P5
| | -
| |
|-
! | "
! | 11 = P8/2
| | M2/4
| | P5
| | P4/3
| | P12/5
| | P12/7
| |
| |
| |
| |
| |
| |
|-
! | 23
! | 23 = P8
| | P4/10
| | P4/5
| | P11/11
| | P12/9
| | P4/2
| | P12/6
| | ccP4/8
| | ccP4/7
| | P12/4
| | P5
| | P11/3
|-
! | 24
! | 24 = P8
| | P4/10
| | -
| | -
| | -
| | P4/2
| | -
| | P5/2
| | -
| | -
| | -
| | c4P5/10
|-
! | "
! | 12 = P8/2
| | M2/4
| | -
| | -
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
|-
! | "
! | 8 = P8/3
| | P5/2
| | -
| | P4/2
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 6 = P8/4
| | P4/2
| | -
| | -
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 4 = P8/6
| | P4/2
| | -
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | "
! | 3 = P8/8
| | P5
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
! |
! |
! | 1
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
|}
Larger edos can have very awkward pergens. For example, 31edo with G = 14/31 is (P8, c5P4/12). It's much simpler to think of the generator as the downfifth = 17\31, and the pergen as the pseudopergen (P8, v5).

'''EDO-pair names'''

Just as a pair of edos and a prime subgroup can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo & N'-edo, m = GCD (N,N'). The period P equals both (N/m)\N and (N'/m)\N'. For example, for 12edo and 16edo, m = 4, and the period is both 3\12 and 4\16.

For each edo, find the nearest '''edomapping''' (patent val) for the 2.3 subgroup. If the edo has a "b" wart, e.g. 13b-edo, use the second-nearest edomapping. Form a 2x2 matrix from these edomappings. Let d be the determinant of this matrix. If d = m, -m or 0, the generator is the 5th, and the pergen is simply (P8/m, P5).

For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.

If |d| ≠ m or 0, P5 is not the generator, and we must find the multigen. Take the ratio of the two edos N/N' and reduce it by m. In the scale tree ([http://tallkite.com/misc_files/Scale-Tree-Complete.pdf pdf] or [http://tallkite.com/misc_files/Scale-Tree-Complete.jpg jpeg]), let g/g' be the smallest ancestor of this ratio. The generator G maps to both g\N and g'\N'. The two edos come closest to coinciding here.

For example, for 7-edo and 17-edo, m = 1 but d = 2, so G ≠ P5. The smallest ancestor of 7/17 is 2/5. G maps to both 2\7 and 5\17, which are both neutral 3rds. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths. 2\7 = 343¢ and 5\17 = 353¢, and their difference is only 1\N", where N" = LCM (N, N'). This 10¢ difference is the least difference between any 7edo note and any 17edo note, except that the two neutral 6ths differ by the same 10¢, and of course some note pairs coincide exactly, such as 0\7 and 0\17.

Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found from the scale tree, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this new vector with either edomapping is zero. Treating this new vector as a monzo, since the dot product is zero, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for. If the comma is (a, b, c), then a·P8 + b·P5 + c·G = 0, and G = (b-a, -b) / c.

For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2).

To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 (Dicot aka Yoyo). 11/9 also works, it yields 243/242 (Mohajira aka Lulu).

If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a multi-EDO pergen.

If the 1st edo is 7edo, the pergen indicates the natural heptatonic notation for the 2nd edo, e.g. 12edo = unsplit, 15edo = third-4th and 17edo = half-5th.

The closer two edos are in the scale tree, the smaller the splitting fractions in the pergen they make. Examples:

{| class="wikitable" style="text-align:center;"
|-
! |
! | 12-edo
! | 13b-edo
! | 14-edo
! | 15-edo
! | 16-edo
! | 17-edo
! | 18b-edo
! | 19-edo
! | 20-edo
|-
! | 13b-edo
| | (P8, P5/7)
| |
| |
| |
| |
| |
| |
| |
| |
|-
! | 14-edo
| | (P8/2, P5)
| | (P8, P4/6)
| |
| |
| |
| |
| |
| |
| |
|-
! | 15-edo
| | (P8/3, P5)
| | (P8, c5P5/12)
| | (P8, P4/6)
| |
| |
| |
| |
| |
| |
|-
! | 16-edo
| | (P8/4, P5)
| | (P8, P12/5)
| | (P8/2, P5)
| | (P8, P5/9)
| |
| |
| |
| |
| |
|-
! | 17-edo
| | (P8, P5)
| | (P8, ccP5/11)
| | (P8, P11/4)
| | (P8, P11/3)
| | (P8/2, P4/7)
| |
| |
| |
| |
|-
! | 18b-edo
| | (P8/6, P5)
| | (P8, P12/4)
| | (P8/2, P4/2)
| | (P8/3, P12/4)
| | (P8/2, P5)
| | (P8, P5/10)
| |
| |
| |
|-
! | 19-edo
| | (P8, P5)
| | (P8, P12/10)
| | (P8, P4/2)
| | (P8, P12/6)
| | (P8, P12/5)
| | (P8, P11/3)
| | (P8, P4/8)
| |
| |
|-
! | 20-edo
| | (P8/4, P5)
| | (P8, ccP4/16)
| | (P8/2, P5/4)
| | (P8/5, ^1)
| | (P8/4, P5/3)
| | (P8, P11/4)
| | (P8/2, P4/8)
| | (P8, P4/8)
| |
|-
! | 21-edo
| | (P8/3, P5)
| | (P8, c3P4/9)
| | (P8/7, ^1)
| | (P8/3, P4/3)
| | (P8, P5/3)
| | (P8, P11/6)
| | (P8/3, P5/2)
| | (P8, P11/3)
| | (P8, P5/12)
|-
! | 22-edo
| | (P8/2, P5)
| | (P8, c3P4/15)
| | (P8/2, P4/3)
| | (P8, P4/3)
| | (P8/2, P12/5)
| | (P8, P5)
| | (P8/2, P12/7)
| | (P8, P12/5)
| | (P8/2, M2/4)
|-
! | 23-edo
| | (P8, P4/5)
| | (P8, ccP4/8)
| | (P8, P4/2)
| | (P8, P12/12)
| | (P8, P5)
| | (P8, P12/9)
| | (P8, P12/4)
| | (P8, P12/6)
| | (P8, c5P5/16)
|-
! | 24-edo
| | (P8/12, ^1)
| | (P8, c6P4/14)
| | (P8/2, P4/2)
| | (P8/3, P4/2)
| | (P8/8, P5)
| | (P8, P5/2)
| | (P8/6, P4/2)
| | (P8, P4/2)
| | (P8/4, P4/2)
|}

A specific pergen can be converted to an edo pair by finding the range of its generator cents in the [[pergen#Further Discussion-Notating tunings with an arbitrary generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of edos 7, 10 and 17 defines (P8, P5/2).

==Array Keyboards (unfinished)==

Array keyboards have a 2-dimensional layout of keys, and are very appropriate for rank-2 tunings. A good layout can be found from the tuning's pergen. First find an edo N-edo that is compatible with the pergen, then arrange the keys in N columns to the 8ve, with each row usually containing the multigen interval. The unsplit pergen in 7 columns:

{| class="wikitable" style="text-align:center;"
|-
| | D#
| |
| |
| |
| |
| |
| |
| |
|-
| | D
| | E
| | F#
| | G#
| | A#
| |
| |
| |
|-
| | Db
| | Eb
| | F
| | G
| | A
| | B
| | C#
| | D#
|-
| |
| |
| |
| | Gb
| | Ab
| | Bb
| | C
| | D
|-
| |
| |
| |
| |
| |
| |
| |
| | Db
|}
Higher notes are at the top of each column. The rows would actually be angled so that the two D's are at the same horizontal level. The vertical steps are A1 and the horizontal steps are M2, and the keyboard is defined as 7(A1, M2).

The third-4th keyboard is 7(A1/3 = ^1 = vvA1, P4/3 = vM2 = ^^m2).

{| class="wikitable" style="text-align:center;"
|-
| | vD#
| | ^E
| | F#
| | vG#
| | ^A
| | B
| | vC#
| | ^D
|-
| | ^D
| | E
| | vF#
| | ^G
| | A
| | vB
| | ^C
| | D
|-
| | D
| | vE
| | ^F
| | G
| | vA
| | ^B
| | C
| | vD
|-
| | vD
| | ^Eb
| | F
| | vG
| | ^Ab
| | Bb
| | vC
| | ^Db
|}

''Hypothesis: Let the 5th's keyspan (i.e. column-span) be F. In order for the keyboard to have the pitches in order, the fifth must fall between the two Stern-Brocot ancestors of F\N (simplified if possible). For example, an 8-column keyboard has F = 5, the ancestors of 5\8 are 3\5 and 2\3, and the 5th must be between 720¢ and 800¢. Thus the most musically useful N values are 5, 7, 10, 12 and 14.''

(more to come)

==Supplemental materials==

===Notation guide PDF===

This PDF is a rank-2 notation guide that shows the full lattice for the first 32 pergens, up through the quarter-splits block. It also includes the single-split pergens from the fifth-split, sixth-split and seventh-split blocks. It includes alternate EUs for many pergens.

[http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf '''<big>TallKite.com/misc_files/notation guide for rank-2 pergens.pdf</big>''']

{| class="wikitable" style="text-align:center;"
|+Table of contents for the N'''otation Guide for Rank-2 Pergens''' (* indicates a true double)
|-
! colspan="3" |unsplit
! colspan="3" |quarter-splits
! colspan="3" |single-split fifth-splits
! colspan="3" |single-split seventh-splits
|-
!1
|(P8, P5)
|unsplit
!16
|(P8/4, P5)
|quarter-8ve
!33
|(P8/5, P5)
|fifth-8ve
!96
|(P8/7, P5)
|seventh-8ve
|-
! colspan="3" |half-splits
!17
|(P8, P4/4)
|quarter-4th
!34
|(P8, P4/5)
|fifth-4th
!97
|(P8, P4/7)
|seventh-4th
|-
!2
|(P8/2, P5)
|half-8ve
!18
|(P8, P5/4)
|quarter-5th
!35
|(P8, P5/5)
|fifth-5th
!98
|(P8, P5/7)
|seventh-5th
|-
!3
|(P8, P4/2)
|half-4th
!19
|(P8, P11/4)
|quarter-11th
!36
|(P8, P11/5)
|fifth-11th
!99
|(P8, P11/7)
|seventh-11th
|-
!4
|(P8, P5/2)
|half-5th
!20
|(P8, P12/4)
|quarter-12th
!37
|(P8, P12/5)
|fifth-12th
!100
|(P8, P12/7)
|seventh-12th
|-
!5
|(P8/2, P4/2) *
|half-everything *
!21
|(P8/4, P4/2) *
|quarter-8ve, half-4th *
!38
|(P8, ccP4/5)
|fifth-coco-4th
!101
|(P8, ccP4/7)
|seventh-coco-4th
|-
! colspan="3" |third-splits
!22
|(P8/2, M2/4)
|half-8ve, quarter-tone
! colspan="3" |single-split sixth-splits
!102
|(P8, ccP5/7)
|seventh-coco-5th
|-
!6
|(P8/3, P5)
|third-8ve
!23
|(P8/2, P4/4) *
|half-8ve, quarter-4th *
!64
|(P8/6, P5)
|sixth-8ve
!103
|(P8, c3P4/7)
|seventh-trico-4th
|-
!7
|(P8, P4/3)
|third-4th
!24
|(P8/2, P5/4) *
|half-8ve, quarter-5th *
!65
|(P8, P4/6)
|sixth-4th
| colspan="3" rowspan="9" |
|-
!8
|(P8, P5/3)
|third-5th
!25
|(P8/4, P4/3)
|quarter-8ve, third-4th
!66
|(P8, P5/6)
|sixth-5th
|-
!9
|(P8, P11/3)
|third-11th
!26
|(P8/4, P5/3)
|quarter-8ve, third-5th
!67
|(P8, P11/6)
|sixth-11th
|-
!10
|(P8/3, P4/2)
|third-8ve, half-4th
!27
|(P8/4, P11/3)
|quarter-8ve, third-11th
!68
|(P8, P12/6)
|sixth-12th
|-
!11
|(P8/3, P5/2)
|third-8ve, half-5th
!28
|(P8/3, P4/4)
|third-8ve, quarter-4th
!69
|(P8, ccP4/6)
|sixth-coco-4th
|-
!12
|(P8/2, P4/3)
|half-8ve, third-4th
!29
|(P8/3, P5/4)
|third-8ve, quarter-5th
!70
|(P8, ccP5/6)
|sixth-coco-5th
|-
!13
|(P8/2, P5/3)
|half-8ve, third-5th
!30
|(P8/3, P11/4)
|third-8ve, quarter-11th
| colspan="3" rowspan="3" |
|-
!14
|(P8/2, P11/3)
|half-8ve, third-11th
!31
|(P8/3, P12/4)
|third-8ve, quarter-12th
|-
!15
|(P8/3, P4/3) *
|third-everything *
!32
|(P8/4, P4/4) *
|quarter-everything *
|}Screenshots of the first 2 pages:

[[File:pergens_1.png|alt=pergens 1.png|704x948px|pergens 1.png]]

[[File:pergens_2.png|alt=pergens 2.png|704x760px|pergens 2.png]]

===PergenLister===

PergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonic unisons for each one. Alternate EUs are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair.

http://www.tallkite.com/apps/pergenLister.html (written in Jesusonic, runs inside Reaper or ReaJS)

The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. EUs of a 3rd or more are in red.

Screenshots of the first 69 pergens:

[[File:alt-pergenLister_1.png|alt=alt-pergenLister 1.png|800x427px|alt-pergenLister 1.png]]

[[File:alt-pergenLister_2.png|alt=alt-pergenLister 2.png|800x455px|alt-pergenLister 2.png]]

The first 29 pergens supported by 12edo:

[[File:alt-pergenLister_12edo.png|alt=alt-pergenLister 12edo.png|800x449px|alt-pergenLister 12edo.png]]

Some of the pergens supported by 15edo. A red asterisk means partial support.

[[File:alt-pergenLister_15edo.png|alt=alt-pergenLister 15edo.png|800x493px|alt-pergenLister 15edo.png]]

Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.

[[File:alt-pergenLister_19edo.png|alt=alt-pergenLister 19edo.png|800x459px|alt-pergenLister 19edo.png]]

Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:

If z > 0, then y is at least ceiling (x·(3/2 - z/2)) and at most floor (x·5/3)

If z < 0, then y is at least ceiling (x·3/2) and at most floor (x·(5/3 - z/2))

Next we loop through all combinations of x and z in such a way that larger values of x and z come last:

i = 1; loop (maxFraction,

<ul><li>j = 1; loop (i - 1,<ul><li>makeMapping (i, j); makeMapping (i, -j);</li><li>makeMapping (j, i); makeMapping (j, -i);</li><li>j += 1;</li></ul></li><li>);</li><li>makeMapping (i, i); makeMapping (i, -i);</li><li>i += 1;</li></ul>);

The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.

==Various proofs (unfinished)==

Although not yet rigorously proven, the two false-double tests have been empirically verified by pergenLister.

The interval P8/2 has a "ratio" of the square root of 2, which equals 21/2, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the '''pergen matrix''' [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.

Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise a, b, and n could all be reduced by GCD (a,b), and the multigen could be simplified. Since GCD (a, b) = 1 and -am/b is an integer, it follows that m must be a multiple of |b| as well.

Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|) = |b| · r. If r = 1, then GCD (m, n) = |b|, and vice versa, which is the proposed test for a false double.

If m = |b|, is the pergen explicitly false? Does splitting (a,b) into n generators also split P8 into m periods?

(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5

Because n is a multiple of b, n/b is an integer

M/b = (n/b)·M/n = (n/b)·G 
(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)

Let c and d be the bezout pair of a+b and b, with c·(a+b) + d·b = 1

Since the pergen is a double-split, m > 1, therefore |b| > 1, therefore c ≠ 0

c·(a+b)·P8 = c·b·((n/b)·G - P5) 
(1 - d·b)·P8 = c·b·((n/b)·G - P5) 
P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5) 
P8/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)

Therefore P8 is split into m periods 
Therefore if m = |b|, the pergen is explicitly false

Assume the pergen is a false double, and there's a comma C that splits both P8 and (a,b) appropriately. Can we prove r = 1? Let Q = the higher prime that C uses. Express P, G and C as monzos of the prime subgroup 2.3.Q, by expanding the 2x2 pergen matrix to a 3x3 matrix A:

P = (1/m, 0, 0) 
G = (a/n, b/n, 0) 
C = (u, v, w)

Here u, v and w are integers. If GCD (u, v, w) > 1, simplify C so that it = 1. The inverse of A expresses 2, 3 and Q in terms of P, G and C. If C is tempered out, the C column can be discarded, making the usual 3x2 period-generator mapping. However, if C is not tempered out, the inverse of A is a 3x3 period-generator-comma mapping, which is simply a change of basis. For example, 5-limit JI can be generated by 2/1, 3/2 and 81/80. Here is the inverse of A:

2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) 
3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C) 
Q = Q/1 = ((av-bu)m/wb, -vn/wb, 1/w) · (P, G, C)

Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. Every pergen except the unsplit one requires |w| > 1, so the last column almost always has a fraction. To avoid fractions in the first two columns, A must be unimodular '''''[I think, not sure]''''', and we have wb/mn = ±1, and w = ±mn/b. Substituting for w, we have:

2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) 
3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C) 
Q = Q/1 = (±(av-bu)/n, ±(-v)/m, ±b/mn) · (P, G, C)

For v/m to be an integer, v must equal i·m for some integer i. Likewise, av-bu must equal j·n for some integer j. Thus bu = av - jn = iam - jn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb. Substituting, we get bu = iaprb - jqrb, and u = r(iap - jq). Furthermore, v = im = iprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r > 1, this contradicts the requirement that GCD (u, v, w) = 1, therefore r must be 1, and GCD (m, n) = |b|, and all false doubles pass the false-double test.

Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q?

Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b.

a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)

Thus a·P8 splits into b parts, and since m = |b|, a·P8 splits into m parts. Proceed as before with a bezout pair to find the monzo for P8/m.

Next, assume the pergen isn't explicitly false. The unreduced form is (P8/m, (n - am, -bm) / mn). Substituting in m = pb and n = qb, with p and q coprime, we get (P8/m, (q - ap, -pb) / pqb).

Assume the pergen is a true double, and r > 1. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.

Given a pergen (P8/m, (a,b)/n), how many parts is an arbitrary interval (a',b') split into?

(a',b') = (a'·b, b'·b) / b = (a'·b - a·b', 0) / b + (a·b', b'·b) / b = (a'·b - a·b')·P8 / b + b'·(a,b) / b = (a'·b - a·b')·(m/b)·P + b'·(n/b)·G

Therefore (a',b') is split into GCD (a'·b - a·b')·(m/b), b'·(n/b)) parts.

If m = 1, then b = ±1, and we have GCD (a' ± a·b', b'·n)

If n = 1, then a = -1 and b = 1, and we have GCD (a'·m + b'·m, b') = GCD (a'·m, b')

If m = 1 and n = 1, we have GCD (a', b') = the naturally occurring split.

If m = n (nth-everything), we have n · GCD (a', b')

The multigen and the arbitrary interval can be expressed as gedras:

(a,b) = [k,s] = (-11k+19s, 7k-12s)

(a',b') = [k',s'] = (-11k'+19s', 7k'-12s')

a'·b - a·b' works out to be k·s' - k'·s, and we have GCD ((k·s' - k'·s)·m/b, b'·n/b)

If s is a multiple of n (happens when EU is an A1) and s' is a multiple of n, let s = x·n and s' = y·n

GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')

Thus every such interval is split, e.g. the half-5th pergen splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. This is an easy way to determine if an interval is split or not by the pergen.

To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts

if r > 1, it's a true double

a·P8 = (a,0) = (a,b) - (0,b) = M - b·P12

M = n·G = qrb·G

a·P8 = qrb·G - b·P12 = b·(qr·G - P12)

Let c and d be the bezout pair of a and b, with c·a + d·b = 1

If |b| = 1, let c = 1 and d = ±a, to avoid c = 0

ca·P8 = cb·(qr·G - P12)

(1 - d·b)·P8 = c·b·(qr·G - P12)

P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) = b · (d,-c) + bcqr·G

== Glossary ==
to find the definition, use control-F to search for the first (bolded) occurrence of the word on this page.

pergen 
split 
multigen 
ups and downs (the ^ and v symbols) 
higher prime (any prime > 3) 
color depth 
dependent/independent 
square mapping 
lifts and drops (the / and \ symbols) 
enharmonic unison, EU 
uninflected 
genchain 
perchain 
compound (increased by an octave) 
single-split, double-split 
single-pair, double-pair (number of new accidentals in the notation) 
true double, false double 
explicitly false 
unreduced 
alternate vs. equivalent (generator or period) 
mapping comma 
keyspan 
stepspan 
gedra 
count 
mid 
edomapping 
upspan 
liftspan

chain number 
single-chain 
multi-chain 
arrow comma

==Miscellaneous Notes==

=== Combining pergens ===
Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). If adding a comma to a temperament doesn't change the pergen, it's a strong extension, otherwise it's a weak extension.

General rules for combining pergens:

<ul><li>(P8/m, M/n) + (P8, P5) = (P8/m, M/n)</li>
<li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li>
<li>(P8/m, P5) + (P8/m', P5) = (P8/m", P5), where m" = LCM (m,m')</li>
<li>(P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n')</li></ul>

However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.

If a false double pergen can be broken down into two simpler ones, that may help with finding a double-pair notation with an EU of a 2nd or less. For example, sixth-4th's single pair notation has an EU of a 4th. But since sixth-4th is half-4th plus third-4th, and those two have a good EU, sixth-4th can be notated with one pair from half-4th and another from third-4th.

=== Expanding gedras ===
Gedras can be expanded to 5-limit or higher by including another keyspan that is compatible with 7 and 12, such as 9 or 16. But a more useful approach is for the third number to be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]:

k = 12a + 19b + 28c + 34d 
s = 7a + 11b + 14c + 20d 
g = -c 
r = -d

a = -11k + 19s - 4g + 6r 
b = 7k - 12s + 4g - 2r 
c = -g 
d = -r

Gedras can also be expanded by adding an entry for ups/downs. The entry shows the '''upspan''', which is the number of ups the interval has. Downed intervals have a negative upspan. An entry for '''liftspan''' can also be added. For example, vM3 = [4,2,-1], and ^^\4 = [5,3,2,-1].

=== Height of a pergen ===
The LCM of the pergen's two splitting fractions could be called the '''height''' of the pergen. For example, (P8, P5) has height 1, and (P8/2, M2/4) has height 4. In single-pair notation, the EU's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.

=== Generalizing the pergen ===
See [[User:AthiTrydhen/Abstract pergens]]

=== Credits ===
Pergens were discovered by [[KiteGiedraitis|Kite Giedraitis]] in 2017, and developed with the help of [[Praveen Venkataramana]].

== Addenda (late 2023) ==
=== New terminology===
All temperaments have a '''chain number''', which is the number of fifthchains in the temperament's lattice. Any (P8, P5) temperament has a chain number of 1, and is '''single-chain'''. All other pergens are '''multi-chain'''. For example, Porcupine/Triyoti has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Saguguti has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m * n / f, where M is the multigen and f is the absolute value of M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.

===The EU(s) define the pergen===
The pergen can be derived directly from the EU(s). Thus the EU(s) define both the pergen and the notation. An EU can be thought of as a comma in the 2.3.^ subgroup. One derives a mapping from this comma as one would for any 3-prime comma, and one derives the pergen by inverting the first 2 columns of the mapping.

For example, if the EU is vvd2, the 2.3.^ monzo is [19 -12 -2]. There are many possible mappings, but only one that gives a canonical pergen: [(2 2 7) (0 1 -6)]. Discarding the last column and inverting gives us [(1/2 0) (-1 1)] = (P8/2, P5). Another example: vvA1 = [-11 7 -2]. The mapping [(1 1 -2) (0 2 7)] reduces to [(1 1) (0 2)], which inverts to [(1 0) (-1/2 1/2)] = (P8, P5/2).

One can explore the universe of possible EUs, and thus possible pergen notations, more easily if using the gedra format, expressing the EU as a combination of A1's, d2's and arrows. Thus vvA1 = [1 0 -2], v3m2 = [1 1 -3], etc. Unlike a conventional 2.3.^ monzo, the first two numbers in a A1.d2.^1 monzo are fairly small, and the second number is never negative (since it's an EU). And we can require that the first two numbers be coprime (see the next section). All this facilitates one's search.

===Simplifying a "squared" EU===
Consider an uninflected EU of AA1. AA1 is "squared" in the sense that AA1 = A1 + A1, thus both numbers in its ratio are square numbers. If it had an even upspan (the number of ups), it would obviously be an invalid EU, for if v4AA1 = 0¢, then so does vvA1, and v4AA1 could be replaced with vvA1. So the upspan must be odd.

Consider an EU of v3AA1. The pergen is (P8, P4/3). Here are the [[twin squares]].

<math>
\begin{array} {rrr}
\text{P8} \\
\text{^m2} \\
\text{vvvAA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & {\color {Green}0} \\
8 & -5 & {\color {Green}1} \\
\hline
{\color {Red}-22} & {\color {Red}14} & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & {\color {Red}2} \\
0 & -3 & {\color {Red}-14} \\
\hline
{\color {Green}0} & {\color {Green}-1} & -5 \\
\end{array} \right]
</math>

The two red numbers in the lower left are both even, because AA1 is a squared ratio. As a result, the two red numbers in the upper right must both be even to ensure their rows' dot products with the EU are zero, which is an even number. If we halve all the red numbers and double all the green numbers, all the various row dot products will be unchanged, and the twin squares will remain valid:

<math>
\begin{array} {rrr}
\text{P8} \\
\text{^^m2} \\
\text{vvvA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & {\color {Green}0} \\
8 & -5 & {\color {Green}2} \\
\hline
{\color {Red}-11} & {\color {Red}7} & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & {\color {Red}1} \\
0 & -3 & {\color {Red}-7} \\
\hline
{\color {Green}0} & {\color {Green}-2} & -5 \\
\end{array} \right]
</math>

But while the EU has become simpler, the generator has become more complex in that it is dup not up. This is remedied by adding the EU to it. Changes are in red:

<math>
\begin{array} {rrr}
\text{P8} \\
\text{vM2} \\
\text{vvvA1} \\
\end{array}
\left[ \begin{array} {rrr}
1 & 0 & 0 \\
{\color {Red}-3} & {\color {Red}2} & {\color {Red}-1} \\
\hline
-11 & 7 & -3 \\
\end{array} \right]
\longleftrightarrow
\left[ \begin{array} {rrr}
1 & 2 & 1 \\
0 & -3 & -7 \\
\hline
{\color {Red}0} & {\color {Red}1} & {\color {Red}2} \\
\end{array} \right]
</math>

Following this procedure, it's always possible to simplify a squared (or cubed, etc.) EU.

===Arrow commas===
The '''[[arrow]] comma''' is the ratio that equals the up [[arrow]]. This term overlaps a lot with the term mapping comma, but it isn't quite identical to it. For 5-limit temperaments the arrow comma is almost always 81/80, and for 7-limit it's almost always 64/63. But other commas can occur.

Consider Triyoti/Porcupine which is (P8, P4/3). The vanishing comma or '''VC''' is 250/243, which has a 2.3.5 monzo of [2 -5 3]. The EU is v3A1, which has a 2.3.^ monzo of [-11 7 -3]. The arrow comma or '''AC''' equals an up, therefore it vanishes when downed. The downed AC (or '''vAC''') can be expressed as a 2.3.5.^ monzo. For Triyoti/Porcupine with an EU of v3A1, the vAC is v(81/80) or [-4 4 -1 -1].

===The three commas ===
Thus when we consider a single-comma temperament along with its notation, there are three commas of interest, the VC, the vAC and the EU. In a 5-limit rank-2 temperament, they can all be expressed as 2.3.5.^ monzos.

Any two of these 3 commas determines the third comma. Thus we can approach temperaments and notations from 6 different angles. We can start with any one of these three commas and give it a specific monzo. Then we can choose one of the other two commas, experiment with a variety of specific monzos and examine the results. Let's start with specifying the VC, experimenting with the vAC and seeing what the EU turns out to be.

The EU always equals the VC (possibly inverted) plus or minus some number of vAC's. That number is whatever is needed to eliminate the higher prime. The VC must be inverted if the resulting EU would otherwise have a negative stepspan, or is a diminished unison.

In our Triyoti example, 250/243 plus 3 downed syntonic commas = v3A1. As 2.3.5.^ monzos, we have [1 -5 3 0] + 3·[-4 4 -1 -1] = [-11 7 0 -3]. Note the zeros. The VC always has a zero arrow-count and the EU always has a zero prime-5-count.

Visualizing the three commas in the lattice, one starts at the VC and heads towards the row of 3-limit intervals via the vAC. Wherever one lands is the uninflected EU. One can try other AC's besides 81/80. The AC's prime-5-count must be ±1, so [[135/128|Layobi]] (135/128) or [[Schisma|Layo]] (the schisma) are possibilities. But either of these would lead to a very remote EU (v3AA1 and v3d44 respectively), making a very awkward notation.

Next let's specify the AC, experiment with the VC and see what the EU turns out to be. For 5-limit temperaments, one can require that the AC always be 81/80, and derive the EU (and thus the notation) from the VC. For example, Sagugu/Diaschismic has VC = [11 -4 -2 0]. Subtracting two vAC's makes [19 -12 0 2] = ^^d2. This is in fact the recommended EU for (P8/2, P5).

More examples: Laquinyoti/Magic is (P8, P12/5) and has VC = [-10 -1 5 0]. Adding five vAC's makes [-30 19 0 -5]. This appears to be a AA7 but is actually a negative 2nd. We invert to get [30 -19 0 5] = ^5dd2. Guguti/Bug has VC = 27/25 = [0 3 -2 0]. Subtracting two vAC's makes [8 -5 0 2] = ^^m2. Again, this is the recommended EU.

Let's try a 7-limit temperament with the obvious vAC of 64/63 = [6 -2 -1 -1]. Zozo/Semaphore has VC = 49/48 = [-4 -1 2 0]. Adding two vAC's makes [8 -5 0 -2] = vvm2.{{todo|complete section|comment=apply to multi-comma temperaments|inline=1}}

== Addenda (late 2024) ==

=== Chord names ===
When naming chords, it's very convenient to have the freedom to rename an aug 4th as a dim 5th, or a minor 10th as an aug ninth. Thus for some pergens, an extra pair of accidentals is used. Some examples:

* [[Chords of meantone]] (P8, P5) (^1 = -d2 = pythagorean comma)
* [[Chords of diaschismic]] (P8/2, P5)
* [[Chords of hemififths]] (P8, P5/2) (/1 = vm2 = ~81/80 = ~64/63)
* [[Chords of porcupine]] (P8, P4/3)
* [[Chords of magic]] (P8, P12/5) (/1 = ^^d2)

=== Frequency of imperfect pergens ===
Imperfect pergens occur when there are multiple genchains (i.e. the octave is split), and the fifth is on a different genchain than the tonic, and also on a different perchain. How often do they occur? In order to answer that, we need to survey all pergens in order. But the question of how to do that depends on how they are sorted. The pergenLister app sorts them by the size of the larger denominator. Using this order, pergenLister finds about 4% of all pergens are imperfect. But they can also be sorted by their canonical mappings [(a b) (0 c)]. If sorted by a (octave fraction), and then by |c| (perfect multigen's fraction), more complex pergens appear sooner, and the percentage rises to about 25%.

This table lists all pergens with an unsplit octave up to the fifth-splits. In each column, the pergens are sorted by the size of the generator. The generator is listed followed by a, b and c from its mapping. All pergens with an unsplit octave are perfect.
{| class="wikitable"
|+Pergens of the form (P8, x), showing generator and mapping (a = 1)
!unsplit
!half-splits
!third-splits
!quarter-splits
!fifth-splits
!sixth-splits
|-
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (1 1 1)
|P4/2 (1 2 -2)
|P4/3 (1 2 -3)
|P4/4 (1 2 -4)
|P4/5 (1 2 -5)
|P4/6 (1 2 -6)
|-
|
|P5/2 (1 1 2)
|P5/3 (1 1 3)
|P5/4 (1 1 4)
|P5/5 (1 1 5)
|P5/6 (1 1 6)
|-
|
|
|P11/3 (1 3 -3)
|P11/4 (1 3 -4)
|P11/5 (1 3 -5)
|P11/6 (1 3 -6)
|-
|
|
|
|P12/4 (1 0 4)
|P12/5 (1 0 5)
|P12/6 (1 0 6)
|-
|
|
|
|
|ccP4/5 (1 4 -5)
|ccP4/6 (1 4 -6)
|-
|
|
|
|
|
|ccP5/6 (1 -1 6)
|}
Of all the half-octave pergens, half of every other column (i.e. 25%) are imperfect. Imperfect pergens occur whenever b is not a multiple of a.
{| class="wikitable"
|+Pergens of the form (P8/2, x), showing generator and mapping (a = 2)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (2 2 1)
|'''M2/4 (2 3 2)'''
|P4/3 (2 4 -3)
|'''M2/8 (2 3 4)'''
|P4/5 (2 4 -5)
|'''M2/12 (2 3 6)'''
|-
|
|P4/2 (2 4 -2)
|P5/3 (2 2 3)
|P4/4 (2 4 -4)
|P5/5 (2 2 5)
|P4/6 (2 4 -6)
|-
|
|
|P11/3 (2 6 -3)
|P5/4 (2 2 4)
|P11/5 (2 6 -5)
|P5/6 (2 2 6)
|-
|
|
|
|'''cm7/8 (2 5 -4)'''
|P12/5 (2 0 5)
|P11/6 (2 6 -6)
|-
|
|
|
|
|ccP4/5 (2 8 -5)
|'''cm7/12 (2 5 -6)'''
|-
|
|
|
|
|
|'''cM9/12 (2 1 6)'''
|}
Note that some of these pergens, when put in mingen form, become imperfect. For example, (P8/2, P11/3) becomes (P8/2, M2/6). Also note that for many of these pergens, the generators are comma-sized, and MOS scales will either be very "hard" (L/s very large) or else will contain very many notes per octave. For example, to bring the L/s ratio down to about 5, (P8/2, M2/4) needs a 16 note scale, and (P8/2, P11/3) needs a 28 note scale!

Of all the third-octave pergens, two-thirds of every third column (2/9 or 22%) are imperfect:
{| class="wikitable"
|+Pergens of the form (P8/3, x), showing generator and mapping (a = 3)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
|-
|P5 (3 3 1)
|P4/2 (3 6 -2)
|'''m3/9 (3 5 -3)'''
|P4/4 (3 6 -4)
|P4/5 (3 6 -5)
|'''m3/18 (3 5 -6)'''
|-
|
|P5/2 (3 3 2)
|'''M6/9 (3 4 3)'''
|P5/4 (3 3 4)
|P5/5 (3 3 5)
|'''M6/18 (3 4 6)'''
|-
|
|
|P4/3 (3 6 -3)
|P11/4 (3 9 -4)
|P11/5 (3 9 -5)
|P4/6 (3 6 -6)
|-
|
|
|
|P12/4 (3 0 4)
|P12/5 (5 0 5)
|P5/6 (3 3 6)
|-
|
|
|
|
|ccP4/5 (3 12 -5)
|'''ccm3/18 (3 7 -6)'''
|-
|
|
|
|
|
|'''ccM6/18 (3 2 6)'''
|}
Of all the quarter-octave pergens, imperfection occurs in half of every 4th column and 3/4 of every 4th column (5/16 or 31.25%).
{| class="wikitable"
|+Pergens of the form (P8/4, x), showing generator and mapping (a = 4)
!c = ±1
!c = ±2
!c = ±3
!c = ±4
!c = ±5
!c = ±6
!c = ±7
!c = ±8
|-
|P5 (4 4 1)
|'''m6/8 (4 7 -2)'''
|P4/3 (4 8 -3)
|'''M2/8 (4 6 4)'''
|P4/5 (4 8 -5)
|P4/6 (4 8 -6)
|P4/7 (4 8 -7)
|'''M2/16 (4 6 8)'''
|-
|
|P4/2 (4 8 -2)
|P5/3 (4 4 3)
|'''m6/16 (4 7 -4)'''
|P5/5 (4 4 5)
|P5/6 (4 4 6)
|P5/7 (4 4 7)
|'''m6/32 (4 7 -8)'''
|-
|
|
|P11/3 (4 12 -3)
|'''M10/16 (4 5 4)'''
|P11/5 (4 12 -5)
|P11/6 (4 12 -6)
|P11/7 (4 12 -7)
|'''M10/32 (4 5 8)'''
|-
|
|
|
|P4/4 (4 8 -4)
|P12/5 (4 0 5)
|'''m6/24 (4 7 -6)'''
|P12/7 (4 0 7)
|P4/8 (4 8 -8)
|-
|
|
|
|
|ccP4/5 (4 16 -5)
|'''M10/24 (4 5 6)'''
|ccP4/7 (4 16 -7)
|P5/8 (4 4 8)
|-
|
|
|
|
|
|'''ccm6/24 (4 9 -6)'''
|ccP5/7 (4 -4 7)
|'''ccm6/32 (4 9 -8)'''
|-
|
|
|
|
|
|
|c3P4/7 (4 20 -7)
|'''cm7/16 (4 10 -8)'''
|-
|
|
|
|
|
|
|
|'''c3M3/32 (4 3 8)'''
|}
Percentage of imperfect pergens in each category:
{| class="wikitable"
|+
!(P8, x)
!(P8/2, x)
!(P8/3, x)
!(P8/4, x)
!(P8/5, x)
!(P8/6, x)
!(P8/7, x)
|-
|none
|1/4
|2/9
|5/16
|4/25
|5/12
|6/49
|-
|0%
|25%
|22.22%
|31.25%
|16%
|41.67%
|12.24%
|}

== Addenda (Spring 2026) ==

WORK IN PROGRESS

As one stacks generators and octave-reduces, at some point one overshoots or undershoots the octave by an interval of about a quartertone or less. This small interval is the pergen's initial comma. For example, (P8, P5)'s initial comma is the pythagorean comma, its next comma is Mercator's comma, etc. Each comma has a certain genspan, here 12 and 53. The genspan of the initial comma limits the size of a scale one can construct, assuming one wants to avoid overly-small steps. Thus one can have a pythagorean scale of up to 12 notes, but a 13-note scale will have a very small step. Note that the genspan gives the maximum notes per period, not per octave.

The table below lists the initial comma of various pergens. "±" indicates a tippy pergen. "c" is the difference between the fifth and 7\12. "abs(6c)" means the absolute value of 6c. The dim 2nd is a pythagorean comma.
{| class="wikitable sortable"
|+Initial comma of each pergen
!#
!pergen
!interval
!cents
!genspan
!max
!min
!comments
|-
!1
!(P8, P5)
|±d2
|abs(12c)
|±12G
|12
|
|
|-
!2
!(P8/2, P5)
|±d2/2
|abs(6c)
|±6G
|12
|
|
|-
!3
!(P8, P4/2)
|m2/2
|50¢ - 2.5c
|5G
|5
|
|
|-
!4
!(P8, P5/2)
|A1/2
|50¢ + 3.5c
|7G
|7
|
|
|-
!5
!(P8/2, P4/2)
| colspan="2" | ''same as #3 (P8, P4/2)''
|5G
|10
|
|
|-
!6
!(P8/3, P5)
|±d2/3
|abs(4c)
|±4G
|12
|
|
|-
!7
!(P8, P4/3)
|A1/3
|33.3¢ + 2.33c
| -7G
|7
|
|
|-
!8
!(P8, P5/3)
|m2/3
|33.3¢ - 1.67c
| -5G
|5
|
|
|-
!9
!(P8, P11/3)
|M2/3
|66.7¢ + 0.67c
|2G
|2
|15
|
|-
!10
!(P8/3, P4/2)
|A2/6
|50¢ + 1.5c
|3G
|9
|
|
|-
!11
!(P8/3, P5/2)
|m3/6
|50¢ - 0.5c
|1G
|3
|
|
|-
!12
!(P8/2, P4/3)
| colspan="2" | ''same as #7 (P8, P4/3)''
| -7G
|14
|
|
|-
!13
!(P8/2, P5/3)
| colspan="2" | ''same as #8 (P8, P5/3)''
| -5G
|10
|
|
|-
!14
!(P8/2, P11/3)
|M2/6
|33.3¢ + 0.33c
|1G
|2
|30
|
|-
!15
!(P8/3, P4/3)
| colspan="2" | ''same as #7 (P8, P4/3)''
| -7G
|21
|
|
|-
!16
!(P8/4, P5)
|±d2/4
|abs(3c)
|±3G
|12
|
|
|-
!17
!(P8, P4/4)
| colspan="2" | ''same as #3 (P8, P4/2)''
|10G
|10
|
|
|-
!18
!(P8, P5/4)
|A1/4
|25¢ + 1.75c
|7G
|7
|
|
|-
!
!(P8, P5)
|
|
|
|
|
|
|-
!
!(P8, P5)
|
|
|
|
|
|
|-
!
!(P8, P5)
|
|
|
|
|
|
|-
!
!(P8, P5)
|
|
|
|
|
|
|-
!
!(P8, P5)
|
|
|
|
|
|
|-
!
!(P8, P5)
|
|
|
|
|
|
|-
!
!(P8, P5)
|
|
|
|
|
|
|-
!
!(P8, P5)
|
|
|
|
|
|
|-
!
!(P8, P5)
|
|
|
|
|
|
|-
!
!(P8, P5)
|
|
|
|
|
|
|-
!
!(P8, P5)
|
|
|
|
|
|
|}
Note the similarity of the initial comma to the EU divided by the height.

The initial comma of (P8, P11/3) is a rather large 67¢, but if there are more than 2 notes per 8ve, the L/s ratio becomes enormous!

Note the unusability of certain pergens such as (P8/2, P11/3).
[[Category:Regular temperament theory]]
[[Category:Notation]]
[[Category:Pages with proofs]]

Small comma

2026-04-29T05:39:57Z

TallKite: added new commas

A '''small comma''' is a [[comma]] whose size is approximately between 3.5 and 30 cents. These intervals are in the range from just noticeable up to usable as melodic steps. The actual perception of course varies. In [[Sagittal notation]], intervals in the smaller part of this category are [[kleisma (interval region)|kleismas]], and intervals in the larger part of this category are [[comma (interval region)|commas]]<ref>[https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>.

For commas over 100 cents in size, see [[Large comma]]; for commas in between 30 and 100 cents in size, see [[Medium comma]]; and for commas under 3.5 cents in size, see [[Unnoticeable comma]].

Due to the wide range of sizes, cents values here and on the other commas pages are listed to 5 significant digits, instead of to a fixed number of decimal places as is the [[Xenharmonic Wiki: Conventions|convention]] elsewhere on the wiki.

The commas might be numerous, but there is no need to memorise all the names. For pretty much all use cases, it is perfectly acceptable to just refer to a comma by the monzo or ratio, whichever is simpler.

== List of commas, by prime limit ==
=== 3-limit commas ===
{{See also| Table of 3-limit commas }}

{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cents]]
! Named by
|-
| 241-comma
| 241wama
| 241wM
|
| {{Monzo| 382 -241 }}
| 28.845
|
|-
| 65-comma, Pythagorean septimal comma
| 65wama, Thequiwama
| 65wM
|
| {{Monzo| -103 65 }}
| 27.075
|
|-
| [[Pythagorean comma]]
| Lalawama, Poma
| LLwM
| 531441 / 524288
| {{Monzo| -19 12 }}
| 23.460
|
|-
| [[41-comma]], Pythagorean countercomma, countercomp comma
| 41wama, Fowewama
| 41wM
| <abbr title="36893488147419103232 / 36472996377170786403">(40 digits)</abbr>
| {{Monzo| 65 -41 }}
| 19.845
| [[Flora Canou]] (2022)
|-
| [[94-comma]], garistearn comma
| 94wama, Fosebiwama
| 94wM
|
| {{Monzo| 149 -94 }}
| 16.230
| [[User:Xenllium|Xenllium]] (2021)
|-
| 147-comma
| 147wama
| 147wM
|
| {{Monzo| 233 -147 }}
| 12.615
|
|-
| 200-comma, Pythagorean integer-cent ET comma
| 200wama
| 200wM
|
| {{Monzo| 317 -200 }}
| 8.9998
|
|-
| 253-comma
| 253wama
| 253wM
|
| {{Monzo| 401 -253 }}
| 5.3848
|
|-
| [[Mercator's comma]], 53-comma
| 53wama, Fithewama
| 53wM
| <abbr title="19383245667680019896796723 / 19342813113834066795298816">(52 digits)</abbr>
| {{Monzo| -84 53 }}
| 3.6150
|
|}

=== 5-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cents]]
! Named by
|-
| [[Magic comma]], small diesis
| Laquinyoma
| L5yM
| 3125 / 3072
| {{Monzo| -10 -1 5 }}
| 29.614
|
|-
| [[Triscordial comma]]
| Tribila-triyoma
| 6L3yM
| <abbr title="18761829412124890125 / 18446744073709551616">(40 digits)</abbr>
| {{Monzo| -64 36 3 }}
| 29.321
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Hendecatonic comma]]
| Trisa-leguma
| 3s11gM
| 8796093022208 / 8649755859375
| {{Monzo| 43 -11 -11 }}
| 29.044
|
|-
| [[Devil's tridecalimma]]
| Lala-theguma
| LL13gM
| 2541865828329 / 2500000000000
| {{Monzo| -11 26 -13 }}
| 28.752
|
|-
| [[Anthoine comma]]
| Trila-quinquadyoma
| 3L20yM
| 286102294921875 / 281474976710656
| {{Monzo| -48 1 20 }}
| 28.229
|
|-
| [[Tetracot comma]], minimal diesis
| Saquadyoma
| s4yM
| 20000 / 19683
| {{Monzo| 5 -9 4 }}
| 27.660
|
|-
| [[Biscordial comma]]
| Quadla-yoyoma
| 4LyyM
| 571919811374025 / 562949953421312
| {{Monzo| -49 28 2 }}
| 27.367
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Semaja comma]]
| Lala-neyoma
| LL19yM
| 19073486328125 / 18786186952704
| {{Monzo| -33 -7 19 }}
| 26.276
| [[Petr Pařízek]] (2011)
|-
| [[Quanic comma]]
| Sepsa-quinyoma
| 7s5yM
|
| {{Monzo| 74 -54 5 }}
| 25.999
|
|-
| [[Roda]], rodan comma
| Sasa-triyoma
| ss3yM
| 131072000 / 129140163
| {{Monzo| 20 -17 3 }}
| 25.706
|
|-
| [[Gracecordial comma]]
| Trilayoma
| 3LyM
| 17433922005 / 17179869184
| {{Monzo| -34 20 1 }}
| 25.414
|
|-
| [[Trisedodge comma]]
| Saquintriguma
| s15gM
| 30958682112 / 30517578125
| {{Monzo| 19 10 -15 }}
| 24.844
| [[Petr Pařízek]] (2011)
|-
| [[Birds comma]]
| Quadsa-thiweguma
| 4s31gM
|
| {{Monzo| 72 0 -31 }}
| 24.275
|
|-
| [[Neuk comma]]
| Trisa-yoyoma
| 3syyM
| 858993459200 / 847288609443
| {{Monzo| 35 -25 2 }}
| 23.752
| [[User:CompactStar|CompactStar]] (2023)
|-
| [[Maja comma]]
| Saseyoma
| s17yM
| 762939453125 / 753145430616
| {{Monzo| -3 -23 17 }}
| 22.368
| [[Petr Pařízek]] (2011)
|-
| [[Satin comma]]
| Quinbisa-triyoma
| 10s3yM
|
| {{Monzo| 104 -70 3 }}
| 22.091
|
|-
| [[Misneb comma]]
| Quadla-quintriyoma
| 4L15yM
| <abbr title="145964630126953125 / 144115188075855872">(36 digits)</abbr>
| {{Monzo| -57 14 15 }}
| 22.076
| [[Petr Pařízek]] (2011)
|-
| [[Syntonic comma]], Didymus comma, meantone comma
| Guma
| gM
| 81 / 80
| {{Monzo| -4 4 -1 }}
| 21.506
|
|-
| [[Maquila comma]]
| Trisa-seguma
| 3s17gM
| 562949953421312 / 556182861328125
| {{Monzo| 49 -6 -17 }}
| 20.937
| [[Petr Pařízek]] (2011)
|-
| [[Sfourth comma]]
| Lala-neguma
| LL19gM
| 617673396283947 / 610351562500000
| {{Monzo| -5 31 -19 }}
| 20.644
| [[Petr Pařízek]] (2011)
|-
| [[Diaschisma]]
| Saguguma
| sggM
| 2048 / 2025
| {{Monzo| 11 -4 -2 }}
| 19.553
| Hermann von Helmholtz, Alexander Ellis (1875)
|-
| [[Countermeantone comma]]
| Quinquadguma
| 20gM
| 96402615118848 / 95367431640625
| {{Monzo| 10 23 -20 }}
| 18.691
| [[Petr Pařízek]] (2012)
|-
| [[Ditonma]]
| Lala-theyoma
| LL13yM
| 1220703125 / 1207959552
| {{Monzo| -27 -2 13 }}
| 18.168
| [[Petr Pařízek]] (2011)
|-
| [[Misty comma]]
| Sasa-triguma
| ss3gM
| 67108864 / 66430125
| {{Monzo| 26 -12 -3 }}
| 17.599
| [[Paul Erlich]] (2002)
|-
| [[Quintile comma]]
| Trila-quinguma
| 3L5gM
| 847288609443 / 838860800000
| {{Monzo| -28 25 -5 }}
| 17.306
| See the dedicated page.
|-
| [[Enneadecacot comma]]
| Tribisa-neguma
| 6s19gM
| <abbr title="604462909807314587353088 / 598546211414337158203125">(48 digits)</abbr>
| {{monzo| 79 -22 -19 }}
| 17.029
| [[User:CompactStar|CompactStar]] (2024)
|-
| [[Oquatonic comma]]
| Quadla-sepquadyoma
| 4L28yM
| <abbr title="37252902984619140625 / 36893488147419103232">(40 digits)</abbr>
| {{Monzo| -65 0 28 }}
| 16.784
| [[Petr Pařízek]] (2011)
|-
| [[Undim comma]]
| Trisa-quadguma
| 3s4gM
| 2199023255552 / 2179240250625
| {{Monzo| 41 -20 -4 }}
| 15.645
| [[Petr Pařízek]] (2011)
|-
| [[Graviton]], gravity comma
| Lala-tribiguma
| LL6gM
| 129140163 / 128000000
| {{Monzo| -13 17 -6 }}
| 15.353
| [[Mike Battaglia]] (2011)
|-
| [[Majvam comma]]
| Sasa-lebiguma
| ss22gM
| <abbr title="2404631929946112 / 2384185791015625">(32 digits)</abbr>
| {{Monzo| 40 7 -22 }}
| 14.783
| [[Petr Pařízek]] (2011)
|-
| [[Quartonic comma]]
| Saleyoma
| s11yM
| 390625000 / 387420489
| {{Monzo| 3 -18 11 }}
| 14.261
|
|-
| [[Untritonic comma]]
| Quadla-tritriyoma
| 4L9yM
| <abbr title="2270041927734375 / 2251799813685248">(32 digits)</abbr>
| {{Monzo| -51 19 9 }}
| 13.968
| [[Petr Pařízek]] (2011)
|-
| [[Quindromeda comma]]
| Quinsa-quinguma
| 5s5gM
| <abbr title="72057594037927936 / 71489976421753125">(34 digits)</abbr>
| {{Monzo| 56 -28 -5 }}
| 13.691
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Sensipent comma]], medium semicomma
| Sepguma
| 7gM
| 78732 / 78125
| {{Monzo| 2 9 -7 }}
| 13.399
|
|-
| [[Copper comma]]
| Theneyoma
| 41L29yM
|
| {{Monzo| -481 261 29 }}
| 13.353
| [[Eliora]] (2023)
|-
| [[Counterwürschmidt comma]]
| Trisa-twetheguma
| 3s23gM
| <abbr title="36028797018963968 / 35762786865234375">(34 digits)</abbr>
| {{Monzo| 55 -1 -23 }}
| 12.830
| [[Flora Canou]] (2023)
|-
| [[Tertiosec comma]]
| Laquadtribiyoma
| 6L24yM
|
| {{Monzo| -89 21 24 }}
| 12.584
| [[Petr Pařízek]] (2012)
|-
| [[Submajor comma]]
| Trila-quadbiyoma
| 3L8yM
| 69198046875 / 68719476736
| {{Monzo| -36 11 8 }}
| 12.015
|
|-
| [[Würschmidt comma]]
| Saquadbiguma
| s8gM
| 393216 / 390625
| {{Monzo| 17 1 -8 }}
| 11.445
|
|-
| [[Bicommatic comma]]
| Quadla-quinbiguma
| 4L10gM
| <abbr title="1350851717672992089 / 1342177280000000000">(38 digits)</abbr>
| {{Monzo| -37 38 -10 }}
| 11.153
|
|-
| [[Counterhanson comma]]
| Quinquinyoma
| 25yM
| <abbr title="298023223876953125 / 296148833645101056">(36 digits)</abbr>
| {{Monzo| -20 -24 25 }}
| 10.923
| [[Petr Pařízek]] (2012)
|-
| [[Countritonic comma]]
| Quadsa-tritriyoma
| 4s9yM
| <abbr title="16777216000000000 / 16677181699666569">(34 digits)</abbr>
| {{Monzo| 33 -34 9 }}
| 10.353
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Semicomma]], Fokker's comma
| Lasepyoma
| L7yM
| 2109375 / 2097152
| {{Monzo| -21 3 7 }}
| 10.061
|
|-
| [[Heptacot comma]]
| Sepsa-sepguma
| 7s7gM
|
| {{Monzo| 86 -44 -7 }}
| 9.7840
| [[Tristan Bay]] (2024)
|-
| [[Escapade comma]]
| Sasa-tritriguma
| ss9gM
| 4294967296 / 4271484375
| {{Monzo| 32 -7 -9 }}
| 9.4916
| [[Paul Erlich]] (2002)
|-
| [[Undetritisma]], twentcufo comma
| Trila-leguma
| 3L11gM
| 205891132094649 / 204800000000000
| {{Monzo| -22 30 -11 }}
| 9.1992
|
|-
| [[15625/15552|Kleisma]], semicomma majeur
| Tribiyoma
| 6yM
| 15625 / 15552
| {{Monzo| -6 -5 6 }}
| 8.1073
| {{W|Shohé Tanaka}} (1890)
|-
| [[Quintosec comma]]
| Quadsa-quinbiguma
| 4s10gM
| 140737488355328 / 140126044921875
| {{Monzo| 47 -15 -10 }}
| 7.5378
| [[Petr Pařízek]] (2012)
|-
| 59-5-comma
| Quadbisa-fineguma
| 8s59gM
|
| {{Monzo| 137 0 -59 }}
| 7.4909
|
|-
| [[Unidecma]]
| Laquadtriguma
| L12gM
| 31381059609 / 31250000000
| {{Monzo| -7 22 -12 }}
| 7.2455
|
|-
| [[Mutt comma]]
| Trila-septriyoma
| 3L21yM
| 476837158203125 / 474989023199232
| {{Monzo| -44 -3 21 }}
| 6.7230
| [[Gene Ward Smith]] (2004)
|-
| [[Sulfur comma]]
| Lela-quadquadguma
| 11L16gM
|
| {{Monzo| -115 96 -16 }}
| 6.6607
|
|-
| [[Amity comma]]
| Saquinyoma
| s5yM
| 1600000 / 1594323
| {{Monzo| 9 -13 5 }}
| 6.1536
| [[Gene Ward Smith]] (2001)
|-
| [[Parakleisma]]
| Theguma
| 13gM
| 1224440064 / 1220703125
| {{Monzo| 8 14 -13 }}
| 5.2917
| [[Gene Ward Smith]] (2001)
|-
| [[Gammic comma]]
| Laquinquadyoma
| L20yM
| 95367431640625 / 95105071448064
| {{Monzo| -29 -11 20 }}
| 4.7693
|
|-
| [[Squarschmidt comma]]
| Quadsa-tweneguma
| 4s29gM
| <abbr title="186773283746309210112 / 186264514923095703125">(42 digits)</abbr>
| {{Monzo| 61 4 -29 }}
| 4.7223
| [[Petr Pařízek]] (2012)
|-
| 43-15-comma, Huntian 15-cycle comma
| Quadtrisa-fotheguma
| 12s43gM
|
| {{Monzo| 168 -43 -43 }}
| 4.4453
|
|-
| [[Barium comma]]
| Quadtribila-sepquadbiguma
| 24L56gM
|
| {{Monzo| -225 224 -56 }}
| 4.3522
| [[Eliora]] (2023)
|-
| [[Vulture comma]]
| Sasa-quadyoma
| ss4yM
| 10485760000 / 10460353203
| {{Monzo| 24 -21 4 }}
| 4.1998
| [[Paul Erlich]] (2002)
|-
| [[Dipromethia]]
| Thebila-siweyoma
| 26L61yM
|
| {{Monzo| -335 122 61 }}
| 3.6467
| [[Eliora]] (2023)
|-
| [[Lafa comma]]
| Tribisa-quadtriguma
| 6s12gM
|
| {{Monzo| 77 -31 -12 }}
| 3.6304
| [[Petr Pařízek]] (2011)
|}

=== 7-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cents]]
! Named by
|-
| [[3645/3584|Schismean comma]]
| Laruyoma
| LryM
| 3645 / 3584
| {{Monzo| -9 6 1 -1 }}
| 29.218
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Doublehearted comma]]
| Quadbizoma
| 8zM
| 5764801 / 5668704
| {{Monzo| -5 -11 0 8 }}
| 29.102
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Frostburn comma]]
| Quadru-asepyoma
| 4ra7yM
| 78125 / 76832
| {{Monzo| -5 0 7 -4 }}
| 28.892
|
|-
| [[686/675|Senga]]
| Trizo-aguguma
| 3zaggM
| 686 / 675
| {{Monzo| 1 -3 -2 3 }}
| 27.985
| [[Gene Ward Smith]] (2005)
|-
| 23-21-comma
| Sepla-twethezoma
| 7L23zM
|
| {{Monzo| -101 23 0 23 }}
| 27.961
|
|-
| [[64/63|Septimal comma]], Archytas' comma, Leipziger Komma
| Ruma
| rM
| 64 / 63
| {{Monzo| 6 -2 0 -1 }}
| 27.264
|
|-
| [[Mandos comma]]
| Biruguguma
| 2rggM
| 31104 / 30625
| {{Monzo| 7 5 -4 -2 }}
| 26.868
|
|-
| [[Slither comma]]
| Satritriru-aquadyoma
| s9ra4yM
| 40960000 / 40353607
| {{Monzo| 16 0 4 -9 }}
| 25.822
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Bastille comma]]
|
|
|
| {{Monzo| 1426 0 -596 -15 }}
| 24.638
| [[Eliora]] (2023)
|-
| 33-7/5-comma
| Letrizoguma
| 33zgM
|
| {{Monzo| -16 0 -33 33 }}
| 22.902
|
|-
| Huntian 35-cycle comma
| Quintrisa-tritritribiruguma
| 15s54rgM
|
| {{Monzo| 277 0 -54 -54 }}
| 22.461
|
|-
| [[Blackjackisma]]
| Lasepru-aquadbiyoma
| L7ra8yM
| 854296875 / 843308032
| {{Monzo| -10 7 8 -7 }}
| 22.413
| [[Gene Ward Smith]] (2011)
|-
| [[Squalentine comma]]
| Laquadzo-atriguma
| L4za3gM
| 64827 / 64000
| {{Monzo| -9 3 -3 4 }}
| 22.227
|
|-
| [[875/864|Keema]]
| Zotriyoma
| z3yM
| 875 / 864
| {{Monzo| -5 -3 3 1 }}
| 21.902
| [[Gene Ward Smith]] (2005)
|-
| [[Betelgeuse comma]]
| Satritrizo-aguguma
| s9zaggM
| 40353607 / 39858075
| {{Monzo| 0 -13 -2 9 }}
| 21.391
|
|-
| [[3125/3087|Gariboh comma]]
| Triru-aquinyoma
| 3ra5yM
| 3125 / 3087
| {{Monzo| 0 -2 5 -3 }}
| 21.181
| [[Gene Ward Smith]] (2005)
|-
| [[Secanticornisma]]
| Laruquinguma
| Lr5gM
| 177147 / 175000
| {{Monzo| -3 11 -5 -1 }}
| 21.111
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[2430/2401|Nuwell comma]]
| Quadru-ayoma
| 4rayM
| 2430 / 2401
| {{Monzo| 1 5 1 -4 }}
| 20.785
| [[Gene Ward Smith]] (2005)
|-
| [[Septimagic comma]]
| Saquinzoma
| s5zM
| 537824 / 531441
| {{Monzo| 5 -12 0 5 }}
| 20.670
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Mermisma]]
| Sepruyoma
| 7ryM
| 2500000 / 2470629
| {{Monzo| 5 -1 7 -7 }}
| 20.460
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Negricorn comma]], small quadruple bluish
| Saquadzoguma
| s4zgM
| 153664 / 151875
| {{monzo| 6 -5 -4 4 }}
| 20.274
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Tolerant comma]]
| Sazoyoyoma
| szyyM
| 179200 / 177147
| {{Monzo| 10 -11 2 1 }}
| 19.948
|
|-
| [[Icosipentatonic comma]], 25-36/35-comma
| Quinquinruguma
| 25rgM
|
| {{Monzo| 49 50 -25 -25 }}
| 19.260
|
|-
| [[Valenwuer comma]]
| Sarutribiguma
| sr6gM
| 110592 / 109375
| {{Monzo| 12 3 -6 -1 }}
| 19.157
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Buzzardsma]], buzzard comma
| Saquadruma
| s4rM
| 65536 / 64827
| {{Monzo| 16 -3 0 -4 }}
| 18.831
| See the page.
|-
| Huntian 21-cycle comma
| Quadbisa-sepquadruma
| 8s28rM
|
| {{Monzo| 123 -28 0 -28 }}
| 18.135
|
|-
| [[Mirwomo comma]]
| Labizoyoma
| L2zyM
| 33075 / 32768
| {{Monzo| -15 3 2 2 }}
| 16.144
|
|-
| [[Catasyc comma]]
| Laruquadbiyoma
| Lr8yM
| 390625 / 387072
| {{Monzo| -11 -3 8 -1 }}
| 15.819
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Compass comma]]
| Quinruyoyoma
| 5ryyM
| 9765625 / 9680832
| {{monzo| -6 -2 10 -5 }}
| 15.098
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Sensibeta comma]]
| Satrizo-aquinyoma
| s3za5yM
| 1071875 / 1062882
| {{monzo| -1 -12 5 3 }}
| 14.586
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Trimyna comma]]
| Quinzoguma
| 5zgM
| 50421 / 50000
| {{monzo| -4 1 -5 5 }}
| 14.516
|
|-
| [[245/243|Sensamagic comma]]
| Zozoyoma
| zzyM
| 245 / 243
| {{monzo| 0 -5 1 2 }}
| 14.191
| [[Gene Ward Smith]] (2010)
|-
| [[126/125|Starling comma]], septimal semicomma
| Zotriguma
| z3gM
| 126 / 125
| {{Monzo| 1 2 -3 1 }}
| 13.795
|
|-
| [[Vermeil comma]], 34-49/48-comma
| Quinla-sequadzoma
| 5L68zM
|
| {{Monzo| -137 -34 0 68 }}
| 13.692
| [[User:Perry.k|Perry.k]] (2026)
|-
| [[4000/3969|Octagar comma]]
| Rurutriyoma
| rr3yM
| 4000 / 3969
| {{Monzo| 5 -4 3 -2 }}
| 13.469
| [[Gene Ward Smith]] (2005)
|-
| [[1728/1715|Orwellisma]]
| Triru-aguma
| 3ragM
| 1728 / 1715
| {{Monzo| 6 3 -1 -3 }}
| 13.074
|
|-
| [[Mynaslender comma]]
| Sepru-ayoma
| 7rayM
| 829440 / 823543
| {{Monzo| 11 4 1 -7 }}
| 12.352
| [[User:Xenllium|Xenllium]] (2021)
|-
| 35-7/5-comma
| Sepquinruyoma
| 35ryM
|
| {{monzo| 17 0 35 -35 }}
| 12.073
|
|-
| [[Chromatisma]]
| Trisa-triru-aquadquadyoma
| 3s3ra16yM
| <abbr title="640000000000000000 / 635585924776181463">(36 digits)</abbr>
| {{Monzo| 22 -32 16 -3 }}
| 11.982
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Mistisma]]
| Sazoquadguma
| sz4gM
| 458752 / 455625
| {{Monzo| 16 -6 -4 1 }}
| 11.841
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Bronzisma]]
| Satriru-aguguma
| s3raggM
| 2097152 / 2083725
| {{Monzo| 21 -5 -2 -3 }}
| 11.120
| [[User:Xenllium|Xenllium]] (2021)
|-
| 34-jubilismic comma
| Sequadzoguma
| 68zgM
|
| {{Monzo| -33 0 -68 68 }}
| 10.829
|
|-
| [[Fynn's comma]], 26-7-comma, Hunt 7-cycle comma
| Quadsa-thebiruma
| 4s26rM
| <abbr title="9444732965739290427392 / 9387480337647754305649">(44 digits)</abbr>
| {{Monzo| 73 0 0 -26 }}
| 10.526
| [[Fynn Cerulean]] (2026) for ''Fynn's comma''
|-
| [[Septiness comma]]
| Sasasepruma
| ss7rM
| 67108864 / 66706983
| {{Monzo| 26 -4 0 -7 }}
| 10.399
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[31-comma temperaments|31-35-comma]]
| Tritrila-thiwezoyoma
| 9L31zyM
|
| {{Monzo| -159 0 31 31 }}
| 9.3282
|
|-
| [[Quince comma]]
| Lasepzo-aguguma
| L7zaggM
| 823543 / 819200
| {{Monzo| -15 0 -2 7 }}
| 9.1539
|
|-
| [[Uniwiz comma]]
| Quadzoyoma
| 4zyM
| 1500625 / 1492992
| {{Monzo| -11 -6 4 4 }}
| 8.8285
|
|-
| [[Historisma]]
| Latribizoguma
| L6zgM
| 257298363 / 256000000
| {{Monzo| -14 7 -6 6 }}
| 8.7582
| [[ArrowHead294]] (2024)
|-
| [[1029/1024|Gamelisma]]
| Latrizoma
| L3zM
| 1029 / 1024
| {{Monzo| -10 1 0 3 }}
| 8.4327
|
|-
| [[Varunisma]]
| Labizoguguma
| L2zggM
| 321489 / 320000
| {{Monzo| -9 8 -4 2 }}
| 8.0370
|
|-
| [[225/224|Marvel comma]], septimal kleisma
| Ruyoyoma
| ryyM
| 225 / 224
| {{Monzo| -5 2 2 -1 }}
| 7.7115
| [[Gene Ward Smith]] (2003)
|-
| [[Dimcomp comma]]
| Quadruyoyoma
| 4ryyM
| 390625 / 388962
| {{Monzo| -1 -4 8 -4 }}
| 7.3861
|
|-
| [[Cataharry comma]]
| Labiruguma
| L2rgM
| 19683 / 19600
| {{Monzo| -4 9 -2 -2 }}
| 7.3158
| [[Gene Ward Smith]] (2005)
|-
| [[Procyon comma]]
| Sasepzo-atriguma
| s7za3gM
| 823543 / 820125
| {{Monzo| 0 -8 -3 7 }}
| 7.2002
|
|-
| [[Qiqi comma]]
| Sepruyoyoma
| 7ryyM
| 48828125000 / 48629390607
| {{Monzo| 3 -10 14 -7 }}
| 7.0606
| [[User:Angekallikoita|Ange]] (2025)
|-
| [[Mirkwai comma]]
| Quinru-aquadyoma
| 5ra4yM
| 16875 / 16807
| {{Monzo| 0 3 4 -5 }}
| 6.9903
| [[Gene Ward Smith]] (2005)
|-
| [[Canousma]]
| Saquadzo-atriyoma
| s4za3yM
| 4802000 / 4782969
| {{Monzo| 4 -14 3 4 }}
| 6.8748
| [[Flora Canou]] (2020)
|-
| [[Triwellisma]]
| Tribizo-asepguma
| 6za7gM
| 235298 / 234375
| {{Monzo| 1 -1 -7 6 }}
| 6.8044
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Stearnsma]]
| Latribiruma
| L6rM
| 118098 / 117649
| {{Monzo| 1 10 0 -6 }}
| 6.5946
|
|-
| [[10976/10935|Hemimage comma]]
| Satrizo-aguma
| s3zagM
| 10976 / 10935
| {{Monzo| 5 -7 -1 3 }}
| 6.4790
| [[Gene Ward Smith]] (2005)
|-
| [[3136/3125|Hemimean comma]]
| Zozoquinguma
| zz5gM
| 3136 / 3125
| {{Monzo| 6 0 -5 2 }}
| 6.0832
| [[Gene Ward Smith]] (2005)
|-
| [[5120/5103|Hemifamity comma]], 5/7-kleisma
| Saruyoma
| sryM
| 5120 / 5103
| {{Monzo| 10 -6 1 -1 }}
| 5.7578
| [[Gene Ward Smith]] (2005)
|-
| [[Parkleiness comma]]
| Zotritriguma
| z9gM
| 1959552 / 1953125
| {{Monzo| 7 7 -9 1 }}
| 5.6875
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Octaphore comma]], enneagari comma
| Sasa-quadbizoma
| ss8zM
| 94450499584 / 94143178827
| {{Monzo| 14 -23 0 8 }}
| 5.6422
| [[User:Unque|Unque]] (2024)
|-
| [[Linus comma]]
| Saquinbizoguma
| s10zgM
| 578509309952 / 576650390625
| {{Monzo| 11 -10 -10 10 }}
| 5.5719
| [[Gene Ward Smith]] (2011)
|-
| [[Reiwa comma]]
| Saquadru-asepyoma
| s4ra7yM
| 1280000000 / 1275989841
| {{monzo| 14 -12 7 -4 }}
| 5.4324
| [[Flora Canou]] (2022)
|-
| [[6144/6125|Porwell comma]]
| Sarurutriguma
| srr3gM
| 6144 / 6125
| {{Monzo| 11 1 -3 -2 }}
| 5.3620
| [[Gene Ward Smith]], [[Petr Pařízek]] (2010)
|-
| [[Acromagic comma]]
| Sasa-sepzo-aquadguma
| ss7za4gM
| 26985857024 / 26904200625
| {{Monzo| 15 -16 -4 7 }}
| 5.2466
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Cartoonisma]]
| Satritrizo-asepbiguma
| s9za14gM
| 165288374272 / 164794921875
| {{Monzo| 12 -3 -14 9 }}
| 5.1762
| [[Eliora]] (2023)
|-
| [[Hemfiness comma]]
| Saquadru-atriyoma
| s4ra3yM
| 4096000 / 4084101
| {{Monzo| 15 -5 3 -5 }}
| 5.0366
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Acrodec comma]]
| Sasa-tribizo-aquadbiguma
| ss6za8gM
| 7710244864 / 7688671875
| {{Monzo| 16 -9 -8 6 }}
| 4.8507
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Hewuermera comma]]
| Satribiru-aguma
| s6ragM
| 589824 / 588245
| {{Monzo| 16 2 -1 -6 }}
| 4.6408
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Hearts comma]]
| Trila-quadzoma
| 3L4zM
| 34451725707 / 34359738368
| {{Monzo| -35 15 0 4 }}
| 4.6286
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Lokisma]], loki comma
| Sasa-bizotriguma
| ss2z3gM
| 102760448 / 102515625
| {{Monzo| 21 -8 -6 2 }}
| 4.1295
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Garischisma]], septimal schisma
| Sasaruma
| ssrM
| 33554432 / 33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.8041
|
|-
| [[Wadisma]]
| Latritrizo-ayoma
| L9zayM
| 201768035 / 201326592
| {{Monzo| -26 -1 1 9 }}
| 3.7919
|
|-
| [[Septimal enneadeca]]
| Quinla-neruma
| 5L19rM
| <abbr title="1570042899082081611640534563 / 1566652225014704215735402496">(56 digits)</abbr>
| {{Monzo| -37 57 0 -19 }}
| 3.7428
| [[Godtone]] (2024)
|-
| [[Quasiorwellisma]]
| Sazoquinbiguma
| sz10gM
| 29360128 / 29296875
| {{Monzo| 22 -1 -10 1 }}
| 3.7338
|
|}

=== 11-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[Dew comma]]
| Saloma
| s1oM
| 180224 / 177147
| {{Monzo| 14 -11 0 0 1 }}
| 29.812
| [[User:CompactStar|CompactStar]] (2023)
|-
| [[Thuja comma]]
| Saquinlu-ayoma
| s5(1u)yM
| 163840 / 161051
| {{Monzo| 15 0 1 0 -5 }}
| 29.724
|
|-
| [[625/616|Quadrikite comma]]
| Luruquadyoma
| 1ur4yM
| 625 / 616
| {{Monzo| -3 0 4 -1 -1 }}
| 25.111
| [[Praveen Venkataramana]], [[Lumi Pakkanen]] (2022)
|-
| [[1350/1331|Large tetracot diesis]]
| Trilu-ayoyoma
| 3(1u)yyM
| 1350 / 1331
| {{Monzo| 1 3 2 0 -3 }}
| 24.539
| [[User:ArrowHead294|ArrowHead294]] (2024)
|-
| [[Sensmus comma]]
| Salozoguma
| s1ozgM
| 1232 / 1215
| {{Monzo| 4 -5 -1 1 1 }}
| 24.055
|
|-
| [[Sevnothrush comma]]
| Loquinguma
| 1o5gM
| 3168 / 3125
| {{Monzo| 5 2 -5 0 1 }}
| 23.659
|
|-
| [[245/242|Frostma]]
| Biluzo-ayoma
| 2(1uz)yM
| 245 / 242
| {{Monzo| -1 0 1 2 -2 }}
| 21.330
| [[Praveen Venkataramana]] (2022)
|-
| [[Distarma]]
| Trilozoma
| 3(1o)zM
| 9317 / 9216
| {{Monzo|-10 -2 0 1 3}}
| 18.869
| [https://twitter.com/Lilly__Flores Lilly Flores] (2023)
|-
| [[1617/1600|Antimisma]]
| Lobizoguma
| 1o2zgM
| 1617 / 1600
| {{Monzo| -6 1 -2 2 1 }}
| 18.297
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[99/98|Mothwellsma]]
| Loruruma
| 1orrM
| 99 / 98
| {{Monzo| -1 2 0 -2 1 }}
| 17.576
|
|-
| [[1610510/1594323|Fifthchromisma]]
| Saquinlo-ayoma
| s5(1o)yM
| 1610510 / 1594323
| {{Monzo| 1 -13 1 0 5 }}
| 17.488
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[100/99|Ptolemisma]]
| Luyoyoma
| 1uyyM
| 100 / 99
| {{Monzo| 2 -2 2 0 -1 }}
| 17.399
|
|-
| [[Hemimin comma]]
| Trilu-azoma
| 3(1u)zM
| 1344 / 1331
| {{Monzo| 6 1 0 1 -3 }}
| 16.827
|
|-
| [[Betarabian comma]]
| Laloloma
| L1ooM
| 264627 / 262144
| {{Monzo| -18 7 0 0 2 }}
| 16.321
| [[Dawson Berry]] (2020)
|-
| [[Biyatisma]]
| Lologuma
| 1oogM
| 121 / 120
| {{Monzo| -3 -1 -1 0 2 }}
| 14.367
| [[Gene Ward Smith]] (2010)
|-
| [[Absinthma]]
| Luluruyoma
| 1uuryM
| 2560 / 2541
| {{Monzo| 9 -1 1 -1 -2 }}
| 12.897
| [[User:Jerdle|Jerdle]] (2023)
|-
| [[2835/2816|35/11 kleisma]]
| Laluzoyoma
| L1uzyM
| 2835 / 2816
| {{Monzo| -8 4 1 1 -1 }}
| 11.642
|
|-
| [[Aphrowe comma]]
| Trilo-aruruma
| 3(1o)rrM
| 1331 / 1323
| {{Monzo| 0 -3 0 -2 3 }}
| 10.437
| [[Lériendil]] (2024)
|-
| [[2200/2187|Small tetracot diesis]]
| Saloyoyoma
| s1oyyM
| 2200 / 2187
| {{Monzo| 3 -7 2 0 1 }}
| 10.260
| [[User:ArrowHead294|ArrowHead294]]
|-
| [[Valinorsma]]
| Loruguguma
| 1orggM
| 176 / 175
| {{Monzo| 4 0 -2 -1 1 }}
| 9.8646
|
|-
| [[Pentacircle comma]]
| Saluzoma
| s1uzM
| 896 / 891
| {{Monzo| 7 -4 0 1 -1 }}
| 9.6880
|
|-
| [[Alpharabian comma]]
| Laquadloma
| L4(1o)M
| 131769 / 131072
| {{Monzo| -17 2 0 0 4 }}
| 9.1818
| [[Dawson Berry]] (2020)
|-
| [[Orgonisma]]
| Satrilu-aruruma
| s3(1u)rrM
| 65536 / 65219
| {{Monzo| 16 0 0 -2 -3 }}
| 8.3944
|
|-
| [[Quindecic comma]]
| Sasa-quintriloruma
| ss15(1or)M
|
| {{Monzo| 14 -15 0 -15 15 }}
| 8.0555
|
|-
| [[117649/117128]]
| Bilulutrizoma
| 2(1uu3z)M
| 117649 / 117128
| {{Monzo| -3 0 0 6 -4 }}
| 7.6837
|
|-
| [[Topsy comma]]
| Quadlo-atrizo-asepguma
| 4(1o)3za7gM
| 5021863 / 5000000
| {{Monzo| -6 0 -7 3 4 }}
| 7.5535
|
|-
| [[4375/4356|Fantares comma]]
| Luluzoquadyoma
| 1uuz4yM
| 4375 / 4356
| {{Monzo| -2 -2 4 1 -2 }}
| 7.5349
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Semicanousma]]
| Quadlo-aguma
| 4(1o)gM
| 14641 / 14580
| {{Monzo| -2 -6 -1 0 4 }}
| 7.2281
| [[Flora Canou]] (2020)
|-
| [[243/242|Rastma]]
| Luluma
| 1uuM
| 243 / 242
| {{Monzo| -1 5 0 0 -2 }}
| 7.1391
|
|-
| [[3388/3375|Myhemiwell comma]]
| Lolozotriguma
| 1ooz3gM
| 3388 / 3375
| {{Monzo| 2 -3 -3 1 2 }}
| 6.6556
|
|-
| [[Pythrabian comma]]
| Trisaloma
| 3s1oM
| 94489280512 / 94143178827
| {{Monzo| 33 -23 0 0 1 }}
| 6.3529
| [[Dawson Berry]] (2021)
|-
| [[Semiporwellisma]]
| Saluluguma
| s1uugM
| 16384 / 16335
| {{Monzo| 14 -3 -1 0 -2 }}
| 5.1854
| [[Flora Canou]] (2021)
|-
| [[Octatonic comma]], undecimal octatonic comma
| Quadbiluma
| 8(1u)M
| 214990848 / 214358881
| {{Monzo| 15 8 0 0 -8 }}
| 5.0965
| [[Eliora]] (2023)
|-
| [[385/384|Keenanisma]]
| Lozoyoma
| 1ozyM
| 385 / 384
| {{Monzo| -7 -1 1 1 1 }}
| 4.5026
| [[Paul Erlich]] (2001)
|-
| [[Trimitone comma]]
| Lalotriguma
| L1o3gM
| 8019 / 8000
| {{Monzo| -6 6 -3 0 1 }}
| 4.1068
| [[User:Godtone|Godtone]] (2021)
|-
| [[4-cent comma]]
| Lutritryoma
| 1u9yM
| 1953125 / 1948617
| {{Monzo| 0 -11 9 0 -1 }}
| 4.0004
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[441/440|Werckisma]]
| Luzozoguma
| 1uzzgM
| 441 / 440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.9302
|
|-
| [[1375/1372|Moctdel comma]]
| Lotriruyo
| 1o3ryM
| 1375 / 1372
| {{Monzo| -2 0 3 -3 1 }}
| 3.7814
|
|-
| [[Unisquarisma]], unisquary comma
| Trilu-aquadzo-ayoma
| 3(1u)4zayM
| 12005 / 11979
| {{Monzo| 0 -2 1 4 -3 }}
| 3.7535
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[6250/6237|Liganellus comma]], liganellisma
| Luruquinyoma
| 1ur5yM
| 6250 / 6237
| {{Monzo| 1 -4 5 -1 -1 }}
| 3.6047
| [[Dawson Berry]] (2020)
|}

=== 13-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color Name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[1600/1573|Cameratasma]]
|
|
| 1600/1573
| {{Monzo| 6 0 2 0 -2 -1 }}
| 29.464
| [[Phlub]] (2026)
|-
| [[Lovecraft comma]]
| Thothotriluma
| 3oo3(1u)M
| 1352/1331
| {{Monzo| 3 0 0 0 -3 2 }}
| 27.101
|
|-
| [[65/64|Wilsorma]]
| Thoyoma
| 3oyM
| 65/64
| {{Monzo| -6 0 1 0 0 1 }}
| 26.841
|
|-
| [[Hyperpyth comma]]
| Quadtho-aquinguma
| 4(3o)5gM
| 28561/28125
| {{Monzo| 0 -2 -5 0 0 4 }}
| 26.632
|
|-
| [[66/65|Winmeanma]]
| Thuloguma
| 3u1ogM
| 66/65
| {{Monzo| 1 1 -1 0 1 -1 }}
| 26.432
|
|-
| [[343/338|Sooty fox comma]]
| Thuthutrizoma
| 3uu3zM
| 343/338
| {{Monzo| -1 0 0 3 0 -2 }}
| 25.422
| [[Budjarn Lambeth]] (2024)
|-
| [[Tetris comma]]
| Sathoma
| s3oM
| 6656/6561
| {{Monzo| 9 -8 0 0 0 1 }}
| 24.888
| [[Godtone]] (2024)
|-
| [[507/500|Large semisixthma]]
| Thothotriguma
| 3oo3gM
| 507/500
| {{Monzo| -2 1 -3 0 0 2 }}
| 24.069
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[78/77|Negustma]]
| Tholuruma
| 3o1urM
| 78/77
| {{Monzo| 1 1 0 -1 -1 1 }}
| 22.339
|
|-
| [[Greater tendoneutralisma]]
| Laquadbithoma
| L8(3o)M
| 815730721 / 805306368
| {{Monzo| -28 -1 0 0 0 8 }}
| 22.266
| [[Godtone]] (2024)
|-
| [[2025/2002|Beyoncisma]]
|
|
| 2025/2002
| {{Monzo| -1 4 2 -1 -1 -1 }}
| 19.776
| [[Phlub]] (2026)
|-
| [[91/90|Biome comma, superleap comma]]
| Thozoguma
| 3ozgM
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.130
|
|-
| [[8281/8192|Diahuntmisma]]
| Labithozoma
| L2(3oz)M
| 8281/8192
| {{Monzo| -13 0 0 2 0 2 }}
| 18.707
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[512/507|Tridecimal neutral thirds comma]]
| Thuthuma
| 3uuM
| 512/507
| {{Monzo| 9 -1 0 0 0 -2 }}
| 16.990
|
|-
| [[105/104|Animist comma]]
| Thuzoyoma
| 3uzyM
| 105/104
| {{Monzo| -3 1 1 1 0 -1 }}
| 16.567
|
|-
| [[28812/28561|Tesseract comma]]
| Quadthuzoma
| 4(3uz)M
| 28812/28561
| {{Monzo| 2 1 0 4 0 -4 }}
| 15.148
| [[User:Unque|Unque]] (2025)
|-
| [[832/825|Tholuguguma]]
| Tholuguguma
| 3o1uggM
| 832/825
| {{Monzo| 6 -1 -2 0 -1 1 }}
| 14.627
|
|-
| [[Secorian comma]]
| Sathuzoma
| s3uzM
| 28672 / 28431
| {{Monzo| 12 -7 0 1 0 -1 }}
| 14.613
|
|-
| [[3159/3136|Mosaic comma]]
| Lathoruruma
| L3orrM
| 3159/3136
| {{Monzo| -6 5 0 -2 0 1}}
| 12.651
|
|-
| [[275/273|Gassorma]]
| Thuloruyoyoma
| 3u1oryyM
| 275/273
| {{Monzo| 0 -1 2 -1 1 -1 }}
| 12.637
|
|-
| [[144/143|Grossma]]
| Thuluma
| 3u1uM
| 144/143
| {{Monzo| 4 2 0 0 -1 -1 }}
| 12.064
|
|-
| [[24167/24000|Tritho-alotriguma]]
| Tritho-alotriguma
| 3(3o)1o3gM
| 24167/24000
| {{Monzo| -6 -1 -3 0 1 3}}
| 12.005
|
|-
| [[Lesser tendoneutralisma]]
| Sasa-quadtrithuma
| ss12(3u)M
| 70368744177664 / 69894255367443
| {{Monzo| 46 -1 0 0 0 -12 }}
| 11.713
| [[Godtone]] (2024)
|-
| [[1701/1690|Kuhnausma]]
|
|
| 1701/1690
| {{Monzo| -1 5 -1 1 0 -2 }}
| 11.232
| [[Phlub]] (2026)
|-
| [[Dinos comma]]
| Lathuthuquinguma
| L3uu5gM
| 531441/528125
| {{Monzo| 0 12 -5 0 0 -2 }}
| 10.836
| [[Dummy Index]] (2024)
|-
| [[169/168|Buzurgisma, dhanvantarisma]]
| Thothoruma
| 3oorM
| 169/168
| {{Monzo| -3 -1 0 -1 0 2 }}
| 10.274
| [[Margo Schulter]] (2012) for ''buzurgisma''
|-
| [[3042/3025|Diagassormisma]]
| Bitholuguma
| 2(3o1ug)M
| 3042/3025
| {{Monzo| 1 2 -2 0 -2 2 }}
| 9.7020
| [[User:Xenllium|Xenllium]] (2022)
|-
| Greater nelindic comma
| Thothoquinru-ayoyoma
| 3oo5rayyM
| 16900/16807
| {{Monzo| 2 0 2 -5 0 2 }}
| 9.5532
| [[User:Kaiveran|Kaiveran]] (2019)
|-
| [[1287/1280|Catadictma]]
| Thologuma
| 3o1ogM
| 1287/1280
| {{Monzo| -8 2 -1 0 1 1 }}
| 9.4419
|
|-
| [[Glacier comma]]
| Quinthuma
| 5(3u)M
| 373248/371293
| {{Monzo| 9 6 0 0 0 -5 }}
| 9.0917
| [[ArrowHead294]] (2024)
|-
| [[196/195|Mynucuma]]
| Thuzozoguma
| 3uzzgM
| 196/195
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.8554
|
|-
| [[1625/1617|Sopreisma]]
| Tholururutriyoma
| 3urr3yM
| 1625/1617
| {{Monzo| 0 -1 3 -2 -1 1 }}
| 8.5440
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[640/637|Huntma]], lesser nelindic comma
| Thururuyoma
| 3urryM
| 640/637
| {{Monzo| 7 0 1 -2 0 -1 }}
| 8.1342
| [[User:Kaiveran|Kaiveran]] (2019) for ''lesser nelindic comma''
|-
|Sonoma
|Lala-tritritho-aquadyoma
|LL9(3o)4yM
|6627812108125/
6597069766656
|{{Monzo|-41 -1 4 0 0 9}}
|8.0488
|[https://x.com/vib_gen/status/2038852033244246443 Vib, Misohito Nakai] (2026)
|-
| [[2197/2187|Threedie]]
| Satrithoma
| s3(3o)M
| 2197/2187
| {{Monzo| 0 -7 0 0 0 3 }}
| 7.8980
|
|-
| [[Tridecimal nakaisma]]
| Quinsa-quadtritrithu-azoma
| 5s36(3u)zM
|
| {{Monzo| 132 -1 0 1 0 -36 }}
| 7.8751
| [[User:原井玉葱郎|Misohito Nakai]] (2025)
|-
| [[4394/4375|Hebrewsma]]
| Tritho-aruquadguma
| 3(3o)r4gM
| 4394/4375
| {{Monzo| 1 0 -4 -1 0 3 }}
| 7.5022
| [[Eliora]] (2022)
|-
| [[1188/1183|Kestrel comma]]
| Thuthuloruma
| 3uu1orM
| 1188/1183
| {{Monzo| 2 3 0 -1 1 -2 }}
| 7.3017
| [[Gene Ward Smith]] (2011)
|-
| [[30E comma|2D9 comma]]
| Thotriyoma
| 3o3yM
| 131625/131072
| {{Monzo|-17 4 3 0 0 1}}
| 7.2888
| [https://twitter.com/Regret_March/status/1709762093749252209 Figreflekt] (2023) but [https://twitter.com/Figreflekt/status/1710195052520337680 revised later]{{dead link}}
|-
| [[Brontesisma]]
| Trithu-azozoyoma
| 3(3u)zzM
| 2205/2197
| {{Monzo| 0 2 1 2 0 -3 }}
| 6.2925
| [[Francium]] (2025)
|-
| [[Praveensma]]
| Thoquadzoma
| 3o4zM
| 31213/31104
| {{Monzo| -7 -5 0 4 0 1 }}
| 6.0563
| [[Praveen Venkataramana]] (2022)
|-
| [[1573/1568|Lambeth comma]]
| Thobiloruma
| 3o2(1or)M
| 1573/1568
| {{Monzo| -5 0 0 -2 2 1 }}
| 5.5117
| [[Flora Canou]] (2022)
|-
| [[325/324|Marveltwin comma]]
| Thoyoyoma
| 3oyyM
| 325/324
| {{Monzo| -2 -4 2 0 0 1 }}
| 5.3351
|
|-
| [[Valerisma]], Hunt 13-cycle comma
| Laquinbithoma
| L10(3o)M
| 137858491849 / 137438953472
| {{Monzo| -37 0 0 0 0 10 }}
| 5.2766
| [[Mason Green]] (2016)
|-
| [[351/350|Ratwolfsma]]
| Thoruguguma
| 3orggM
| 351/350
| {{Monzo| -1 3 -2 -1 0 1 }}
| 4.9393
|
|-
| [[352/351|Major minthma, major gentle comma]], 11/13-kleisma
| Thuloma
| 3u1oM
| 352/351
| {{Monzo| 5 -3 0 0 1 -1 }}
| 4.9253
| See the page.
|-
| [[364/363|Minor minthma, minor gentle comma]]
| Tholuluzoma
| 3o1uuzM
| 364/363
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.7627
| See the page.
|-
| [[847/845|Cuthbert comma]]
| Bithulo-azoguma
| 2(3u1o)zgM
| 847/845
| {{Monzo| 0 0 -1 1 2 -2 }}
| 4.0928
|
|}

=== 17-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[2048/2023|Susurrisma, susurration comma]]
| Susuruma
| 17uurM
| 2048/2023
| {{Monzo| 11 0 0 -1 0 0 -2 }}
| 21.263
| [[User:Jerdle|Jerdle]] (2026)
|-
| [[85/84|Monk comma]]
| Soruyoma
| 17oryM
| 85/84
| {{Monzo| -2 -1 1 -1 0 0 1 }}
| 20.488
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[289/286|Lum comma]]
| Sosothuluma
| 17oo3u1uM
| 289/286
| {{Monzo| -1 0 0 0 -1 -1 2 }}
| 18.065
|
|-
| [[2197/2176|Mey comma]]
| Sutrithov
| 17u3(3o)M
| 2197/2176
| {{Monzo| -7 0 0 0 0 3 -1 }}
| 16.628
|
|-
| [[429/425|Middle semisixthma]]
| Suthologuguma
| 17u3o1oggM
| 429/425
| {{Monzo| 0 1 -2 0 1 1 -1 }}
| 16.218
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[4131/4096|Septendecimal comma]], Hunt flat 2 comma
| Lasoma
| L17oM
| 4131/4096
| {{Monzo| -12 5 0 0 0 0 1 }}
| 14.730
| [[Flora Canou]] (2020) for ''septendecimal comma''
|-
| [[120/119|Lynchisma]]
| Suruyoma
| 17uryM
| 120/119
| {{Monzo| 3 1 1 -1 0 0 -1 }}
| 14.487
| [[User:Xenllium|Xenllium]] (2022)
|-
| 23-17-comma, 23 semitone comma
| Trila-twethesoma
| 3L23(17o)M
|
| {{Monzo| -94 0 0 0 0 0 23 }}
| 13.974
|
|-
| [[136/135|Diatisma]], diatic comma, fiventeen comma, septendecimal semicomma
| Soguma
| 17ogM
| 136/135
| {{Monzo| 3 -3 -1 0 0 0 1 }}
| 12.777
| See the page.
|-
| [[154/153|Augustma]]
| Sulozoma
| 17u1ozM
| 154/153
| {{Monzo| 1 -2 0 1 1 0 -1 }}
| 11.278
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[170/169|Major naiadma]]
| Sothuthuyoma
| 17o3uuyM
| 170/169
| {{Monzo| 1 0 1 0 0 -2 1 }}
| 10.214
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2187/2176|Septendecimal schisma]]
| Lasuma
| L17uM
| 2187/2176
| {{Monzo| -7 7 0 0 0 0 -1 }}
| 8.7296
| [[Plainsound Music Edition]] (2020)
|-
| [[1452/1445|Small semisixthma]]
| Susulologuma
| 17uu1oogM
| 1452/1445
| {{Monzo| 2 1 -1 0 2 0 -2 }}
| 8.3664
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[Mean thirds comma]]
| Lasosoyoma
| L17ooyM
| 1053405/1048576
| {{Monzo|-20 6 1 0 0 0 2}}
| 7.9545
| [[User:Jerdle|Jerdle]] (2022)
|-
| [[221/220|Minor naiadma]]
| Sotholuguma
| 17o3o1ugM
| 221/220
| {{Monzo| -2 0 -1 0 -1 1 1 }}
| 7.8514
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2057/2048|Blume comma]]
| Sololoma
| 17o1ooM
| 2057/2048
| {{monzo| -11 0 0 0 2 0 1 }}
| 7.5913
| [[Douglas Blumeyer]]
|-
| [[256/255|Charisma]], charic comma, septendecimal kleisma
| Suguma
| 17ugM
| 256/255
| {{Monzo| 8 -1 -1 0 0 0 -1 }}
| 6.7759
| See the page.
|-
| [[273/272|Tannisma, prototannisma]]
| Suthozoma
| 17u3ozM
| 273/272
| {{Monzo| -4 1 0 1 0 1 -1}}
| 6.3532
| [[Scott Dakota]] (2017) for ''tannisma'' [[Flora Canou]] (2023) for ''prototannisma''
|-
| [[289/288|Semitonisma]], septendecimal semitones comma, septendecimal 6-cent comma
| Sosoma
| 17ooM
| 289/288
| {{Monzo| -5 -2 0 0 0 0 2 }}
| 6.0008
| [[Flora Canou]] (2023) ''for semitonisma''
|-
| [[375/374|Ursulisma]]
| Sulutriyoma
| 17u1u3yM
| 375/374
| {{Monzo| -1 1 3 0 -1 0 -1 }}
| 4.6228
| [[Dawson Berry]], [[User:VIxen|VIxen]] (2023)
|-
| [[442/441|Seminaiadma]]
| Sothoruruma
| 17o3orrM
| 442/441
| {{Monzo| 1 -2 0 -2 0 1 1 }}
| 3.9213
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[80-17-comma]], 17-ripple <strike>integer cents</strike> comma{{clarify}}
| Lesa-quinquadquadsuma
| 11s80(17u)M
|
| {{Monzo| 327 0 0 0 0 0 -80 }}
| 3.5672
| [[Eliora]] (2022) for ''80-17-comma''
|}

=== 19-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[135/133|Nuruyoma]]
| Nuruyoma
| 19uryM
| 135/133
| {{Monzo| 0 3 1 -1 0 0 0 -1 }}
| 25.84
|
|-
| [[76/75|Large undevicesimal 1/9-tone]]
| Noguguma
| 19oggM
| 76/75
| {{Monzo| 2 -1 -2 0 0 0 0 1 }}
| 22.931
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[77/76|Small undevicesimal 1/9-tone]]
| Nulozoma
| 19u1ozM
| 77/76
| {{Monzo| -2 0 0 1 1 0 0 -1 }}
| 22.631
| [[User:Xenllium|Xenllium]] (2023)
|-
|[[Lanuma]]
|Lanuma
|L19uM
|19683/19456
|{{Monzo| -10 9 0 0 0 0 0 -1}}
|20.082
|[[Kite Giedraitis]] (2026)
|-
| [[96/95|19th-partial chroma]]
| Nuguma
| 19ugM
| 96/95
| {{Monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.128
| [[User:Flirora|Flirora]] (2021)
|-
| [[Ume comma]]
| Nutrisoma
| 19u3(17o)M
| 4913/4864
| {{Monzo| -8 0 0 0 0 0 3 -1 }}
| 17.353
|
|-
| [[729/722|Undevicesimal diaschisma]]
| Lanunuma
| L19uuM
| 729/722
| {{Monzo| -1 6 0 0 0 0 0 -2 }}
| 16.704
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[6912/6859|Deviaug comma]]
| Trinuma
| 3(19u)M
| 6912/6859
| {{Monzo| 8 3 0 0 0 0 0 -3 }}
| 13.326
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[133/132|Minithirdma]]
| Noluzoma
| 19o1uzM
| 133/132
| {{Monzo| -2 -1 0 1 -1 0 0 1 }}
| 13.066
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[153/152|Ganassisma]], Ganassi's comma
| Nusoma
| 19u17oM
| 153/152
| {{Monzo| -3 2 0 0 0 0 1 -1 }}
| 11.352
| [[User:Flirora|+merlan #flirora]] (2021)
|-
| [[171/170|Malcolmisma]]
| Nosuguma
| 19o17ugM
| 171/170
| {{Monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.154
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[131072/130321|Undevicesimal diminished comma]], Hunt 19-cycle comma
| Saquadnuma
| s4(19u)M
| 131072 / 130321
| {{Monzo| 17 0 0 0 0 0 0 -4 }}
| 9.9479
|
|-
| [[Eye comma]]
| Nubisoluma
| 19u2(17o1u)M
| 2312/2299
| {{Monzo| 3 0 0 0 -2 0 2 -1 }}
| 9.7619
|
|-
| [[363/361|Godzillisma]]
| Binuloma
| 2(19u1o)M
| 363/361
| {{Monzo| 0 1 0 0 2 0 0 -2 }}
| 9.5649
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[190/189|Cotylisma]]
| Noruyoma
| 19oryM
| 190/189
| {{Monzo| 1 -3 1 -1 0 0 0 1 }}
| 9.1358
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[209/208|Yama comma]]
| Nothuloma
| 19o3u1oM
| 209/208
| {{Monzo| -4 0 0 0 1 -1 0 1 }}
| 8.3033
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[210/209|Spleen comma]]
| Nuluzoyoma
| 19u1uzyM
| 210/209
| {{Monzo| 1 1 1 1 -1 0 0 -1 }}
| 8.2637
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1083/1078|Bihendrixma]]
| Nonolururuma
| 19oo1urrM
| 1083/1078
| {{Monzo| -1 1 0 -2 -1 0 0 2 }}
| 8.0113
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[286/285|Chthonisma]]
| Nuthologuma
| 19u3o1ogM
| 286/285
| {{Monzo| 1 -1 -1 0 1 1 0 -1 }}
| 6.0639
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[324/323|Photisma]]
| Nusuma
| 19u17uM
| 324/323
| {{Monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.3516
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[343/342|Nutrisma]]
| Nutrizoma
| 19u3zM
| 343/342
| {{Monzo| -1 -2 0 3 0 0 0 -1 }}
| 5.0547
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Triraptor comma]]
| Trinuso-azoguma
| 3(19u17o)zgM
| 34391/34295
| {{Monzo|0 0 -1 1 0 0 3 -3}}
| 4.8394
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[361/360|Go comma]], dudon comma
| Nonoguma
| 19oogM
| 361/360
| {{Monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.8023
| [[User:Xenwolf|Xenwolf]] (2013) for ''go comma''
|-
| [[400/399|Devichroma]]
| Nuruyoyoma
| 19uryyM
| 400/399
| {{Monzo| 4 -1 2 -1 0 0 0 -1 }}
| 4.3335
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[456/455|Abnobisma]]
| Nothuruguma
| 19o3urgM
| 456/455
| {{Monzo| 3 1 -1 -1 0 -1 0 1 }}
| 3.8007
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[476/475|Hedwigma]]
| Nusozoguguma
| 19u17ozggM
| 476/475
| {{Monzo| 2 0 -2 1 0 0 1 -1 }}
| 3.6409
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[495/494|Eulalisma]]
| Nuthuloyoma
| 19u3u1oyM
| 495/494
| {{Monzo| -1 2 1 0 1 -1 0 -1 }}
| 3.5010
| [[User:Xenllium|Xenllium]] (2023)
|}

=== 23-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[187/184|Twethusoloma]]
| Twethusoloma
| 23u17o1oM
| 187/184
| 2.11.17.23 {{monzo| -3 1 1 -1 }}
| 27.999
|
|-
| [[69/68|Large vicesimotertial 1/8-tone]]
| Twethosuma
| 23o17uM
| 69/68
| 2.3.17.23 {{monzo| -2 1 -1 1 }}
| 25.274
|[[User:Xenllium|Xenllium]] (2023)
|-
| [[70/69|Small vicesimotertial 1/8-tone]]
| Twethuzoyoma
| 23uzyM
| 70/69
| 2.3.5.7.23 {{monzo| 1 -1 1 1 -1 }}
| 24.910
| [[User:Xenllium|Xenllium]] (2023)
|-
|[[Twethuluma]]
|Twethuluma
|23u1uM
|256/253
|2.7.11.23 {{monzo| 8 -1 -1 }}
|20.408
|[[Kite Giedraitis]] (2026)
|-
| [[92/91|Undinisma]]
| Twethothuruma
| 23o3urM
| 92/91
| 2.7.13.23 {{monzo| 2 -1 -1 1 }}
| 18.921
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[736/729|23-limit Tenney/Cage comma]]
| Satwethoma
| s23oM
| 736/729
| 2.3.23 {{monzo| 5 -6 1 }}
| 16.544
| [[Stephen Weigel]] (2023)
|-
| [[115/114|Yarmanisma]]
| Twethonuyoma
| 23o19uyM
| 115/114
| 2.3.5.19.23 {{monzo| -1 -1 1 -1 1 }}
| 15.120
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[161/160|Major kirnbergerisma]]
| Twethozoguma
| 23ozgM
| 161/160
| 2.5.7.23 {{monzo| -5 -1 1 1 }}
| 10.787
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[162/161|Minor kirnbergerisma]]
| Twethuruma
| 23urM
| 162/161
| 2.3.7.23 {{monzo| 1 4 -1 -1 }}
| 10.720
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[208/207|Vicetone comma]]
| Twethuthoma
| 23u3oM
| 208/207
| 2.3.13.23 {{monzo| 4 -2 1 -1 }}
| 8.3433
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[231/230|Major neutravicema]]
| Twethulozoguma
| 23u1ozgM
| 231/230
| {{monzo| -1 1 -1 1 1 0 0 0 -1 }}
| 7.5108
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Vicesimotertial schisma]]
| Lala-twethuma
| LL23uM
| 387420489 / 385875968
| 2.3.23 {{monzo| -24 18 -1 }}
| 6.9157
|
|-
| [[253/252|Middle neutravicema]]
| Twetholoruma
| 23o1orM
| 253/252
| 2.3.7.11.23 {{monzo| -2 -2 -1 1 1 }}
| 6.8564
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[276/275|Minor neutravicema]]
| Twetholuguguma
| 23o1uggM
| 276/275
| 2.3.5.11.23 {{monzo| 2 1 -2 -1 1 }}
| 6.2840
| [[User:Xenllium|Xenllium]] (2023)
|-
| 21-23-comma
| Trisa-septritwethuma
| 3s21(23u)M
| 281474976710656 / 280462473659039
| 2.23 {{monzo| 95 -21 }}
| 6.2387
|
|-
| [[300/299|Major naiadvicema]]
| Twethuthuyoyoma
| 23u3uyyM
| 300/299
| 2.3.5.13.23 {{monzo| 2 1 2 -1 -1 }}
| 5.7804
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[323/322|Major semivicema]]
| Twethunosoruma
| 23u19o17orM
| 323/322
| 2.7.17.19.23 {{monzo| -1 -1 1 1 -1 }}
| 5.3682
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[391/390|Minor naiadvicema]]
| Twethosothuguma
| 23o17o3ugM
| 391/390
| {{monzo| -1 -1 -1 0 0 -1 1 0 1 }}
| 4.4334
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[392/391|Minor semivicema]]
| Twethusuzozoma
| 23u17uzzM
| 392/391
| 2.7.17.23 {{monzo| 3 2 -1 -1 }}
| 4.4221
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[460/459|Scanisma, vicewolf comma]]
| Twethosuyoma
| 23o17uyM
| 460/459
| 2.3.5.17.23 {{monzo| 2 -3 1 -1 1 }}
| 3.7676
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[484/483|Pittsburghisma]]
| Twethuloloruma
| 23u1oorM
| 484/483
| 2.3.7.11.23 {{monzo| 2 -1 -1 2 -1 }}
| 3.5806
| [[User:Xenllium|Xenllium]] (2023)
|}

=== 29-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| Classical mediant of Didymus' and Archytas' commas
| Twenothuluyoma
| 29o3u1uyM
| 145/143
| 5.11.13.29 {{monzo| 1 -1 -1 1 }}
| 24.045
|
|-
| [[88/87|Farewell comma]]
| Twenuloma
| 29u1oM
| 88/87
| 2.3.11.29 {{monzo| 3 -1 1 -1 }}
| 19.786
|
|-
| [[116/115|Sironisma]]
| Twenotwethuguma
| 29o23ugM
| 116/115
| 2.5.23.29 {{monzo| 2 -1 -1 1 }}
| 14.989
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[117/116|Lomisma]]
| Twenuthoma
| 29u3oM
| 117/116
| 2.3.13.29 {{monzo| -2 2 1 -1 }}
| 14.860
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[145/144|29th-partial chroma]]
| Twenoyoma
| 29oyM
| 145/144
| 2.3.5.29 {{monzo| -4 -2 1 1 }}
| 11.981
| [[User:Flirora|Flirora]] (2023)
|-
| [[175/174|Major chthonovinema]]
| Twenuzoyoyoma
| 29uzyyM
| 175/174
| 2.3.5.7.29 {{monzo| -1 -1 2 1 -1 }}
| 9.9211
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[204/203|Kallistisma]]
| Twenusoruma
| 29u17orM
| 204/203
| 2.3.7.17.29 {{monzo| 2 1 -1 1 -1 }}
| 8.5073
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[232/231|Major paravinema]]
| Twenoluruma
| 29o1urM
| 232/231
| 2.3.7.11.29 {{monzo| 3 -1 -1 -1 1 }}
| 7.4783
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[Jackpot comma]]
| Laseptwenoma
| L7(29o)M
| 17249876309 / 17179869184
| 2.29 {{monzo| -34 7 }}
| 7.0404
| [[Eliora]] (2024)
|-
| [[261/260|Major vinetonema]]
| Twenothuguma
| 29o3ugM
| 261/260
| 2.3.5.13.29 {{monzo| -2 2 -1 -1 1 }}
| 6.6458
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[290/289|Brunisma]]
| Twenosusuyoma
| 29o17uuyM
| 290/289
| 2.5.17.29 {{monzo| 1 1 -2 1 }}
| 5.9801
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[320/319|Minor paravinema]]
| Twenuluyoma
| 29u1uyM
| 320/319
| 2.5.11.29 {{monzo| 6 1 -1 -1 }}
| 5.4186
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[378/377|Major semivinema]]
| Twenuthuzoma
| 29u3uzM
| 378/377
| 2.3.7.13.29 {{monzo| 1 3 1 -1 -1 }}
| 4.5861
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[406/405|Minor semivinema]]
| Twenozoguma
| 29ozgM
| 406/405
| 2.3.5.7.29 {{monzo| 1 -4 -1 1 1 }}
| 4.2694
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[494/493|Minor vinetonema]]
| Twenunosuthoma
| 29u19o17u3oM
| 494/493
| 2.13.17.19.29 {{monzo| 1 1 -1 1 -1 }}
| 3.5081
| [[User:Xenllium|Xenllium]] (2024)
|}

=== 31-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[63/62|Co-archytas comma]]
| Thiwuzoma
| 31uzM
| 63/62
| 2.3.7.31 {{monzo| -1 2 1 -1 }}
| 27.700
| [[Xenwolf]] (2020)
|-
| [[93/92|Tricema]]
| Thiwotwethuma
| 31o23uM
| 93/92
| 2.3.23.31 {{monzo| -2 1 -1 1 }}
| 18.716
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[125/124|Twizzler]]
| Thiwutriyoma
| 31u3yM
| 125/124
| 2.5.31 {{monzo| -2 3 -1 }}
| 13.906
| [[Xenwolf]] (2020)
|-
| [[155/154|Scyllisma]]
| Thiwoluruyoma
| 31o1uryM
| 155/154
| 2.5.7.11.31 {{monzo| -1 1 -1 -1 1 }}
| 11.205
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[156/155|Xanthippisma]]
| Thiwuthoguma
| 31u3ogM
| 156/155
| 2.3.5.13.31 {{monzo| 2 1 -1 1 -1 }}
| 11.133
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[187/186|Lambertisma]]
| Thiwusoloma
| 31u17o1oM
| 187/186
| 2.3.11.17.31 {{monzo| -1 -1 1 1 -1 }}
| 9.2828
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[217/216|Tricesimoprimal kleisma]]
| Thiwozoma
| 31ozM
| 217/216
| 2.3.7.31 {{monzo| -3 -3 1 1 }}
| 7.9965
|
|-
| [[Doctorsma]]
| Latrithiwu-athuquadzoma
| L3(31u)3u4zM
| 388962/387283
| 2.3.7.13.31 {{Monzo|1 4 4 -1 -3}}
| 7.4892
| [[User:Stavats|Stavats]] (2026)
|-
| [[248/247|Lameisma]]
| Thiwonuthuma
| 31o19u3uM
| 248/247
| 2.13.19.31 {{monzo| 3 -1 -1 1 }}
| 6.9949
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[280/279|Tricetone comma]]
| Thiwuzoyoma
| 31uzyM
| 280/279
| 2.3.5.7.31 {{monzo| 3 -2 1 1 -1 }}
| 6.1940
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Junebug comma]]
| Thiwutwenotwethunusotholuzoyoma
| 31u29o23u19u17o3o1uzyM
| 448630/447051
| {{monzo| 1 -1 1 1 -1 1 1 -1 -1 1 -1 }}
| 6.1040
| [[Tristan Bay]] (2024)
|-
| [[341/340|Californisma]]
| Thiwosuloguma
| 31o17u1ogM
| 341/340
| 2.5.11.17.31 {{monzo| -2 -1 1 -1 1 }}
| 5.0844
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[342/341|Endymisma]]
| Thiwunoluma
| 31u19o1uM
| 342/341
| 2.3.11.19.31 {{monzo| 1 2 -1 1 -1 }}
| 5.0695
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[435/434|Chinthisma]]
| Thiwutwenoruyoma
| 31u29oryM
| 435/434
| 2.3.5.7.29.31 {{monzo| -1 1 1 -1 1 -1 }}
| 3.9844
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[465/464|Alektisma]]
| Thiwotwenuyoma
| 31o29uyM
| 465/464
| 2.3.5.29.31 {{monzo| -4 1 1 -1 1 }}
| 3.7271
| [[User:Xenllium|Xenllium]] (2025)
|}

=== Higher-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[714984/704969|Lightyear comma]]
|
| 89u331o3M
| 714984/704969
| 2.3.31.89 {{monzo| 3 1 3 -3 }}
| 24.421
| [[Budjarn Lambeth]] (2023)
|-
| [[82/81|41-limit Johnston comma]]
| Fowoma
| 41oM
| 82/81
| 2.3.41 {{monzo| 1 -4 1 }}
| 21.242
| [[Stephen Weigel]] (2023)
|-
| [[2883/2848|Lilac comma]]
|
| 89u31ooM
| 2883/2848
| 2.3.31.89 {{monzo| -5 1 2 -1 }}
| 21.146
| [[Budjarn Lambeth]] (2023)
|-
| [[86/85|43-limit 10th-tone]], large quadracesimotertial 1/10-tone
| Fothosuguma
| 43o17ugM
| 86/85
| 2.5.17.43 {{monzo| 1 -1 -1 1 }}
| 20.249
| [[Mister Shaf]] (2025)
|-
| [[87/86|43-limit 10th-tone]], small quadracesimotertial 1/10-tone
| Fothutwenoma
| 43u29oM
| 87/86
| 2.3.29.43 {{monzo| -1 1 1 -1 }}
| 20.014
|
|-
| [[89/88|Tailwind comma]]
|
| 89o1uM
| 89/88
| 2.11.89 {{monzo| -3 -1 1 }}
| 19.562
| [[Budjarn Lambeth]] (2023)
|-
| [[389/385|Rebbe comma]]
|
| 389o1urgM
| 389/385
| 5.7.11.389 {{monzo| -1 -1 -1 1 }}
| 17.8794
| [[Mister Shaf]] (2025)
|-
| [[8277/8192|Lilly pilly comma]]
|
| 89o31oM
| 8277/8192
| 2.3.31.89 {{monzo| -13 1 1 1 }}
| 17.871
| [[Budjarn Lambeth]] (2023)
|-
| [[8000/7921|Incisor comma]]
|
| 89uu3yM
| 8000/7921
| 2.5.89 {{monzo| 6 3 -2 }}
| 17.181
| [[Budjarn Lambeth]] (2023)
|-
| [[129/128|43-limit Johnston comma]]
| Fothoma
| 43oM
| 129/128
| 2.3.43 {{monzo| -7 1 1 }}
| 13.473
| [[Stephen Weigel]] (2023)
|-
| [[979/972|Basement comma]]
|
| 89o1oM
| 979/972
| 2.3.11.89 {{monzo| -2 -5 1 1 }}
| 12.423
| [[Budjarn Lambeth]] (2023)
|-
| [[226/225|Reversed marvel comma]]
|
| 113oggM
| 226/225
| 2.3.5.113 {{monzo| 1 -2 -2 1 }}
| 7.6773
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Sidereal comma]]
|
| 73u61ogM
| 366/365
| 2.3.5.61.73 {{monzo| 1 1 -1 1 -1 }}
| 4.7366
| [[User:Frostburn|Frostburn]] (2024)
|-
| [[381/380|Five feet comma]]
|
| 127o19ugM
| 381/380
| 2.3.5.19.127 [-2 1 -1 -1 1⟩
| 4.5499
| [[Eliora]] (2022)
|-
| [[481/480|Semaphorisma]]
| Thisothoguma
| 37o3ogM
| 481/480
| 2.3.5.13.37 {{monzo| -5 -1 -1 1 1 }}
| 3.6030
| [[User:Kaiveran|Kaiveran]] (2023)
|}

== List of irrational commas ==
For intervals expressible as edosteps, see [[Interval size measure]]. We skip them here for brevity.

{| class="wikitable"
! Name(s)
! colspan="2" | [[Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[Caffeinterval]]
|
|
| 2((7/12) - (1/sqrt(3)))
|
| 7.1797
| [[User:R-4981|R-4981]] (2023)
|}

== See also ==
* [[Comma and diesis]]

== References ==

[[Category:Small commas| ]] 
[[Category:Lists of commas]]
{{Todo|complete table|add color name}}

Unnoticeable comma

2026-04-29T04:31:37Z

TallKite: updated comma color names

'''Unnoticeable commas''' are very small intervals. These [[comma]]s are called "unnoticeable" because, being equal to or less than 3.5{{cent}}, they are smaller than the average peak [[just-noticeable difference]] (JND) of human pitch perception, as illustrated by the research of [[Aaron Andrew Hunt]]<ref>[http://musictheory.zentral.zone/huntsystem2.html#2 H-Pi Instruments | ''Hunt System Scale §The JND'']</ref>. It is improbable that even a trained listener would be able to notice these intervals, and as such they are a prime target for psychoacoustically informed [[microtempering]]. (However, a considerably larger comma can be unnoticeable in an [[adaptive just intonation|adaptive]] tuning context. Instead of one large pitch shift of the entire comma, there can be many small pitch shifts of a fraction of a comma, one per chord change. Given this, a noticeable 3-limit comma that arguably deserves inclusion is the [[mercator comma]], corresponding to using [[53edo]] for the circle of fifths.) In [[Sagittal notation]], intervals in the smaller part of this category are [[schismina]]s, and intervals in the larger part of this category are [[schisma (interval region)|schismas]]<ref>[https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>.

For commas over 100{{c}} in size, see [[Large comma]]; for commas in between 30–100{{c}} in size, see [[Medium comma]]; and for commas between 3.5–30{{c}} in size, see [[Small comma]].

Due to the wide range of sizes, cents values here and on the other commas pages are listed to 5 significant digits, instead of to a fixed number of decimal places as is the [[conventions|convention]] elsewhere on the wiki.

The commas might be numerous, but there is no need to memorize all the names. For pretty much all use cases, it is perfectly acceptable to just refer to a comma by the monzo or ratio, whichever is simpler.

== List of commas, by prime limit ==
=== 3-limit commas ===
{{See also| Table of 3-limit commas }}

{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Qian's large comma]]
| 359wama
| 359wM
| <abbr title="1934329451767021421190980423006270754962252499447372679942534652297789463068718331568475476554301659845788554312924531179306109686817232569946089263295619210341718686733067/1932268761508629172347675945465993672149463664853217499328617625725759571144780212268096883290961288981231808015751088588682539330521493827871454336733540374348490407411712">(344 digits)</abbr>
| {{Monzo| -569 359 }}
| 1.8453
| Chen Yingshi (2009)
|-
| [[Qian's small comma]], sasktel comma
| 306wama
| 306wM
| <abbr title="99895953610111751404211111353381321783955140565279076827493022708011895642232499843849795298031743077114461795885011932654335221737225129801285632/99793888233710926097676673961542382339552034110870991187709058567130998942396826836880350287497238272034603157195937657211050782186192219658614729">(292 digits)</abbr>
| {{Monzo| 485 -306 }}
| 1.7697
| Chen Yingshi (2009) for ''Qian's small comma''
|-
| [[Satanic comma]]
| 665wama
| 665wM
| <abbr title="193034257116813465350415306746516837350333763798962117430788985740786485136987943002905988649011085058426719117038711696606024631330152759176330399379617346789616335692978372064681236597226671488585092334981423081811727458166457361300251189808300631437024118571790058070714566731059066970852059271394655662607817543843/193025830561934107162947985381047541665608072055952185017491682078771915023799273387871154500424503798663213600460826789274033295999330021731389427128542432710187362934652673115221889249890533772697227171395058697282798274445240687006095271729621464100656563293799180557568945517759802372156455525060659659679134121984">(636 digits)</abbr>
| {{Monzo| -1054 665 }}
| 0.075575
| [[Marc Jones]] (1990)
|-
| 15601-comma
| 15601wama
| 15601wM
| (14888 digits)
| {{Monzo| 24727 -15601 }}
| 0.031499
|
|-
| 31867-comma
| 31867wama
| 31867wM
| (30410 digits)
| {{Monzo| -50508 31867 }}
| 0.012577
|
|-
| [[Archangelic comma]]
| 190537wama
| 190537wM
| (181820 digits)
| {{monzo| 301994 -190537 }}
| 0.00011162
| [[Dawson Berry]] (2022)
|}

=== 5-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Dodifo comma]]
| Quadla-sepquinyoma
| 4L35yM
| <abbr title="2910383045673370361328125 / 2904698108822600835661824">(50 digits)</abbr>
| {{Monzo| -67 -9 35 }}
| 3.3850
| [[Petr Pařízek]] (2011)
|-
| [[Vishnuzma]], semisuper comma
| Sasepbiguma
| s14gM
| 6115295232 / 6103515625
| {{Monzo| 23 6 -14 }}
| 3.3380
| [[Gene Ward Smith]] (2001), for ''semisuper comma''
|-
| [[Enneadeca]], 19-tone-comma
| Neyoma
| 19yM
| 19073486328125 / 19042491875328
| {{Monzo| -14 -19 19 }}
| 2.8155
| [[Gene Ward Smith]] (2001)
|-
| [[Vavoom comma]]
| Quinla-seyoma
| 5L17yM
| <abbr title="295578376007080078125 / 295147905179352825856">(42 digits)</abbr>
| {{Monzo| -68 18 17 }}
| 2.5232
| [[Paul Erlich]] (2002)
|-
| [[Alphatricot comma]]
| Quadsa-triyoma
| 4s3yM
| 68719476736000 / 68630377364883
| {{Monzo| 39 -29 3 }}
| 2.2461
| [[Paul Erlich]] (2002)
|-
| [[Schisma]]
| Layoma
| LyM
| 32805 / 32768
| {{Monzo| -15 8 1 }}
| 1.9537
| [[Hermann von Helmholtz]], [[Alexander Ellis]] (1875)
|-
| [[Aluminium comma]]
| Sepsa-theguma
| 7s13gM
| <abbr title="4951760157141521099596496896 / 4946966739525117513427734375">(56 digits)</abbr>
| {{Monzo| 92 -39 -13 }}
| 1.6767
| [[Eliora]] (2023)
|-
| [[Counterschisma]]
| Tribilaguma
| 6LgM
| <abbr title="2954312706550833698643 / 2951479051793528258560">(44 digits)</abbr>
| {{Monzo| -69 45 -1 }}
| 1.6613
| [[Gene Ward Smith]] (2003)
|-
| [[Neon comma]]
| Laquinquinbiguma
| L50gM
| <abbr title="88900702359186211632409599176343552 / 88817841970012523233890533447265625">(70 digits)</abbr>
| {{monzo| 21 60 -50 }}
| 1.6144
| [[Eliora]] (2024)
|-
| [[Septendecima]]
| Lala-sebiyoma
| LL34yM
| <abbr title="582076609134674072265625 / 581595589965365114830848">(48 digits)</abbr>
| {{Monzo| -52 -17 34 }}
| 1.4313
|
|-
| [[Luna comma]], hemithirds comma
| Sasa-quintriguma
| ss15gM
| 274877906944 / 274658203125
| {{Monzo| 38 -2 -15 }}
| 1.3843
| [[Gene Ward Smith]] (2002)
|-
| [[Minortone comma]], minortonma
| Trila-seguma
| 3L17gM
| <abbr title="50031545098999707 / 50000000000000000">(34 digits)</abbr>
| {{Monzo| -16 35 -17 }}
| 1.0919
| [[Gene Ward Smith]] (2002)
|-
| [[Ennealimma]]
| Satritribiyoma
| s18yM
| 7629394531250 / 7625597484987
| {{Monzo| 1 -27 18 }}
| 0.86183
| [[Paul Erlich]], [[Gene Ward Smith]] (2001)
|-
| Astro comma
| Tribisa-thiweguma
| 6s31gM
| <abbr title="2475880078570760549798248448 / 2474715001881122589111328125">(56 digits)</abbr>
| {{Monzo| 91 -12 -31 }}
| 0.81486
| [[Paul Erlich]] (2002)
|-
| [[Gaster comma]]
| Quadbila-neguma
| 8L19gM
| <abbr title="22528399544939174411840147874772641 / 22517998136852480000000000000000000">(70 digits)</abbr>
| {{Monzo| -70 72 -19 }}
| 0.79950
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Niobium comma]]
|
|
|
| {{Monzo| -875 492 41 }}
| 0.72269
| [[Eliora]] (2023)
|-
| [[Kwazy comma]]
| Quadla-quadquadyoma
| 4L16yM
| <abbr title="9010162353515625 / 9007199254740992">(32 digits)</abbr>
| {{Monzo| -53 10 16 }}
| 0.56943
| [[Paul Erlich]], [[Gene Ward Smith]] (2002)
|-
| Whoosh
| Saletriguma
| s33gM
| <abbr title="116450459770592056836096 / 116415321826934814453125">(48 digits)</abbr>
| {{Monzo| 37 25 -33 }}
| 0.52246
| [[Paul Erlich]] (2002)
|-
| Egads
| Setriyoma
| 51yM
| <abbr title="444089209850062616169452667236328125 / 444002166576103304796646509039845376">(72 digits)</abbr>
| {{Monzo| -36 -52 51 }}
| 0.33936
| [[Paul Erlich]] (2002)
|-
| [[Monzisma]]
| Quinsa-yoyoma
| 5syyM
| <abbr title="450359962737049600 / 450283905890997363">(36 digits)</abbr>
| {{Monzo| 54 -37 2 }}
| 0.29240
| [[Margo Schulter]] (2001)
|-
| Fortune
| Tritrila-sepbiyoma
| 9L14yM
| <abbr title="162285243890121480027996826171875 / 162259276829213363391578010288128">(66 digits)</abbr>
| {{Monzo| -107 47 14 }}
| 0.27703
| [[Paul Erlich]] (2002)
|-
| Gross
| Quinbisa-foseguma
| 10s47gM
| <abbr title="22300745198530623141535718272648361505980416 / 22297583945629639856633730232715606689453125">(88 digits)</abbr>
| {{Monzo| 144 -22 -47 }}
| 0.24543
| [[Paul Erlich]] (2002)
|-
| Senior
| Quadla-sepquinguma
| 4L35gM
| <abbr title="381520424476945831628649898809 / 381469726562500000000000000000">(60 digits)</abbr>
| {{Monzo| -17 62 -35 }}
| 0.23007
| [[Paul Erlich]] (2002)
|-
| [[Counterquectisma]], deltapion
| Quintritrilayoma
| 45LyM
|
| {{Monzo| -500 314 1 }}
| 0.18399
| [[Eliora]] (2023)
|-
| [[Septarium comma]]
| Sasepquadtriguma
| s84gM
| <abbr title="51704256436926231056548749215693807357721577836111615492096 / 51698788284564229679463043254372678347863256931304931640625">(118 digits)</abbr>
| {{Monzo| 73 77 -84 }}
| 0.18310
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Quectisma]]
|
|
|
| {{Monzo| 554 -351 1 }}
| 0.10841
| [[User:CompactStar|CompactStar]] (2023)
|-
| Raider
| Tritrisa-thiseyoma
| 9s37yM
| <abbr title="171798691840000000000000000000000000000000000000 / 171792506910670443678820376588540424234035840667">(96 digits)</abbr>
| {{Monzo| 71 -99 37 }}
| 0.062327
| [[Gene Ward Smith]] (2002)
|-
| Pirate
| Quinla-sepsepyoma
| 5L49yM
| <abbr title="17763568394002504646778106689453125 / 17763086495282268024161967871623168">(70 digits)</abbr>
| {{Monzo| -90 -15 49 }}
| 0.046966
| [[Paul Erlich]] (2002)
|-
| Viking
| Nela-siweyoma
| 19L61yM
|
| {{Monzo| -251 69 61 }}
| 0.031605
| [[Gene Ward Smith]] (2002)
|-
| [[Kirnberger's atom]]
| Sepbisa-quadtriguma
| 14s12gM
| <abbr title="2923003274661805836407369665432566039311865085952 / 2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
| {{Monzo| 161 -84 -12 }}
| 0.015361
|
|-
| Selenia
|
|
|
| {{Monzo| -433 -137 280 }}
| 0.0047636
|
|-
| Titania
|
|
|
| {{Monzo| 2746 -521 -827 }}
| 0.0031829
|
|-
| Quark
|
|
|
| {{Monzo| -573 237 85 }}
| 8.8361 × 10-4
|
|-
| Scamp
|
|
|
| {{Monzo| -5836 4293 -417 }}
| 3.3472 × 10-5
|
|-
| Rover
|
|
|
| {{Monzo| 1634 1502 -1729 }}
| 2.7513 × 10-5
|
|-
| Rascal
|
|
|
| {{Monzo| -7470 2791 1312 }}
| 5.9596 × 10-6
|
|}

=== 7-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Decovulture comma]]
| Sasa-biruguguma
| ss2rggM
| 67108864 / 66976875
| {{Monzo| 26 -7 -4 -2 }}
| 3.4083
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Rainy comma]]
| Laquinzo-atriyoma
| L5za3yM
| 2100875/2097152
| {{Monzo| -21 0 3 5 }}
| 3.0706
| [[User:Flirora|+merlan #flirora]] (2020)
|-
| [[Pontiqak comma]]
| Lazozotritriyoma
| Lzz9yM
| 95703125 / 95551488
| {{Monzo| -17 -6 9 2 }}
| 2.7452
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Pessoalisma]]
| Sasa-tribiru-aguguma
| ss6raggM
| 2147483648 / 2144153025
| {{Monzo| 31 -6 -2 -6 }}
| 2.6871
| [[Dawson Berry]] (2023)
|-
| [[Mitonisma]]
| Laquadzo-aguma
| L4zagM
| 5250987/5242880
| {{Monzo| -20 7 -1 4 }}
| 2.6749
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Horwell comma]]
| Lazoquinyoma
| Lz5yM
| 65625/65536
| {{Monzo| -16 1 5 1 }}
| 2.3495
| [[Gene Ward Smith]] (2005)
|-
| [[Forge comma]]
| Lala-trizo-aquinguma
| LL3za5gM
| 1640558367 / 1638400000
| {{Monzo| -19 14 -5 3 }}
| 2.2792
| [[Dawson Berry]] (2022)
|-
| [[109-7-comma]]
|
|
|
| {{Monzo| -306 0 0 109 }}
| 2.0238
|
|-
| [[Neptunisma]]
| Laruruleyoma
| Lrr11yM
| 48828125 / 48771072
| {{monzo| -12 -5 11 -2 }}
| 2.0240
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Slendroschisma]], slendric schisma
| Sasa-quinbiruma
| ss10rM
| 68719476736 / 68641485507
| {{monzo| 36 -5 0 -10 }}
| 1.9659
| [[User:Xenllium|Xenllium]] (2024), modified by [[Flora Canou]] (2025)
|-
| [[Septimal ennealimma]]
| Tritrizoma
| 9zM
| 40353607 / 40310784
| {{Monzo| -11 -9 0 9 }}
| 1.8382
| [[Eliora]] (2021)
|-
| [[Meter]]
| Latriru-asepyoma
| L3ra7yM
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.6283
|
|-
| [[Scheme comma]]
| Lala-rutriguma
| LLr3gM
| 14348907 / 14336000
| {{Monzo| -14 15 -3 -1}}
| 1.5580
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[Breeze comma]]
| Laquadru-atriyoma
| L4ra3yM
| 2460375 / 2458624
| {{Monzo| -10 9 3 -4 }}
| 1.2325
| [[Tristan Bay]] (2024)
|-
| [[Wizma]]
| Quinzo-ayoyoma
| 5zayyM
| 420175/419904
| {{Monzo| -6 -8 2 5 }}
| 1.1170
|
|-
| [[Supermatertisma]]
| Lasepru-atritriyoma
| L7ra9yM
| 52734375 / 52706752
| {{Monzo| -6 3 9 -7 }}
| 0.90708
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Trienstonisma]]
| Laquinru-aguma
| L5ragM
| 43046721 / 43025920
| {{monzo| -9 16 -1 -5 }}
| 0.83677
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[2401/2400|Breedsma]]
| Bizozoguma
| 2zzgM
| 2401/2400
| {{monzo| -5 -1 -2 4 }}
| 0.72120
| [[Gene Ward Smith]] (2004)
|-
| [[Gariatom]]
| Quintrila-tribizoma
| 15L6zM
|
| {{monzo| -169 96 0 6 }}
| 0.63552
| [[Tristan Bay]] (2023)
|-
| 171-9/7-comma
| Quadtribisa-netritrizoma
| 24s171zM
|
| {{monzo| 62 -342 0 171 }}
| 0.61971
|
|-
| [[Supermasesquartisma]]
| Laquadbiru-aquinyoma
| L8ra5yM
| 184528125 / 184473632
| {{monzo| -5 10 5 -8 }}
| 0.51133
| [[User:Xenllium|Xenllium]] (2021)
|-
| 571-7-comma
|
|
|
| {{monzo| 1603 0 0 -571 }}
| 0.40741
|
|-
| [[Ragisma]]
| Zoquadyoma
| z4yM
| 4375/4374
| {{Monzo| -1 -7 4 1 }}
| 0.39576
|
|-
| [[Septiruthenia]], septimal ruthenia
| Nela-lequadzoma
| 19L44zM
|
| {{Monzo| -263 88 0 44 }}
| 0.37996
| [[Eliora]] (2022)
|-
| [[Akjaysma]]
| Trisa-sepruguma
| 3s7rgM
| 140737488355328 / 140710042265625
| {{Monzo| 47 -7 -7 -7 }}
| 0.33765
| [[Aaron Krister Johnson]] (2011)
|-
| [[Landscape comma]]
| Trizoguguma
| 3zggM
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.32544
| [[Yahya Abdal-Aziz]] (2005)
|-
| [[Izar comma]], bapbo schismina
| Saquadtrizo-asepguma
| s12za7gM
| 13841287201 / 13839609375
| {{Monzo| 0 -11 -7 12 }}
| 0.20987
|
|-
| [[Nanisma]]
| Quinbisaruma
| 10srM
| <abbr title="649037107316853453566312041152512 / 648966242035284859600333477874109">(66 digits)</abbr>
| {{Monzo| 109 -67 0 -1 }}
| 0.18904
| [[Margo Schulter]] (2002)
|-
| laleruyo (171&1547&3125)
| Laleruyoma
| L11ryM
| 3955078125 / 3954653486
| {{Monzo| -1 4 11 -11 }}
| 0.18588
|
|-
| 87-fold starling comma
| Twenetrizotriguma
|
|
| {{Monzo| 86 174 -261 87 }}
| 0.14469
|
|-
| [[Revopentisma]]
| Sasa-neruma
| ss19rM
| <abbr title="11399736556781568 / 11398895185373143">(34 digits)</abbr>
| {{Monzo| 47 4 0 -19 }}
| 0.12778
| [[Eliora]] (2023)
|-
| [[Starscape comma]]
| Latritriru-ayoma
| L9rayM
| 645700815 / 645657712
| {{Monzo| -4 17 1 -9 }}
| 0.11557
|
|-
| [[Nommisma]]
| Quinla-zoyoyoma
| 5LzzyM
| <abbr title="36030948116563575 / 36028797018963968">(34 digits)</abbr>
| {{Monzo| -55 30 2 1 }}
| 0.10336
| [[Flora Canou]] (2023)
|-
| [[Euzenius comma]]
| Sabiruquinyoma
| s2r5yM
| 78125000 / 78121827
| {{Monzo| 3 -13 10 -2 }}
| 0.070314
|
|-
| [[Exodia comma]]
| Trila-quadbizo-aleyoma
| 3L8za11yM
| 281484423828125 / 281474976710656
| {{Monzo| -48 0 11 8 }}
| 0.058104
|
|-
| [[Septrongisma]]
| Lala-sepru-atritriguma
| LL7ra9gM
| 205891132094649 / 205885750000000
| {{Monzo| -7 30 -9 -7 }}
| 0.045256
| [[User:Xenllium|Xenllium]] (2026)
|-
| 171&1547&4973 comma
| Satwethezo-atritribiguma
| s23za18gM
| <abbr title="54737494680161832686 / 54736736297607421875">(40 digits)</abbr>
| {{Monzo| 1 -15 -18 23 }}
| 0.023986
|
|-
| [[Technologisma]]
| Trisa-quinbiru-aguma
| 3s10ragM
| <abbr title="2251799813685248 / 2251783932057135">(32 digits)</abbr>
| {{Monzo| 51 -13 -1 -10 }}
| 0.012210
| [[User:Godtone|Godtone]] (2024)
|-
| Termite
| Satritribiru-athiseyoma
| s18ra37yM
| <abbr title="37252902984619140625000000000 / 37252879910233655318543787489">(58 digits)</abbr>
| {{Monzo| 9 -28 37 -18 }}
| 0.0010723
|
|-
| Neutrino
|
|
|
| {{Monzo| 1889 -2145 138 424 }}
| 1.6361 × 10-10
|
|}

=== 11-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Lifthrasirsma]]
| Sasa-biluguma
| ss2(1ug)M
| 536870912 / 535869675
| {{Monzo| 29 -11 -2 0 -2 }}
| 3.2317
| [[Flora Canou]] (2023)
|-
| [[540/539|Swetisma]]
| Lururuyoma
| 1urryM
| 540/539
| {{Monzo| 2 3 1 -2 -1 }}
| 3.2090
| [[Manuel Op de Coul]] (2004)
|-
| [[Anthill comma]]
| Satrilo-ayoyoma
| s3(1o)yyM
| 532400/531441
| {{Monzo| 4 -12 2 0 3 }}
| 3.1212
| [[Budjarn Lambeth]] (2024)
|-
| [[4000/3993|Wizardharry comma]], pine comma
| Triluyoma
| 3(1uy)M
| 4000/3993
| {{Monzo| 5 -1 3 0 -3 }}
| 3.0323
| [[User:Godtone|Godtone]] (2023) for ''pine comma''
|-
| 23-11/7-comma
| Twetheluzoma
| 23(1uz)M
| <abbr title="896819112839771466727424 / 895430243255237372246531">(48 digits)</abbr>
| {{Monzo| 15 0 0 23 -23 }}
| 2.6832
|
|-
| [[Symbiotic comma]]
| Salozoma
| s1ozM
| 19712/19683
| {{Monzo| 8 -9 0 1 1 }}
| 2.5488
| [[Dawson Berry]] (2020)
|-
| [[5632/5625|Vishdel comma]]
| Saloquadguma
| s1o4gM
| 5632/5625
| {{Monzo| 9 -2 -4 0 1 }}
| 2.1531
|
|-
| [[Nexus comma]], nexisma
| Tribiloma
| 6(1o)M
| 1771561/1769472
| {{Monzo| -16 -3 0 0 6 }}
| 2.0427
| [[Dawson Berry]] (2020)
|-
| [[Reef comma]]
| Salubizoguma
| s1u2zgM
| 200704/200475
| {{Monzo| 12 -6 -2 2 -1 }}
| 1.9764
| [[Dawson Berry]] (2022)
|-
| [[41503/41472|Argyria]], tinge
| Lolotrizoma
| 1oo3zM
| 41503/41472
| {{Monzo| -9 -4 0 3 2 }}
| 1.2936
| [[Gayle Young]] (2018) and [[Todd Harrop]] (2020) for ''tinge'' [[Lériendil]] (2024) for ''argyria''
|-
| [[Alpharabian schisma]]
| Trisa-tritriloma
| 3s9(1o)M
| 618121839509504 / 617673396283947
| {{Monzo| 18 -31 0 0 9 }}
| 1.2565
| [[Dawson Berry]] (2024)
|-
| [[Olympia]]
| Salururuma
| s1urrM
| 131072/130977
| {{Monzo| 17 -5 0 -2 -1 }}
| 1.2552
| [[Flora Canou]] (2021)
|-
| [[37-11-comma]], 11-cycle schisma
| Quinsa-thiseluma
| 5s37(1u)M
| <abbr title="340282366920938463463374607431768211456 / 340039485861577398992406882305761986971">(78 digits)</abbr>
| {{Monzo| 128 0 0 0 -37 }}
| 1.2361
|
|-
| [[Seascape comma]], undecimal hemifourths comma
| Bilozoguguma
| 2(1ozgg)M
| 160083/160000
| {{Monzo| -8 3 -4 2 2 }}
| 0.89784
| [[User:Tristanbay|Tristan Bay]] (2024) for ''seascape comma''
|-
| [[Sesdecal comma]]
| Laquadlu-asepyoma
| L4(1u)7yM
| 234375/234256
| {{Monzo| -4 1 7 0 -4 }}
| 0.87923
|
|-
| [[Triagnoshenisma]]
| Trila-trilo-aguma
| 3L3(1o)gM
| 171885556953 / 171798691840
| {{Monzo| -35 17 -1 0 3 }}
| 0.87513
| [[Dawson Berry]], [[User:Frostburn|Frostburn]] (2024)
|-
| [[Frameshift comma]]
| Quadla-triluma
| 4L3(1u)M
| 22876792454961 / 22866405883904
| {{Monzo| -34 28 0 0 -3 }}
| 0.78620
| [[Dawson Berry]] (2021)
|-
| [[Chrysia]]
| Quadlo-atriruma
| 4(1o)3rM
| 43923/43904
| {{Monzo| -7 1 0 -3 4 }}
| 0.74905
| [[VIxen]] (2025)
|-
| [[Sossmarvel comma]]
| Trila-lusepruyoyoma
| 3L1u7ryyM
| <abbr title="9730975341796875 / 9726998192586752">(32 digits)</abbr>
| {{Monzo| -30 13 14 -7 -1 }}
| 0.70772
| [[Dawson Berry]] (2022)
|-
| [[3025/3024|Lehmerisma]]
| Loloruyoyoma
| 1ooryyM
| 3025/3024
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57240
| [[Gene Ward Smith]] (2004)
|-
| [[Ptolemi-nicema]]
| Quinbisa-twethetriluyoyoma
| 10s69(1uyy)M
|
| {{Monzo| 137 -138 138 0 -69 }}
| 0.56437
| [[User:Eliora|Eliora]] (2023)
|-
| [[Elysia]]
| Bilutrizoma
| 2(1u3z)M
| 117649/117612
| {{Monzo| -2 -5 0 6 -2 }}
| 0.54455
| [[Lériendil]] (2024)
|-
| [[Quartisma]]
| Saquinlu-azoma
| s5(1u)zM
| 117440512 / 117406179
| {{Monzo| 24 -6 0 1 -5 }}
| 0.50619
| [[Dawson Berry]] (2020)
|-
| [[9801/9800|Kalisma]], Gauss' comma
| Biloruguma
| 2(1org)M
| 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| 0.17665
| [[Margo Schulter]] (2000) [[Gene Ward Smith]] (2004) for ''Gauss' comma''
|-
| [[151263/151250|Odiheim comma]]
| Luluquinzo-aquadguma
| 1uu5za4gM
| 151263/151250
| {{Monzo| -1 2 -4 5 -2 }}
| 0.14879
|
|-
| [[Countercentisma]]
|
|
| (3776 digits)
| {{Monzo| -1 -3300 2700 0 -300 }}
| 0.14187
| [[User:CompactStar|CompactStar]] (2024)
|-
| [[Spoob]]
| Tribiluzozoguma
| 6(1uzzg)M
| 27682574402 / 27680640625
| {{Monzo| 1 0 -6 12 -6 }}
| 0.12094
| [[User:Eliora|Eliora]] (2023)
|-
| [[Luxma]]
| Saquinlu-aquadguma
| s5(1u)4gM
| 100663296/100656875
| {{Monzo|25 1 -4 0 -5}}
| 0.11043
| [[User:Tristanbay|Tristan Bay]] (2026)
|-
| [[Syntonoschisma]]
| Trisa-lusepyoma
| 3s1u7yM
| 83886080000000 / 83881572334857
| {{Monzo|30 -27 7 0 -1}}
| 0.093031
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Parimo]]
| Satribilo-aguma
| s6(1o)gM
| 1771561/1771470
| {{Monzo|-1 -11 -1 0 6}}
| 0.088931
|
|-
| Parisma
| Laquadlu-aruruguma
| L4(1u)rrgM
| 14348907 / 14348180
| {{Monzo|-2 15 -1 -2 -4}}
| 0.087717
|
|-
| [[Blare comma]]
| Laquadquadlo-aquadtrizoma
| L16(1o)12zM
| <abbr title="636003407850068828189211361 / 635974777627126753067532288">(54 digits)</abbr>
| {{Monzo|-51 -24 0 12 16}}
| 0.077935
| [[User:Akselai|Akselai]] (2024)
|-
| Ultimo
| Quadlo-asepru-ayoyoma
| 4(1o)7rayyM
| 3294225/3294172
| {{Monzo|-2 2 2 -7 4}}
| 0.027854
|
|-
| Parismina
| Sasa-quinbilo-azozoma
| ss10(1o)zzM
| 2541867610898 / 2541865828329
| {{Monzo|1 -26 0 2 10}}
| 0.0012141
|
|}

=== 13-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Wilschisma]]
| Sathoyoma
| s3oyM
| 532480/531441
| {{Monzo| 13 -12 1 0 0 1 }}
| 3.3814
| [[Flora Canou]] (2021)
|-
| Bean
| Sathuquinluma
| s3u5(1u)M
| 2097152/2093663
| {{Monzo| 21 0 0 0 -5 -1 }}
| 2.8826
|
|-
| [[625/624|Tunbarsma]]
| Thuquadyoma
| 3u4yM
| 625/624
| {{Monzo| -4 -1 4 0 0 -1 }}
| 2.7722
|
|-
| [[Fifthless mohohomma]]
| Thuthululuyoma
| 3uu1uuyM
| 20480/20449
| {{Monzo| 12 0 1 0 -2 -2 }}
| 2.6225
| [[User:原井玉葱郎|Misohito Nakai]] (2025)
|-
| [[676/675|Island comma]]
| Bithoguma
| 2(3og)M
| 676/675
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.5629
| [[Mike Battaglia]] (2011)
|-
| [[729/728|Squbema]]
| Lathuruma
| L3urM
| 729/728
| {{Monzo| -3 6 0 -1 0 -1 }}
| 2.3764
|
|-
| [[2200/2197|Petrma]]
| Trithu-aloyoyoma
| 3(3u)1oyyM
| 2200/2197
| {{Monzo| 3 0 2 0 1 -3 }}
| 2.3624
|
|-
| [[Tridecimal eighth-octave comma]]
| Thotrilo-aguma
| 3o3(1o)gM
| 17303/17280
| {{Monzo| -7 -3 -1 0 3 1 }}
| 2.3028
| [[Budjarn Lambeth]] (2024)
|-
| [[Sinarabian comma]]
| Lathotriluma
| L3o3(1u)M
| 85293/85184
| {{Monzo| -6 8 0 0 -3 1 }}
| 2.2138
| [[Dawson Berry]] (2025)
|-
| [[1575/1573|Nicola]]
| Thululuzoyoyoma
| 3u1uuzyyM
| 1575/1573
| {{Monzo| 0 2 2 1 -2 -1 }}
| 2.1998
|
|-
| [[Navicular comma]]
| Trithu-aluzoma
| 3(3u)1uzM
| 24192/24167
| {{Monzo| 7 3 0 1 -1 -3 }}
| 1.7900
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1001/1000|Sinbadma]]
| Tholozotriguma
| 3o1oz3gM
| 1001/1000
| {{Monzo| -3 0 -3 1 1 1 }}
| 1.7303
|
|-
| [[4459/4455|Tristanisma]]
| Tholutrizo-aguma
| 3o1u3zagM
| 4459/4455
| {{Monzo| 0 -4 -1 3 -1 1}}
| 1.5537
| [[User:Tristanbay|Tristan Bay]] (2024)
|-
| [[Tridecapyth comma]]
| Trisathoma
| 3s3oM
| 3489660928 / 3486784401
| {{Monzo| 28 -20 0 0 0 1 }}
| 1.4276
| [[Dawson Berry]] (2021)
|-
| [[Cantonisma]]
| Trithoru-ayoma
| 3(3or)yM
| 10985/10976
| {{Monzo| -5 0 1 -3 0 3 }}
| 1.4190
| [[Margo Schulter]] (2013)
|-
| [[Punctisma]]
| Sathutrizoguma
| s3u3zgM
| 43904/43875
| {{Monzo| 7 -3 -3 3 0 -1 }}
| 1.1439
| [[User:Jerdle|Jerdle]], [[Dawson Berry]] (2023)
|-
| [[Neguschisma]]
| Lala-thulozoma
| LL3u1ozM
| 13640319 / 13631488
| {{Monzo| -20 11 0 1 1 -1 }}
| 1.1212
| [[User:Tristanbay|Tristan Bay]] (2024)
|-
| [[1716/1715|Lummic comma]]
| Tholotriru-aguma
| 3o1o3ragM
| 1716/1715
| {{Monzo| 2 1 -1 -3 1 1 }}
| 1.0092
|
|-
| [[Pseudovishnuzma]]
| Sasa-thozosepbiguma
| ss3oz14gM
| 6106906624 / 6103515625
| {{Monzo| 26 0 -14 1 0 1 }}
| 0.96157
| [[User:Eliora|Eliora]] (2023)
|-
| [[Sercloreminisma]]
| Bithuthuzo-aguma
| 2(3uuz)gM
| 142884/142805
| {{Monzo| 2 6 -1 2 0 -4 }}
| 0.95746
| [[Francium]] (2024)
|-
| [[2080/2079|Ibnsinma, sinaisma]]
| Tholuruyoma
| 3o1uryM
| 2080/2079
| {{Monzo| 5 -3 1 -1 -1 1 }}
| 0.83252
| [[Margo Schulter]], [[Gene Ward Smith]] (2012) [[Flora Canou]] (2023)
|-
| [[Phaotic comma]], phaotisma
| Sathotriyoma
| s3u3yM
| 256000/255879
| {{Monzo| 11 -9 3 0 0 -1 }}
| 0.81847
| [[Dawson Berry]] (2021)
|-
| [[Kuragesma]]
| Tritho-aquadlu-ayoma
| 3(3o)4(1u)gM
| 43940/43923
| {{Monzo| 2 -1 1 0 -4 3 }}
| 0.66993
| [[User:原井玉葱郎|Misohito Nakai]] (2025)
|-
| [[Barbadisma]]
| Quadla-thuyoma
| 4L3uyM
| 114383962274805 / 114349209288704
| {{Monzo| -43 28 1 0 0 -1 }}
| 0.52608
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[4096/4095|Minisma]]
| Sathuruguma
| s3urgM
| 4096/4095
| {{Monzo| 12 -2 -1 -1 0 -1 }}
| 0.42272
| [[Flora Canou]] (2026)
|-
| [[4225/4224|Leprechaun comma]]
| Thotholuyoyoma
| 3oo1uyyM
| 4225/4224
| {{Monzo| -7 -1 2 0 -1 2 }}
| 0.40981
| [[Margo Schulter]] (2012)
|-
| [[6656/6655|Jacobin comma]]
| Thotrilu-aguma
| 3o3(1u)gM
| 6656/6655
| {{Monzo| 9 0 -1 0 -3 1 }}
| 0.26012
| [[Gene Ward Smith]] (2014)
|-
| [[Catasma]]
| Latrithuyoyoma
| L3(3uyy)M
| 140625/140608
| {{Monzo| -6 2 6 0 0 -3 }}
| 0.20930
| [[Tristan Bay]] (2023)
|-
| [[492128/492075|13^3⋅7/25 schismina]]
| Satritho-azoguguma
| s3(3o)zggM
| 492128/492075
| {{Monzo| 5 -9 -2 1 0 3 }}
| 0.18646
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[Harmonisma]]
| Thuthutrilo-aruma
| 3uu3(1o)rM
| 10648/10647
| {{Monzo| 3 -2 0 -1 3 -2 }}
| 0.16260
| [[Margo Schulter]] (2002)
|-
| [[Pentonisma]]
| Saquinthuzoguma
| s5(3uzg)M
| 281974669312 / 281950621875
| {{Monzo| 24 -5 -5 5 0 -5 }}
| 0.14765
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Pontigailimma]]
| Thururuquinguma
| 3urr5gM
| 1990656/1990625
| {{Monzo| 13 5 -5 -2 0 -1 }}
| 0.026960
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[Grossmisma]]
| septholo-azoguma
| 7(3o1o)zgM
| <abbr title="8559537565427849 / 8559456430325760">(32 digits)</abbr>
| {{Monzo| -30 -13 -1 1 7 7 }}
| 0.016410
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Chalmersia]]
| Lathotholuruguguma
| L3oo1urggM
| 123201/123200
| {{Monzo| -6 6 -2 -1 -1 2 }}
| 0.01405
| [[Gene Ward Smith]] (2003)
|-
| [[Lanasma]]
| Trila-septrithu-aquinquadbizoma
| 3L21(3u)40zM
| <abbr title="6366805760909027985741435139224001 / 6366804434232663711262864979263488">(68 digits)</abbr>
| {{Monzo| -33 -1 0 40 0 -21 }}
| 3.6074 × 10-4
| [[User:Angekallikoita|Ange]] (2025)
|}

=== 17-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Quinticular comma]]
| Saquinsoma
| s5(17o)M
| 1419857/1417176
| {{Monzo| -3 -11 0 0 0 0 5 }}
| 3.2720
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[561/560|Monardisma]]
| Soloruguma
| 17o1orgM
| 561/560
| {{Monzo| -4 1 -1 -1 1 0 1 }}
| 3.0887
| [[Scott Dakota]] (2023)
|-
| [[595/594|Dakotisma]]
| Soluzoyoma
| 17o1uzyM
| 595/594
| {{Monzo| -1 -3 1 1 -1 0 1 }}
| 2.9121
| [[Praveen Venkataramana]] (2022)
|-
| [[715/714|September comma]]
| Sutholoruyoma
| 17u3o1oryM
| 715/714
| {{Monzo| -1 -1 1 -1 1 1 -1 }}
| 2.4230
| [[Scott Dakota]] (2021)
|-
| [[833/832|Horizma, horizon comma]]
| Sothuzozoma
| 17o3uzzM
| 833/832
| {{Monzo| -6 0 0 2 0 -1 1 }}
| 2.0796
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[936/935|Ainisma, ainic comma]]
| Sutholuguma
| 17u3o1ugM
| 936/935
| {{Monzo| 3 2 -1 0 -1 1 -1 }}
| 1.8506
| [[Dawson Berry]] (2020)
|-
| [[2025/2023|Fidesma]]
| Susuruyoyoma
| 17uuryyM
| 2025/2023
| {{Monzo| 0 4 2 -1 0 0 -2 }}
| 1.7107
| [[Dawson Berry]] (2023)
|-
| [[1089/1088|Twosquare comma]]
| Suloloma
| 17u1ooM
| 1089/1088
| {{Monzo| -6 2 0 0 2 0 -1 }}
| 1.5905
| [[User:Kaiveran|Kaiveran]] (2018)
|-
| [[1156/1155|Quadrantonisma]]
| Sosoluruguma
| 17oo1urgM
| 1156/1155
| {{Monzo| 2 -1 -1 -1 -1 0 2 }}
| 1.4983
| [[Flora Canou]] (2023)
|-
| [[1225/1224|Noellisma]]
| Subizoyoma
| 17u2zyM
| 1225/1224
| {{Monzo| -3 -2 2 2 0 0 -1 }}
| 1.4138
| [[Flora Canou]] (2022)
|-
| [[1275/1274|Cimbrisma]]
| Sothubiruyoma
| 17o3u2ryM
| 1275/1274
| {{Monzo| -1 1 2 -2 0 -1 1 }}
| 1.3584
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1701/1700|Palingenetic comma, palingenesis]]
| Suzoguguma
| 17uzggM
| 1701/1700
| {{Monzo| -2 5 -2 1 0 0 -1 }}
| 1.0181
| [[Dawson Berry]] (2021)
|-
| [[Laser comma]]
| Lasorutriyoma
| L17or3yM
| 57375/57344
| {{Monzo| -13 3 3 -1 0 0 1 }}
| 0.93564
| [[User:Jerdle|Jerdle]] (2023)
|-
| [[2058/2057|Xenisma]]
| Sululutrizoma
| 17u1uu3zM
| 2058/2057
| {{Monzo| 1 1 0 3 -2 0 -1 }}
| 0.84143
| [[Margo Schulter]] (2000)
|-
| [[11016/11011|Cyclops comma]]
| Sothululuruma
| 17o3u1uurM
| 11016/11011
| {{Monzo| 3 4 0 -1 -2 -1 1 }}
| 0.78596
| [[Eliora]] (2022)
|-
| [[24576/24565|Mavka comma]], archagallisma
| Trisu-aguma
| 3(17u)gM
| 24576/24565
| {{Monzo| 13 1 -1 0 0 0 -3 }}
| 0.77506
| [[Eliora]] (2022) for ''mavka comma''
|-
| [[2431/2430|Heptacircle comma]]
| Sothologuma
| 17o3o1ogM
| 2431/2430
| {{Monzo| -1 -5 -1 0 1 1 1 }}
| 0.71230
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2500/2499|Sperasma]]
| Subiruyoyoma
| 17u2ryyM
| 2500/2499
| {{Monzo| 2 -1 4 -2 0 0 -1 }}
| 0.69263
| [[Dawson Berry]] (2023)
|-
| [[2601/2600|Sextantonisma]]
| Sosothuguguma
| 17oo3uggM
| 2601/2600
| {{Monzo| -3 2 -2 0 0 -1 2 }}
| 0.66573
| [[Flora Canou]] (2023)
|-
| [[Semisixthmisma]]
| Trisu-athutriloma
| 3(17u)3u3(1o)M
| 63888/63869
| {{Monzo| 4 1 0 0 3 -1 -3 }}
| 0.51494
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[4914/4913|Baladisma]]
| Trisu-athozoma
| 3(17u)3ozM
| 4914/4913
| {{Monzo| 1 3 0 1 0 1 -3 }}
| 0.35234
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[5832/5831|Chlorisma]]
| Sutriruma
| 17u3rM
| 5832/5831
| {{Monzo| 3 6 0 -3 0 0 -1 }}
| 0.29688
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[Galileisma]]
| Lalesu-aguma
| L11(17u)gM
| 171382426877952 / 171359481538165
| {{Monzo| 14 21 -1 0 0 0 -11 }}
| 0.23180
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Centisma]]
|
|
|
| 2.3.17 {{Monzo| -1001 -400 400 }}
| 0.16345
| [[CompactStar]] (2023)
|-
| [[Flashma]]
| Sotholuzotriguma
| 17o3o1uz3gM
| 12376/12375
| {{Monzo| 3 -2 -3 1 -1 1 1 }}
| 0.13989
| [[User:Kaiveran|Kaiveran]] (2016)
|-
| [[Sparkisma]]
| Sululuruyoyoma
| 17u1uuryyM
| 14400/14399
| {{Monzo| 6 2 2 -1 -2 0 -1 }}
| 0.12023
| [[User:Kaiveran|Kaiveran]] (2016)
|-
| [[Insanobromisma]]
| Sepquinsuyoyoma
| 35(17uyy)M
|
| {{Monzo| 36 -35 70 0 0 0 -35 }}
| 0.095608
| [[Eliora]] (2023)
|-
| [[1257795/1257728|Large triquarterisma]]
| Latrisulo-azoyoma
| L3(17u1o)zyM
| 1257795/1257728
| {{Monzo| -8 3 1 1 3 0 -3 }}
| 0.092222
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[471648/471625|Small triquarterisma]]
| Triso-alutriruguma
| 3(17o)1u3(rg)M
| 471648/471625
| {{Monzo| 5 1 -3 -3 -1 0 3 }}
| 0.084426
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[28561/28560|Pisanoisma]]
| Suquadtho-aruguma
| 17u4(3o)rgM
| 28561/28560
| {{Monzo| -4 -1 -1 -1 0 4 -1 }}
| 0.060616
| [[Budjarn Lambeth]] (2024)
|-
| [[E-shaped comma]]
| Susuthoquadzoma
| 17uu3o4zM
| 31213/31212
| {{Monzo| -2 -3 0 4 0 1 -2 }}
| 0.055466
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Lateral comma]]
| Sasuthotholoyoma
| s17u3oo1oyM
| 37180/37179
| {{Monzo| 2 -7 1 0 1 2 -1 }}
| 0.046564
| [[User:Akselai|Akselai]] (2024)
|-
| [[Clevelandisma]]
| Sotribizoguma
| 17o6(zg)M
| 2000033/2000000
| {{Monzo| -7 0 -6 6 0 0 1 }}
| 0.028565
| [[Lériendil]] (2024)
|-
| [[Scintillisma]]
| Lasuthuluquadzo-aguma
| L17u3u1u4zagM
| 194481/194480
| {{Monzo| -4 4 -1 4 -1 -1 -1 }}
| 0.0089018
| [[Margo Schulter]] (2012)
|-
| [[Aksial comma]]
| Sotritho-aquinru-aguma
| 17o3(3o)5ragM
| 336141/336140
| {{Monzo| -2 2 -1 -5 0 3 1 }}
| 0.0051503
| [[User:Akselai|Akselai]] (2024)
|}

=== 19-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[513/512|Undevicesimal schisma]], undevicesimal formal comma, Boethius' comma
| Lanoma
| L19oM
| 513/512
| 2.3.19 {{Monzo| -9 3 1 }}
| 3.3780
| Plainsound Music Edition (2020)<ref>[https://marsbat.space/pdfs/HEJI2legend+series.pdf The Helmholtz-Ellis JI Pitch Notation (HEJI)]</ref> for ''undevicesimal schisma''
|-
| [[6144/6137|Langwisma]]
| Nunusuma
| 19uu17uM
| 6144/6137
| {{Monzo| 11 1 0 0 0 0 -1 -2 }}
| 1.9736
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[969/968|Kingfisher comma]]
| Nosoluluma
| 19o17o1uuM
| 969/968
| {{Monzo| -3 1 0 0 -2 0 1 1 }}
| 1.7875
| [[Budjarn Lambeth]] (2025)
|-
| [[Mercurial comma]]
| Quinnosu-abiruyoma
| 5(19o17u)rryyM
| 557122275 / 556583944
| {{Monzo| -3 2 2 -2 0 0 -5 5 }}
| 1.6736
| [[User:Yourmusic Productions|Yourmusic Productions]] (2020)
|-
| [[1216/1215|Password, Eratosthenes' comma]]
| Sanoguma
| s19ogM
| 1216/1215
| {{Monzo| 6 -5 -1 0 0 0 0 1 }}
| 1.4243
|
|-
| [[1331/1330|Solvejgsma]]
| Nutrilo-aruguma
| 19u3(1o)rgM
| 1331/1330
| {{Monzo| -1 0 -1 -1 3 0 0 -1 }}
| 1.3012
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1445/1444|Aureusma]]
| Nunusosoyoma
| 19uu17ooyM
| 1445/1444
| {{Monzo| -2 0 1 0 0 0 2 -2 }}
| 1.1985
| [[Flora Canou]] (2021)
|-
| [[1521/1520|Pinkanberry]]
| Nuthothoguma
| 19u3oogM
| 1521/1520
| {{Monzo| -4 2 -1 0 0 2 0 -1 }}
| 1.1386
| [[User:Kaiveran|Kaiveran]] (2021)
|-
| [[1540/1539|Kevolisma]]
| Nulozoyoma
| 19u1ozyM
| 1540/1539
| {{Monzo| 2 -4 1 1 1 0 0 -1 }}
| 1.1245
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3213/3211|Cobaltomenisma]]
| Nusothuthuzoma
| 19u17o3uuzM
| 3213/3211
| {{Monzo| 0 3 0 1 0 -2 1 -1 }}
| 1.0780
| [[Francium]] (2025)
|-
| [[1729/1728|Ramanujanisma]]
| Nothozoma
| 19o3ozM
| 1729/1728
| {{Monzo| -6 -3 0 1 0 1 0 1 }}
| 1.0016
| [[Frédéric Gagné]] (2021)
|-
| [[3971/3969|Heartlandisma]]
| Nonoloruruma
| 19oo1orrM
| 3971/3969
| {{Monzo| 0 -4 0 -2 1 0 0 2 }}
| 0.87216
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[830297/829939|Minthtone schismina]]
| Trinuso-abitholuma
| 3(19u17o)2(3o1u)M
| 830297/829939
| {{Monzo| 0 0 0 0 -2 2 3 -3 }}
| 0.74662
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[2376/2375|Trichthonisma]]
| Nulotriguma
| 19u1o3gM
| 2376/2375
| {{Monzo| 3 3 -3 0 1 0 0 -1 }}
| 0.72879
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Crawma]]
| Nuquadso-atrithuma
| 19u4(17o)3(3u)M
| 83521/83486
| {{Monzo| -1 0 0 0 0 -3 4 -1 }}
| 0.72564
| [[groundfault]] (2023)
|-
| [[2432/2431|Blumeyer comma]]
| Nosuthuluma
| 19o17u3u1uM
| 2432/2431
| {{Monzo| 7 0 0 0 -1 -1 -1 1 }}
| 0.71200
| [[Douglas Blumeyer]] (2015)
|-
| [[93347/93312|Trilute comma]]
| Notrisoma
| 19o3(17o)M
| 93347/93312
| {{Monzo| -7 -6 0 0 0 0 3 1 }}
| 0.64924
| [[Xenllium]] (2025)
|-
| [[2926/2925|Neovulture comma, neovulturisma]]
| Nothulozoguguma
| 19o3u1ozggM
| 2926/2925
| {{Monzo| 1 -2 -2 1 1 -1 0 1 }}
| 0.59177
| [[Xenllium]] (2023)
|-
| [[3136/3135|Neomirkwai comma, neomirkwaisma]]
| Nuluzozoguma
| 19u1uzzgM
| 3136/3135
| {{Monzo| 6 -1 -1 2 -1 0 0 -1 }}
| 0.55214
| [[Xenllium]] (2023)
|-
| [[116640/116603|Large tridevisemma]]
| Trinu-asuyoma
| 3(19u)17uyM
| 116640/116603
| {{Monzo| 5 6 1 0 0 0 -1 -3 }}
| 0.54926
| [[Xenllium]] (2025)
|-
| [[3250/3249|Martebisma]]
| Nunuthotriyoma
| 19uu3o3yM
| 3250/3249
| {{Monzo| 1 -2 3 0 0 1 0 -2 }}
| 0.53277
| [[Xenllium]] (2023)
|-
| [[48013/48000|Small tridevisemma]]
| Trino-azotriguma
| 3(19o)z3gM
| 48013/48000
| {{Monzo| -7 -1 -3 1 0 0 0 3 }}
| 0.46881
| [[Xenllium]] (2025)
|-
| [[4200/4199|Neosatanisma]]
| Nusuthuzoyoyoma
| 19u17u3uzyyM
| 4200/4199
| {{Monzo| 3 1 2 1 0 -1 -1 -1 }}
| 0.41225
| [[Xenllium]] (2023)
|-
| [[176000/175959|Triseptichrome comma]]
| Nulotriruyoma
| 19u1o3(ry)M
| 176000/175959
| {{Monzo| 7 -3 3 -3 1 0 0 -1 }}
| 0.40335
| [[Xenllium]] (2025)
|-
| [[5776/5775|Neovish comma, neovishma]]
| Nonoluruguguma
| 19oo1urggM
| 5776/5775
| {{Monzo| 4 -1 -2 -1 -1 0 0 2 }}
| 0.29975
| [[Xenllium]] (2023)
|-
| [[5929/5928|Manzanisma]]
| Nuthubilozoma
| 19u3u2(1oz)M
| 5929/5928
| {{Monzo| -3 -1 0 2 2 -1 0 -1 }}
| 0.29202
| [[Xenllium]] (2025)
|-
| [[5985/5984|Neogrendel comma, neogrendelisma]]
| Nosuluzoyoma
| 19o17u1uzyM
| 5985/5984
| {{Monzo| -5 2 1 1 -1 0 -1 1 }}
| 0.28929
| [[Xenllium]] (2023)
|-
| BMO schismina
| Sabinothuma
| s2(19o3u)M
| 369664/369603
| {{Monzo| 10 -7 0 0 0 -2 0 2 }}
| 0.28570
|
|-
| [[6175/6174|Neonewtisma]]
| Nothotriru-ayoyoma
| 19o3o3rayyM
| 6175/6174
| {{Monzo| -1 -2 2 -3 0 1 0 1 }}
| 0.28038
| [[Xenllium]] (2023)
|-
| [[6860/6859|Devicubisma]]
| Trinuzo-ayoma
| 3(19uz)yM
| 6860/6859
| {{Monzo| 2 0 1 3 0 0 0 -3 }}
| 0.25238
| [[Xenllium]] (2025)
|-
| [[Undevicesimal counterschisma]]
| Seplanuma
| 7L19uM
| <abbr title=" 717897987691852588770249 / 717799705396186072481792">(48 digits)</abbr>
| 2.3.19 {{Monzo| -75 50 -1 }}
| 0.23703
| [[Xenllium]] (2025)
|-
| Frouggie comma
| Nusuquinthu-aquadloma
| 19u17u5(3u)4(1o)M
| 119939072 / 119927639
| {{Monzo| 13 0 0 0 4 -5 -1 -1 }}
| 0.16503
| [[Douglas Blumeyer]] (2020)
|-
| [[12636/12635|Padriellisma]]
| Nunuthoruguma
| 19uu3orgM
| 12636/12635
| {{Monzo| 2 5 -1 -1 0 1 0 -2 }}
| 0.13701
| [[Xenllium]] (2025)
|-
| [[Lakisma]]
| Saquadnoso-aguma
| s4(19o17o)gM
| 10884540241 / 10883911680
| {{Monzo| -12 -12 -1 0 0 0 4 4 }}
| 0.09998
| [[User:Flirora|+merlan #flirora]] (2021)
|-
| [[Aubertisma]]
| Nosothutrilu-arutriyoma
| 19o17o3u3(1u)r3yM
| 121125/121121
| {{monzo| 0 1 3 -1 -3 -1 1 1 }}
| 0.057173
| [[Francium]] (2025)
|-
| Pollar comma
| Nunusuquintho-aluluma
| 19uu17u5(3o)1uuM
| 742586/742577
| {{Monzo| 1 0 0 0 -2 5 -1 -2 }}
| 0.020982
| [[Douglas Blumeyer]] (2020)
|-
| [[Decimillisma]], 19-limit decimill
| Sanosorurutriguma
| s19o17orr3gM
| 165376/165375
| {{Monzo| 9 -3 -3 -2 0 0 1 1 }}
| 0.010469
| [[Flora Canou]] (2021), for ''decimillisma''
|-
| [[65/19 atom]]
| Sasa-nuthoyoma
| ss19u3oyM
| 272629760 / 272629233
| {{Monzo| 22 -15 1 0 0 1 0 -1 }}
| 0.0033465
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[Devicisma]]
| Nunusothutrilo-azoguma
| 19uu17o3u3(1o)zgM
| 633556/633555
| {{Monzo| 2 -3 -1 1 3 -1 1 -2 }}
| 0.0027326
| [[Xenllium]] (2023)
|-
| [[11859211/11859210|Tredekisma]]
| Quadno-athoquadlu-azoguma
| 19o43o1u4zgM
| 11859211/11859210
| {{Monzo| -1 -4 -1 1 -4 1 0 4 }}
| 0.000146
| [[Eufalesio]] (2026)
|}

=== 23-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[507/506|Laodicisma]]
| Twethuthotholuma
| 23u3oo1uM
| 507/506
| 2.3.11.13.23 {{Monzo| -1 1 -1 2 -1 }}
| 3.4180
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[529/528|Preziosisma]]
| Bitwetho-aluma
| 23oo1uM
| 529/528
| 2.3.11.23 {{Monzo| -4 -1 -1 2 }}
| 3.2758
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[576/575|Worcester comma]]
| Twethuguguma
| 23uggM
| 576/575
| 2.3.5.23 {{Monzo| 6 2 -2 -1 }}
| 3.0082
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[9765625/9750528]]
| Labitwethuquinyoma
| L23uu10yM
| 9765625/9750528
| 2.3.5.23 {{Monzo| -11 -2 10 -2 }}
| 2.6784
|
|-
| [[736/735|Harvardisma]]
| Twethoruruguma
| 23orrgM
| 736/735
| 2.3.5.7.23 {{Monzo| 5 -1 -1 -2 1 }}
| 2.3538
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[760/759|Squadronisma]]
| Twethunoluyoma
| 23u19o1uyM
| 760/759
| {{Monzo| 3 -1 1 0 -1 0 0 1 -1 }}
| 2.2794
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[875/874|Nymphisma]]
| Twethunuzotriyoma
| 23u19uz3yM
| 875/874
| 2.5.7.19.23 {{Monzo| -1 3 1 -1 -1 }}
| 1.9797
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[897/896|Lysistratisma]]
| Twethothoruma
| 23o3orM
| 897/896
| 2.3.7.13.23 {{Monzo| -7 1 -1 1 1 }}
| 1.9311
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3014656/3011499|23/17-schisma]]
| Sasa-twethosuma
| ss23o17uM
| 3014656/3011499
| 2.3.17.23 {{monzo| 17 -11 -1 1 }}
| 1.8139
|
|-
| [[2187/2185|Vashegyitisma]]
| Latwethunuguma
| L23u19ugM
| 2187/2185
| 3.5.19.23 {{monzo| 7 -1 -1 -1 }}
| 1.5839
| [[Francium]] (2025)
|-
| [[1105/1104|Fragarisma]]
| Twethusothoyoma
| 23u17o3oyM
| 1105/1104
| {{Monzo| -4 -1 1 0 0 1 1 0 -1 }}
| 1.5674
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[7942/7935|Brigade comma]]
| Bitwethuno-aloguma
| 23uu19oo1ogM
| 7942/7935
| {{Monzo| 1 -1 -1 0 1 0 0 2 -2 }}
| 1.5266
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1197/1196|Rodessisma]]
| Twethunothuzoma
| 23u19o3uzM
| 1197/1196
| {{Monzo| -2 2 0 1 0 -1 0 1 -1 }}
| 1.4469
| [[Lériendil]] (2025)
|-
| [[1288/1287|Triaphonisma]], santisma
| Twethothuluzoma
| 23o3u1uzM
| 1288/1287
| {{Monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 1.3446
| [[User:Xenllium|Xenllium]] (2023) for ''santisma''
|-
| [[1496/1495|Turkisma]]
| Twethusothuloguma
| 23u17o3u1ogM
| 1496/1495
| {{Monzo| 3 0 -1 0 1 -1 1 0 -1 }}
| 1.1576
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[7429/7425|Gordaitisma]]
| Twethonosoluguguma
| 23o19o17o1uggM
| 7429/7425
| {{monzo| 0 -3 -2 0 -1 0 1 1 1 }}
| 0.93240
| [[Francium]] (2025)
|-
| [[1863/1862|Antinousisma]]
| Twethonururuma
| 23o19urrM
| 1863/1862
| 2.3.7.19.23 {{Monzo| -1 4 -2 -1 1 }}
| 0.92952
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Kaifsma]]
| Twethunosutholuzozoguma
| 23u19o17u3o1uzzgM
| 193648/193545
| {{Monzo| 4 -2 -1 2 -1 1 -1 1 -1 }}
| 0.92108
| [[User:Kaiveran|Kaiveran]] (2022)
|-
| [[2024/2023|Artifisma]], insincere comma
| Twethosusuloruma
| 23o17uu1orM
| 2024/2023
| 2.7.11.17.23 {{Monzo| 3 -1 1 -2 1 }}
| 0.85556
| [[User:Xenllium|Xenllium]] (2023) for ''insincere comma''
|-
| [[2025/2024|Cupcake comma]], cupcakesma
| Latwethuluyoyoma
| L23u1uyyM
| 2025/2024
| 2.3.5.11.23 {{Monzo| -3 4 2 -1 -1 }}
| 0.85514
| See the page.
|-
| [[2185/2184|Guangdongisma]]
| Twethonothuruyoma
| 23o19o3uryM
| 2185/2184
| {{Monzo| -3 -1 1 -1 0 -1 0 1 1 }}
| 0.79251
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2300/2299|Travellisma]]
| Twethonubiluyoma
| 23o19u1uuyyM
| 2300/2299
| 2.5.11.19.23 {{Monzo| 2 2 -2 -1 1 }}
| 0.75287
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2646/2645|Biyativice comma]]
| Bitwethuzo-aguma
| 23uuzzgM
| 2646/2645
| 2.3.5.7.23 {{Monzo| 1 3 -1 2 -2 }}
| 0.65441
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2737/2736|Kotkisma]]
| Twethonusozoma
| 23o19u17ozM
| 2737/2736
| {{Monzo| -4 -2 0 1 0 0 1 -1 1 }}
| 0.63265
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Kwadransma]]
| Quadtwethuma
| 4(23u)M
| 279936/279841
| {{Monzo| 7 7 0 0 0 0 0 0 -4 }}
| 0.58762
| [[Lériendil]] (2024)
|-
| [[3060/3059|Vicious comma]], viciousma
| Twethunusoruyoma
| 23u19u17oryM
| 3060/3059
| {{Monzo| 2 2 1 -1 0 0 1 -1 -1 }}
| 0.56586
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3381/3380|Mikkolisma]], seminaiadvice comma
| Twethothuthuzozoguma
| 23o3uuzzgM
| 3381/3380
| {{Monzo| -2 1 -1 2 0 -2 0 0 1 }}
| 0.51212
| See the page.
|-
| [[6877/6875|Grossvice comma]]
| Bitwetho-atholuquadguma
| 23oo3o1u4gM
| 6877/6875
| {{Monzo| 0 0 -4 0 -1 1 0 0 2 }}
| 0.50356
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3520/3519|Vicedim comma]]
| Twethusuloyoma
| 23u17u1oyM
| 3520/3519
| {{Monzo| 6 -2 1 0 1 0 -1 0 -1 }}
| 0.49190
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3888/3887|Shoalma]], vicetride comma
| Twethuthuthuma
| 23u3uuM
| 3888/3887
| 2.3.13.23 {{Monzo| 4 5 -2 -1 }}
| 0.44533
| See the page.
|-
| [[8075/8073|Hagendorfisma]]
| Twethunosothuyoyoma
| 23u19o17o3uyyM
| 8075/8073
| {{monzo| 0 -3 2 0 0 -1 1 1 -1 }}
| 0.42884
| [[Francium]] (2025)
|-
| [[4693/4692|Viceaug comma]]
| Twethunonosuthoma
| 23u19oo17u3oM
| 4693/4692
| {{Monzo| -2 -1 0 0 0 1 -1 2 -1 }}
| 0.36894
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[4761/4760|Demiquartervice comma]]
| Bitwetho-asuruguma
| 23oo17urgM
| 4761/4760
| {{Monzo| -3 2 -1 -1 0 0 -1 0 2 }}
| 0.36367
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[5083/5082|Broadviewsma]]
| Twethosotholuluruma
| 23o17o3o1uurM
| 5083/5082
| {{Monzo| -1 -1 0 -1 -2 1 1 0 1 }}
| 0.34063
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[8625/8624|Beerglass comma]]
| Twetholururutriyoma
| 23o1urr3yM
| 8625/8624
| {{Monzo| -4 1 3 -2 -1 0 0 0 1 }}
| 0.20073
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Galeaclolusisma]]
| Twethususutholuquadyoma
| 23u17uu3o1u4yM
| 73125/73117
| {{Monzo| 0 2 4 0 -1 1 -2 0 -1 }}
| 0.18941
| [[Francium]] (2025)
|-
| [[10626/10625|Demiglace comma]]
| Twethosulozoquadguma
| 23o17u1oz4gM
| 10626/10625
| {{Monzo| 1 1 -4 1 1 0 -1 0 1 }}
| 0.16293
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Vicetertisma]]
| Tritwethu-athothoma
| 3(23u)3ooM
| 12168/12167
| 2.3.13.23 {{Monzo| 3 2 2 -3 }}
| 0.14228
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Joshuavoisma]]
| Twethusutholozoyoyoma
| 23u17u3o1ozyyM
| 25025/25024
| {{monzo| -6 0 2 1 1 1 -1 0 -1 }}
| 0.06918
| [[Francium]] (2024)
|-
| [[Diarithmedia]]
| Bitwethozo-aguma
| 23oozzgM
| 25921/25920
| 2.3.5.7.23 {{monzo| -6 -4 -1 2 2 }}
| 0.066790
| [[Flora Canou]] (2023), modified by [[Lériendil]] (2024)
|-
| [[Jeffbenisma]]
| Labitwethu-anutholuzoyoma
| L23uu19u3o1uzyM
| 110565/110561
| {{monzo| 0 5 1 1 -1 1 0 -1 -2 }}
| 0.062633
| [[Francium]] (2025)
|}

=== 29-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[551/550|Minor chthonovinema]]
| Twenonoluguguma
| 29o19o1uggM
| 551/550
| 2.5.11.19.29 {{monzo| -1 -2 -1 1 1 }}
| 3.1448
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[552/551|Sigelindisma]]
| Twenutwethonuma
| 29u23o19uM
| 552/551
| 2.3.19.23.29 {{monzo| 3 1 -1 1 -1 }}
| 3.1391
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[609/608|Vineyard comma]]
| Twenonuzoma
| 29o19uzM
| 609/608
| 2.3.7.19.29 {{monzo| -5 1 1 -1 1 }}
| 2.8451
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[638/637|Moirisma]]
| Twenothuloruruma
| 29o3u1orrM
| 638/637
| 2.7.11.13.29 {{monzo| 1 -2 1 -1 1 }}
| 2.7157
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[726/725|Joellisma]]
| Twenubiloguma
| 29u1ooggM
| 726/725
| 2.3.5.11.29 {{monzo| 1 1 -2 2 -1 }}
| 2.3863
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[783/782|Norisma]]
| Twenotwethusuma
| 29o23u17uM
| 783/782
| 2.3.17.23.29 {{monzo| -1 3 -1 -1 1 }}
| 2.2124
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[784/783|Biminorisma]], spoogalactic comma
| Twenuzozoma
| 29uzzM
| 784/783
| 2.3.7.29 {{monzo| 4 -3 2 -1 }}
| 2.2096
| [[Scott Dakota]] for ''biminorisma''
|-
| [[841/840|Arabellisma]]
| Bitweno-aruguma
| 29oorgM
| 841/840
| 2.3.5.7.29 {{monzo| -3 -1 -1 -1 2 }}
| 2.0598
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1015/1014|Christisma]]
| Twenothuthuzoyoma
| 29o3uuzyM
| 1015/1014
| {{monzo| -1 -1 1 1 0 -2 0 0 0 1 }}
| 1.7065
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1045/1044|Michelisma]]
| Twenunoloyoma
| 29u19o1oyM
| 1045/1044
| {{monzo| -2 -2 1 0 1 0 0 1 0 -1 }}
| 1.6575
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1276/1275|Ucclisma]]
| Twenosuloguguma
| 29o17u1oggM
| 1276/1275
| {{monzo| 2 -1 -2 0 1 0 -1 0 0 1 }}
| 1.3573
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1450/1449|Raimondisma]]
| Twenotwethuruyoyoma
| 29o23uryyM
| 1450/1449
| {{monzo| 1 -2 2 -1 0 0 0 0 -1 1 }}
| 1.1944
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1596/1595|Itzigsohnisma]]
| Twenunoluzoguma
| 29u19o1uzgM
| 1596/1595
| {{monzo| 2 1 -1 1 -1 0 0 1 0 -1 }}
| 1.0851
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1625/1624|Norcma]]
| Twenuthorutriyoma
| 29u3or3yM
| 1625/1624
| 2.5.7.13.29 {{monzo| -3 3 -1 1 -1 }}
| 1.0657
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1683/1682|Castafiorisma]]
| Bitwenu-asoloma
| 29uu17o1oM
| 1683/1682
| 2.3.11.17.29 {{monzo| -1 2 1 1 -2 }}
| 1.0290
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2001/2000|Major discoverisma]]
| Twenotwethotriguma
| 29o23o3gM
| 2001/2000
| 2.3.5.23.29 {{monzo| -4 1 -3 1 1 }}
| 0.86540
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2002/2001|Minor discoverisma]]
| Twenutwethutholozoma
| 29u23u3o1ozM
| 2002/2001
| {{monzo| 1 -1 0 1 1 1 0 0 -1 -1 }}
| 0.86497
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2176/2175|Donarisma]]
| Twenusoguguma
| 29u17oggM
| 2176/2175
| 2.3.5.17.29 {{monzo| 7 -1 -2 1 -1 }}
| 0.79579
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2205/2204|Glinkisma]]
| Twenunuzozoyoma
| 29u19uzzyM
| 2205/2204
| {{monzo| -2 2 1 2 0 0 0 -1 0 -1 }}
| 0.78532
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2262/2261|Mitidikisma]]
| Twenonusuthoruma
| 29o19u17u3orM
| 2262/2261
| {{monzo| 1 1 0 -1 0 1 -1 -1 0 1 }}
| 0.76552
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2465/2464|Laservine comma]]
| Twenosoluruyoma
| 29o17o1uryM
| 2465/2464
| {{monzo| -5 0 1 -1 -1 0 1 0 0 1 }}
| 0.70247
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2640/2639|Hällströmisma]]
| Twenuthuloruyoma
| 29u3u1oryM
| 2640/2639
| {{monzo| 4 1 1 -1 1 -1 0 0 0 -1 }}
| 0.65589
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2755/2754|Avicema]]
| Twenonosuyoma
| 29o19o17uyM
| 2755/2754
| {{monzo| -1 -4 1 0 0 0 -1 1 0 1 }}
| 0.62851
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2784/2783|Domeykisma]]
| Twenotwethululuma
| 29o23u1uuM
| 2784/2783
| 2.3.11.23.29 {{monzo| 5 1 -2 -1 1 }}
| 0.62196
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[9251/9248|Helevenisma]]
| Bitwenosu-aloma
| 29oo17uu1oM
| 9251/9248
| 2.11.17.29 {{monzo| -5 1 -2 2 }}
| 0.56151
| [[Zhea Erose]] (2023)
|-
| [[3249/3248|Musashinisma]]
| Twenunonoruma
| 29u19oorM
| 3249/3248
| 2.3.7.19.29 {{monzo| -4 2 -1 2 -1 }}
| 0.53293
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[3451/3450|Mentorisma]]
| Twenotwethusozoguguma
| 29o23u17ozggM
| 3451/3450
| {{monzo| -1 -1 -2 1 0 0 1 0 -1 1 }}
| 0.50173
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Bronxisma]]
| Twenununusolozoma
| 29u19uu17o1ozM
| 10472/10469
| {{monzo| 3 0 0 1 1 0 1 -2 0 -1 }}
| 0.49603
| [[Francium]] (2024)
|-
| [[3510/3509|Veederisma]]
| Twenutholuluyoma
| 29u3o1uuyM
| 3510/3509
| {{monzo| 1 3 1 0 -2 1 0 0 0 -1 }}
| 0.49330
| [[Francium]] (2024)
|-
| [[4641/4640|Vinecute comma]]
| Twenusothozoguma
| 29u17o3ozgM
| 4641/4640
| {{monzo| -5 1 -1 1 0 1 1 0 0 -1 }}
| 0.37307
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[4785/4784|Petrovsma]]
| Twenotwethuthuloyoma
| 29o23u3u1oyM
| 4785/4784
| {{monzo| -4 1 1 0 1 -1 0 0 -1 1 }}
| 0.36184
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[4901/4900|Large grapevine]]
| Twenothothobiruguma
| 29o3oorrggM
| 4901/4900
| 2.5.7.13.29 {{monzo| -2 -2 -2 2 1 }}
| 0.35328
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Mynucuvine comma]]
| Labitwenu-athuyoma
| L29uu3uyM
| 10935/10933
| 3.5.13.29 {{monzo| 7 1 -1 -2 }}
| 0.31667
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[5888/5887|Vinocular comma]]
| Bitwenu-atwethoruma
| 29uu23orM
| 5888/5887
| 2.7.23.29 {{monzo| 8 -1 1 -2 }}
| 0.29405
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[5916/5915|Woudisma]]
| Twenosothuthuruguma
| 29o17o3uurgM
| 5916/5915
| {{monzo| 2 1 -1 -1 0 -2 1 0 0 1 }}
| 0.29266
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[7425/7424|Small grapevine]]
| Latwenuloyoyoma
| L29u1oyyM
| 7425/7424
| 2.3.5.11.29 {{monzo| -8 3 2 1 -1 }}
| 0.23318
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[8671/8670|Vinous comma]], vinousma
| Twenotwethosusuthoguma
| 29o23o17uu3ogM
| 8671/8670
| {{monzo| -1 -1 -1 0 0 1 -2 0 1 1 }}
| 0.19967
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[9802/9801|Kakisma]]
| Twenobitholuma
| 29o3oo1uuM
| 9802/9801
| {{Monzo| 1 -4 0 0 -2 2 0 0 0 1 }}
| 0.17663
| [[Lériendil]] (2025)
|-
| [[10557/10556|Rowlandisma]]
| Twenutwethosothuruma
| 29u23o17o3urM
| 10557/10556
| {{monzo| -2 3 0 -1 0 -1 1 0 1 -1 }}
| 0.16400
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Uma]], umic comma
| Twenotwethoquadru-aguma
| 29o23o4rgM
| 12006/12005
| {{monzo| 1 2 -1 -4 0 0 0 0 1 1 }}
| 0.14420
|
|-
| [[Vuillafansisma]]
| Twenunosoluyoma
| 29u19o17o1uyM
| 25840/25839
| {{monzo| 4 -4 1 0 -1 0 1 1 0 -1 }}
| 0.067000
| [[Francium]] (2024)
|-
| [[Odyssey comma]]
| Bitwenotwetho-athulurutriguma
| 29oo23oo3u1ur3gM
| 4004001/4004000
| {{monzo| -5 2 -3 -1 -1 -1 0 0 2 2 }}
| 0.00043238
| [[Tristan Bay]] (2024)
|}

=== 31-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[496/495|Navatonisma]]
| Thiwoluguma
| 31o1ugM
| 496/495
| 2.3.5.11.31 {{monzo| 4 -2 -1 -1 1 }}
| 3.4939
| [[User:FilterNashi|FilterNashi]] (2025)
|-
| [[528/527|Rezisma]]
| Thiwusuloma
| 31u17u1oM
| 528/527
| 2.3.11.17.31 {{monzo| 4 1 1 -1 -1 }}
| 3.2820
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[589/588|Croatisma]]
| Thiwonoruruma
| 31o19orrM
| 589/588
| 2.3.7.19.31 {{monzo| -2 -1 -2 1 1 }}
| 2.9418
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[621/620|Owowhatsthisma]]
| Thiwutwethoguma
| 31u23ogM
| 621/620
| 2.3.5.23.31 {{monzo| -2 3 -1 1 -1 }}
| 2.7901
| [[HEHEHE I AM A SUPAHSTAR SAGA]] (2021)
|-
| [[651/650|Antiklisma]]
| Thiwothuzoguguma
| 31o3uzggM
| 651/650
| 2.3.5.7.13.31 {{monzo| -1 1 -2 1 -1 1 }}
| 2.6614
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[714/713|Ululisma]]
| Thiwutwethusozoma
| 31u23u17ozM
| 714/713
| 2.3.7.17.23.31 {{monzo| 1 1 1 1 -1 -1 }}
| 2.4264
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[961/960|Tricesimoprimal quartertones comma]]
| Bithiwo-aguma
| 31oogM
| 961/960
| 2.3.5.31 {{monzo| -6 -1 -1 2 }}
| 1.8024
| [[Godtone]] (2024)
|-
| [[1024/1023|Kibisma]]
| Thiwuluma
| 31u1uM
| 1024/1023
| 2.3.11.31 {{Monzo| 10 -1 -1 -1 }}
| 1.6915
| See the page.
|-
| [[2233/2232|Kuznetsovisma]]
| Thiwotwenolozoma
| 31u29o1ozM
| 2233/2232
| 2.3.7.11.29.31 {{monzo| -3 -2 1 1 1 -1 }}
| 0.77547
| [[Francium]] (2024)
|-
| [[3969/3968|Yunzee comma]]
| Lathiwuzozoma
| L31uzzM
| 3969/3968
| 2.3.7.31 {{monzo| -7 4 2 -1 }}
| 0.43624
| [[User:Kaiveran|Kaiveran]] (2021)
|-
| [[4186/4185|Tamashimisma]]
| Thiwutwethothozoguma
| 31u23o3ozgM
| 4186/4185
| 2.3.5.7.13.23.31 {{monzo| 1 -3 -1 1 1 1 -1 }}
| 0.41363
| [[Francium]] (2025)
|-
| [[4992/4991|Kalmanisma]]
| Thiwutwethuthoruma
| 31u23u3orM
| 4992/4991
| 2.3.7.13.23.31 {{monzo| 7 1 -1 1 -1 -1 }}
| 0.34684
| [[Francium]] (2024)
|-
| [[5797/5796|Bivojisma]]
| Thiwotwethusoloruma
| 31o23u17o1orM
| 5797/5796
| 2.3.7.11.17.23.31 {{monzo| -2 -2 -1 1 1 -1 1 }}
| 0.29867
| [[Francium]] (2025)
|-
| [[6076/6075|Large ricegrain]]
| Sathiwobizoguma
| s31ozzggM
| 6076/6075
| 2.3.5.7.31 {{monzo| 2 -5 -2 2 1 }}
| 0.28495
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[6480/6479|Scarlattisma]]
| Thiwunuluyoma
| 31u19u1uyM
| 6480/6479
| 2.3.5.11.19.31 {{monzo| 4 4 1 -1 -1 -1 }}
| 0.26719
| [[Francium]] (2024)
|-
| [[6728/6727|Sushi comma]]
| Bithiwutweno-aruma
| 31uu29oorM
| 6728/6727
| 2.7.29.31 {{monzo| 3 -1 2 -2 }}
| 0.25734
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[16337/16335|Brown rice comma]]
| Bithiwo-asoluluguma
| 31oo17o1uugM
| 16337/16335
| 3.5.11.17.31 {{monzo| -3 -1 -2 1 2 }}
| 0.21195
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[8464/8463|Polishookisma]]
| Thiwubitwetho-athuruma
| 31u23oo3urM
| 8464/8463
| 2.3.7.13.23.31 {{monzo| 4 -1 -1 -1 2 -1 }}
| 0.20455
| [[Francium]] (2024)
|-
| [[8960/8959|Small ricegrain]]
| Thiwususuzoyoma
| 31u17uuzyM
| 8960/8959
| 2.5.7.17.31 {{monzo| 8 1 1 -2 -1 }}
| 0.19323
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Acronymisma]]
| Thiwotrithu-azoma
| 31o3(3u)zM
| 17577/17576
| 2.3.7.13.31 {{monzo| -3 4 1 -3 1 }}
| 0.098497
| [[User:Lériendil|Lériendil]] (2025)
|-
| [[Totziensisma]]
| Thiwotwetholurutriguma
| 31o23o1ur3gM
| 19251/19250
| 2.3.5.7.11.23.31 {{monzo| -1 3 -3 -1 -1 1 1 }}
| 0.089932
| [[Francium]] (2024)
|-
| [[Honeybrookisma]]
| Thiwobitwenu-atwethuthoma
| 31o29uu23u3oM
| 19344/19343
| 2.3.13.23.29.31 {{monzo| 4 1 1 -1 -2 1 }}
| 0.089500
| [[Francium]] (2024)
|-
| [[Tricecubisma]]
| Trithiwu-anozozoma
| 3(31u)19ozzM
| 29792/29791
| 2.7.19.31 {{monzo| 5 2 1 -3 }}
| 0.058112
| [[User:Xenllium|Xenllium]] (2025)
|}

=== 37-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[666/665|Beastisma]]
| Thisonuruguma
| 37o19urgM
| 666/665
| 2.3.5.7.19.37 {{monzo| 1 2 -1 -1 -1 1 }}
| 2.6014
| [[Francium]] (2025)
|-
| [[667/666|Denisisma]]
| Thisutwenotwethoma
| 37u29o23oM
| 667/666
| 2.3.23.29.37 {{monzo| -1 -2 1 1 -1 }}
| 2.5975
| [[Francium]] (2025)
|-
| [[703/702|Noemisma]]
| Thisonothuma
| 37o19o3uM
| 703/702
| 2.3.13.19.37 {{monzo| -1 -3 -1 1 1 }}
| 2.4644
| [[Francium]] (2025)
|-
| [[704/703|Minimyna]]
| Thisunuloma
| 37u19u1oM
| 704/703
| 2.11.19.37 {{monzo| 6 1 -1 -1 }}
| 2.4609
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[741/740|Botolphisma]]
| Thisunothoguma
| 37u19o3ogM
| 741/740
| 2.3.5.13.19.37 {{monzo| -2 1 -1 1 1 -1 }}
| 2.3379
| [[Francium]] (2025)
|-
| [[851/850|Zeissisma]]
| Thisotwethosuguguma
| 37o23o17uggM
| 851/850
| 2.5.17.23.37 {{monzo| -1 -2 -1 1 1 }}
| 2.0355
| [[Francium]] (2025)
|-
| [[925/924|Alphonsinisma]]
| Thisoluruyoyoma
| 37o1uryyM
| 925/924
| 2.3.5.7.11.37 {{monzo| -2 -1 2 -1 -1 1 }}
| 1.8726
| [[Francium]] (2025)
|-
| [[1036/1035|Ganymedisma]]
| Thisotwethuzoguma
| 37o23uzgM
| 1036/1035
| 2.3.5.7.23.37 {{monzo| 2 -2 -1 1 -1 1 }}
| 1.6719
| [[Francium]] (2025)
|-
| [[1184/1183|Gaeisma]]
| Thisothuthuruma
| 37o3uurM
| 1184/1183
| 2.7.13.37 {{monzo| 5 -1 -2 1}}
| 1.4628
| [[Francium]] (2025)
|-
| [[1332/1331|Marconisma]]
| Thisotriluma
| 37o3(1u)M
| 1332/1331
| 2.3.11.37 {{monzo| 2 2 -3 1 }}
| 1.3002
| [[Francium]] (2024)
|-
| [[1480/1479|Aunusisma]]
| Thisotwenusuyoma
| 37o29u17uyM
| 1480/1479
| 2.3.5.17.29.37 {{monzo| 3 -1 1 -1 -1 1 }}
| 1.1702
| [[Francium]] (2024)
|-
| [[1665/1664|Gabisma]]
| Thisothuyoma
| 37o3uyM
| 1665/1664
| 2.3.5.13.37 {{monzo| -7 2 1 -1 1 }}
| 1.0401
| [[Francium]] (2025)
|-
| [[1666/1665|Gentisma]]
| Thisusozozoguma
| 37u17ozzgM
| 1666/1665
| 2.3.5.7.17.37 {{monzo| 1 -2 -1 2 1 -1 }}
| 1.0395
| [[Francium]] (2025)
|-
| [[1702/1701|Kalaharisma]]
| Sathisotwethoruma
| S37o23orM
| 1702/1701
| 2.3.7.23.37 {{monzo| 1 -5 -1 1 1 }}
| 1.0175
| [[Francium]] (2025)
|-
| [[1925/1924|Misericorde]]
| Thisuthulozoyoyoma
| 37u3u1ozyyM
| 1925/1924
| 2.5.7.11.13.37 {{monzo| -2 2 1 1 -1 -1 }}
| 0.89958
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[2109/2108|Dhotelisma]]
| Thisothiwunosuma
| 37o31u19o17uM
| 2109/2108
| 2.3.17.19.31.37 {{monzo| -2 1 -1 1 -1 1 }}
| 0.82107
| [[Francium]] (2025)
|-
| [[2146/2145|Stentorisma]]
| Thisotwenothuluguma
| 37o29o3u1ugM
| 2146/2145
| 2.3.5.11.13.29.37 {{monzo| 1 -1 -1 -1 -1 1 1 }}
| 0.80691
| [[Francium]] (2024)
|-
| [[2553/2552|Viljevisma]]
| Thisotwenutwetholuma
| 37o29u23o1uM
| 2553/2552
| 2.3.11.23.29.37 {{monzo| -3 1 -1 1 -1 1 }}
| 0.67825
| [[Francium]] (2024)
|-
| [[2850/2849|Mozhaiskisma]]
| Thisunoluruyoyoma
| 37u19o1uryyM
| 2850/2849
| 2.3.5.7.11.19.37 {{monzo| 1 1 2 -1 -1 1 -1 }}
| 0.60756
| [[Francium]] (2025)
|-
| [[3146/3145|Datonisma]]
| Thisusutholologuma
| 37u17u3o1oogM
| 3146/3145
| 2.5.11.13.17.37 {{monzo| 1 -1 2 1 -1 -1 }}
| 0.55038
| [[Francium]] (2025)
|-
| [[3220/3219|Murayamisma]]
| Thisutwenutwethozoyoma
| 37u29u23ozyM
| 3220/3219
| 2.3.5.7.23.29.37 {{monzo| 2 -1 1 1 1 -1 -1 }}
| 0.53773
| [[Francium]] (2025)
|-
| [[3256/3255|Daguerrisma]]
| Thisothiwuloruguma
| 37o31u1orgM
| 3256/3255
| 2.3.5.7.11.31.37 {{monzo| 3 -1 -1 -1 1 -1 1 }}
| 0.53179
| [[Francium]] (2025)
|-
| [[3367/3366|Alexisma]]
| Thisosutholuzoma
| 37o17u3o1uzM
| 3367/3366
| 2.3.7.11.13.17.37 {{monzo| -1 -2 1 -1 1 -1 1 }}
| 0.51425
| [[Francium]] (2025)
|-
| [[3553/3552|Meranisma]]
| Thisunosoloma
| 37u19o17o1oM
| 3553/3552
| 2.3.11.17.19.37 {{monzo| -5 -1 1 1 1 -1 }}
| 0.48733
| [[Francium]] (2024)
|-
| [[3626/3625|Ohsakisma]]
| Thisotwenuzozotriguma
| 37o29uzz3gM
| 3626/3625
| 2.5.7.29.37 {{monzo| 1 -3 2 -1 1 }}
| 0.47752
| [[Francium]] (2025)
|-
| [[3627/3626|Sayersisma]]
| Thisuthiwothoruruma
| 37u31o3orrM
| 3627/3626
| 2.3.7.13.31.37 {{monzo| -1 2 -2 1 1 -1 }}
| 0.47738
| [[Francium]] (2025)
|-
| [[3774/3773|Megumisma]]
| Thisosolutriruma
| 37o17o1u3rM
| 3774/3773
| 2.3.7.11.17.37 {{monzo| 1 1 -3 -1 1 1 }}
| 0.45879
| [[Francium]] (2025)
|-
| [[4625/4624|Shchedrinisma]]
| Thisosusutriyoma
| 37o17uu3yM
| 4625/4624
| 2.5.17.37 {{monzo| -4 3 -2 1 }}
| 0.37436
| [[Francium]] (2024)
|-
| [[5292/5291|Bullionisma]]
| Thisuthuluzozoma
| 37u3u1uzzM
| 5292/5291
| 2.3.7.11.13.37 {{monzo| 2 3 2 -1 -1 -1 }}
| 0.32717
| [[Eliora]] (2023)
|-
| [[5440/5439|Teraosma]]
| Thisusoruruyoma
| 37u17orryM
| 5440/5439
| 2.3.5.7.17.37 {{monzo| 6 -1 1 -2 1 -1 }}
| 0.31827
| [[Francium]] (2025)
|-
| [[7105/7104|Yousyozanisma]]
| Thisutwenozozoyoma
| 37u29ozzyM
| 7105/7104
| 2.3.5.7.29.37 {{monzo| -6 -1 1 2 1 -1 }}
| 0.24368
| [[Francium]] (2025)
|-
| [[7696/7695|Liebisma]]
| Sathisonuthoguma
| S37o19u3ogM
| 7696/7695
| 2.3.5.13.19.37 {{monzo| 4 -4 -1 1 -1 1 }}
| 0.22497
| [[Francium]] (2025)
|-
| [[8991/8990|Solidarity comma]]
| Thisothiwutwenuguma
| 37o31u29ugM
| 8991/8990
| 2.3.5.29.31.37 {{monzo| -1 5 -1 -1 -1 1 }}
| 0.19256
| [[Francium]] (2025)
|-
| [[9177/9176|Donsaarisma]]
| Thisuthiwutwethonozoma
| 37u31u23o19oM
| 9177/9176
| 2.3.7.19.23.31.37 {{monzo| -3 1 1 1 1 -1 -1 }}
| 0.18866
| [[Francium]] (2025)
|-
| [[9251/9250|Harchisma]]
| Thisubitweno-alotriguma
| 37u29oo1o3gM
| 9251/9250
| 2.5.11.29.37 {{monzo| -1 -3 1 2 -1 }}
| 0.18715
| [[Francium]] (2025)
|-
| [[9361/9360|Friesachisma]]
| Thisotwethothuloguma
| 37o23o3u1ogM
| 9361/9360
| 2.3.5.11.13.23.37 {{monzo| -4 -2 -1 1 -1 1 1}}
| 0.18495
| [[Francium]] (2024)
|-
| [[Zangarisma]]
| Thisososoluma
| 37o17oo1uM
| 10693/10692
| 2.3.11.17.37 {{monzo| -2 -5 -1 2 1 }}
| 0.16191
| [[Francium]] (2024)
|-
| [[23275/23273|Sunrise comma]]
| Bithisu-anosubizoyoma
| 37uu19o17uzzyyM
| 23275/23273
| 5.7.17.19.37 {{monzo| 2 2 -1 1 -2 }}
| 0.14877
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Sinachopoulosisma]]
| Thisotwethonunuluzoma
| 37o23o19uu1uzM
| 11914/11913
| 2.3.7.11.19.23.37 {{monzo| 1 -1 1 -1 -2 1 1 }}
| 0.14532
| [[Francium]] (2024)
|-
| [[12321/12320|Zurakowskisma]]
| Bithiso-aluruguma
| 37oo1urgM
| 12321/12320
| 2.3.5.7.11.37 {{monzo| -5 2 -1 -1 -1 2 }}
| 0.14052
| [[Francium]] (2025)
|-
| [[13690/13689|Lesleymartin comma]]
| Bithisothu-ayoma
| 37oo3uuyM
| 13690/13689
| 2.3.5.13.37 {{monzo| 1 -4 1 -2 2 }}
| 0.12646
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Berylisma]]
| Quadthisoluma
| 4(37o1u)M
| 1874161/1874048
| 2.11.37 {{monzo| -7 -4 4 }}
| 0.10439
| [[User:Jerdle|Jerdle]] (2023)
|-
| [[Genzelisma]]
| Thisotwenonusoguma
| 37o29o19u17ogM
| 18241/18240
| 2.3.5.17.19.29.37 {{monzo| -6 -1 -1 1 -1 1 1 }}
| 0.094912
| [[Francium]] (2024)
|-
| [[33264/33263|Maryrogersisma]]
| Thisuthiwutwenulozoma
| 37u31u29u1ozM
| 33264/33263
| 2.3.7.11.29.31.37 {{monzo| 4 3 1 1 -1 -1 -1 }}
| 0.052046
| [[Francium]] (2025)
|-
| [[Jouvisma]]
| Thisotwethotholuluzoguma
| 37o23o3o1uuzgM
| 77441/77440
| 2.5.7.11.13.23.37 {{monzo| -7 -1 1 -2 1 1 1 }}
| 0.022356
| [[Francium]] (2024)
|}

=== 41-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[575/574|Renatisma]]
| Fowutwethoruyoyoma
| 41u23oryyM
| 575/574
| 2.5.7.23.41 {{monzo| -1 2 -1 1 -1 }}
| 3.0135
| [[Francium]] (2025)
|-
| [[616/615|Ellisma]]
| Fowulozoguma
| 41u1ozgM
| 616/615
| 2.3.5.7.11.41 {{monzo| 3 -1 -1 1 1 -1 }}
| 2.8127
| [[Francium]] (2024)
|-
| [[780/779|Wiesentisma]]
| Fowunuthoyoma
| 41u19u3oyM
| 780/779
| 2.3.5.13.19.41 {{monzo| 2 1 1 1 -1 -1 }}
| 2.2210
| [[Francium]] (2025)
|-
| [[1025/1024|Kilobytisma]]
| Fowoyoyoma
| 41oyyM
| 1025/1024
| 2.5.41 {{Monzo| -10 2 1 }}
| 1.6898
| [[CompactStar]] (2024)
|-
| [[1026/1025|Ingridisma]]
| Fowunoguguma
| 41u19oggM
| 1026/1025
| 2.3.5.19.41 {{monzo| 1 3 -2 1 -1 }}
| 1.6882
| [[Francium]] (2025)
|-
| [[1190/1189|Pelagisma]]
| Fowutwenusozoyoma
| 41u29u17ozyM
| 1190/1189
| 2.5.7.17.29.41 {{monzo| 1 1 1 1 -1 -1 }}
| 1.4554
| [[Francium]] (2025)
|-
| [[1518/1517|Rovaniemisma]]
| Fowuthisutwetholoma
| 41u37u23o1oM
| 1518/1517
| 2.3.11.23.37.41 {{monzo|1 1 1 1 -1 -1 }}
| 1.1408
| [[Francium]] (2025)
|-
| [[1682/1681|Shaftesburisma]]
| Bifowutwenoma
| 41uu29ooM
| 1682/1681
| 2.29.41 {{monzo| 1 2 -2 }}
| 1.0296
| [[Budjarn Lambeth]] (2023)
|-
| [[2255/2254|Qinghaisma]]
| Fowotwethuloruruyoma
| 41o23u1orryM
| 2255/2254
| 2.5.7.11.23.41 {{monzo| -1 1 -2 1 -1 1 }}
| 0.76790
| [[Francium]] (2025)
|-
| [[2871/2870|Schoberisma]]
| Fowutwenoloruguma
| 41u29o1orgM
| 2871/2870
| 2.3.5.7.11.29.41 {{monzo| -1 2 -1 -1 1 1 -1 }}
| 0.60311
| [[Francium]] (2025)
|-
| [[3773/3772|Smithsonianisma]]
| Fowutwethulotrizoma
| 41u23u1o3zM
| 3773/3772
| 2.7.11.23.41 {{monzo| -2 3 1 -1 -1 }}
| 0.45891
| [[Francium]] (2025)
|-
| [[4060/4059|Deipylosisma]]
| Fowutwenoluzoyoma
| 41u29o1uzyM
| 4060/4059
| 2.3.5.7.11.29.41 {{monzo| 2 -2 1 1 -1 1 -1 }}
| 0.42646
| [[Francium]] (2025)
|-
| [[4675/4674|Ohbokisma]]
| Fowunusoloyoyoma
| 41u19u17o1oyyM
| 4675/4674
| 2.3.5.11.17.19.41 {{monzo| -1 -1 2 1 1 -1 -1 }}
| 0.37036
| [[Francium]] (2025)
|-
| [[4921/4920|Volontisma]]
| Fowuthisonozoguma
| 41u37o19ozgM
| 4921/4920
| 2.3.5.7.19.37.41 {{monzo| -3 -1 -1 1 1 1 -1 }}
| 0.35184
| [[Francium]] (2025)
|-
| [[5577/5576|Priestlisma]]
| Fowusuthotholoma
| 41u17u3oo1oM
| 5577/5576
| 2.3.11.13.17.41 {{monzo| -3 1 1 2 -1 -1 }}
| 0.31045
| [[Francium]] (2025)
|-
| [[6069/6068|Cevolanisma]]
| Fowuthisusosozoma
| 41u37u17oozM
| 6069/6068
| 2.3.7.17.37.41 {{monzo| -2 1 1 2 -1 -1 }}
| 0.28528
| [[Francium]] (2025)
|-
| [[6930/6929|Bedanisma]]
| Fowuthuthulozoyoma
| 41u3uu1ozyM
| 6930/6929
| 2.3.5.7.11.13.41 {{monzo| 1 2 1 1 1 -2 -1 }}
| 0.24984
| [[Francium]] (2025)
|-
| [[7176/7175|Kunijisma]]
| Fowutwethothoruguguma
| 41u23o3orggM
| 7176/7175
| 2.3.5.7.13.23.41 {{monzo| 3 1 -2 -1 1 1 -1 }}
| 0.24127
| [[Francium]] (2025)
|-
| [[8569/8568|Mamelisma]]
| Fowonosuloruma
| 41o19o17u1orM
| 8569/8568
| 2.3.7.11.17.19.41 {{monzo| -3 -2 -1 1 -1 1 1 }}
| 0.20205
| [[Francium]] (2025)
|-
| [[9472/9471|Brugesisma]]
| Fowuthisoluruma
| 41u37o1urM
| 9472/9471
| 2.3.7.11.37.41 {{monzo| 8 -1 -1 -1 1 -1 }}
| 0.18278
| [[Francium]] (2025)
|-
| [[Etampesisma]]
| Fowutwethunotholuzoma
| 41u23u19o3o1uzM
| 10374/10373
| 2.3.7.11.13.19.23.41 {{monzo| 1 1 1 -1 1 1 -1 -1 }}
| 0.16689
| [[Francium]] (2024)
|-
| [[11440/11439|Massironisma]]
| Fowuthiwutholoyoma
| 41u31u3o1oyM
| 11440/11439
| 2.3.5.11.13.31.41 {{monzo| 4 -2 1 1 1 -1 -1 }}
| 0.15134
| [[Francium]] (2025)
|-
| [[15376/15375|Martakisma]]
| Fowubithiwo-atriguma
| 41u31oo3gM
| 15376/15375
| 2.3.5.31.41 {{monzo| 4 -1 -3 2 -1 }}
| 0.11260
| [[Francium]] (2025)
|-
| [[Canupisma]]
| Fowutwenuthotrizo-aguma
| 41u29u3o3zagM
| 17836/17835
| 2.3.5.7.13.29.41 {{monzo| 2 -1 -1 3 1 -1 -1 }}
| 0.097067
| [[Francium]] (2024)
|-
| [[76384/76383|Vernonisma]]
| Fowuthiwotwethulozoma
| 41u31o23u1ozM
| 76384/76383
| 2.3.7.11.23.31.41 {{monzo| 5 -4 1 1 -1 1 -1 }}
| 0.022665
| [[Francium]] (2025)
|-
| [[Mebisma]]
| Safowuthiwuluguguma
| s41u31u1uggM
| 1048576/1048575
| 2.3.5.11.31.41 {{Monzo| 20 -1 -2 -1 -1 -1 }}
| 0.0016510
| See the page.
|}

=== 43-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[646/645|Kastalisma]]
| Fothunosoguma
| 43u19o17ogM
| 646/645
| 2.3.5.17.19.43 {{monzo| 1 -1 -1 1 1 -1 }}
| 2.6820
| [[Francium]] (2025)
|-
| [[990/989|Yerkesisma]]
| Fothutwethuloyoma
| 43u23u1oyM
| 990/989
| 2.3.5.11.23.43 {{monzo| 1 2 1 1 -1 -1 }}
| 1.7496
| [[Francium]] (2025)
|-
| [[1333/1332|Cevelonisma]]
| Fothothisuthiwoma
| 43o37u31oM
| 1333/1332
| 2.3.31.37.43 {{monzo| -2 -2 1 -1 1 }}
| 1.2992
| [[Francium]] (2025)
|-
| [[1377/1376|Roberbauxisma]]
| Lafothusoma
| L43u17oM
| 1377/1376
| 2.3.17.43 {{monzo| -5 4 1 -1}}
| 1.2577
| [[Francium]] (2025)
|-
| [[1463/1462|Nordenmarkisma]]
| Fothunosulozoma
| 43u19o17uozM
| 1463/1462
| 2.7.11.17.19.43 {{monzo| -1 1 1 -1 1 -1 }}
| 1.1838
| [[Francium]] (2025)
|-
| [[Magnetisma]]
| Tritrila-quinquadtrifothutwenoma
| 9L60(43u29o)M
|
| 2.3.29.43 {{monzo| -61 60 60 -60 }}
| 0.86936
| [[User:Eliora|Eliora]] (2022)
|-
| [[2925/2924|Beattisma]]
| Fothusuthoyoyoma
| 43u17u3oyyM
| 2925/2924
| 2.3.5.13.17.43 {{monzo| -2 2 2 1 -1 -1 }}
| 0.59198
| [[Francium]] (2025)
|-
| [[3312/3311|Pedersenisma]]
| Fothutwetholuruma
| 43u23o1urM
| 3312/3311
| 2.3.7.11.23.43 {{monzo| 4 2 -1 -1 1 -1 }}
| 0.52279
| [[Francium]] (2025)
|-
| [[4000/3999|Hipparchusisma]]
| Fothuthiwutriyoma
| 43u31u3yM
| 4000/3999
| 2.3.5.31.43 {{monzo| 5 -1 3 -1 -1 }}
| 0.43286
| [[Francium]] (2025)
|-
| [[4301/4300|Boydenisma]]
| Fothutwethosologuguma
| 43u23o17o1oggM
| 4301/4300
| 2.5.11.17.23.43 {{monzo| -2 -2 1 1 1 -1 }}
| 0.40257
| [[Francium]] (2025)
|-
| [[4774/4773|Hobetsisma]]
| Fothuthisuthiwolozoma
| 43u37u31o1ozM
| 4774/4773
| 2.3.7.11.31.37.43 {{monzo| 1 -1 1 1 1 -1 -1 }}
| 0.36268
| [[Francium]] (2025)
|-
| [[5720/5719|Halweaverisma]]
| Fothunutholoruyoma
| 43u19u3o1oryM
| 5720/5719
| 2.5.7.11.13.19.43 {{monzo| 3 1 -1 1 1 -1 -1 }}
| 0.30269
| [[Francium]] (2025)
|-
| [[7225/7224|Huntressisma]]
| Fothusosoruyoyoma
| 43u17ooryyM
| 7225/7224
| 2.3.5.7.17.43 {{monzo| -3 -1 2 -1 2 -1 }}
| 0.23963
| [[Francium]] (2025)
|-
| [[7956/7955|Yajinisma]]
| Fothuthisusothoguma
| 43u37u17o3ogM
| 7956/7955
| 2.3.5.13.17.37.43 {{monzo| 2 2 -1 1 1 -1 -1 }}
| 0.21761
| [[Francium]] (2025)
|-
| [[9504/9503|Lionelisma]]
| Fothusuthuloma
| 43u17u3u1oM
| 9504/9503
| 2.3.11.13.17.43 {{monzo| 5 3 1 -1 -1 -1 }}
| 0.18217
| [[Francium]] (2025)
|-
| [[9633/9632|Coturisma]]
| Fothunothothoruma
| 43u19o3oorM
| 9633/9632
| 2.3.7.13.19.43 {{monzo| -5 1 -1 2 1 -1 }}
| 0.17973
| [[Francium]] (2025)
|-
| [[Girardisma]]
| Fothunoloyoyoma
| 43u19o1oyyM
| 10450/10449
| 2.3.5.11.19.43 {{monzo| 1 -5 2 1 1 -1 }}
| 0.16567
| [[Francium]] (2024)
|-
| [[Kaguyisma]]
| Fothutwethusoluyoma
| 43u23u17o1uyM
| 10880/10879
| 2.5.11.17.23.43 {{monzo| 7 1 -1 1 -1 -1 }}
| 0.15913
| [[Francium]] (2024)
|-
| [[Manheimisma]]
| Fothutwenosuloloyoma
| 43u29o17u1ooyM
| 17545/17544
| 2.3.5.11.17.29.43 {{monzo| -3 -1 1 2 -1 1 -1 }}
| 0.098677
| [[Francium]] (2024)
|-
| [[Dimigenes comma]]
| Fothosepyoma
| 43o7yM
| 3359375/3359232
| 2.3.5.43 {{monzo| -9 -8 7 1 }}
| 0.073696
| [[User:Akselai|Akselai]] (2024)
|-
| [[27048/27047|Jangongisma]]
| Fothuthisutwethosuzozoma
| 43u37u23o17uzzM
| 27048/27047
| 2.3.7.17.23.37.43 {{monzo| 3 1 2 -1 1 -1 -1 }}
| 0.064007
| [[Francium]] (2025)
|-
| [[29241/29240|Locquirecisma]]
| Fothunonosuguma
| 43u19oo17ugM
| 29241/29240
| 2.3.5.17.19.43 {{monzo| -3 4 -1 -1 2 -1 }}
| 0.059207
| [[Francium]] (2025)
|}

=== Higher-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[7936/7921|Lily comma]]
|
| 89uu31oM
| 7936/7921
| 2.31.89 {{monzo| 8 1 -2 }}
| 3.2753
| [[Budjarn Lambeth]] (2023)
|-
| [[Molar comma]]
|
| 3(89o)1u3gM
| 704969/704000
| 2.5.11.89 {{monzo| -9 -3 -1 3 }}
| 2.3813
| [[Budjarn Lambeth]] (2023)
|-
| [[750/749|Ancient Chinese tempering comma]]{{Clarify}}
|
| 107ur3yM
| 750/749
| 2.3.5.7.107 {{monzo| 1 1 3 -1 -1 }}
| 2.3099
|
|-
| [[1176/1175|Lucidorisma]]
| Fosubizoguma
| 47uzzggM
| 1176/1175
| 2.3.5.7.47 {{monzo| 3 1 -2 2 -1 }}
| 1.4728
| [[Francium]] (2025)
|-
| [[2520/2519|Platonisma]]
|
| 229u1uzyM
| 2520/2519
| 2.3.5.7.11.229 {{monzo| 3 2 1 1 -1 -1 }}
| 0.68713
| [[User:Eliora|Eliora]] (2022)
|-
| [[5041/5040|Third brown pair comma]], 19th highly compositema
|
| 71oorgM
| 5041/5040
| 2.3.5.7.71 {{monzo| -4 -2 -1 -1 2 }}
| 0.34347
| [[Eliora]] (2022)
|-
| [[7777/7776|Pulsar comma]]
|
| 101o1ozM
| 7777/7776
| 2.3.7.11.101 {{monzo| -5 -5 1 1 1 }}
| 0.22262
| [[Eliora]] (2022)
|-
| Palimilli
|
| 1003001o23o1uM
| 23069023 / 23068672
| 2.11.23.1003001 {{monzo| -21 -1 1 1 }}
| 0.026341
| [[User:Kaiveran|Kaiveran]] (2021)
|-
| [[Sulbasutrisma]]
|
| 577oo17uuM
| 332929/332928
| 2.3.17.577 {{monzo| -7 -2 -2 2 }}
| 0.0052000
| [[User:2^67-1|Cole]] (2024)
|-
| [[Zudilisma]]
|
| 4L397u23urM
| 68630377364883 / 68630356164608
| 2.3.7.23.397 {{monzo| -30 29 -1 -1 -1 }}
| 0.00053479
| [[Budjarn Lambeth]] (2023)
|-
| [[Borcherdsma]]
|
| 71u3(59u)47o31o 29o19o3u1uur5yM
| 160561400000 / 160561399999
| 2.5.7.11.13.19.29.31.47.59.71 {{monzo| 6 5 -1 -2 -1 1 1 1 1 -3 -1 }}
| 1.0783 × 10-8
| [[User:Akselai|Akselai]] (2024)
|}

== Notes ==
<references />

== See also ==
* [[Comma and diesis]]

[[Category:Unnoticeable commas| ]] 
[[Category:Lists of commas]]
{{Todo|complete table|add color name}}

135/128

2026-04-29T04:12:49Z

TallKite: updated the comma color name

{{Infobox Interval
| Name = ptolemaic chromatic semitone, major limma, major chroma, mavila comma
| Color name = Ly1, layo unison, LybM, layobima
| Sound = jid_135_128_pluck_adu_dr220.mp3
| Comma = yes
}}

The [[5-limit]] interval '''135/128''', about 92.2 [[cent]]s in size, is called the '''ptolemaic chromatic semitone''', the '''major limma''' or the '''major chroma'''. It is a [[syntonic comma]] away from the Pythagorean chromatic semitone [[2187/2048]], and so is tuned justly in [[1/7-comma meantone]]. Flattening by another syntonic comma reaches the even simpler [[25/24]]. In regular 5-limit diatonic systems, it is the chromatic semitone that compliments [[16/15]], as the two semitones add up to [[9/8]].

== Temperaments ==
If 135/128 is treated as a comma to be [[tempering out|tempered out]], it may be called the '''mavila comma'''. It represents the difference between three [[4/3|perfect fourth]]s and a [[5/4|just major third]] (plus an [[octave]]), or the difference between [[9/8]] and [[16/15]]. Tempering it out results in the [[mavila]] temperament.

135/128 is very close to one step of [[13edo]], in fact being a {{w|Continued fraction|semiconvergent}}. [[Aluminium]] temperament realizes this through a regular temperament lens.

== See also ==
* [[256/135]] – its [[octave complement]]
* [[Aluminium comma]] - the difference between a stack of 13 instances of this interval and [[2/1]]
* [[Gallery of just intervals]]
* [[Medium comma]]
* [[:File:Ji-135-128-csound-foscil-220hz.mp3]] – another sound example

[[Category:Semitone]]
[[Category:Chroma]]
[[Category:Mavila]]
[[Category:Meantone]]
[[Category:Commas named after musical traditions]]

Garischisma

2026-04-29T04:10:51Z

TallKite: updated the comma color name

{{Infobox Interval
| Ratio = 33554432/33480783
| Name = garischisma, septimal schisma
| Color name = ssr2, sasaru 2nd, ssrM, sasaruma
| Comma = yes
}}

The '''garischisma''' or '''septimal schisma''' is a very [[small comma|small]] [[7-limit]] [[comma]] measuring about 3.80 [[cent]]s. It can be considered the septimal counterpart of the [[schisma]]: whereas the schisma is the difference between the [[Pythagorean comma]] and the [[syntonic comma]], the garischisma is the difference between the [[septimal comma]] and the Pythagorean comma. Equivalently, it can be described as the amount by which a stack of two [[2187/2048|apotomes]] falls short of a [[8/7|septimal whole tone]]. It is also the difference between the [[aberschisma]] and the schisma in the full 7-limit.

It factors into the following [[11-limit|11-]] and [[13-limit]] commas:
* The [[19712/19683|symbiotic comma]] and [[131072/130977|olympia]];
* The [[wilschisma]] and the [[minisma]].

== Temperaments ==
[[Tempering out]] this comma in the full 7-limit leads to the rank-3 '''garischismic''' temperament, or in the [[2.3.7 subgroup]], the rank-2 '''gary''' temperament. See [[Garischismic family]] for the [[temperament families and clans|family]] of rank-3 temperaments where it is tempered out. See [[Garischismic clan]] for the [[temperament families and clans|clan]] of rank-2 temperaments where it is tempered out.

== Etymology ==
Its name comes from the temperament [[garibaldi]] and the schisma, as it is a similarly sized interval that alongside it, defines garibaldi.

[[Category:Garischismic]]
[[Category:Commas named by combining multiple temperament names]]

Schisma

2026-04-29T04:09:52Z

TallKite: updated the comma color name

{{interwiki
| de = 32805/32768
| en = 32805/32768
| es =
| ja =
}}
{{Infobox Interval
| Ratio = 32805/32768
| Name = schisma
| Color name = LyM, layoma
| Comma = yes
}}
{{Wikipedia| Schisma }}

The '''schisma''', '''32805/32768''', is a small interval about 2 [[cent]]s. It arises as the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])4/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])3/([[64/45]]).

== History and etymology ==
''Schisma'' is a borrowing of Ancient Greek, meaning "split". The term was first used by [[Boethius]] (6th century), in his ''De institutione musica'', using it to refer to half of the [[Pythagorean comma]]. The modern sense was introduced by [[Helmholtz]]' ''On the Sensations of Tone'', in particular the translation by [[Alexander Ellis]], where it is spelled ''skhisma''. Since it is extremely close to the [[superparticular]] ratio [[887/886]] {{nowrap|(2-1 443-1 887)}}, it is used interchangably with this interval in some of Helmholtz' writing.

== Temperaments ==
{{main|Schismatic family}}
Tempering out this comma gives a [[5-limit]] microtemperament called [[schismic|schismatic, schismic or helmholtz]], which if extended to larger [[subgroup]]s leads to the [[schismatic family]] of temperaments.

== Other intervals ==

Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the [[Pythagorean comma]] and [[septimal comma]] is called the [[septimal schisma]]. Other examples are [[undevicesimal schisma]] and [[Alpharabian schisma]].

== Trivia ==
The schisma explains how the greatly composite numbers 1048576 (220) and 104976 (184) look alike in decimal. The largest common power of two between these numbers is 25, (when 1049760 is written to equalize) and when reduced by that, 1049760/1048576 becomes 32805/32768.

It's also very close in size—about 0.0013¢ off—from the difference between 3/2 and 7\12, which is about 1.9550009¢. Tempering out this difference instead results in [[atomic]], an extremely high accuracy temperament.

== See also ==
* [[Pythagorean tuning]]
* [[Unnoticeable comma]]

[[Category:Schismic]]
[[Category:Commas named for their regular temperament properties]]

5120/5103

2026-04-29T04:08:31Z

TallKite: updated the comma color name

{{Infobox Interval
| Name = argent comma, 5/7-kleisma
| Color name = sry1, saruyo 1sn, sryM, saruyoma
| Comma = yes
}}
'''5120/5103''', the '''argent comma''', '''5/7-kleisma''', '''hemifamity comma''', or '''aberschisma''', is a [[small comma|small]] [[7-limit]] [[comma]] measuring about {{frac|5|3|4}}{{cent}}. It is the difference between [[7/5]] (small septimal tritone) and [[1024/729]] (Pythagorean diminished fifth), or between [[10/7]] (large septimal tritone) and [[729/512]] (Pythagorean augmented fourth). It is also the difference between [[15/14]] and [[2187/2048]], between [[21/20]] and [[256/243]], between [[80/63]] and [[81/64]], etc., hence, between the septimal comma of [[64/63]] and the syntonic comma of [[81/80]], as well as between [[schisma|32805/32768]], the schisma, and [[225/224]], the marvel comma.

== Temperaments ==
Tempering it out leads to the rank-3 [[hemifamity]] temperament, which splits the septimal quartertone [[36/35]] into two equal parts, each representing 81/80~64/63. Rank-2 temperaments that temper it out include [[hemififths]], [[amity]], and [[garibaldi]] to name three. [[Edo]]s that temper it out include [[12edo|12]], {{EDOs| 29, 34, 41, 46, 53, 58, 87, 94, 99, 111, 140, 145, 152, 239, 292, 391 }}, etc. See [[Hemifamity family]] for the rank-3 family where it is tempered out. See [[Hemifamity temperaments]] for a collection of rank-2 temperaments where it is tempered out.

== Sagittal notation ==
In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | !( }} and is called the '''7/5 kleisma''', or '''7/5k''' for short, because the simplest interval it notates is 7/5, as for example in B-F{{nbhsp}}{{sagittal | !( }}. The upward version is called '''5/7k''' or '''7/5k up''' and is represented by {{sagittal| |( }}.

== Etymology ==
The name ''hemifamity'' was given by [[Gene Ward Smith]] in 2005 as a contraction of ''hemififths'' and ''amity''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12900.html Yahoo! Tuning Group | ''Seven limit comma names from pairs of temperament names'']</ref>.

The name ''aberschisma'', was coined by [[User:Tristanbay|Tristan Bay]] in 2024 in reference to [[groundfault]]'s [[aberrismic theory]], since 5120/5103 is the difference between two of the most common aberrismas in just intonation (81/80 and 64/63).

In 2025, [[User:FloraC|FloraC]] et al. gave this comma the name ''argent comma'' due to the fifth size from the [[argent temperament]] being essentially optimal for tuning the full 7-limit temperament (and the 2.3.7/5 subgroup temperament) that tempers only it out.

== Notes ==

[[Category:Hemifamity]]

250/243

2026-04-29T04:07:41Z

TallKite: updated the comma color name

{{Interwiki
| de = 250/243
| en = 250/243
| es =
| ja =
}}
{{Infobox Interval
| Name = porcupine comma, maximal diesis
| Color name = y31, triyo 1sn, y3M, triyoma
| Comma = yes
}}
'''250/243''' is known as the '''porcupine comma''' or the '''maximal diesis'''. Measuring about 49{{cent}}, it is a [[medium comma]]. It is the amount by which two [[10/9|minor whole tones]] exceed a minor third, that is, (10/9)2/(6/5). It is also the difference between [[25/24]] and [[81/80]], the two smallest [[5-limit]] [[superparticular]] ratios, and between three syntonic commas and the [[2187/2048|Pythagorean apotome]], putting it on the [[Syntonic–chromatic equivalence continuum]].

== Temperaments ==
Tempering it out leads to the [[5-limit]] [[porcupine]] temperament. See [[porcupine family]] for the family of rank-2 temperaments where it is tempered out.

== Approximation ==
250/243 is very close to one step of [[24edo]], which is the quarter tone that is exactly the half of [[12edo]] semitone. Therefore, if 250/243 is not tempered and instead is treated as an identity where it is equated with 1/24th of the octave, it serves as the period in the [[chromium]] temperament. Thus in the framework of this temperament and the tuning systems associated with it, [[Eliora]] proposes the name ''chromium quartertone''.

[[Category:Porcupine]]
[[Category:Commas named after compositions]]

49/48

2026-04-29T04:05:49Z

TallKite: updated the comma color name

{{interwiki
| de = 49/48
| en = 49/48
| es =
| ja =
}}
{{Infobox Interval
| Name = large septimal diesis, large septimal sixth-tone, slendro diesis, semaphore comma, semaphoresma
| Color name = zz2, zozo 2nd, zzM, zozoma
| Sound = Ji-49-48-csound-foscil-220hz.mp3
| Comma = yes
}}
{{Wikipedia|Septimal diesis}}
'''49/48''', the '''large septimal diesis''' (a.k.a. '''large septimal sixth-tone''' or '''slendro diesis'''), is a [[7-limit]] [[superparticular]] ratio spanning the small distance between a subminor third ([[7/6]]) and a supermajor second ([[8/7]]) or between the supermajor sixth ([[12/7]]) and the harmonic seventh ([[7/4]]). Measuring about 35.7{{cent}}, it is a [[medium comma]]; however, in classical Western music, this interval is not known as a [[comma]] as it is not tempered out in [[12edo|12tet]].

This interval has a function similar to [[25/24]] in that it separates the [[7/6]] and [[8/7]] intervals in a [[6:7:8]] triad, similarly to how [[25/24]] separates [[5/4]] and [[6/5]] in a [[4:5:6]] triad. The 6:7:8 triad consists of odd [[harmonic]]s [[1/1|1]], [[3/1|3]], and [[7/1|7]] [[octave reduced]] to span the [[4/3|perfect fourth]], while the 4:5:6 triad consists of odd harmonics 1, 3, and 5 octave reduced to span the [[3/2|perfect fifth]]. In that regard, tempering out 49/48 can be considered a form of [[exotemperament|exotempering]] that neutralizes the 6:7:8 chord and equates it with its inverse [[21:24:28|1/(8:7:6)]], just like how [[dicot]], which tempers out 25/24, neutralizes the 4:5:6 chord and equates it with its inverse [[10:12:15|1/(6:5:4)]].

== Temperaments ==
49/48 is [[tempered out]] in [[15edo]] and [[19edo]], where 7/6 and 8/7 are equated, and the fourth is split in a perfect half. 3/1 is also split into two [[7/4]]~[[12/7]]'s. In the 2.3.7 [[subgroup]], this is known as the [[semaphore]] temperament, and the comma is thus known as the '''semaphore comma''' or '''semaphoresma'''.

''It cannot be tempered out if all of the consonances of the 7-odd-limit are distinct'', but it ''can'' be equated with other commas; for example:
* (49/48)/([[81/80]]) = [[245/243]]
* (49/48)/([[64/63]]) = [[1029/1024]]
* (49/48)/([[3125/3072]]) = [[3136/3125]]
* (49/48)/([[50/49]]) = [[2401/2400]]
* ([[128/125]])/(49/48) = [[6144/6125]]
* ([[36/35]])/(49/48) = [[1728/1715]]

See [[Semaphoresmic family]] for the rank-3 family where it is tempered out. See [[Semaphoresmic clan]] for the rank-2 clan where it is tempered out.

== Approximations ==
{{interval edo approximation|min_edo=5}}

== See also ==
* [[Medium comma]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:Semaphore]]
[[Category:Semaphoresmic]]
[[Category:Commas named for how they divide the fourth]]
[[Category:Commas named after musical traditions]]

2048/2025

2026-04-29T04:04:00Z

TallKite: updated the comma color name

{{interwiki
| de = 2048/2025
| en = diaschisma
| es =
| ja =
}}
{{Infobox Interval
| Ratio = 2048/2025
| Name = diaschisma
| Color name = sgg2, sagugu 2nd, sggM, saguguma
| Comma = yes
}}
{{Wikipedia| Diaschisma }}

'''2048/2025''', the '''diaschisma''', a [[comma]] of 19.553 [[cent]]s, is the size of a [[pythagorean comma]] minus two [[schisma]]s, from which it derives its name. It may also be defined as the difference between four [[3/2|just perfect fifths]] plus two [[5/4|just major thirds]] and three octaves, the difference between a Pythagorean minor seventh ([[16/9]]) and a just augmented sixth ([[225/128]]), as the difference between two classic diatonic semitones ([[16/15]]) and the major whole tone ([[9/8]]), that is, (16/15)2/(9/8), or as the difference between the 5-limit tritone [[45/32]] and its octave complement [[64/45]].

== Temperaments ==
Tempering it out leads to the [[diaschismic family]] of temperaments. See [[Diaschismic family]] for the rank-2 temperament family where it is tempered out, especially [[Srutal archagall]] which takes advantage of this comma's relation to [[256/255]] and [[289/288]] to make it as efficient and natural as possible. See [[Diaschismic rank-3 family]] for the rank-3 temperament family where it is tempered out.

=== Significance ===
Pařízek's diaschisma pump [https://web.archive.org/web/20201127014513/http://micro.soonlabel.com/petr_parizek/pp_pump_examples/pump_1.ogg play] [https://luphoria.com/xenpaper/#(osc%3Asawtooth8)(bpm%3A90)_%7Br220hz%7D%0A%23_This_is_a_transcription_of_Pařízek's_diaschisma_comma_pump.png%0A%23_Just_intonation%2C_with_pitch_drifting.%0A2%3A3%3A5--%7Br5%2F4%7D_2%3A3%3A4-_%7Br%603%2F2%7D_2%3A3%3A5_%7Br%603%2F2%7D_2%3A5%3A6--_%7Br5%2F4%7D_2%3A3%3A5-_%7Br%603%2F2%7D_2%3A5%3A6_%7Br3%2F2%7D_%0A2%3A3%3A5--%7Br5%2F4%7D_2%3A3%3A4-_%7Br%603%2F2%7D_2%3A3%3A5_%7Br%603%2F2%7D_2%3A5%3A6--_%7Br5%2F4%7D_2%3A3%3A5-_%7Br%603%2F2%7D_2%3A5%3A6_%7Br3%2F2%7D%0A2%3A3%3A5--%7Br5%2F4%7D_2%3A3%3A4-_%7Br%603%2F2%7D_2%3A3%3A5_%7Br%603%2F2%7D_2%3A5%3A6--_%7Br5%2F4%7D_2%3A3%3A5-_%7Br%603%2F2%7D_2%3A5%3A6_%7Br3%2F2%7D_%0A2%3A3%3A5--%7Br5%2F4%7D_2%3A3%3A4-_%7Br%603%2F2%7D_2%3A3%3A5_%7Br%603%2F2%7D_2%3A5%3A6--_%7Br5%2F4%7D_2%3A3%3A5-_%7Br%603%2F2%7D_2%3A5%3A6_%7Br3%2F2%7D_%0A2%3A3%3A5--%0A%23_We_are_now_4_diaschismas_from_where_we_started.%0A...%0A2%3A3%3A5-_%7Br220hz%7D_2%3A3%3A5-%0A..%0A%23_12edo%2C_the_comma_is_tempered_out_therefore_there_is_no_pitch_drifting.%0A%0A%7B12edo%7D%0A%5B0%2C7%2C16%5D--_%7Br4%7D_%5B0%2C7%2C12%5D-_%7Br%607%7D_%5B0%2C7%2C16%5D_%7Br%607%7D_%5B0%2C16%2C19%5D--_%7Br4%7D_%5B0%2C7%2C16%5D-_%7Br%607%7D_%5B0%2C16%2C19%5D_%7Br7%7D%5B0%2C7%2C16%5D--_%7Br4%7D_%5B0%2C7%2C12%5D-_%7Br%607%7D_%5B0%2C7%2C16%5D_%7Br%607%7D_%5B0%2C16%2C19%5D--_%7Br4%7D_%5B0%2C7%2C16%5D-_%7Br%607%7D_%5B0%2C16%2C19%5D_%7Br7%7D%0A%5B0%2C7%2C16%5D--_%7Br4%7D_%5B0%2C7%2C12%5D-_%7Br%607%7D_%5B0%2C7%2C16%5D_%7Br%607%7D_%5B0%2C16%2C19%5D--_%7Br4%7D_%5B0%2C7%2C16%5D-_%7Br%607%7D_%5B0%2C16%2C19%5D_%7Br7%7D%0A%5B0%2C7%2C16%5D--_%7Br4%7D_%5B0%2C7%2C12%5D-_%7Br%607%7D_%5B0%2C7%2C16%5D_%7Br%607%7D_%5B0%2C16%2C19%5D--_%7Br4%7D_%5B0%2C7%2C16%5D-_%7Br%607%7D_%5B0%2C16%2C19%5D_%7Br7%7D%0A%5B0%2C7%2C16%5D--_%0A%23The_root_hasn't_changed_from_where_we_started. xenpaper] – a [[comma pump]] progression that requires the diaschisma to be tempered out (i.e. equates two notes that are separated by a diaschisma).
[[File:Parizek diaschisma comma pump.png|thumb|Pařízek's diaschisma comma pump example in JI (notated with HEJI) and 12edo.]]
In the progression, the bassline moves as follows:
D (up 5/4) F# (down 4/3) C# (down 4/3) G# (up 5/4) C (down 4/3) G (up 3/2) D (*).
If we ignore octaves,
* the first three steps (cumulatively D to G#) moves us up by the tritone [[45/32]];
* the last three steps (cumulatively G# to D) are the same moves as the first three, moving up by the tritone 45/32 a second time.
In pure JI, since 45/32 is flat of 600c, each cycle of this progression (*) would shift the tonic down by the diaschisma, which is (2/1) / (45/32)2 = 2048/2025. The fact that the D we come back to is exactly the same as the first D, indicates that that their difference, the diaschisma, is tempered out. To carry out this tempering-out (assuming octaves are kept pure), the basic 5-limit intervals, 5/4 and 3/2, are adjusted, or tempered, such that a stack of two 45/32 tritones is sharpened up to the octave 2/1.

This also tells us that if a system tempers out the diaschisma, it has an interval that is equal to exactly half of an octave‚ namely the tempered 45/32 tritone. Thus all edos (such as [[12edo]], [[22edo]], [[34edo]] and [[46edo]]) and mos scale structures (such as the mos scales of [[diaschismic family|diaschismic]] and [[pajara]]) that temper out the diaschisma split the octave into two equal parts; in particular, all diaschismic edos are even-numbered edos.

== Etymology ==
The modern sense of the term is due to {{w|Hermann von Helmholtz}} and {{w|Alexander John Ellis}} in 1875 when the English translation of ''{{w|Sensations of Tone}}'' was first published.

2048/2025 was earlier referred to as the ''diminished comma'' and ''comma minor'' by {{w|Jean-Philippe Rameau}} (1683-1764). However in modern (1875 onwards) music theory the term ''diaschisma'' is almost always used.

There have been other intervals besides 2048/2025 that were called ''diaschisma'' in the [[Ancient Greek]], Roman and [[historical temperaments|medieval]] periods, however those alternate meanings of the word fell out of use centuries ago.

== See also ==
* [[Small comma]]

[[Category:Diaschismic]]
[[Category:Commas named for their regular temperament properties]]

225/224

2026-04-29T04:01:52Z

TallKite: updated the comma color name

{{interwiki
| de = 225/224
| en = 225/224
| es =
| ja =
}}
{{Infobox Interval
| Name = marvel comma, septimal kleisma
| Color name = ryy-2, ruyoyo negative 2nd, ryyM, ruyoyoma
| Comma = yes
}}
{{Wikipedia|Septimal kleisma}}

The interval of '''225/224''', the '''marvel comma''', otherwise known as the '''septimal kleisma''', is a [[7-limit]] [[superparticular]] [[comma]]. It pops up as the difference between a 7-limit ratio and a 5-limit ratio. For example, it's the difference between [[16/15]] and [[15/14]], and between [[7/5]] and [[45/32]]. Moreover, it can be seen as the amount by which [[8/7]] exceeds a stack of two {{nowrap|[[16/15]]'s}}, or as the amount by which a stack of two {{nowrap|[[5/4]]'s}} exceeds [[14/9]]. It's also the difference between [[75/64]] and [[7/6]], and between [[25/24]], the classical chromatic semitone, and [[28/27]], the septimal third-tone.

As a comma with a single power of 7 in it, it is tremendously useful in terms of bringing prime 7 into the framework of [[5-limit]] [[just intonation|JI]]; tempering it out maps [[7/4]] to the classic augmented sixth, [[225/128]] and enables all of the aforementioned equivalences.

In terms of commas, it is the difference between [[81/80]] and [[126/125]] and is tempered out alongside these two commas in [[septimal meantone]]. In the 11-limit it factors neatly into ([[385/384]])([[540/539]]), and in the 13-limit, ([[351/350]])([[625/624]]) or ([[325/324]])([[729/728]]).

== Temperaments ==
Tempering out this comma alone in the 7-limit leads to the [[marvel]] temperament, which enables [[marvel chords]]. See [[Marvel family]] for the family of rank-3 temperaments where it is tempered out. See [[Marvel temperaments]] for a collection of rank-2 temperaments where it is tempered out.

== Approximation ==
If we do not temper out this interval and instead repeatedly stack (and octave-reduce) it, we almost return to the starting point at the 311th step, meaning [[311edo]] is a [[consistent circle]] of 225/224's. Note that this is not true for 226/225 or 224/223, the adjacent superparticulars, as they accumulate too much error to close into a circle in 311edo.

== Etymology ==
''Marvel comma'' was named after the corresponding temperament, marvel, which was named by [[Gene Ward Smith]] in 2002–2003.

== See also ==
* [[Marvel]]
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Marvel]]
[[Category:Commas with unknown etymology]]

64/63

2026-04-29T04:00:48Z

TallKite: updated the comma color name

{{interwiki
| de = 64/63
| en = 64/63
}}
{{Infobox Interval
| Name = septimal comma, Archytas' comma
| Color name = r1, ru unison, rM, ruma
| Sound = Ji-64-63-csound-foscil-220hz.mp3
| Comma = yes
}}
{{Wikipedia|Septimal comma}}
'''64/63''', the '''septimal comma''' (also '''Archytas' comma''', or more simply and systematically the '''archytas comma''' or '''archy comma'''), is a [[small comma|small]] [[7-limit]] [[superparticular]] [[comma]] which separates [[9/8]] and [[8/7]] and has the eighth square number as a numerator. It can be considered the [[2.3.7 subgroup|2.3.7-]][[subgroup]] equivalent of the [[syntonic comma]], and seperates complex pythagorean intervals from simpler 7-limit ones. For example, it is the difference between [[32/27]] and [[7/6]], and the difference between [[81/64]] and [[9/7]]. Since its numerator is a power of 2, it is a [[Mersenne comma]].

== Temperaments ==
[[Tempering out]] this comma leads to [[superpyth]] temperament (sometimes called ''archy'' in the 2.3.7-subgroup), which equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]]. This means that the just dominant seventh chord, [[36:45:54:64|1–5/4–3/2–16/9]], and the harmonic seventh chord, [[4:5:6:7|1–5/4–3/2–7/4]], are equated to the same chord. Equal temperaments tempering out 64/63 include {{EDOs| 12, 15, 17, 22, 27, 37, 49 and 59 }}.

Archytas' comma is similar to Didymus' or the syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths [[octave reduction|octave reduced]] equal a relatively consonant major third. In the case of 81/80, the major third is [[5/4]], while with Archytas' comma, the major third is [[9/7]].

If one is using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both [[9/8]] and [[8/7]]: if a stack of four fifths (octave-reduced) reaches the interval 9/7, and a stack of two fifths reaches 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equated, however, as a result of the generation process.

See [[Archytas family]] for the family of rank-3 temperaments where it is tempered out. See [[Archytas clan]] for the clan of rank-2 temperaments where it is tempered out.

== Comma pumps ==
The septimal version of the common vi–ii–V–I progression, which uses the 6:7:9 subminor and 14:18:21 supermajor triads, requires that 64/63 be tempered out in order to avoid shifting the root. If 64/63 is not tempered out and intervals are kept pure, the root in the final I chord will be 64/63 higher than the root in the vi chord.
{{todo|add sound example}}

== Notation ==
This interval is significant in the [[Functional Just System]] and [[Helmholtz–Ellis notation]] as the septimal formal comma which translates a Pythagorean interval to a nearby septimal interval.

=== Sagittal notation ===
In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | !) }} and is called the '''7 comma''', or '''7C''' for short, because the simplest interval it notates is 7/1 (equiv. 7/4), as for example in G–F{{nbhsp}}{{sagittal | !) }}. The upward version is called '''1/7C''' or '''7C up''' and is represented by {{sagittal| |) }}.

== Approximation ==
If one wants to treat Archytas' comma as a musical interval in its own right as opposed to tempering it out, you will find that it acts as a sort of chroma – specifically, it functions as a septimal equivalent of [[55/54]], from which it differs by a [[385/384|keenanisma]], or of [[56/55]], from which it differs by a [[441/440|werckisma]]. In addition, its incredible proximity to 1/44th of the octave – to the point where the [[septimal ruthenia|44-64/63 comma]] is tempered out in edos as large as tens of thousands – enables the tuning of [[ruthenium]] temperament. As a result, the major second of [[22edo]] is a good approximation to [[17/15]], due to it being the [[mediant]] of [[9/8]] and [[8/7]], so that the ~7:8:9 chord is much more accurately a 17/15–17/15 chord, with the outer interval as 9/7, by tempering out [[2025/2023]].

== See also ==
* [[Septimal comma]] (disambiguation page)
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Commas named for their regular temperament properties]]
[[Category:Commas named after polymaths]]

3200/3159

2026-04-29T03:59:03Z

TallKite:

{{Infobox interval| Color name = s3uyy1, sathuyoyo unison, s3uyyM, sathuyoyoma}}
'''3200/3159''' is the amount by which the [[320/243]] grave fourth exceeds the [[13/10]] ultramajor third.

== Temperaments ==
It is tempered out in the 2.3.5.13 version of [[tetracot]] temperament, among others.

81/80

2026-04-29T03:55:36Z

TallKite:

{{Interwiki
| en = 81/80
| de = 81/80
}}
{{Infobox Interval
| Name = syntonic comma, Didymus' comma, meantone comma, Ptolemaic comma
| Color name = g1, gu unison, gM, guma
| Comma = yes
| Sound = audacity pluck 81 80.wav
}}
{{Wikipedia|Syntonic comma}}

The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is the difference between many [[3-limit]] and [[5-limit]] ratios in [[just intonation]]. Adding or subtracting this comma to/from any complex 3-limit [[ratio]] (such as [[32/27]] or [[81/64]]) creates a 5-limit ratio with a much lower odd-limit (such as [[6/5]] or [[5/4]]). Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. For example, the pythagorean major triad, [[64:81:96]], is quite dissonant, but flattening the 81/64 major third by 81/80 leads to the much more consonant [[4:5:6]] chord, with a 5/4 major third in place of 81/64. However, adding/subtracting this comma to/from the [[4/3|perfect fourth]], [[3/2|fifth]], or [[2/1|octave]] creates a wolf interval of [[odd limit]] 27 or higher, such as the [[40/27]] wolf fifth. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to make intervals and chords pure in one key will create wolf intervals in others; thus, for those who wish to avoid such wolves in composition, [[tempering out]] 81/80 is desirable. This leads to [[meantone]] temperament, which equates the complex pythagorean intervals with the simpler 5-limit ones. This also equates [[10/9]] with [[9/8]], giving a tuning for the [[tone|whole tone]] which is intermediate between them; hence the name "meantone".

81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]], and in fact 81/80 is a [[square superparticular]], being the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]].

== Comma pumps ==
The familiar vi–ii–V–I progression requires that 81/80 be tempered out in order for the root in the vi chord to be the same as the root in the final I chord. If 81/80 is not tempered out, the new root will be 81/80 lower than the original root.

A passage ([https://youtu.be/DO7yTiM-YJk?si=e4wVU4IlbITCAaNG&t=325 listen]) from [[Ben Johnston]]'s 9th string quartet, near the end of movement 1, makes a sudden and prominent use of the 81/80 comma, which demonstrates how a simple progression with held common tones can quickly lead to severe interference [[Beat|beating]], rupturing the diatonic collection routinely associated with the [[5-limit]] and exposing "C major" as anything but simple.

[[Monroe Golden]]'s ''Incongruity'' uses just-intonation chord progressions that exploit this comma<ref>[http://untwelve.org/interviews/golden UnTwelve's interview to Monroe Golden]</ref>.

[https://x.com/its_adamneely/status/1249700624003989508 Adam Neely's harmonization] of ''the licc'' pumps upward by 81/80 every measure. After 9 iterations, D modulates nearly to E.

== Temperaments ==
See [[Meantone family #Extensions]] for a discussion on possible extensions.

== Relations to other 5-limit intervals ==
81/80 is the difference between a large number of intervals of the 5-limit, so that if tempered, it simplifies the structure of the 5-limit drastically. For some differences in higher limits, see [[#Relations to other superparticular ratios]]. A few important ones are that 81/80 is:
* The amount by which [[2187/2048]] exceeds [[135/128]].
* The amount by which [[25/24]] exceeds [[250/243]].
* The amount by which [[135/128]] exceeds [[25/24]].
* The amount by which [[648/625]] exceeds [[128/125]].
* The amount by which [[128/125]] exceeds [[2048/2025]].
* The amount by which [[27/25]] exceeds [[16/15]].
* The amount by which [[16/15]] exceeds [[256/243]].

== Notation ==
This interval is significant in the [[Functional Just System]] and [[Helmholtz-Ellis notation]] as the classical (5-limit) formal comma which translates a Pythagorean interval to a nearby classical interval.

=== Ben Johnston's notation ===
In [[Ben Johnston's notation]], this interval is denoted with "+" and its reciprocal with "-".

=== Sagittal notation ===
In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | \! }} and is called the '''5 comma''', or '''5C''' for short, because the simplest interval it notates is 5/1 (equiv. 5/4), as for example in C–E{{nbhsp}}{{sagittal | \! }}. The upward version is called '''1/5C''' or '''5C up''' and is represented by {{sagittal| /| }}.

== Approximation ==
If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out, one can easily use it in melodies as either an {{w|appoggiatura}}, an {{w|acciaccatura}}, or a quick passing tone. It is also very easy to exploit in [[comma pump]] modulations, as among the [[Meantone comma pump examples|known examples]] of this kind of thing are familiar-sounding chord progressions. Furthermore, not tempering out 81/80 both allows wolf intervals like [[40/27]] and [[27/20]] to be deliberately exploited as dissonances to be resolved, and it also allows one to contrast intervals like 5/4 and [[81/64]]. The [[barium]] temperament exploits the comma by setting it equal to exactly 1/56th of the octave, thus tempering out the [[barium comma]] ({{monzo| -225 224 -56 }}).

== Relations to other superparticular ratios ==
Superparticular ratios, like 81/80, can be expressed as products or quotients of other superparticular ratios. Following is a list of such representations ''r''1 ⋅ ''r''2 or ''r''2 / ''r''1 of 81/80, where ''r''1 and ''r''2 are other superparticular ratios.

Names in brackets refer to 7-limit [[meantone family|meantone extensions]], or 11-limit rank-3 temperaments from the [[didymus rank-3 family]] that temper out the respective ratios as commas. (Cases where the meantone comma is expressed as a difference, rather than a product, usually correspond to [[exotemperament]]s.)

{| class="wikitable mw-collapsible mw-collapsed"
|+ Relations between 81/80 and other superparticular ratios
|-
! Limit
! ''r''1 ⋅ ''r''2
! ''r''2 / ''r''1
|-
| 5
| -
| 9/8 ⋅ 9/10
|-
| 7
| 126/125 ⋅ 225/224 (septimal meantone)
| 21/20 ⋅ 27/28 (sharptone), 36/35 ⋅ 63/64 (dominant)
|-
| 11
| 99/98 ⋅ 441/440 (euterpe), 121/120 ⋅ 243/242 (urania)
| 33/32 ⋅ 54/55 (thalia), 45/44 ⋅ 99/100 (calliope)
|-
| 13
| 91/90 ⋅ 729/728, 105/104 ⋅ 351/350
| 27/26 ⋅ 39/40, 65/64 ⋅ 324/325, 66/65 ⋅ 351/352, 78/77 ⋅ 2079/2080
|-
| 17
| 85/84 ⋅ 1701/1700
| 51/50 ⋅ 135/136
|-
| 19
| 96/95 ⋅ 513/512, 153/152 ⋅ 171/170
| 57/56 ⋅ 189/190, 76/75 ⋅ 1215/1216, 77/76 ⋅ 1539/1540
|-
| 23
| 161/160 ⋅ 162/161
| 69/68 ⋅ 459/460
|-
| 29
| 117/116 ⋅ 261/260
| -
|-
| 31
| 93/92 ⋅ 621/620
| 63/62 ⋅ 279/280
|-
| 37
| 111/110 ⋅ 297/296
| 75/74 ⋅ 999/1000
|-
| 41
| 82/81 ⋅ 6561/6560
| 41/40 ⋅ 81/82
|-
| 43
| 86/85 ⋅ 1377/1376, 87/86 ⋅ 1161/1160, 129/128 ⋅ 216/215
| -
|-
| 47
| 141/140 ⋅ 189/188
| -
|-
| 53
| -
| 54/53 ⋅ 159/160
|-
| 59
| -
| -
|-
| 61
| -
| 61/60 ⋅ 243/244
|-
| 67
| 135/134 ⋅ 201/200
| -
|-
| 71
| -
| 71/70 ⋅ 567/568, 72/71 ⋅ 639/640
|-
| 73
| -
| 73/72 ⋅ 729/730
|-
| 79
| -
| 79/78 ⋅ 3159/3160, 80/79 ⋅ 6399/6400
|-
| 83
| 83/82 ⋅ 3321/3320, 84/83 ⋅ 2241/2240
| -
|-
| 89
| 89/88 ⋅ 891/890, 90/89 ⋅ 801/800
| -
|-
| 97
| 97/96 ⋅ 486/485
| -
|-
| 101
| 101/100 ⋅ 405/404
| -
|-
| 103
| -
| -
|-
| 107
| 108/107 ⋅ 321/320
| -
|}

== See also ==
* [[160/81]] – its [[octave complement]]
* [[40/27]] – its [[fifth complement]]
* [[1ed81/80]] – its equal multiplication
* [[Pythagorean comma]]
* [[64/63]] – the septimal comma or Archytas' comma
* [[Small comma]]
* [[List of superparticular intervals]]

== Notes ==

[[Category:Meantone]]
[[Category:Commas named for their regular temperament properties]]
[[Category:Commas named after polymaths]]
[[Category:Commas named for the intervals they stack]]

Gallery of just intervals

2026-04-29T00:54:47Z

TallKite: updated comma color names

The following is a '''gallery of many just intervals''' that contributors to this wiki have found notable. It is of course not meant to be comprehensive, and is not the result of a systematic search.

Do not be intimidated by all the common names: there is no need to memorise the names. For pretty much all use cases, is perfectly acceptable - preferred, even - to just refer to an interval by its [[ratio]] (e.g. 3/2, 5/4).

== Introduction ==
In [[just intonation]], a musical [[interval]] is specified as a ratio of two frequencies. When two (or more) pitches are sounded that are in simple proportions to one another, there is a "fusing" quality to the sound which is often described as pleasing; hence the interest in tuning the pitches of musical systems according to such proportions. There is much debate as to what "[[consonance]]" means in a musical system, but in just intonation, it is generally assumed that lower numbers in frequency ratios lead to greater consonance.

In the actual performance of a piece of music, the number of factors involved are enormous, and it is not often helpful to reduce a musical experience to a one-dimensional description of "consonance versus dissonance." Hence the need for this gallery, to give life to conversation about what an interval means beyond the numerical description: "5/3" or "21/16" or what have you.

What follows is a gallery of just intervals in ascending order from [[1/1]] to [[2/1]] and beyond. No such list could possibly be complete (as there are infinite possible ratios), so please add intervals of interest as you see fit. Any rational interval is welcome, as long as the wiki author has some interest in it.

This page lists links to dedicated pages for each interval, under the heading "frequency ratio".

Said dedicated pages may include: descriptions of common usage, technical notes, poetry, links, reservations, complaints, chords or compositions that feature it, edos that approximate it, intervals that are functionally (or emotionally) related to it, nicknames, love letters, fan art, etc. Readers are encouraged to add such things to the interval pages. If your contribution is unconventional, add your name or pseudonym so future readers know where the idea originated.

== Gallery of just intervals ==
See also [[list of superparticular intervals]] and [http://www.huygens-fokker.org/docs/intervals.html List of intervals (Huygens-Fokker foundation)]

{| class="wikitable sortable right-3"
|-
! Frequency Ratio
! Cents Value (7 sig. dig.)
! [[Sagittal notation|Sagittal notation]]
! [[Color notation|Color Name]]
! [[Functional Just System|FJS Name]]
! class="unsortable" | Some common names
|-
| [[1/1]]
| 0.000000
| <ref group=note>The [[Sagittal]] column shows a pitch-class-sensitive [[Sagittal_notation#Athenian|Athenian]] [[Sagittal_notation#Evo|Evo]] notation with 1/1 = C. A white notehead {{sagittal|nhhf|size=300%}} indicates exact notation while black {{sagittal|nhbl|size=300%}} indicates approximation (typically within 2 ¢).</ref>&numsp;{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| w1, wa unison
| P1&numsp;&numsp;<ref group=note>If any [[FJS]] names are missing, please [https://misotanni.github.io/fjs/en/calc.html add] them.</ref>
| unity, perfect prime, Tonic, unison
|-
| [[32805/32768]]
| 1.953721
| {{sagittal| \! }}{{sagittal| # |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| Ly-2, Layoma
| d-25
| schisma ({{Monzo|-15, 8, 1}})
|-
| [[256/255]]
| 6.775876
| {{sagittal| |( |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 17ug1, Suguma
| P185
| charisma, charic comma, septendecimal kleisma
|-
| [[225/224]]
| 7.711523
| {{sagittal| ~!( }}{{sagittal| # |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| ryy-2, Ruyoyoma
| d-2257
| septimal kleisma, marvel comma ({{Monzo|-5, 2, 2, -1}})
|-
| [[126/125]]
| 13.79477
| {{sagittal| (|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| zg32, Zotriguma
| d27125
| septimal semicomma, starling comma ({{Monzo|1, 2, -3, 1}})
|-
| [[100/99]]
| 17.39948
| {{sagittal| ~|( |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 1uyy1, Luyoyoma
| A12511
| Ptolemy's comma, ptolemisma
|-
| [[99/98]]
| 17.57613
| {{sagittal| !( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| 1orr-2, Loruruma
| m-21149
| mothwellsma
|-
| [[2048/2025]]
| 19.55257
| {{sagittal| //| |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| sgg2, Saguguma
| d225
| diaschisma ({{Monzo|11, -4, -2}})
|-
| [[81/80]]
| 21.50629
| {{sagittal| /| |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| g1, Guma
| P15
| meantone comma, syntonic comma, Didymus comma
|-
| [[531441/524288]]
| 23.46001
| {{sagittal| # |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| LLw-2, Lalawama
| d-2
| Pythagorean comma, ditonic comma ({{Monzo|-19, 12}})
|-
| [[66/65]]
| 26.43157
| {{sagittal| |) |size=300%}}{{sagittal|ntCbl|size=400%}}&numsp;&numsp;
| 3u1og1, Thuloguma
| P11165
| winmeanma
|-
| [[65/64]]
| 26.84138
| {{sagittal| |) |size=300%}}{{sagittal|ntCbl|size=400%}}&numsp;&numsp;
| 3oy1, Thoyoma
| P165
| wilsorma, 13th-partial chroma
|-
| [[64/63]]
| 27.26409
| {{sagittal| |) |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| r1, Ruma
| P17
| septimal comma, Archytas' comma
|-
| [[3125/3072]]
| 29.61357
| {{sagittal| |( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| Ly5-2, Laquinyoma
| dd-23125
| magic comma, small diesis ({{Monzo|-10, -1, 5}})
|-
| [[50/49]]
| 34.97562
| {{sagittal| )|( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| rryy-2, Biruyoma
| d-22549
| small septimal diesis, small septimal sixth-tone, septimal tritonic diesis, jubilisma
|-
| [[49/48]]
| 35.69681
| {{sagittal| \!/ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| zz2, Zozoma
| m249
| large septimal diesis, large septimal sixth-tone, slendro diesis, semaphoresma
|-
| [[45/44]]
| 38.90577
| {{sagittal|(|(|size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 1uy1, Luyoma
| A1511
| undecimal 1/5 tone
|-
| [[128/125]]
| 41.05886
| {{sagittal| (|\ |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| g32, Triguma
| d2125
| diesis, minor diesis, augmented comma, enharmonic comma ({{Monzo|7, 0, -3}})
|-
| [[525/512]]
| 43.40834
| {{sagittal| //| |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| Lzyy1, Lazoyoyoma
| A1175
| Avicenna's enharmonic diesis, avicennma ({{Monzo|-9, 1, 2, 1}})
|-
| [[1053/1024]]
| 48.34767
| {{sagittal| (!/ |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| L3o1, Lathoma
| P113
| tridecimal quartertone, tridecimal flattone comma
|-
| [[36/35]]
| 48.77038
| {{sagittal| /|) |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| rg1, Ruguma
| P135
| septimal quarter tone, double comma
|-
| [[250/243]]
| 49.16614
| {{sagittal| (!/ |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntCbl|size=400%}}&numsp;&numsp;
| y31, Triyoma
| A1125
| porcupine comma ({{Monzo|1, -5, 3}})
|-
| [[59049/57344]]
| 50.72410
| {{sagittal| |) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| Lr-2, Laruma
| d-27
| Harrison's comma ({{Monzo|-13, 10, 0, -1}})
|-
| [[100/97]]
| 52.73202
| {{sagittal| (!) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntCbl|size=400%}}&numsp;&numsp;
| 97uyy1, ninety-suyoyo unison
| A12597
| shrutar quarter tone
|-
| [[33/32]]
| 53.27294
| {{sagittal| /|\ |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 1o1, ilo unison
| P111
| undecimal quarter tone, undecimal diesis, al-Farabi's 1/4-tone, [[octave-reduced]] 33rd [[harmonic]]
|-
| [[648/625]]
| 62.56515
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDbl|size=400%}}&numsp;&numsp;
| g42, quadgu 2nd
| d2625
| diminished comma, major diesis ({{Monzo|8, 4, -4}})
|-
| [[28/27]]
| 62.96090
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| z2, zo 2nd
| m27
| small septimal chroma, septimal third-tone, subminor second, septimal minor second
|-
| [[27/26]]
| 65.33734
| {{sagittal| (|\ |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 3u1, thu unison
| A113
| small tridecimal third tone
|-
| [[26/25]]
| 67.90023
| {{sagittal| \! |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 3ogg2, thogugu 2nd
| d21325
| large tridecimal third tone
|-
| [[25/24]]
| 70.67243
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| yy1, yoyo unison
| A125
| chroma, classic chromatic semitone, Zarlinian semitone
|-
| [[24/23]]
| 73.68065
| {{sagittal| ~!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 23u1, twethu unison
| m223
| lesser vicesimotertial semitone
|-
|[[23/22]]
|76.95641
| {{sagittal| (!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|23o1u2, twetholu 2nd
|A12311
|greater vicesimotertial semitone
|-
| [[68/65]]
| 78.11403
| {{sagittal| )!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 17o3ug2, sothugu 2nd
| m21765
| valentine semitone
|-
| [[22/21]]
| 80.53704
| {{sagittal| (! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 1or1, loru unison
| P1117
| undecimal minor semitone
|-
| [[64/61]]
| 83.11520
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDbl|size=400%}}&numsp;&numsp;
| 61u2, siwu 2nd
| m261
| harry minor semitone
|-
| [[21/20]]
| 84.46719
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| zg2, zogu 2nd
| m275
| minor semitone, large septimal chroma
|-
| [[20/19]]
| 88.80070
| {{sagittal| !) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 19uy1, nuyo unison
| A1519
| small undevicesimal semitone, undevicesimal chroma, Eratosthenes' semitone
|-
| [[256/243]]
| 90.22500
| {{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| sw2, sawa 2nd
| m2
| Pythagorean limma, Pythagorean diatonic semitone, Pythagorean minor second ({{Monzo|8, -5}})
|-
| [[135/128]]
| 92.17872
| {{sagittal| \! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| Ly1, layo unison
| A15
| major limma ({{Monzo|-7, 3, 1}})
|-
| [[19/18]]
| 93.60301
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 19o2, ino 2nd
| m219
| large undevicesimal semitone, undevicesimal limma, Boethius' semitone
|-
| [[18/17]]
| 98.95459
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 17u1, su unison
| A117
| small septendecimal semitone, Arabic lute index finger
|-
| [[17/16]]
| 104.9554
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 17o2, iso 2nd
| m217
| large septendecimal semitone, octave-reduced 17th harmonic
|-
| [[16/15]]
| 111.7313
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| g2, gu 2nd
| m25
| classic diatonic semitone, classic minor second, octave-reduced 15th subharmonic
|-
| [[2187/2048]]
| 113.6850
| {{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| Lw1, lawa unison
| A1
| apotome, Pythagorean chromatic semitone, Pythagorean augmented unison ({{Monzo|-11, 7}})
|-
| [[77/72]]
| 116.2338
| {{sagittal| |) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1oz2, lozo 2nd
| m277
| undecimal secor
|-
| [[15/14]]
| 119.4428
| {{sagittal| |( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| ry1, ruyo unison
| A157
| septimal diatonic semitone
|-
| [[14/13]]
| 128.2982
| {{sagittal| (|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 3uz2, thuzo 2nd
| M2713
| tridecimal 2/3-tone, trienthird, tridecimal supraminor second
|-
| [[27/25]]
| 133.2376
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| gg2, gugu 2nd
| m225
| large limma
|-
| [[13/12]]
| 138.5727
| {{sagittal| (!/ |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 3o2, tho 2nd
| m213
| tridecimal neutral second, tridecimal 2/3-tone
|-
| [[243/224]]
| 140.9491
| {{sagittal| |) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| Lr1, laru unison
| A17
| septimal subtone ({{Monzo|-5, 5, 0, -1}})
|-
| [[88/81]]
| 143.4980
| {{sagittal| /|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1o2, ilo 2nd
| m211
| undecimal subtone
|-
| [[25/23]]
| 144.3531
| {{sagittal| (!) |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 23uyy1, twethuyoyo unison
| M25,523
| small vicesimotertial neutral second
|-
| [[49/45]]
| 147.4281
| {{sagittal| (! |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| zzg3, zozogu 3rd
| d3495
| swetismic neutral second
|-
| [[12/11]]
| 150.6371
| {{sagittal| \!/ |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1u2, lu 2nd
| M211
| small undecimal neutral second, 3/4-tone
|-
| [[35/32]]
| 155.1396
| {{sagittal| \!) |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| zy2, zoyo 2nd
| M235
| septimal neutral second
|-
|[[23/21]]
|157.4934
| {{sagittal| //| |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|23or2, twethoru 2nd
|A1237
|large vicesimotertial neutral second
|-
| [[78/71]]
| 162.7861
| {{sagittal| \\! |size=300%}}{{sagittal|ntDbl|size=400%}}&numsp;&numsp;
| 71u3o2, seventy-wutho 2nd
| m21371
| porcupine neutral second
|-
| [[11/10]]
| 165.0042
| {{sagittal| (!( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1og2, logu 2nd
| m2115
| undecimal submajor second, large undecimal neutral second, 4/5-tone, Ptolemy's second
|-
| [[54/49]]
| 168.2132
| {{sagittal| /|\ |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| rr1, ruru unison
| A149
| Zalzal's mujannab
|-
| [[21/19]]
| 173.2679
| {{sagittal| (! |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 19uz2, nuzo 2nd
| M2719
| large neutral undevicesimal second
|-
| [[10/9]]
| 182.4037
| {{sagittal| \! |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| y2, yo 2nd
| M25
| classic (whole) tone, minor (whole) tone
|-
| [[49/44]]
| 186.3339
| {{sagittal| |( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1uzz3, luzozo 3rd
| m34911
| mothwellsmic major second
|-
| [[39/35]]
| 187.3430
| {{sagittal| ~!( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 3org2, thorugu 2nd
| m21335
| animist major second
|-
| [[6272/5625]]
| 188.4870
| {{sagittal| (| |size=300%}}{{sagittal| b |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| szzg44, sabizogugu 4th
| ddd449625
| double marvelous second, double marvelous (whole) tone
|-
| [[19/17]]
| 192.5576
| {{sagittal| )!( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 19o17u2, nosu 2nd
| M21917
| quasi-meantone
|-
| [[28/25]]
| 196.1985
| {{sagittal| ~|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| zgg3, zogugu 3rd
| d3725
| middle major second
|-
| [[55/49]]
| 199.9798
| {{sagittal| !) |size=300%}}{{sagittal| x |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 1orry1, loruruyo unison
| A15549
| werckismic tone
|-
| [[64/57]]
| 200.5320
| {{sagittal| !( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 19u2, inu 2nd
| M219
| quasi-tempered whole tone, octave-reduced 57th subharmonic
|-
| [[9/8]]
| 203.9100
| {{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| w2, wa 2nd
| M2
| Pythagorean (whole) tone, major (whole) tone, octave-reduced 9th harmonic
|-
| [[26/23]]
| 212.2533
| {{sagittal| (| |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 23u3o2, twethutho 2nd
| d31323
| vicesimotertial whole tone
|-
| [[17/15]]
| 216.6867
| {{sagittal| (|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 17og3, sogu 3rd
| d3175
| septendecimal whole tone, septendecimal eventone
|-
| [[25/22]]
| 221.3095
| {{sagittal| ~|( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1uyy2, luyoyo 2nd
| A22511
| ptolemismic whole tone, undecimal acute whole tone
|-
| [[729/640]]
| 225.4163
| {{sagittal| /| |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| Lg2, lagu 2nd
| M25
| acute whole tone
|-
| [[57/50]]
| 226.8406
| {{sagittal| /|) |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 19ogg3, nogugu 3rd
| d31925
| devichromic supermajor second
|-
| [[8/7]]
| 231.1741
| {{sagittal| |) |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| r2, ru 2nd
| M27
| supermajor second, septimal whole tone, octave-reduced 7th subharmonic
|-
| [[63/55]]
| 235.1043
| {{sagittal| (!) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1uzg3, luzogu 3rd
| m3755
| werckismic supermajor second
|-
| [[55/48]]
| 235.6767
| {{sagittal| (| |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1oy2, loyo 2nd
| M255
| keenanismic supermajor second
|-
| [[224/195]]
| 240.0295
| {{sagittal| (|) |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 3uzg3, thuzogu 3rd
| m3765
| quasi-tempered 1/5-octave, "5EDO"-esque tone
|-
| [[23/20]]
| 241.9606
| {{sagittal| (|( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 23og3, twethogu 3rd
| M2235
| vicesimotertial inframinor third, vicesimotertial ultramajor second, vicesimotertial semifourth
|-
| [[15/13]]
| 247.7411
| {{sagittal| //| |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 3uy2, thuyo 2nd
| A2513
| tridecimal semifourth, tridecimal ultramajor second, tridecimal inframinor third
|-
| [[97/84]]
| 249.1145
| {{sagittal| //| |size=300%}}{{sagittal|ntDbl|size=400%}}&numsp;&numsp;
| 97or2, ninety-soru 2nd
| M2977
| homothetic semifourth
|-
| [[81/70]]
| 252.2680
| {{sagittal| /|) |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| rg2, rugu 2nd
| M235
| septimal semi-augmented second, septimal ultramajor second
|-
| [[22/19]]
| 253.8049
| {{sagittal| /|) |size=300%}}{{sagittal|ntDbl|size=400%}}&numsp;&numsp;
| 19u1o2, nulo 2nd
| M21119
| undevicesimal semifourth, minimal minor third, godzilla semifourth
|-
| [[64/55]]
| 262.3683
| {{sagittal| (! |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1ug3, lugu 3rd
| m355
| keenanismic subminor third, octave-reduced 55th subharmonic
|-
| [[7/6]]
| 266.8709
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| z3, zo 3rd
| m37
| subminor third, septimal minor third
|-
| [[90/77]]
| 270.0799
| {{sagittal| \!) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1ury2, luruyo 2nd
| A2577
| swetismic subminor third
|-
| [[62/53]]
| 271.5310
| {{sagittal| \! |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 53u31o2, fithu-thiwo 2nd
| M23153
| orwell subminor third
|-
| [[75/64]]
| 274.5824
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| yy2, yoyo 2nd
| A225
| classic augmented second
|-
| [[27/23]]
| 277.5907
| {{sagittal| ~!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| twethu 2nd, 23u2
| m323
| vicesimotertial augmented second
|-
| [[20/17]]
| 281.3583
| {{sagittal| (!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 17uy2, suyo 2nd
| A2517
| septendecimal augmented second, septendecimal minor third, diatismic minor third
|-
| [[13/11]]
| 289.2097
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 3o1u3, tholu 3rd
| m31311
| tridecimal minor third
|-
| [[45/38]]
| 292.7107
| {{sagittal| !) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 19uy2, nuyo 2nd
| A2519
| Eratosthenes' minor third
|-
| [[32/27]]
| 294.1350
| {{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| w3, wa 3rd
| m3
| Pythagorean minor third, octave-reduced 27th subharmonic
|-
| [[19/16]]
| 297.5130
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 19o3, ino 3rd
| m319
| otonal minor third, octave-reduced 19th harmonic
|-
| [[289/243]]
| 300.1358
| {{sagittal| (| |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 17oo4, soso 4th
| dd417, 17
| semitonismic quasi-tempered minor third
|-
| [[25/21]]
| 301.8465
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| ryy2, ruyoyo 2nd
| A2257
| quasi-tempered minor third
|-
| [[61/51]]
| 309.9744
| {{sagittal| )!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDbl|size=400%}}&numsp;&numsp;
| 61o17u2, siwosu 2nd
| A26117
| myna third
|-
| [[6/5]]
| 315.6413
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| g3, gu 3rd
| m35
| classic minor third, just minor third
|-
| [[77/64]]
| 320.1438
| {{sagittal| |) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1oz3, lozo 3rd
| m377
| keenanismic minor third, octave-reduced 77th harmonic
|-
| [[135/112]]
| 323.3528
| {{sagittal| |( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| ry2, ruyo 2nd
| A257
| large septimal minor third, marvelous minor third ({{Monzo|-4, 3, 1, -1}})
|-
| [[35/29]]
| 325.5624
| {{sagittal| (| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 29uzy3, twenuzoyo 3rd
| M33529
| doublewide minor third
|-
| [[23/19]]
| 330.7613
| {{sagittal| )|( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 23o19u3, twethonu 3rd
| A22319
| vicesimotertial supraminor third
|-
| [[17/14]]
| 336.1295
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEbl|size=400%}}&numsp;&numsp;
| 17or3, soru 3rd
| m3175
| septendecimal supraminor third
|-
| [[175/144]]
| 337.5433
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| zyy3, zoyoyo 3rd
| M35,5,7
| keemic minor third
|-
| [[73/60]]
| 339.5208
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEbl|size=400%}}&numsp;&numsp;
| 73og3, seventy-thogu 3rd
| m3735
| amity supraminor third
|-
| [[28/23]]
| 340.5516
| {{sagittal| \\! |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 23uz3, twethuzo 3rd
| d4723
| vicesimotertial neutral third
|-
| [[39/32]]
| 342.4827
| {{sagittal| (!/ |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 3o3, tho 3rd
| m313
| lesser tridecimal neutral third, octave-reduced 39th harmonic
|-
| [[128/105]]
| 342.9054
| {{sagittal| /|) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| rg3, rugu 3rd
| m335
| quasi-tempered 2/7-octave, 105th subharmonic
|-
| [[625/512]]
| 345.2549
| {{sagittal| |) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| Ly42, laquadyo 2nd
| AA2625
| 5-limit neutral third ({{Monzo|-9, 0, 4}})
|-
| [[11/9]]
| 347.4079
| {{sagittal| /|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1o3, ilo 3rd
| m311
| undecimal neutral third
|-
| [[60/49]]
| 350.6169
| {{sagittal| (| |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| rry2, ruruyo 2nd
| A2549
| smaller septimal neutral third, (purple 3rd)
|-
| [[49/40]]
| 351.3381
| {{sagittal| (! |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| zzg4, zozogu 4th
| d4495
| larger septimal neutral third, (purple 3rd)
|-
| [[27/22]]
| 354.5471
| {{sagittal| \!/ |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1u3, lu 3rd
| M311
| rastmic neutral third
|-
| [[16/13]]
| 359.4723
| {{sagittal| (|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 3u3, thu 3rd
| M313
| greater tridecimal neutral third, octave reduced 13subharmonic
|-
|[[21/17]]
|365.8255
| {{sagittal| \\! |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
|17uz3, suzo 3rd
|M3717
|septendecimal submajor third
|-
|[[26/21]]
|369.7468
| {{sagittal| (!( |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
|thoru 3rd, 3or3
|m3137
|tridecimal submajor third
|-
|[[51/41]]
|377.8480
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
|41u17o4, fowuso 4th
|d41741
|
|-
|[[56/45]]
|378.6022
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
|zg4, zogu 4th
|d475
|narrow perde segah, marvelous major third
|-
| [[71/57]]
| 380.2285
| {{sagittal| !) |size=300%}}{{sagittal|ntEbl|size=400%}}&numsp;&numsp;
| 71o19u3, seventy-wonu 3rd
| M37119
| witchcraft major third
|-
| [[76/61]]
| 380.6282
| {{sagittal| b |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 61u19o4, siwuno 4th
| d41961
| magic major third
|-
| [[96/77]]
| 381.8112
| {{sagittal| !) |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1ur3, luru 3rd
| M377
| undecimal perde segah, keenanismic major third
|-
| [[5/4]]
| 386.3137
| {{sagittal| \! |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| y3, yo 3rd
| M35
| classic major third, just major third, octave-reduced 5th harmonic
|-
| [[44/35]]
| 396.1783
| {{sagittal| )!( |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1org3, lorugu 3rd
| n3115,7
| fwiwismic major third
|-
| [[161/128]]
| 397.1003
| {{sagittal| )!( |size=300%}}{{sagittal|ntEbl|size=400%}}&numsp;&numsp;
| 23oz4, twethozo 4th
| M3161
| just/Pythagorean major third meantone, octave-reduced 161th harmonic
|-
| [[34/27]]
| 399.0904
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 17o4, iso 4th
| d417
| septendecimal quasi-tempered major third
|-
| [[63/50]]
| 400.1085
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| zgg4, zogugu 4th
| d4725
| quasi-tempered major third
|-
| [[24/19]]
| 404.4420
| {{sagittal| !( |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 19u3, inu 3rd
| M319
| Boethius' major third
|-
| [[81/64]]
| 407.8200
| {{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| Lw3, lawa 3rd
| M3
| Pythagorean major third, octave-reduced 81st harmonic
|-
| [[19/15]]
| 409.2443
| {{sagittal| |) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 19og4, nogu 4th
| d4195
| Eratosthenes' major third
|-
| [[33/26]]
| 412.7453
| {{sagittal| |( |size=300%}}{{sagittal|ntEbl|size=400%}}&numsp;&numsp;
| 3u1o3, thulo 3rd
| M31113
| tridecimal major third
|-
| [[80/63]]
| 413.5778
| {{sagittal| |( |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| ry3, ruyo 3rd
| M357
| 5/7-kleismic major third
|-
| [[14/11]]
| 417.5080
| {{sagittal| (| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1uz4, luzo 4th
| P4711
| undecimal major third, undecimal diminished fourth
|-
| [[23/18]]
| 424.3643
| {{sagittal| ~|( |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 23o4, twetho 4th
| M323
| vicesimotertial diminished fourth
|-
| [[32/25]]
| 427.3726
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| gg4, gugu 4th
| d425
| classic diminished fourth
|-
| [[41/32]]
| 429.0624
| {{sagittal| /| |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 41o3, fowo 3rd
| M341
| octave-reduced 41st harmonic, quadracesimoprimal supermajor third
|-
| [[77/60]]
| 431.8751
| {{sagittal| /|) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1ozg4, lozogu 4th
| d4775
| swetismic supermajor third
|-
| [[9/7]]
| 435.0841
| {{sagittal| |) |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| r3, ru 3rd
| M37
| supermajor third, septimal major third
|-
| [[31/24]]
| 443.0806
| {{sagittal| \!/ |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 31o3, thiwo 3rd
| M331
| sensi supermajor third
|-
| [[22/17]]
| 446.3625
| {{sagittal| (|\ |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 17u1o3, sulo 3rd
| M31117
| septendecimal supermajor third
|-
| [[35/27]]
| 449.2746
| {{sagittal| \!) |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| zy4, zoyo 4th
| P435
| septimal semidiminished fourth
|-
| [[83/64]]
| 450.0473
| {{sagittal| \!) |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 83o4
| P483
| octave-reduced 83rd harmonic, octacesimotertial harmonic semisixth
|-
| [[13/10]]
| 454.2139
| {{sagittal| \\! |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 3og4, thogu 4th
| d4135
| tridecimal semisixth, Barbados third, tridecimal 9/4 tone
|-
| [[30/23]]
| 459.9944
| {{sagittal| (!( |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 23uy3, twethuyo 3rd
| P4523
| vicesimotertial ultramajor third
|-
| [[64/49]]
| 462.3482
| {{sagittal| /|\ |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| rr3, ruru 3rd
| M349
| septatonic major third
|-
| [[17/13]]
| 464.4277
| {{sagittal| (! |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 17o3u4, sothu 4th
| P41713
| septendecimal sub-fourth
|-
| [[21/16]]
| 470.7809
| {{sagittal| !) |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| z4, zo 4th
| P47
| sub-fourth, narrow fourth, 8ve-reduced 21st harmonic
|-
| [[25/19]]
| 475.1144
| {{sagittal| \!) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 19uyy3, nuyoyo 3rd
| A35,519
| undevicesimal augmented third, undevicesimal grave fourth
|-
| [[95/72]]
| 479.9167
| {{sagittal| ~!( |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 19oy4, noyo 4th
| P45,19
| undevicesimal quasi-tempered 2/5-octave, undevicesimal "[[5edo|5EDO]]"-esque fourth
|-
| [[128/97]]
| 480.1046
| {{sagittal| ~!( |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 97u4
| P497
| nonacesimoseptimal subharmonic fourth
|-
| [[33/25]]
| 480.6455
| {{sagittal| ~!( |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1ogg4, logugu 4th
| d41125
| ptolemismic fourth, "5EDO"-esque fourth
|-
| [[160/121]]
| 483.6778
| {{sagittal| ~!( |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 1uuy4, luluyo 4th
| A4511,11
| narrow biyatismic fourth
|-
| [[324/245]]
| 483.8545
| {{sagittal| (!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| rrg3, rurugu 3rd
| M35,7,7
| sensamagic fourth
|-
| [[85/64]]
| 491.2691
| {{sagittal| !( |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 17oy4, soyo 4th
| P485
| archagall fourth, septendecimal harmonic fourth
|-
| [[117/88]]
| 493.1197
| {{sagittal| !( |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 3o1u4, tholu 4th
| P41311
| minthmic fourth ({{Monzo|-3, 2, 0, 0, -1, 1}})
|-
| [[4/3]]
| 498.0450
| {{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| w4, wa 4th
| P4
| just perfect fourth, octave-reduced 3rd subharmonic, diatessaron
|-
| [[171/128]]
| 501.4230
| {{sagittal| |( |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 19o4, ino 4th
| P419
| undevicesimal harmonic fourth
|-
| [[75/56]]
| 505.7565
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| ryy3, ruyoyo 3rd
| A3257
| marvelous fourth
|-
| [[121/90]]
| 512.4122
| {{sagittal| ~|( |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 1oog4, lologu 4th
| d411,115
| wide biyatismic fourth
|-
| [[35/26]]
| 514.6120
| {{sagittal| ~|( |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 3uzy4, thuzoyo 4th
| A45,713
| animist fourth
|-
| [[128/95]]
| 516.1733
| {{sagittal| ~|( |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 19ug4, nugu 4th
| P45,19
| undevicesimal subharmonic fourth
|-
| [[27/20]]
| 519.5513
| {{sagittal| /| |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| g4, gu 4th
| P45
| acute fourth, wolf fourth
|-
| [[875/648]]
| 519.9470
| {{sagittal| /| |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| zy34, zotriyo 4th
| A45,5,5,7
| maviloid generator superfourth
|-
| [[23/17]]
| 523.3189
| {{sagittal| # |size=300%}}{{sagittal|ntEbl|size=400%}}&numsp;&numsp;
| 23o17u4, twethosu 4th
| A32317
| vicesimotertial acute fourth
|-
| [[19/14]]
| 528.6871
| {{sagittal| (| |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 19or4, noru 4th
| P4197
| hendrix fourth
|-
| [[34/25]]
| 532.3280
| {{sagittal| (|) |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 17ogg5, sogugu 5th
| dd5175,5
| vengeance superfourth
|-
| [[49/36]]
| 533.7418
| {{sagittal| \!/ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| zz5, zozo 5th
| d549
| Arabic lute acute fourth
|-
| [[15/11]]
| 536.9508
| {{sagittal| (|( |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1uy4, luyo 4th
| A4511
| sub-augmented fourth, undecimal augmented fourth
|-
| [[26/19]]
| 543.0146
| {{sagittal| //| |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 19u3o4, nutho 4th
| P41319
| undevicesimal super-fourth
|-
| [[48/35]]
| 546.8154
| {{sagittal| /|) |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| rg4, rugu 4th
| P435
| septimal super-fourth
|-
| [[11/8]]
| 551.3179
| {{sagittal| /|\ |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1o4, ilo 4th
| P411
| super-fourth, paramajor fourth, undecimal semi-augmented fourth, octave-reduced 11th harmonic, Alphorn-Fa
|-
| [[18/13]]
| 563.3823
| {{sagittal| (|\ |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 3u4, thu 4th
| A413
| tridecimal augmented fourth
|-
| [[25/18]]
| 568.7174
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| yy4, yoyo 4th
| A425
| classic augmented fourth, pental augmented fourth ({{Monzo|-1 -2 2}})
|-
| [[32/23]]
| 571.7257
| {{sagittal| ~!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 23u4, twethu 4th
| d523
| vicesimotertial narrow tritone, octave-reduced 23rd subharmonic
|-
| [[88/63]]
| 578.5820
| {{sagittal| (! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1or4, loru 4th
| P4117
| pentacircle diminished fifth
|-
| [[7/5]]
| 582.5122
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| zg5, zogu 5th
| d575
| augmented fourth, septimal tritone, Huygen's tritone
|-
| [[108/77]]
| 585.7212
| {{sagittal| !) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1ur4, luru 4th
| A477
| swetismic augmented fourth ({{Monzo|2, 3, 0, -1, -1}})
|-
| [[1024/729]]
| 588.2700
| {{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| sw5, sawa 5th
| d5
| Pytharogean diminished fifth
|-
| [[45/32]]
| 590.2237
| {{sagittal| \! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| y4, yo 4th
| A45
| smaller pental tritone, diatonic tritone ({{Monzo|-5 2 1}})
|-
| [[128/91]]
| 590.6464
| {{sagittal| \! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 3ur4, thuru 4th
| A491
| smaller huntmic tritone, 91st subharmonic
|-
| [[38/27]]
| 591.6480
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| ino 5th, 19o5
| d519
| smaller undevicesimal tritone
|-
| [[55/39]]
| 595.1490
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 3u1oy4, thuloyo 4th
| A45513
| smaller gassormic tritone
|-
| [[24/17]]
| 596.9996
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 17u4, su 4th
| A417
| smaller septendecimal tritone
|-
| [[181/128]]
| 599.8151
| {{sagittal| )!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 181o4
| A4181
| octave-reduced 181st harmonic, otonal quasi-tempered tritone
|-
| [[99/70]]
| 600.0883
| {{sagittal| )!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1org4, lorugu 4th
| P41135
| homothetic quasi-tempered tritone
|-
| [[256/181]]
| 600.1849
| {{sagittal| )|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 181u5
| d5181
| octave-reduced 181st subharmonic, utonal quasi-tempered tritone
|-
| [[17/12]]
| 603.0004
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 17o5, iso 5th
| d517
| larger septendecimal tritone
|-
| [[78/55]]
| 604.8510
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 3o1ug5, tholugu 5th
| d51355
| larger gassormic tritone
|-
| [[27/19]]
| 608.3520
| {{sagittal| !) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| inu 4th, 19u4
| A419
| larger undevicesimal tritone
|-
| [[91/64]]
| 609.3536
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 3oz5, thozo 5th
| d591
| larger huntmic tritone, octave-reduced 91st harmonic
|-
| [[64/45]]
| 609.7763
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| g5, gu 5th
| d55
| larger pental tritone, diatonic tritone ({{Monzo|6 -2 -1}})
|-
| [[729/512]]
| 611.7300
| {{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| Lw4, lawa 4th
| A4
| Pythagorean tritone
|-
| [[10/7]]
| 617.4878
| {{sagittal| |( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| ry4, ruyo 4th
| A457
| diminished fifth, Euler's tritone, superaugmented fourth
|-
| [[23/16]]
| 628.2743
| {{sagittal| ~|( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 23o5, twetho 5th
| A423
| vicesimotertial superaugmented fourth, octave-reduced 23rd harmonic
|-
| [[36/25]]
| 631.2826
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| gg5, gugu 5th
| d525
| pental diminished fifth, classic diminshed fifth ({{Monzo|2 2 -2}})
|-
| [[13/9]]
| 636.6177
| {{sagittal| (!/ |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 3o5, tho 5th
| d513
| tridecimal diminished fifth
|-
| [[16/11]]
| 648.6821
| {{sagittal| \!/ |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 1u5, lu 5th
| P511
| sub-fifth, paraminor fifth, octave-reduced 11th subharmonic
|-
| [[35/24]]
| 653.1846
| {{sagittal| \!) |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| zy5, zoyo 5th
| P535
| septimal sub-fifth
|-
| [[19/13]]
| 656.9854
| {{sagittal| \\! |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 19o3u5, nothu 5th
| P51913
| undevicesimal sub-fifth
|-
| [[22/15]]
| 663.0492
| {{sagittal| (!( |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 1og5, logu 5th
| d5115
| undecimal diminished fifth, semidiminished fifth
|-
| [[72/49]]
| 666.2589
| {{sagittal| /|\ |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| rr4, ruru 4th
| A449
| septimal catafifth
|-
| [[25/17]]
| 667.6720
| {{sagittal| (!) |size=300%}}{{sagittal| x |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 17uyy4, suyoyo 4th
| AA45,517
| vengeance subfifth
|-
| [[81/55]]
| 670.1883
| {{sagittal| (! |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 1ug5, lugu 5th
| P555
| undecimal catafifth
|-
| [[28/19]]
| 671.3129
| {{sagittal| (! |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 19uz5, nuzo 5th
| P5719
| hendrix fifth
|-
| [[34/23]]
| 676.6811
| {{sagittal| bb |size=300%}}{{sagittal|ntAbl|size=400%}}&numsp;&numsp;
| 23u17o5, twethuso 5th
| d61723
| vicesimotertial grave fifth
|-
| [[262144/177147]]
| 678.4950
| {{sagittal| bb |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| sasawa 6th, ssw6
| d6
| Pythagorean wolf fifth
|-
| [[40/27]]
| 680.4487
| {{sagittal| \!|size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| y5, yo 5th
| P55
| grave fifth
|-
| [[95/64]]
| 683.8267
| {{sagittal| ~!( |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 19oy5, noyo 5th
| P55,19
| undevicesimal harmonic fifth
|-
| [[52/35]]
| 685.3880
| {{sagittal| ~!( |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 3org5, thorugu 5th
| d51335
| animist fifth
|-
| [[180/121]]
| 687.5878
| {{sagittal| ~!( |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 1uuy5, luluyo 5th
| A5511,11
| narrow biyatismic fifth
|-
| [[112/75]]
| 694.2435
| {{sagittal| ~|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| zgg6, zogugu 6th
| d6725
| marvelous fifth
|-
| [[121/81]]
| 694.8159
| {{sagittal| !( |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 1oo5, lolo 5th
| d511,11
| rastmic fifth, Alpharabian narrow fifth
|-
| [[323/216]]
| 696.6034
| {{sagittal| /| |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntAbl|size=400%}}&numsp;&numsp;
| 19o17o6, noso 6th
| d617,19
| undevicesimal meantone fifth
|-
| [[256/171]]
| 698.5770
| {{sagittal| !( |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 19u5, inu 5th
| P519
| undevicesimal subharmonic fifth
|-
| [[16384/10935]]
| 700.0013
| {{sagittal| /| |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| sg6, sagu 6th
| d65
| Kirnberger's fifth ("[[12edo|12-EDO]]"-esque fifth)
|-
| [[3/2]]
| 701.9550
| {{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| w5, wa 5th
| P5
| [[just perfect fifth]], octave-reduced 3rd harmonic, diapente
|-
| [[182/121]]
| 706.7177
| {{sagittal| |) |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| 3o1uuz6, tholuluzo 6th
| m67,1311,11
| tridecimal gentle fifth ({{Monzo|1, 0, 0, 1, -2, 1}})
|-
| [[176/117]]
| 706.8803
| {{sagittal| |( |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 3u1o5, thulo 5th
| P51113
| minthmic fifth ({{Monzo|4, -2, 0, 0, 1, -1}})
|-
| [[128/85]]
| 708.7309
| {{sagittal| |( |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 17ug5, sugu 5th
| P585
| archagall fifth, septendecimal subharmonic fifth
|-
| [[245/162]]
| 716.1455
| {{sagittal| (|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| zzy6, zozoyo 6th
| m65,7,7
| sensamagic fifth
|-
| [[121/80]]
| 716.3222
| {{sagittal| ~|( |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 1oog5, lologu 5th
| d51215
| wide biyatismic fifth
|-
| [[50/33]]
| 719.3545
| {{sagittal| ~|( |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 1uyy5, luyoyo 5th
| A52511
| ptolemismic fifth, "5EDO"-esque fifth
|-
| [[97/64]]
| 719.8954
| {{sagittal| ~|( |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 97o5
| P597
| nonacesimoseptimal harmonic fifth
|-
| [[144/95]]
| 720.0833
| {{sagittal| ~|( |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 19ug5, nugu 5th
| P55,19
| undevicesimal quasi-tempered 3/5-octave, undevicesimal "5EDO"-esque fifth
|-
| [[38/25]]
| 724.8856
| {{sagittal| /|) |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| 19ogg6, nogugu 6th
| d6195,5
| undevicesimal diminished sixth, undevicesimal acute fifth
|-
| [[32/21]]
| 729.2191
| {{sagittal| |) |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| r5, ru 5th
| P57
| super-fifth, wide fifth, octave-reduced 21st subharmonic
|-
| [[26/17]]
| 735.5723
| {{sagittal| (| |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 17u3o5, sutho 5th
| P51317
| septendecimal super-fifth
|-
| [[49/32]]
| 737.6518
| {{sagittal| \!/ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| zz6, zozo 6th
| m649
| superduper fifth, octave-reduced 49th harmonic
|-
| [[23/15]]
| 740.0056
| {{sagittal| (|( |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 23og6, twethogu 6th
| P5235
| vicesimotertial ultraminor sixth
|-
| [[20/13]]
| 745.7861
| {{sagittal| //| |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 3uy5, thuyo 5th
| A5513
| tridecimal semitenth, Barbados sixth, ratwolf wolf fifth
|-
| [[128/83]]
| 749.9527
| {{sagittal| /|) |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 83u5
| P583
| 83rd subharmonic, octacesimotertial subharmonic semitenth
|-
| [[17/11]]
| 753.6375
| {{sagittal| (!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| 17o1u6, solu 6th
| m61711
| septendecimal subminor sixth
|-
| [[14/9]]
| 764.9159
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| z6, zo 6th
| m67
| subminor sixth, septimal minor sixth
|-
| [[25/16]]
| 772.6274
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| yy5, yoyo 5th
| A525
| pental augmented fifth, classic augmented fifth, otonal minor sixth, octave-reduced 25th harmonic
|-
| [[36/23]]
| 775.6357
| {{sagittal| ~!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| 23u5, twethu 5th
| m623
| vicesimotertial augmented fifth
|-
| [[11/7]]
| 782.4920
| {{sagittal| (! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 1or5, loru 5th
| P5117
| undecimal subminor sixth, undecimal augmented fifth
|-
| [[63/40]]
| 786.4222
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| zg6, zogu 6th
| m675
| narrow minor sixth
|-
| [[52/33]]
| 787.2547
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAbl|size=400%}}&numsp;&numsp;
| 3o1u6, tholu 6th
| m61311
| tridecimal minor sixth
|-
| [[30/19]]
| 790.7557
| {{sagittal| !) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 19uy5, nuyo 5th
| A5519
| small undevicesimal minor sixth, Eratosthenes' minor sixth
|-
| [[128/81]]
| 792.1800
| {{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| sw6, sawa 6th
| m6
| Pythagorean minor sixth, octave-reduced 81st subharmonic
|-
| [[19/12]]
| 795.5580
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| 19o6, ino 6th
| m619
| large undevicesimal minor sixth, Boethius' minor sixth
|-
| [[27/17]]
| 800.9096
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 17u5, su 5th
| A517
| septendecimal quasi-tempered minor sixth
|-
|[[35/22]]
|803.8217
| {{sagittal| )|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|1uzy6, luzoyo 6th
|n65,711
|fwiwismic minor sixth
|-
|[[8/5]]
|813.6863
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|g6, gu 6th
|m65
|classic minor sixth, just minor sixth, octave-reduced 5th subharmonic
|-
|[[21/13]]
|830.2532
| {{sagittal| (|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|3uz6, thuzo 6th
|M6713
|tridecimal supraminor sixth
|-
|[[34/21]]
|834.1745
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|17or6, soru 6th
|m6177
|septendecimal supraminor sixth
|-
|[[13/8]]
|840.5277
| {{sagittal| (!/ |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|3o6, tho 6th
|m613
|lesser tridecimal neutral sixth, octave-reduced 13th harmonic
|-
|[[80/49]]
|848.6619
| {{sagittal| (| |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
|rry5, ruruyo aug 5th
|A5549
|(purple 6th)
|-
|[[49/30]]
|849.3832
| {{sagittal| (! |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|zzg7, zozogu 7th
|d7495
|(purple 6th)
|-
|[[18/11]]
|852.5921
| {{sagittal| \!/ |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|1u6, lu 6th
|M611
|undecimal neutral sixth
|-
|[[105/64]]
|857.0946
| {{sagittal| \!) |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|zy6, zoyo 6th
|M635
|quasi-tempered 5/7-octave, octave-reduced 105th harmonic
|-
|[[23/14]]
|859.4484
| {{sagittal| //| |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
|23or6, twethoru 6th
|A5237
|vicesimotertial neutral sixth
|-
|[[28/17]]
|863.8705
| {{sagittal| \\! |size=300%}}{{sagittal|ntAbl|size=400%}}&numsp;&numsp;
|17uz6, suzo 6th
|M6717
|septendecimal submajor sixth
|-
|[[38/23]]
|869.2387
| {{sagittal| )!( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|23u19o6, twethuno 6th
|d71923
|vicesimotertial submajor sixth
|-
|[[5/3]]
|884.3587
| {{sagittal| \! |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|y6, yo 6th
|M65
|classic major sixth, just major sixth
|-
|[[42/25]]
|898.1535
| {{sagittal| ~|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|zgg7, zogugu 7th
|d7725
|quasi-tempered major sixth
|-
|[[486/289]]
|899.8642
| {{sagittal| (! |size=300%}}{{sagittal| x |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
|17uu5, susu 5th
|AA517, 17
|semitonismic quasi-tempered major sixth
|-
|[[32/19]]
|902.4870
| {{sagittal| !( |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|19u6, inu 6th
|M619
|utonal major sixth, octave-reduced 19th subharmonic
|-
|[[27/16]]
|905.8650
| {{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|w6, wa 6th
|M6
|Pythagorean major sixth, octave-reduced 27th harmonic
|-
| [[76/45]]
| 907.2893
| {{sagittal| |) |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| 19og7, nogu 7th
| d7195
| Eratosthenes' major sixth
|-
|[[22/13]]
|910.7908
| {{sagittal| |( |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|3u1o6, thulo 6th
|M61113
| tridecimal major sixth
|-
|[[17/10]]
|918.6417
| {{sagittal| (|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|17og7, sogu 7th
|d7175
|septendecimal diminished seventh, septendecimal major sixth, diatismic major sixth
|-
|[[46/27]]
|922.4093
| {{sagittal| ~|( |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|twetho 7th, 23o7
|M623
|vicesimotertial diminished seventh, vicesiomtertial supermajor sixth
|-
| [[12/7]]
|933.1291
| {{sagittal| |) |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|r6, ru 6th
| M67
|supermajor sixth, septimal major sixth
|-
|[[55/32]]
|937.6317
| {{sagittal| (| |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|1oy6, loyo 6th
|M655
|keenanismic supermajor sixth, octave-reduced 55th harmonic
|-
|[[19/11]]
|946.1951
| {{sagittal| \!) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBbl|size=400%}}&numsp;&numsp;
|19o1u7, nolu 7th
|m71911
|undevicesimal semitwelfth, maximal major sixth
|-
|[[140/81]]
|947.3196
| {{sagittal| \!) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|zy7, zoyo 7th
|m735
|septimal semidiminished seventh, septimal inframinor seventh
|-
| [[97/56]]
|951.0695
| {{sagittal| //| |size=300%}}{{sagittal|ntAbl|size=400%}}&numsp;&numsp;
|97or6, ninety-soru 6th
|M6977
|homothetic semitwelth
|-
|[[26/15]]
|952.2589
| {{sagittal| \\! |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|3og7, thogu 7th
|d7135
| tridecimal semitwelfth
|-
|[[40/23]]
|958.0394
| {{sagittal| (!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|23uy6, twethuyo 6th
|m7523
|vicesimotertial ultramajor sixth, vicesimotertial inframinor seventh, vicesimotertial semitwelfth
|-
|[[7/4]]
|968.8259
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|z7, zo 7th
|m77
|subminor seventh, septimal minor seventh, harmonic seventh, natural seventh, octave-reduced 7th harmonic
|-
|[[225/128]]
|976.5374
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|Lyy6, layoyo 6th
|A625
|marvel five-limit harmonic seventh, octave-reduced 225th harmonic ({{Monzo|-7, 2, 2}})
|-
|[[44/25]]
|978.6905
| {{sagittal| ~!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|1ogg7, logugu 7th
|d7115,5
|ptolemismic minor seventh, undecimal grave minor seventh
|-
|[[30/17]]
|983.3133
| {{sagittal| (!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|17uy6, suyo 6th
|A6517
|septendecimal minor seventh
|-
|[[23/13]]
|987.7467
| {{sagittal| (! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|23o3u7, twethothu 7th
|A62313
|vicesimotertial minor seventh
|-
|[[71/40]]
|993.3828
|{{sagittal| b |size=300%}}{{sagittal|ntBbl|size=400%}}&numsp;&numsp;
|71oy7
|m7715
| harmonic/just minor seventh meantone
|-
| [[16/9]]
|996.0900
|{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|w7, wa 7th
|m7
|Pythagorean minor seventh, small minor seventh, octave-reduced 9th subharmonic
|-
| [[57/32]]
| 999.4680
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| 19o7, ino 7th
| m719
| quasi-tempered minor seventh, octave-reduced 57th harmonic
|-
|[[25/14]]
|1003.802
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| ryy6, ruyoyo 6th
|A6257
|middle minor seventh
|-
|[[34/19]]
| 1007.442
| {{sagittal| )|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|19u17o7, nuso 7th
|m71719
|quasi-meantone minor seventh
|-
| [[9/5]]
|1017.596
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|g5, gu 7th
|m75
|classic minor seventh, large minor seventh
|-
|[[38/21]]
|1026.732
| {{sagittal| (| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|19or7, noru 7th
|m7197
|minor neutral undevicesimal seventh
|-
|[[29/16]]
|1029.577
| {{sagittal| (| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBbl|size=400%}}&numsp;&numsp;
|29o7, tweno 7th
|m729
|vicesimononal supraminor seventh, octave-reduced 29th harmonic
|-
|[[20/11]]
|1034.996
| {{sagittal| (|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|1uy7, luyo 7th
|M7511
|undecimal supraminor seventh, small undecimal neutral seventh
|-
|[[42/23]]
|1042.507
| {{sagittal| \\! |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|23uz7, twethuzo 7th
|d8723
|small vicesimotertial neutral seventh
|-
|[[64/35]]
|1044.860
| {{sagittal| /|) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|rg7, rugu 7th
|m735
|septimal neutral seventh
|-
|[[11/6]]
|1049.363
| {{sagittal| /|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|1o7, ilo 7th
| m711
|undecimal neutral seventh, 21/4-tone
|-
|[[46/25]]
|1055.647
| {{sagittal| (|) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|twethogugu octave, 23ogg8
|m7235,5
|large vicesimotertial neutral seventh
|-
|[[24/13]]
|1061.427
| {{sagittal| (|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|3u7, thu 7th
|M713
|tridecimal neutral seventh
|-
|[[50/27]]
|1066.762
| {{sagittal| \\! |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|yoyo 7th, yy7
|M75,5
|grave major seventh
|-
|[[13/7]]
|1071.702
| {{sagittal| (!( |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|3or7, thoru 7th
|m7137
|tridecimal submajor seventh, 16/3-tone
|-
|[[28/15]]
|1080.557
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|zg8, zogu octave
|d875
| grave major seventh, octave minus a ruyo aug unison
|-
|[[15/8]]
|1088.269
| {{sagittal| \! |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|y7, yo 7th
|M75
| classic major seventh, just major seventh, octave-reduced 15th harmonic
|-
|[[32/17]]
|1095.045
| {{sagittal| ~!( |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|17u7, su 7th
|M717
| septendecimal major seventh (FJS), septendecimal diminished octave (HEJI), [[octave-reduced]] 17th [[subharmonic]]
|-
|[[17/9]]
|1101.045
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|17o8, iso octave
|d817
| septendecimal major seventh (HEJI), septendecimal diminished octave (FJS)
|-
|[[36/19]]
|1106.397
| {{sagittal| !( |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|19u7, inu 7th
|M719
|undevicesimal major seventh, Boethius' major seventh
|-
| [[243/128]]
| 1109.775
| {{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|Lw7, lawa 7th
|M7
|Pythagorean major seventh, [[octave-reduced]] 243rd harmonic
|-
|[[19/10]]
|1111.199
| {{sagittal| |) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|19og8, nogu 8ve
|d8195
|undevicesimal diminished octave, Eratosthenes' major seventh
|-
|[[40/21]]
|1115.533
| {{sagittal| |( |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|ry7, ruyo 7th
|M757
|septimal acute major seventh
|-
|[[61/32]]
|1116.885
| {{sagittal| |( |size=300%}}{{sagittal|ntBbl|size=400%}}&numsp;&numsp;
|61o7, siwo 7th
|M761
|octave-reduced 61st harmonic
|-
|[[21/11]]
|1119.463
| {{sagittal| (| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|1uz8, luzo 8ve
|P8711
|large undecimal diminished octave, undecimal major seventh
|-
|[[44/23]]
|1123.044
| {{sagittal| (|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|23u1o7, twethulo 7th
|d81123
|small vicesimotertial major seventh
|-
|[[23/12]]
|1126.319
| {{sagittal| ~|( |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|23o8, twetho 8ve
|M723
|large vicesimotertial major seventh
|-
|[[48/25]]
|1129.328
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|gg8, gugu octave
|d825
|classic diminished octave
|-
|[[25/13]]
|1132.100
| {{sagittal| /| |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|3uyy7, thuyoyo 7th
|A75,513
|lesser tridecimal diminished octave
|-
|[[27/14]]
|1137.039
| {{sagittal| |) |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|r7, ru 7th
| M77
|supermajor seventh, septimal major seventh
|-
|[[31/16]]
|1145.036
| {{sagittal| \!/ |size=300%}}{{sagittal|ntCbl|size=400%}}&numsp;&numsp;
|31o7, thiwo 7th
|M731
|tricesimoprimal ultramajor seventh (FJS), tricesimoprimal semidiminished octave (HEJI), octave-reduced 31st harmonic
|-
|[[64/33]]
|1146.727
| {{sagittal| \!/ |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|1u8, lu octave
| P811
|undecimal semidiminished octave, octave-reduced 33rd subharmonic
|-
|[[35/18]]
|1151.240
| {{sagittal| \!) |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|zy8, zoyo octave
|P835
|septimal semidiminished octave
|-
|[[96/49]]
|1164.303
| {{sagittal| /|\ |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|rr7, ruru 7th
|M749
|septimal ultramajor seventh
|-
|[[49/25]]
| 1165.024
| {{sagittal| )!( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
|zzgg9, bizogu 9th
|d94925
|BP eighth
|-
| [[63/32]]
| 1172.736
| {{sagittal| !) |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| z8, zo octave
| P87
| septimal suboctave, octave-reduced 63rd harmonic
|-
| [[160/81]]
| 1178.494
| {{sagittal| \! |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| y8, yo octave
| P85
| syntonic suboctave
|-
| [[255/128]]
| 1193.224
| {{sagittal| !( |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 17oy8, soyo octave
| P885
| charismic suboctave, octave-reduced 255th harmonic
|-
|[[2/1]]
| 1200.000
| {{sagittal|ntChf|size=400%}}&numsp;&numsp;
|w8, wa octave
|P8
|[[Octave|octave]], [[Wikipedia:Diapason|diapason]]
|}
Notes<references group=note/>

=== Intervals larger than an octave ===
{| class="wikitable sortable right-3"
|-
! Frequency Ratio
! Cents Value (7 sig. dig.)
! [[Sagittal notation|Sagittal notation]]
! colspan="2" |[[Color notation|Color Name]]
! FJS Name
! Some common names
|-
| [[27/13]]
| 1265.337
| <ref group=note>The [[Sagittal]] column shows a pitch-class-sensitive [[Sagittal_notation#Athenian|Athenian]] [[Sagittal_notation#Evo|Evo]] notation with 1/1 = C. A white notehead {{sagittal|nhhf|size=300%}} indicates exact notation while black {{sagittal|nhbl|size=300%}} indicates approximation (typically within 2 ¢).</ref>&hairsp;&hairsp;{{sagittal| (|\ |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 3u8
| thu8ve&numsp;&numsp;&numsp;&numsp;&numsp;<ref group=note>In the color names, "co" or "c" stands for compound, {{w|Interval (music) #Compound intervals|a conventional music term}}.</ref>
| A813&numsp;<ref group=note>If any [[FJS]] names are missing, please [https://misotanni.github.io/fjs/en/calc.html add] them.</ref>
| Luxembourg eighth
|-
| [[23/11]]
| 1276.956
| {{sagittal| (!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 23o1u9
| twetholu 9th
| A82311
| vicesimotertial minor ninth
|-
| [[19/9]]
| 1293.603
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 19o9
| ino 9th
| m919
| undevicesimal minor ninth
|-
| [[17/8]]
| 1304.955
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 17o9
| iso 9th
| m917
| septendecimal minor ninth
|-
| [[32/15]]
| 1311.731
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| g9
| gu 9th
| m95
| classic minor ninth
|-
| [[15/7]]
| 1319.443
| {{sagittal| |( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| ry8
| ruyo 8ve
| A857
| septimal minor ninth
|-
| [[13/6]]
| 1338.573
| {{sagittal| (!/ |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 3o9
| tho 9th
| m913
| tridecimal minor ninth, tridecimal neutral ninth
|-
| [[11/5]]
| 1365.004
| {{sagittal| (!( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1o9
| ilo 9th
| m9115
| neutral ninth
|-
| [[20/9]]
| 1382.404
| {{sagittal| \! |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| y9
| yo 9th
| M95
| Ptolemaic ninth
|-
| [[9/4]]
| 1403.910
| {{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| w9
| wa 9th
| M9
| major ninth
|-
| [[25/11]]
| 1421.309
| {{sagittal| ~|( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1uyy9
| luyoyo 9th
| A92511
| undecimal major ninth
|-
| [[16/7]]
| 1431.174
| {{sagittal| |) |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| r9
| ru 9th
| M97
| septimal major ninth
|-
| [[7/3]]
| 1466.871
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| z10
| zo 10th
| m107
| septimal minor tenth
|-
| [[19/8]]
| 1497.513
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 19o3
| ino 10th
|
| undevicesimal minor tenth
|-
| [[12/5]]
| 1515.641
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| g10
| gu 10th
| m105
| classic minor tenth
|-
| [[17/7]]
| 1536.130
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEbl|size=400%}}&numsp;&numsp;
| 17or10
| soru 10th
| m10177
| septendecimal minor tenth
|-
| [[27/11]]
| 1554.547
| {{sagittal| \!/ |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| c1u3
| colu 3rd
| M1011
| rastmic neutral tenth
|-
| [[5/2]]
| 1586.314
| {{sagittal| \! |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| y10
| yo 10th
| M105
| classic major tenth
|-
| [[18/7]]
| 1635.084
| {{sagittal| |) |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| r10
| ru 10th
|
| septimal supermajor tenth
|-
| [[13/5]]
| 1654.214
| {{sagittal| \\! |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 3og11
| thogu 11th
| d11135
| tridecimal eleventh
|-
| [[21/8]]
| 1670.781
| {{sagittal| !) |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| z11
| zo 11th
|
| septimal subeleventh
|-
| [[8/3]]
| 1698.045
| {{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| w11, cw4
| wa 11th, cowa 4th
| P11
| perfect eleventh
|-
| [[19/7]]
| 1728.687
| {{sagittal| (| |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 19or11
| noru 11th
|
| undevicesimal supereleventh
|-
| [[11/4]]
| 1751.318
| {{sagittal| /|\ |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1o11, c1o4
| ilo 11th, colo 4th
| P1111
| undecimal semi-augmented eleventh
|-
| [[14/5]]
| 1782.512
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| zg12, czg5
| zogu 12th, cozogu 5th
| d1275
| short septimal navatone
|-
| [[17/6]]
| 1803.000
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 17o12
| iso 12th
| d1217
| septendecimal navatone
|-
| [[20/7]]
| 1817.488
| {{sagittal| |( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| ry11
| ruyo 11th
| A1157
| tall septimal navatone
|-
| [[23/8]]
| 1828.274
| {{sagittal| ~|( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 23o12
| twetho 12th
| A1123
| trivigintal navatone
|-
| [[35/12]]
| 1853.185
| {{sagittal| \!) |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| zy12, czy5
| zoyo 12th, cozoyo 5th
|
| septimal semi-diminished twelfth
|-
| [[3/1]]
| 1901.955
| {{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| w12, cw5
| wa 12th, cowa 5th
| P12
| 3rd harmonic, perfect twelfth, tritave
|-
| [[28/9]]
| 1964.916
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| z13, cz6
| zo 13th, cozo 6th
|
| septimal subminor thirteenth
|-
| [[19/6]]
| 1995.558
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| 19o13, c19o6
| ino 13th, cono 6th
|
| undevicesimal minor thirteenth
|-
| [[16/5]]
| 2013.686
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| g13, cg6
| gu 13th, cogu 6th
| m135
| classic minor thirteenth
|-
| [[13/4]]
| 2040.528
| {{sagittal| (!/ |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| 3o13, c3o6
| tho 13th, cotho 6th
| m1313
| tridecimal neutral thirteenth
|-
| [[23/7]]
| 2059.448
| {{sagittal| //| |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| c23or13, c23or6
| cotwethoru 6th
|
| trivigintal neutral/submajor thirteenth
|-
| [[10/3]]
| 2084.359
| {{sagittal| \! |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| y13, cy6
| yo 13th, coyo 6th
| M135
| just/classic(al)/ptolemaic major thirteenth
|-
| [[17/5]]
| 2118.642
| {{sagittal| (|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| c17og7
| cosogu 7th
| d14175
| septendecimal major thirteenth
|-
| [[24/7]]
| 2133.129
| {{sagittal| |) |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| r13, cr6
| ru 13th, coru 6th
|
| septimal supermajor thirteenth
|-
| [[7/2]]
| 2168.826
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| cz7
| cozo 7th
| m147
| harmonic fourteenth
|-
| [[32/9]]
| 2196.090
| {{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| cw7
| cowa 7th
|
| 3-limit minor fourteenth
|-
| [[18/5]]
| 2217.596
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| cg7
| cogu 7th
| m145
| 5-limit minor fourteenth
|-
| [[11/3]]
| 2249.363
| {{sagittal| /|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| c1o7
| colo 7th
| m1411
| undecimal neutral fourteenth
|-
| [[15/4]]
| 2288.269
| {{sagittal| \! |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| cy7
| coyo 7th
| M145
| just/classic(al)/ptolemaic major fourteenth
|-
| [[19/5]]
| 2311.199
| {{sagittal| |) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| c19og8
| nogu double 8ve
|
| undevicesimal major fourteenth
|-
| [[35/9]]
| 2351.230
| {{sagittal| \!) |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| czy8
| zoyo double 8ve
|
| septimal semi-augmented fourteenth
|-
| [[4/1]]
| 2400.000
| {{sagittal|ntChf|size=400%}}&numsp;&numsp;
| cw8, ccw1
| wa double 8ve
| P15
| 4th harmonic, two octaves, just double octave
|-
| [[81/20]]
| 2421.506
| {{sagittal| /| |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| cg8, ccg1
| gu double 8ve
| P155
| syntonic double octave
|-
| [[25/6]]
| 2470.672
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| cyy8, ccyy1
| yoyo double 8ve
|
| classic chromatic minor sixteenth
|-
| [[17/4]]
| 2504.955
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c17o9, cc17o2
| coso 9th, cocoso 2nd
| m1617
| septendecimal minor sixteenth
|-
| [[64/15]]
| 2511.731
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| cg9, ccg2
| cogu 9th, cocogu 2nd
| m165
| classic minor sixteenth
|-
| [[13/3]]
| 2538.573
| {{sagittal| (!/ |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c3o9, cc3o2
| cotho 9th, cocotho 2nd
| m1613
| tridecimal minor sixteenth
|-
| [[22/5]]
| 2565.004
| {{sagittal| (!( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c1og9, cc1og2
| coco-logu 2nd
|
| large undecimal neutral sixteenth
|-
| [[9/2]]
| 2603.910
| {{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| cw9, ccw2
| cowa 9th, cocowa 2nd
| M16
| classic large major sixteenth
|-
| [[23/5]]
| 2641.961
| {{sagittal| (|( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| cc23og3
| coco-twethogu 3rd
|
| trivigintal supermajor sixteenth
|-
| [[14/3]]
| 2666.871
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| ccz3
| cocozo 3rd
| m177
| septimal subminor seventeenth
|-
| [[19/4]]
| 2696.513
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| cc19o3
| cocono 3rd
|
| undevicesimal minor seventeenth
|-
| [[24/5]]
| 2715.641
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| ccg3
| cocogu 3rd
|
| classic minor seventeenth
|-
| [[44/9]]
| 2747.408
| {{sagittal| /|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| cc1o3
| cocolo 3rd
|
| undecimal neutral seventeenth
|-
| [[5/1]]
| 2786.314
| {{sagittal| \! |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| ccy3
| cocoyo 3rd
| M175
| 5th harmonic, pentave
|-
| [[36/7]]
| 2835.084
| {{sagittal| |) |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| ccr3
| cocoru 3rd
|
| septimal supermajor seventeenth
|-
| [[21/4]]
| 2870.781
| {{sagittal| !) |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| ccz4
| cocozo 4th
|
| septimal subeighteenth
|-
| [[16/3]]
| 2898.045
| {{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| ccw4
| cocowa 4th
| P18
| perfect eighteenth
|-
| [[27/5]]
| 2919.551
| {{sagittal| /| |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| ccg4
| cocogu 4th
|
| 5-limit supereighteenth
|-
| [[11/2]]
| 2951.318
| {{sagittal| /|\ |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| cc1o4
| cocolo 4th
| P1811
| undecimal semi-augmented eighteenth
|-
| [[28/5]]
| 2982.512
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| cczg5
| coco-zogu 5th
| d1975
| short septimal panchadashatone
|-
| [[45/8]]
| 2990.224
| {{sagittal| \! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| ccy4
| cocoyo 4th
| A185
| quinquadragintimal panchadashatone
|-
| [[17/3]]
| 3003.000
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| cc17o5
| cocoso 5th
| d1917
| septendecimal panchadasatone
|-
| [[40/7]]
| 3017.488
| {{sagittal| |( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| ccry4
| coco-ruyo 4th
| A1857
| tall septimal panchadashatone
|-
| [[23/4]]
| 3028.274
| {{sagittal| ~|( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| cc23o5
| coco-twetho 5th
|
| vicesimotertial panchadashatone
|-
| [[35/6]]
| 3053.185
| {{sagittal| \!) |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| cczy5
| coco-zoyo 5th
|
| septimal semi-diminished nineteenth
|-
| [[6/1]]
| 3101.955
| {{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| ccw5
| cocowa 5th
| P19
| 6th harmonic
|-
| [[55/9]]
| 3133.722
| {{sagittal| (| |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| cc1og5
| coco-logu 5th
|
| undecimal superninteenth
|-
| [[25/4]]
| 3172.627
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| ccyy5
| coco-yoyo 5th
|
| 5-limit subminor twentieth
|-
| [[19/3]]
| 3195.558
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| cc19o6
| cocono 6th
|
| undevicesimal minor twentieth
|-
| [[32/5]]
| 3213.686
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| ccg6
| cocogu 6th
|
| classic minor twentieth
|-
| [[13/2]]
| 3240.528
| {{sagittal| (!/ |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| cc3o6
| cocotho 6th
| m2013
| tridecimal neutral twentieth
|-
| [[20/3]]
| 3284.359
| {{sagittal| \! |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| ccy6
| cocoyo 6th
|
| classic major twentieth
|-
| [[48/7]]
| 3333.129
| {{sagittal| |) |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| ccr6
| cocoru 6th
|
| septimal supermajor twentieth
|-
| [[7/1]]
| 3368.826
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| ccz7
| cocozo 7th
| m217
| 7th harmonic
|-
| [[36/5]]
| 3417.596
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| ccg7
| cocogu 7th
|
| 5-limit minor twentyfirst
|-
| [[22/3]]
| 3449.363
| {{sagittal| /|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| cc1o7
| cocolo 7th
|
| undecimal neutral twentyfirst
|-
| [[15/2]]
| 3488.269
| {{sagittal| \! |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| ccy7
| cocoyo 7th
| M215
| just/classic(al)/ptolemaic major twentyfirst
|-
| [[23/3]]
| 3526.319
| {{sagittal| ~|( |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| c323o1
| twetho triple 8ve
|
| vicesimotertial supermajor twentyfirst
|-
| [[31/4]]
| 3545.036
| {{sagittal| \!/ |size=300%}}{{sagittal|ntCbl|size=400%}}&numsp;&numsp;
| c331o1
| thiwo triple 8ve
|
| 31-limit semi-augmented twentyfirst
|-
| [[8/1]]
| 3600.000
| {{sagittal|ntChf|size=400%}}&numsp;&numsp;
| c3w1
| wa triple 8ve
| P22
| 8th harmonic, three octaves
|-
| [[33/4]]
| 3653.273
| {{sagittal| /|\ |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| c31o1
| ilo triple 8ve
|
| undecimal semi-diminished twentythird
|-
| [[25/3]]
| 3670.672
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| c3yy1
| yoyo triple 8ve
|
| classic chromatic twentythird
|-
| [[17/2]]
| 3704.955
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c317o2
| tricoso 2nd
|
| septendecimal minor twentythird
|-
| [[128/15]]
| 3711.731
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c3g2
| tricogu 2nd
| m235
| classic minor twenty-third
|-
| [[26/3]]
| 3738.573
| {{sagittal| (!/ |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c33o2
| tricotho 2nd
|
| tridecimal neutral twentythird
|-
| [[35/4]]
| 3755.140
| {{sagittal| \!) |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c3zy2
| trico-zoyo 2nd
|
| septimal neutral twentythird
|-
| [[9/1]]
| 3803.910
| {{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c3w2
| tricowa 2nd
| M23
| 9th harmonic
|-
| [[10/1]]
| 3986.314
| {{sagittal| \! |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| c3y3
| tricoyo 3rd
| M245
| 10th harmonic, decade
|-
| [[11/1]]
| 4151.318
| {{sagittal| /|\ |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| c31o4
| tricolo 4th
| P2511
| 11th harmonic
|-
| [[12/1]]
| 4301.955
| {{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| c3w5
| tricowa 5th
| P26
| 12th harmonic
|-
| [[13/1]]
| 4440.528
| {{sagittal| /|) |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| c33o6
| tricotho 6th
| m2713
| 13th harmonic
|-
| [[14/1]]
| 4568.826
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| c3z7
| tricozo 7th
| m287
| 14th harmonic
|-
| [[15/1]]
| 4688.269
| {{sagittal| \! |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| c3y7
| tricoyo 7th
| M285
| 15th harmonic
|-
| [[16/1]]
| 4800.000
| {{sagittal|ntChf|size=400%}}&numsp;&numsp;
| c4w1
| wa quadruple 8ve
| P29
| 16th harmonic, four octaves
|-
| [[17/1]]
| 4904.955
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c417o2
| quadcoso 2nd
| m3017
| 17th harmonic
|-
| [[256/15]]
| 4911.731
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c4g2
| quadcogu 2nd
| m305
| classic minor thirtieth
|-
| [[18/1]]
| 5004.910
| {{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c4w2
| quadcowa 2nd
| M30
| 18th harmonic
|-
| [[19/1]]
| 5097.513
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| c419o3
| quadcono 3rd
| m3119
| 19th harmonic
|-
| [[20/1]]
| 5186.314
| {{sagittal| \! |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| c4y3
| quadcoyo 3rd
| M315
| 20th harmonic
|-
| [[21/1]]
| 5270.781
| {{sagittal| !) |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| c4z4
| quadcozo 4th
| P327
| 21st harmonic
|-
| [[22/1]]
| 5351.318
| {{sagittal| /|\ |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| c41o4
| quadcolo 4th
| P3211
| 22nd harmonic
|-
| [[23/1]]
| 5428.274
| {{sagittal| ~|( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| c423o5
| quadco-twetho 5th
| A3223
| 23rd harmonic
|-
| [[24/1]]
| 5501.955
| {{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| c4w5
| quadcowa 5th
| P33
| 24th harmonic
|-
| [[25/1]]
| 5572.617
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| c4yy5
| quadco-yoyo 5th
| A335,5
| 25th harmonic
|-
| [[27/1]]
| 5705.865
| {{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| c4w6
| quadcowa 6th
| M34
| 27th harmonic
|-
| [[29/1]]
| 5829.577
| {{sagittal| (| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBbl|size=400%}}&numsp;&numsp;
| c429o7
| quadco-tweno 7th
| m3529
| 29th harmonic
|-
| [[31/1]]
| 5945.036
| {{sagittal| \!/ |size=300%}}{{sagittal|ntCbl|size=400%}}&numsp;&numsp;
| c431o7
| quadco-thiwo 7th
| M3531
| 31st harmonic
|-
| [[49/1]]
| 6737.652
| {{sagittal| \!/ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|c5zz6
|quinco-zozo 6th
| M417,7
| 49th harmonic
|}
Notes<references group=note />

== See also ==
* [[List of superparticular intervals]]
* [[:Category:Lists of intervals]] - Other galleries of intervals with different inclusion criteria to this one

== Links ==
* [http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt Regions of the Interval Spectrum] by [[Margo Schulter]] [https://www.webcitation.org/5xeoz4zmC Permalink]
* [http://www.huygens-fokker.org/docs/intervals.html Manuel Op de Coul interval list]
* [http://www.kylegann.com/Octave.html Anantomy of an Octave] by [[Kyle Gann]]
* [http://www.tallkite.com/AlternativeTunings.html Alternative Tunings: Theory, Notation and Practice] by [[Kite Giedraitis]]
* [https://misotanni.github.io/fjs/en/calc.html FJS interval calculators]
* [http://www.huygens-fokker.org/docs/intervals.html Stichting Huygens-Fokker: List of intervals]

[[Category:Lists]]
[[Category:Lists of intervals| ]]
[[Category:Just intonation]]

Small comma

2026-04-29T00:35:54Z

TallKite: removed extraneous -a- delimiters

A '''small comma''' is a [[comma]] whose size is approximately between 3.5 and 30 cents. These intervals are in the range from just noticeable up to usable as melodic steps. The actual perception of course varies. In [[Sagittal notation]], intervals in the smaller part of this category are [[kleisma (interval region)|kleismas]], and intervals in the larger part of this category are [[comma (interval region)|commas]]<ref>[https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>.

For commas over 100 cents in size, see [[Large comma]]; for commas in between 30 and 100 cents in size, see [[Medium comma]]; and for commas under 3.5 cents in size, see [[Unnoticeable comma]].

Due to the wide range of sizes, cents values here and on the other commas pages are listed to 5 significant digits, instead of to a fixed number of decimal places as is the [[Xenharmonic Wiki: Conventions|convention]] elsewhere on the wiki.

The commas might be numerous, but there is no need to memorise all the names. For pretty much all use cases, it is perfectly acceptable to just refer to a comma by the monzo or ratio, whichever is simpler.

== List of commas, by prime limit ==
=== 3-limit commas ===
{{See also| Table of 3-limit commas }}

{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cents]]
! Named by
|-
| 241-comma
| 241wama
| 241wM
|
| {{Monzo| 382 -241 }}
| 28.845
|
|-
| 65-comma, Pythagorean septimal comma
| 65wama, Thequiwama
| 65wM
|
| {{Monzo| -103 65 }}
| 27.075
|
|-
| [[Pythagorean comma]]
| Lalawama, Poma
| LLwM
| 531441 / 524288
| {{Monzo| -19 12 }}
| 23.460
|
|-
| [[41-comma]], Pythagorean countercomma, countercomp comma
| 41wama, Fowewama
| 41wM
| <abbr title="36893488147419103232 / 36472996377170786403">(40 digits)</abbr>
| {{Monzo| 65 -41 }}
| 19.845
| [[Flora Canou]] (2022)
|-
| [[94-comma]], garistearn comma
| 94wama, Fosebiwama
| 94wM
|
| {{Monzo| 149 -94 }}
| 16.230
| [[User:Xenllium|Xenllium]] (2021)
|-
| 147-comma
| 147wama
| 147wM
|
| {{Monzo| 233 -147 }}
| 12.615
|
|-
| 200-comma, Pythagorean integer-cent ET comma
| 200wama
| 200wM
|
| {{Monzo| 317 -200 }}
| 8.9998
|
|-
| 253-comma
| 253wama
| 253wM
|
| {{Monzo| 401 -253 }}
| 5.3848
|
|-
| [[Mercator's comma]], 53-comma
| 53wama, Fithewama
| 53wM
| <abbr title="19383245667680019896796723 / 19342813113834066795298816">(52 digits)</abbr>
| {{Monzo| -84 53 }}
| 3.6150
|
|}

=== 5-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cents]]
! Named by
|-
| [[Magic comma]], small diesis
| Laquinyoma
| L5yM
| 3125 / 3072
| {{Monzo| -10 -1 5 }}
| 29.614
|
|-
| [[Triscordial comma]]
| Tribila-triyoma
| 6L3yM
| <abbr title="18761829412124890125 / 18446744073709551616">(40 digits)</abbr>
| {{Monzo| -64 36 3 }}
| 29.321
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Hendecatonic comma]]
| Trisa-leguma
| 3s11gM
| 8796093022208 / 8649755859375
| {{Monzo| 43 -11 -11 }}
| 29.044
|
|-
| [[Devil's tridecalimma]]
| Lala-theguma
| LL13gM
| 2541865828329 / 2500000000000
| {{Monzo| -11 26 -13 }}
| 28.752
|
|-
| [[Anthoine comma]]
| Trila-quinquadyoma
| 3L20yM
| 286102294921875 / 281474976710656
| {{Monzo| -48 1 20 }}
| 28.229
|
|-
| [[Tetracot comma]], minimal diesis
| Saquadyoma
| s4yM
| 20000 / 19683
| {{Monzo| 5 -9 4 }}
| 27.660
|
|-
| [[Biscordial comma]]
| Quadla-yoyoma
| 4LyyM
| 571919811374025 / 562949953421312
| {{Monzo| -49 28 2 }}
| 27.367
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Semaja comma]]
| Lala-neyoma
| LL19yM
| 19073486328125 / 18786186952704
| {{Monzo| -33 -7 19 }}
| 26.276
| [[Petr Pařízek]] (2011)
|-
| [[Quanic comma]]
| Sepsa-quinyoma
| 7s5yM
|
| {{Monzo| 74 -54 5 }}
| 25.999
|
|-
| [[Roda]], rodan comma
| Sasa-triyoma
| ss3yM
| 131072000 / 129140163
| {{Monzo| 20 -17 3 }}
| 25.706
|
|-
| [[Gracecordial comma]]
| Trilayoma
| 3LyM
| 17433922005 / 17179869184
| {{Monzo| -34 20 1 }}
| 25.414
|
|-
| [[Trisedodge comma]]
| Saquintriguma
| s15gM
| 30958682112 / 30517578125
| {{Monzo| 19 10 -15 }}
| 24.844
| [[Petr Pařízek]] (2011)
|-
| [[Birds comma]]
| Quadsa-thiweguma
| 4s31gM
|
| {{Monzo| 72 0 -31 }}
| 24.275
|
|-
| [[Neuk comma]]
| Trisa-yoyoma
| 3syyM
| 858993459200 / 847288609443
| {{Monzo| 35 -25 2 }}
| 23.752
| [[User:CompactStar|CompactStar]] (2023)
|-
| [[Maja comma]]
| Saseyoma
| s17yM
| 762939453125 / 753145430616
| {{Monzo| -3 -23 17 }}
| 22.368
| [[Petr Pařízek]] (2011)
|-
| [[Satin comma]]
| Quinbisa-triyoma
| 10s3yM
|
| {{Monzo| 104 -70 3 }}
| 22.091
|
|-
| [[Misneb comma]]
| Quadla-quintriyoma
| 4L15yM
| <abbr title="145964630126953125 / 144115188075855872">(36 digits)</abbr>
| {{Monzo| -57 14 15 }}
| 22.076
| [[Petr Pařízek]] (2011)
|-
| [[Syntonic comma]], Didymus comma, meantone comma
| Guma
| gM
| 81 / 80
| {{Monzo| -4 4 -1 }}
| 21.506
|
|-
| [[Maquila comma]]
| Trisa-seguma
| 3s17gM
| 562949953421312 / 556182861328125
| {{Monzo| 49 -6 -17 }}
| 20.937
| [[Petr Pařízek]] (2011)
|-
| [[Sfourth comma]]
| Lala-neguma
| LL19gM
| 617673396283947 / 610351562500000
| {{Monzo| -5 31 -19 }}
| 20.644
| [[Petr Pařízek]] (2011)
|-
| [[Diaschisma]]
| Saguguma
| sggM
| 2048 / 2025
| {{Monzo| 11 -4 -2 }}
| 19.553
| Hermann von Helmholtz, Alexander Ellis (1875)
|-
| [[Countermeantone comma]]
| Quinquadguma
| 20gM
| 96402615118848 / 95367431640625
| {{Monzo| 10 23 -20 }}
| 18.691
| [[Petr Pařízek]] (2012)
|-
| [[Ditonma]]
| Lala-theyoma
| LL13yM
| 1220703125 / 1207959552
| {{Monzo| -27 -2 13 }}
| 18.168
| [[Petr Pařízek]] (2011)
|-
| [[Misty comma]]
| Sasa-triguma
| ss3gM
| 67108864 / 66430125
| {{Monzo| 26 -12 -3 }}
| 17.599
| [[Paul Erlich]] (2002)
|-
| [[Quintile comma]]
| Trila-quinguma
| 3L5gM
| 847288609443 / 838860800000
| {{Monzo| -28 25 -5 }}
| 17.306
| See the dedicated page.
|-
| [[Enneadecacot comma]]
| Tribisa-neguma
| 6s19gM
| <abbr title="604462909807314587353088 / 598546211414337158203125">(48 digits)</abbr>
| {{monzo| 79 -22 -19 }}
| 17.029
| [[User:CompactStar|CompactStar]] (2024)
|-
| [[Oquatonic comma]]
| Quadla-sepquadyoma
| 4L28yM
| <abbr title="37252902984619140625 / 36893488147419103232">(40 digits)</abbr>
| {{Monzo| -65 0 28 }}
| 16.784
| [[Petr Pařízek]] (2011)
|-
| [[Undim comma]]
| Trisa-quadguma
| 3s4gM
| 2199023255552 / 2179240250625
| {{Monzo| 41 -20 -4 }}
| 15.645
| [[Petr Pařízek]] (2011)
|-
| [[Graviton]], gravity comma
| Lala-tribiguma
| LL6gM
| 129140163 / 128000000
| {{Monzo| -13 17 -6 }}
| 15.353
| [[Mike Battaglia]] (2011)
|-
| [[Majvam comma]]
| Sasa-lebiguma
| ss22gM
| <abbr title="2404631929946112 / 2384185791015625">(32 digits)</abbr>
| {{Monzo| 40 7 -22 }}
| 14.783
| [[Petr Pařízek]] (2011)
|-
| [[Quartonic comma]]
| Saleyoma
| s11yM
| 390625000 / 387420489
| {{Monzo| 3 -18 11 }}
| 14.261
|
|-
| [[Untritonic comma]]
| Quadla-tritriyoma
| 4L9yM
| <abbr title="2270041927734375 / 2251799813685248">(32 digits)</abbr>
| {{Monzo| -51 19 9 }}
| 13.968
| [[Petr Pařízek]] (2011)
|-
| [[Quindromeda comma]]
| Quinsa-quinguma
| 5s5gM
| <abbr title="72057594037927936 / 71489976421753125">(34 digits)</abbr>
| {{Monzo| 56 -28 -5 }}
| 13.691
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Sensipent comma]], medium semicomma
| Sepguma
| 7gM
| 78732 / 78125
| {{Monzo| 2 9 -7 }}
| 13.399
|
|-
| [[Copper comma]]
| Theneyoma
| 41L29yM
|
| {{Monzo| -481 261 29 }}
| 13.353
| [[Eliora]] (2023)
|-
| [[Counterwürschmidt comma]]
| Trisa-twetheguma
| 3s23gM
| <abbr title="36028797018963968 / 35762786865234375">(34 digits)</abbr>
| {{Monzo| 55 -1 -23 }}
| 12.830
| [[Flora Canou]] (2023)
|-
| [[Tertiosec comma]]
| Laquadtribiyoma
| 6L24yM
|
| {{Monzo| -89 21 24 }}
| 12.584
| [[Petr Pařízek]] (2012)
|-
| [[Submajor comma]]
| Trila-quadbiyoma
| 3L8yM
| 69198046875 / 68719476736
| {{Monzo| -36 11 8 }}
| 12.015
|
|-
| [[Würschmidt comma]]
| Saquadbiguma
| s8gM
| 393216 / 390625
| {{Monzo| 17 1 -8 }}
| 11.445
|
|-
| [[Bicommatic comma]]
| Quadla-quinbiguma
| 4L10gM
| <abbr title="1350851717672992089 / 1342177280000000000">(38 digits)</abbr>
| {{Monzo| -37 38 -10 }}
| 11.153
|
|-
| [[Counterhanson comma]]
| Quinquinyoma
| 25yM
| <abbr title="298023223876953125 / 296148833645101056">(36 digits)</abbr>
| {{Monzo| -20 -24 25 }}
| 10.923
| [[Petr Pařízek]] (2012)
|-
| [[Countritonic comma]]
| Quadsa-tritriyoma
| 4s9yM
| <abbr title="16777216000000000 / 16677181699666569">(34 digits)</abbr>
| {{Monzo| 33 -34 9 }}
| 10.353
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Semicomma]], Fokker's comma
| Lasepyoma
| L7yM
| 2109375 / 2097152
| {{Monzo| -21 3 7 }}
| 10.061
|
|-
| [[Heptacot comma]]
| Sepsa-sepguma
| 7s7gM
|
| {{Monzo| 86 -44 -7 }}
| 9.7840
| [[Tristan Bay]] (2024)
|-
| [[Escapade comma]]
| Sasa-tritriguma
| ss9gM
| 4294967296 / 4271484375
| {{Monzo| 32 -7 -9 }}
| 9.4916
| [[Paul Erlich]] (2002)
|-
| [[Undetritisma]], twentcufo comma
| Trila-leguma
| 3L11gM
| 205891132094649 / 204800000000000
| {{Monzo| -22 30 -11 }}
| 9.1992
|
|-
| [[15625/15552|Kleisma]], semicomma majeur
| Tribiyoma
| 6yM
| 15625 / 15552
| {{Monzo| -6 -5 6 }}
| 8.1073
| {{W|Shohé Tanaka}} (1890)
|-
| [[Quintosec comma]]
| Quadsa-quinbiguma
| 4s10gM
| 140737488355328 / 140126044921875
| {{Monzo| 47 -15 -10 }}
| 7.5378
| [[Petr Pařízek]] (2012)
|-
| 59-5-comma
| Quadbisa-fineguma
| 8s59gM
|
| {{Monzo| 137 0 -59 }}
| 7.4909
|
|-
| [[Unidecma]]
| Laquadtriguma
| L12gM
| 31381059609 / 31250000000
| {{Monzo| -7 22 -12 }}
| 7.2455
|
|-
| [[Mutt comma]]
| Trila-septriyoma
| 3L21yM
| 476837158203125 / 474989023199232
| {{Monzo| -44 -3 21 }}
| 6.7230
| [[Gene Ward Smith]] (2004)
|-
| [[Sulfur comma]]
| Lela-quadquadguma
| 11L16gM
|
| {{Monzo| -115 96 -16 }}
| 6.6607
|
|-
| [[Amity comma]]
| Saquinyoma
| s5yM
| 1600000 / 1594323
| {{Monzo| 9 -13 5 }}
| 6.1536
| [[Gene Ward Smith]] (2001)
|-
| [[Parakleisma]]
| Theguma
| 13gM
| 1224440064 / 1220703125
| {{Monzo| 8 14 -13 }}
| 5.2917
| [[Gene Ward Smith]] (2001)
|-
| [[Gammic comma]]
| Laquinquadyoma
| L20yM
| 95367431640625 / 95105071448064
| {{Monzo| -29 -11 20 }}
| 4.7693
|
|-
| [[Squarschmidt comma]]
| Quadsa-tweneguma
| 4s29gM
| <abbr title="186773283746309210112 / 186264514923095703125">(42 digits)</abbr>
| {{Monzo| 61 4 -29 }}
| 4.7223
| [[Petr Pařízek]] (2012)
|-
| 43-15-comma, Huntian 15-cycle comma
| Quadtrisa-fotheguma
| 12s43gM
|
| {{Monzo| 168 -43 -43 }}
| 4.4453
|
|-
| [[Barium comma]]
| Quadtribila-sepquadbiguma
| 24L56gM
|
| {{Monzo| -225 224 -56 }}
| 4.3522
| [[Eliora]] (2023)
|-
| [[Vulture comma]]
| Sasa-quadyoma
| ss4yM
| 10485760000 / 10460353203
| {{Monzo| 24 -21 4 }}
| 4.1998
| [[Paul Erlich]] (2002)
|-
| [[Dipromethia]]
| Thebila-siweyoma
| 26L61yM
|
| {{Monzo| -335 122 61 }}
| 3.6467
| [[Eliora]] (2023)
|-
| [[Lafa comma]]
| Tribisa-quadtriguma
| 6s12gM
|
| {{Monzo| 77 -31 -12 }}
| 3.6304
| [[Petr Pařízek]] (2011)
|}

=== 7-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cents]]
! Named by
|-
| [[3645/3584|Schismean comma]]
| Laruyoma
| LryM
| 3645 / 3584
| {{Monzo| -9 6 1 -1 }}
| 29.218
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Doublehearted comma]]
| Quadbizoma
| 8zM
| 5764801 / 5668704
| {{Monzo| -5 -11 0 8 }}
| 29.102
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Frostburn comma]]
| Quadru-asepyoma
| 4ra7yM
| 78125 / 76832
| {{Monzo| -5 0 7 -4 }}
| 28.892
|
|-
| [[686/675|Senga]]
| Trizo-aguguma
| 3zaggM
| 686 / 675
| {{Monzo| 1 -3 -2 3 }}
| 27.985
| [[Gene Ward Smith]] (2005)
|-
| 23-21-comma
| Sepla-twethezoma
| 7L23zM
|
| {{Monzo| -101 23 0 23 }}
| 27.961
|
|-
| [[64/63|Septimal comma]], Archytas' comma, Leipziger Komma
| Ruma
| rM
| 64 / 63
| {{Monzo| 6 -2 0 -1 }}
| 27.264
|
|-
| [[Mandos comma]]
| Biruguguma
| 2rggM
| 31104 / 30625
| {{Monzo| 7 5 -4 -2 }}
| 26.868
|
|-
| [[Slither comma]]
| Satritriru-aquadyoma
| s9ra4yM
| 40960000 / 40353607
| {{Monzo| 16 0 4 -9 }}
| 25.822
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Bastille comma]]
|
|
|
| {{Monzo| 1426 0 -596 -15 }}
| 24.638
| [[Eliora]] (2023)
|-
| 33-7/5-comma
| Letrizoguma
| 33zgM
|
| {{Monzo| -16 0 -33 33 }}
| 22.902
|
|-
| Huntian 35-cycle comma
| Quintrisa-tritritribiruguma
| 15s54rgM
|
| {{Monzo| 277 0 -54 -54 }}
| 22.461
|
|-
| [[Blackjackisma]]
| Lasepru-aquadbiyoma
| L7ra8yM
| 854296875 / 843308032
| {{Monzo| -10 7 8 -7 }}
| 22.413
| [[Gene Ward Smith]] (2011)
|-
| [[Squalentine comma]]
| Laquadzo-atriguma
| L4za3gM
| 64827 / 64000
| {{Monzo| -9 3 -3 4 }}
| 22.227
|
|-
| [[875/864|Keema]]
| Zotriyoma
| z3yM
| 875 / 864
| {{Monzo| -5 -3 3 1 }}
| 21.902
| [[Gene Ward Smith]] (2005)
|-
| [[Betelgeuse comma]]
| Satritrizo-aguguma
| s9zaggM
| 40353607 / 39858075
| {{Monzo| 0 -13 -2 9 }}
| 21.391
|
|-
| [[3125/3087|Gariboh comma]]
| Triru-aquinyoma
| 3ra5yM
| 3125 / 3087
| {{Monzo| 0 -2 5 -3 }}
| 21.181
| [[Gene Ward Smith]] (2005)
|-
| [[Secanticornisma]]
| Laruquinguma
| Lr5gM
| 177147 / 175000
| {{Monzo| -3 11 -5 -1 }}
| 21.111
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[2430/2401|Nuwell comma]]
| Quadru-ayoma
| 4rayM
| 2430 / 2401
| {{Monzo| 1 5 1 -4 }}
| 20.785
| [[Gene Ward Smith]] (2005)
|-
| [[Septimagic comma]]
| Saquinzoma
| s5zM
| 537824 / 531441
| {{Monzo| 5 -12 0 5 }}
| 20.670
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Mermisma]]
| Sepruyoma
| 7ryM
| 2500000 / 2470629
| {{Monzo| 5 -1 7 -7 }}
| 20.460
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Negricorn comma]], small quadruple bluish
| Saquadzoguma
| s4zgM
| 153664 / 151875
| {{monzo| 6 -5 -4 4 }}
| 20.274
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Tolerant comma]]
| Sazoyoyoma
| szyyM
| 179200 / 177147
| {{Monzo| 10 -11 2 1 }}
| 19.948
|
|-
| [[Icosipentatonic comma]], 25-36/35-comma
| Quinquinruguma
| 25rgM
|
| {{Monzo| 49 50 -25 -25 }}
| 19.260
|
|-
| [[Valenwuer comma]]
| Sarutribiguma
| sr6gM
| 110592 / 109375
| {{Monzo| 12 3 -6 -1 }}
| 19.157
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Buzzardsma]], buzzard comma
| Saquadruma
| s4rM
| 65536 / 64827
| {{Monzo| 16 -3 0 -4 }}
| 18.831
| See the page.
|-
| Huntian 21-cycle comma
| Quadbisa-sepquadruma
| 8s28rM
|
| {{Monzo| 123 -28 0 -28 }}
| 18.135
|
|-
| [[Mirwomo comma]]
| Labizoyoma
| L2zyM
| 33075 / 32768
| {{Monzo| -15 3 2 2 }}
| 16.144
|
|-
| [[Catasyc comma]]
| Laruquadbiyoma
| Lr8yM
| 390625 / 387072
| {{Monzo| -11 -3 8 -1 }}
| 15.819
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Compass comma]]
| Quinruyoyoma
| 5ryyM
| 9765625 / 9680832
| {{monzo| -6 -2 10 -5 }}
| 15.098
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Sensibeta comma]]
| Satrizo-aquinyoma
| s3za5yM
| 1071875 / 1062882
| {{monzo| -1 -12 5 3 }}
| 14.586
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Trimyna comma]]
| Quinzoguma
| 5zgM
| 50421 / 50000
| {{monzo| -4 1 -5 5 }}
| 14.516
|
|-
| [[245/243|Sensamagic comma]]
| Zozoyoma
| zzyM
| 245 / 243
| {{monzo| 0 -5 1 2 }}
| 14.191
| [[Gene Ward Smith]] (2010)
|-
| [[126/125|Starling comma]], septimal semicomma
| Zotriguma
| z3gM
| 126 / 125
| {{Monzo| 1 2 -3 1 }}
| 13.795
|
|-
| [[Vermeil comma]], 34-49/48-comma
| Quinla-sequadzoma
| 5L68zM
|
| {{Monzo| -137 -34 0 68 }}
| 13.692
| [[User:Perry.k|Perry.k]] (2026)
|-
| [[4000/3969|Octagar comma]]
| Rurutriyoma
| rr3yM
| 4000 / 3969
| {{Monzo| 5 -4 3 -2 }}
| 13.469
| [[Gene Ward Smith]] (2005)
|-
| [[1728/1715|Orwellisma]]
| Triru-aguma
| 3ragM
| 1728 / 1715
| {{Monzo| 6 3 -1 -3 }}
| 13.074
|
|-
| [[Mynaslender comma]]
| Sepru-ayoma
| 7rayM
| 829440 / 823543
| {{Monzo| 11 4 1 -7 }}
| 12.352
| [[User:Xenllium|Xenllium]] (2021)
|-
| 35-7/5-comma
| Sepquinruyoma
| 35ryM
|
| {{monzo| 17 0 35 -35 }}
| 12.073
|
|-
| [[Chromatisma]]
| Trisa-triru-aquadquadyoma
| 3s3ra16yM
| <abbr title="640000000000000000 / 635585924776181463">(36 digits)</abbr>
| {{Monzo| 22 -32 16 -3 }}
| 11.982
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Mistisma]]
| Sazoquadguma
| sz4gM
| 458752 / 455625
| {{Monzo| 16 -6 -4 1 }}
| 11.841
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Bronzisma]]
| Satriru-aguguma
| s3raggM
| 2097152 / 2083725
| {{Monzo| 21 -5 -2 -3 }}
| 11.120
| [[User:Xenllium|Xenllium]] (2021)
|-
| 34-jubilismic comma
| Sequadzoguma
| 68zgM
|
| {{Monzo| -33 0 -68 68 }}
| 10.829
|
|-
| [[Fynn's comma]], 26-7-comma, Hunt 7-cycle comma
| Quadsa-thebiruma
| 4s26rM
| <abbr title="9444732965739290427392 / 9387480337647754305649">(44 digits)</abbr>
| {{Monzo| 73 0 0 -26 }}
| 10.526
| [[Fynn Cerulean]] (2026) for ''Fynn's comma''
|-
| [[Septiness comma]]
| Sasasepruma
| ss7rM
| 67108864 / 66706983
| {{Monzo| 26 -4 0 -7 }}
| 10.399
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[31-comma temperaments|31-35-comma]]
| Tritrila-thiwezoyoma
| 9L31zyM
|
| {{Monzo| -159 0 31 31 }}
| 9.3282
|
|-
| [[Quince comma]]
| Lasepzo-aguguma
| L7zaggM
| 823543 / 819200
| {{Monzo| -15 0 -2 7 }}
| 9.1539
|
|-
| [[Uniwiz comma]]
| Quadzoyoma
| 4zyM
| 1500625 / 1492992
| {{Monzo| -11 -6 4 4 }}
| 8.8285
|
|-
| [[Historisma]]
| Latribizoguma
| L6zgM
| 257298363 / 256000000
| {{Monzo| -14 7 -6 6 }}
| 8.7582
| [[ArrowHead294]] (2024)
|-
| [[1029/1024|Gamelisma]]
| Latrizoma
| L3zM
| 1029 / 1024
| {{Monzo| -10 1 0 3 }}
| 8.4327
|
|-
| [[Varunisma]]
| Labizoguguma
| L2zggM
| 321489 / 320000
| {{Monzo| -9 8 -4 2 }}
| 8.0370
|
|-
| [[225/224|Marvel comma]], septimal kleisma
| Ruyoyoma
| ryyM
| 225 / 224
| {{Monzo| -5 2 2 -1 }}
| 7.7115
| [[Gene Ward Smith]] (2003)
|-
| [[Dimcomp comma]]
| Quadruyoyoma
| 4ryyM
| 390625 / 388962
| {{Monzo| -1 -4 8 -4 }}
| 7.3861
|
|-
| [[Cataharry comma]]
| Labiruguma
| L2rgM
| 19683 / 19600
| {{Monzo| -4 9 -2 -2 }}
| 7.3158
| [[Gene Ward Smith]] (2005)
|-
| [[Procyon comma]]
| Sasepzo-atriguma
| s7za3gM
| 823543 / 820125
| {{Monzo| 0 -8 -3 7 }}
| 7.2002
|
|-
| [[Qiqi comma]]
| Sepruyoyoma
| 7ryyM
| 48828125000 / 48629390607
| {{Monzo| 3 -10 14 -7 }}
| 7.0606
| [[User:Angekallikoita|Ange]] (2025)
|-
| [[Mirkwai comma]]
| Quinru-aquadyoma
| 5ra4yM
| 16875 / 16807
| {{Monzo| 0 3 4 -5 }}
| 6.9903
| [[Gene Ward Smith]] (2005)
|-
| [[Canousma]]
| Saquadzo-atriyoma
| s4za3yM
| 4802000 / 4782969
| {{Monzo| 4 -14 3 4 }}
| 6.8748
| [[Flora Canou]] (2020)
|-
| [[Triwellisma]]
| Tribizo-asepguma
| 6za7gM
| 235298 / 234375
| {{Monzo| 1 -1 -7 6 }}
| 6.8044
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Stearnsma]]
| Latribiruma
| L6rM
| 118098 / 117649
| {{Monzo| 1 10 0 -6 }}
| 6.5946
|
|-
| [[10976/10935|Hemimage comma]]
| Satrizo-aguma
| s3zagM
| 10976 / 10935
| {{Monzo| 5 -7 -1 3 }}
| 6.4790
| [[Gene Ward Smith]] (2005)
|-
| [[3136/3125|Hemimean comma]]
| Zozoquinguma
| zz5gM
| 3136 / 3125
| {{Monzo| 6 0 -5 2 }}
| 6.0832
| [[Gene Ward Smith]] (2005)
|-
| [[5120/5103|Hemifamity comma]], 5/7-kleisma
| Saruyoma
| sryM
| 5120 / 5103
| {{Monzo| 10 -6 1 -1 }}
| 5.7578
| [[Gene Ward Smith]] (2005)
|-
| [[Parkleiness comma]]
| Zotritriguma
| z9gM
| 1959552 / 1953125
| {{Monzo| 7 7 -9 1 }}
| 5.6875
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Octaphore comma]], enneagari comma
| Sasa-quadbizoma
| ss8zM
| 94450499584 / 94143178827
| {{Monzo| 14 -23 0 8 }}
| 5.6422
| [[User:Unque|Unque]] (2024)
|-
| [[Linus comma]]
| Saquinbizoguma
| s10zgM
| 578509309952 / 576650390625
| {{Monzo| 11 -10 -10 10 }}
| 5.5719
| [[Gene Ward Smith]] (2011)
|-
| [[Reiwa comma]]
| Saquadru-asepyoma
| s4ra7yM
| 1280000000 / 1275989841
| {{monzo| 14 -12 7 -4 }}
| 5.4324
| [[Flora Canou]] (2022)
|-
| [[6144/6125|Porwell comma]]
| Sarurutriguma
| srr3gM
| 6144 / 6125
| {{Monzo| 11 1 -3 -2 }}
| 5.3620
| [[Gene Ward Smith]], [[Petr Pařízek]] (2010)
|-
| [[Acromagic comma]]
| Sasa-sepzo-aquadguma
| ss7za4gM
| 26985857024 / 26904200625
| {{Monzo| 15 -16 -4 7 }}
| 5.2466
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Cartoonisma]]
| Satritrizo-asepbiguma
| s9za14gM
| 165288374272 / 164794921875
| {{Monzo| 12 -3 -14 9 }}
| 5.1762
| [[Eliora]] (2023)
|-
| [[Hemfiness comma]]
| Saquadru-atriyoma
| s4ra3yM
| 4096000 / 4084101
| {{Monzo| 15 -5 3 -5 }}
| 5.0366
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Acrodec comma]]
| Sasa-tribizo-aquadbiguma
| ss6za8gM
| 7710244864 / 7688671875
| {{Monzo| 16 -9 -8 6 }}
| 4.8507
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Hewuermera comma]]
| Satribiru-aguma
| s6ragM
| 589824 / 588245
| {{Monzo| 16 2 -1 -6 }}
| 4.6408
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Hearts comma]]
| Trila-quadzoma
| 3L4zM
| 34451725707 / 34359738368
| {{Monzo| -35 15 0 4 }}
| 4.6286
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Lokisma]], loki comma
| Sasa-bizotriguma
| ss2z3gM
| 102760448 / 102515625
| {{Monzo| 21 -8 -6 2 }}
| 4.1295
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Garischisma]], septimal schisma
| Sasaruma
| ssrM
| 33554432 / 33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.8041
|
|-
| [[Wadisma]]
| Latritrizo-ayoma
| L9zayM
| 201768035 / 201326592
| {{Monzo| -26 -1 1 9 }}
| 3.7919
|
|-
| [[Septimal enneadeca]]
| Quinla-neruma
| 5L19rM
| <abbr title="1570042899082081611640534563 / 1566652225014704215735402496">(56 digits)</abbr>
| {{Monzo| -37 57 0 -19 }}
| 3.7428
| [[Godtone]] (2024)
|-
| [[Quasiorwellisma]]
| Sazoquinbiguma
| sz10gM
| 29360128 / 29296875
| {{Monzo| 22 -1 -10 1 }}
| 3.7338
|
|}

=== 11-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[Dew comma]]
| Saloma
| s1oM
| 180224 / 177147
| {{Monzo| 14 -11 0 0 1 }}
| 29.812
| [[User:CompactStar|CompactStar]] (2023)
|-
| [[Thuja comma]]
| Saquinlu-ayoma
| s5(1u)yM
| 163840 / 161051
| {{Monzo| 15 0 1 0 -5 }}
| 29.724
|
|-
| [[625/616|Quadrikite comma]]
| Luruquadyoma
| 1ur4yM
| 625 / 616
| {{Monzo| -3 0 4 -1 -1 }}
| 25.111
| [[Praveen Venkataramana]], [[Lumi Pakkanen]] (2022)
|-
| [[1350/1331|Large tetracot diesis]]
| Trilu-ayoyoma
| 3(1u)yyM
| 1350 / 1331
| {{Monzo| 1 3 2 0 -3 }}
| 24.539
| [[User:ArrowHead294|ArrowHead294]] (2024)
|-
| [[Sensmus comma]]
| Salozoguma
| s1ozgM
| 1232 / 1215
| {{Monzo| 4 -5 -1 1 1 }}
| 24.055
|
|-
| [[Sevnothrush comma]]
| Loquinguma
| 1o5gM
| 3168 / 3125
| {{Monzo| 5 2 -5 0 1 }}
| 23.659
|
|-
| [[245/242|Frostma]]
| Biluzo-ayoma
| 2(1uz)yM
| 245 / 242
| {{Monzo| -1 0 1 2 -2 }}
| 21.330
| [[Praveen Venkataramana]] (2022)
|-
| [[Distarma]]
| Trilozoma
| 3(1o)zM
| 9317 / 9216
| {{Monzo|-10 -2 0 1 3}}
| 18.869
| [https://twitter.com/Lilly__Flores Lilly Flores] (2023)
|-
| [[1617/1600|Antimisma]]
| Lobizoguma
| 1o2zgM
| 1617 / 1600
| {{Monzo| -6 1 -2 2 1 }}
| 18.297
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[99/98|Mothwellsma]]
| Loruruma
| 1orrM
| 99 / 98
| {{Monzo| -1 2 0 -2 1 }}
| 17.576
|
|-
| [[1610510/1594323|Fifthchromisma]]
| Saquinlo-ayoma
| s5(1o)yM
| 1610510 / 1594323
| {{Monzo| 1 -13 1 0 5 }}
| 17.488
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[100/99|Ptolemisma]]
| Luyoyoma
| 1uyyM
| 100 / 99
| {{Monzo| 2 -2 2 0 -1 }}
| 17.399
|
|-
| [[Hemimin comma]]
| Trilu-azoma
| 3(1u)zM
| 1344 / 1331
| {{Monzo| 6 1 0 1 -3 }}
| 16.827
|
|-
| [[Betarabian comma]]
| Laloloma
| L1ooM
| 264627 / 262144
| {{Monzo| -18 7 0 0 2 }}
| 16.321
| [[Dawson Berry]] (2020)
|-
| [[Biyatisma]]
| Lologuma
| 1oogM
| 121 / 120
| {{Monzo| -3 -1 -1 0 2 }}
| 14.367
| [[Gene Ward Smith]] (2010)
|-
| [[Absinthma]]
| Luluruyoma
| 1uuryM
| 2560 / 2541
| {{Monzo| 9 -1 1 -1 -2 }}
| 12.897
| [[User:Jerdle|Jerdle]] (2023)
|-
| [[2835/2816|35/11 kleisma]]
| Laluzoyoma
| L1uzyM
| 2835 / 2816
| {{Monzo| -8 4 1 1 -1 }}
| 11.642
|
|-
| [[Aphrowe comma]]
| Trilo-aruruma
| 3(1o)rrM
| 1331 / 1323
| {{Monzo| 0 -3 0 -2 3 }}
| 10.437
| [[Lériendil]] (2024)
|-
| [[2200/2187|Small tetracot diesis]]
| Saloyoyoma
| s1oyyM
| 2200 / 2187
| {{Monzo| 3 -7 2 0 1 }}
| 10.260
| [[User:ArrowHead294|ArrowHead294]]
|-
| [[Valinorsma]]
| Loruguguma
| 1orggM
| 176 / 175
| {{Monzo| 4 0 -2 -1 1 }}
| 9.8646
|
|-
| [[Pentacircle comma]]
| Saluzoma
| s1uzM
| 896 / 891
| {{Monzo| 7 -4 0 1 -1 }}
| 9.6880
|
|-
| [[Alpharabian comma]]
| Laquadloma
| L4(1o)M
| 131769 / 131072
| {{Monzo| -17 2 0 0 4 }}
| 9.1818
| [[Dawson Berry]] (2020)
|-
| [[Orgonisma]]
| Satrilu-aruruma
| s3(1u)rrM
| 65536 / 65219
| {{Monzo| 16 0 0 -2 -3 }}
| 8.3944
|
|-
| [[Quindecic comma]]
| Sasa-quintriloruma
| ss15(1or)M
|
| {{Monzo| 14 -15 0 -15 15 }}
| 8.0555
|
|-
| [[117649/117128]]
| Bilulutrizoma
| 2(1uu3z)M
| 117649 / 117128
| {{Monzo| -3 0 0 6 -4 }}
| 7.6837
|
|-
| [[Topsy comma]]
| Quadlo-atrizo-asepguma
| 4(1o)3za7gM
| 5021863 / 5000000
| {{Monzo| -6 0 -7 3 4 }}
| 7.5535
|
|-
| [[4375/4356|Fantares comma]]
| Luluzoquadyoma
| 1uuz4yM
| 4375 / 4356
| {{Monzo| -2 -2 4 1 -2 }}
| 7.5349
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Semicanousma]]
| Quadlo-aguma
| 4(1o)gM
| 14641 / 14580
| {{Monzo| -2 -6 -1 0 4 }}
| 7.2281
| [[Flora Canou]] (2020)
|-
| [[243/242|Rastma]]
| Luluma
| 1uuM
| 243 / 242
| {{Monzo| -1 5 0 0 -2 }}
| 7.1391
|
|-
| [[3388/3375|Myhemiwell comma]]
| Lolozotriguma
| 1ooz3gM
| 3388 / 3375
| {{Monzo| 2 -3 -3 1 2 }}
| 6.6556
|
|-
| [[Pythrabian comma]]
| Trisaloma
| 3s1oM
| 94489280512 / 94143178827
| {{Monzo| 33 -23 0 0 1 }}
| 6.3529
| [[Dawson Berry]] (2021)
|-
| [[Semiporwellisma]]
| Saluluguma
| s1uugM
| 16384 / 16335
| {{Monzo| 14 -3 -1 0 -2 }}
| 5.1854
| [[Flora Canou]] (2021)
|-
| [[Octatonic comma]], undecimal octatonic comma
| Quadbiluma
| 8(1u)M
| 214990848 / 214358881
| {{Monzo| 15 8 0 0 -8 }}
| 5.0965
| [[Eliora]] (2023)
|-
| [[385/384|Keenanisma]]
| Lozoyoma
| 1ozyM
| 385 / 384
| {{Monzo| -7 -1 1 1 1 }}
| 4.5026
| [[Paul Erlich]] (2001)
|-
| [[Trimitone comma]]
| Lalotriguma
| L1o3gM
| 8019 / 8000
| {{Monzo| -6 6 -3 0 1 }}
| 4.1068
| [[User:Godtone|Godtone]] (2021)
|-
| [[4-cent comma]]
| Lutritryoma
| 1u9yM
| 1953125 / 1948617
| {{Monzo| 0 -11 9 0 -1 }}
| 4.0004
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[441/440|Werckisma]]
| Luzozoguma
| 1uzzgM
| 441 / 440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.9302
|
|-
| [[1375/1372|Moctdel comma]]
| Lotriruyo
| 1o3ryM
| 1375 / 1372
| {{Monzo| -2 0 3 -3 1 }}
| 3.7814
|
|-
| [[Unisquarisma]], unisquary comma
| Trilu-aquadzo-ayoma
| 3(1u)4zayM
| 12005 / 11979
| {{Monzo| 0 -2 1 4 -3 }}
| 3.7535
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[6250/6237|Liganellus comma]], liganellisma
| Luruquinyoma
| 1ur5yM
| 6250 / 6237
| {{Monzo| 1 -4 5 -1 -1 }}
| 3.6047
| [[Dawson Berry]] (2020)
|}

=== 13-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color Name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[1600/1573|Cameratasma]]
|
|
| 1600/1573
| {{Monzo| 6 0 2 0 -2 -1 }}
| 29.464
| [[Phlub]] (2026)
|-
| [[Lovecraft comma]]
| Thothotriluma
| 3oo3(1u)M
| 1352/1331
| {{Monzo| 3 0 0 0 -3 2 }}
| 27.101
|
|-
| [[65/64|Wilsorma]]
| Thoyoma
| 3oyM
| 65/64
| {{Monzo| -6 0 1 0 0 1 }}
| 26.841
|
|-
| [[Hyperpyth comma]]
| Quadtho-aquinguma
| 4(3o)5gM
| 28561/28125
| {{Monzo| 0 -2 -5 0 0 4 }}
| 26.632
|
|-
| [[66/65|Winmeanma]]
| Thuloguma
| 3u1ogM
| 66/65
| {{Monzo| 1 1 -1 0 1 -1 }}
| 26.432
|
|-
| [[343/338|Sooty fox comma]]
| Thuthutrizoma
| 3uu3zM
| 343/338
| {{Monzo| -1 0 0 3 0 -2 }}
| 25.422
| [[Budjarn Lambeth]] (2024)
|-
| [[Tetris comma]]
| Sathoma
| s3oM
| 6656/6561
| {{Monzo| 9 -8 0 0 0 1 }}
| 24.888
| [[Godtone]] (2024)
|-
| [[507/500|Large semisixthma]]
| Thothotriguma
| 3oo3gM
| 507/500
| {{Monzo| -2 1 -3 0 0 2 }}
| 24.069
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[78/77|Negustma]]
| Tholuruma
| 3o1urM
| 78/77
| {{Monzo| 1 1 0 -1 -1 1 }}
| 22.339
|
|-
| [[Greater tendoneutralisma]]
| Laquadbithoma
| L8(3o)M
| 815730721 / 805306368
| {{Monzo| -28 -1 0 0 0 8 }}
| 22.266
| [[Godtone]] (2024)
|-
| [[2025/2002|Beyoncisma]]
|
|
| 2025/2002
| {{Monzo| -1 4 2 -1 -1 -1 }}
| 19.776
| [[Phlub]] (2026)
|-
| [[91/90|Biome comma, superleap comma]]
| Thozoguma
| 3ozgM
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.130
|
|-
| [[8281/8192|Diahuntmisma]]
| Labithozoma
| L2(3oz)M
| 8281/8192
| {{Monzo| -13 0 0 2 0 2 }}
| 18.707
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[512/507|Tridecimal neutral thirds comma]]
| Thuthuma
| 3uuM
| 512/507
| {{Monzo| 9 -1 0 0 0 -2 }}
| 16.990
|
|-
| [[105/104|Animist comma]]
| Thuzoyoma
| 3uzyM
| 105/104
| {{Monzo| -3 1 1 1 0 -1 }}
| 16.567
|
|-
| [[28812/28561|Tesseract comma]]
| Quadthuzoma
| 4(3uz)M
| 28812/28561
| {{Monzo| 2 1 0 4 0 -4 }}
| 15.148
| [[User:Unque|Unque]] (2025)
|-
| [[832/825|Tholuguguma]]
| Tholuguguma
| 3o1uggM
| 832/825
| {{Monzo| 6 -1 -2 0 -1 1 }}
| 14.627
|
|-
| [[Secorian comma]]
| Sathuzoma
| s3uzM
| 28672 / 28431
| {{Monzo| 12 -7 0 1 0 -1 }}
| 14.613
|
|-
| [[3159/3136|Mosaic comma]]
| Lathoruruma
| L3orrM
| 3159/3136
| {{Monzo| -6 5 0 -2 0 1}}
| 12.651
|
|-
| [[275/273|Gassorma]]
| Thuloruyoyoma
| 3u1oryyM
| 275/273
| {{Monzo| 0 -1 2 -1 1 -1 }}
| 12.637
|
|-
| [[144/143|Grossma]]
| Thuluma
| 3u1uM
| 144/143
| {{Monzo| 4 2 0 0 -1 -1 }}
| 12.064
|
|-
| [[24167/24000|Tritho-alotriguma]]
| Tritho-alotriguma
| 3(3o)1o3gM
| 24167/24000
| {{Monzo| -6 -1 -3 0 1 3}}
| 12.005
|
|-
| [[Lesser tendoneutralisma]]
| Sasa-quadtrithuma
| ss12(3u)M
| 70368744177664 / 69894255367443
| {{Monzo| 46 -1 0 0 0 -12 }}
| 11.713
| [[Godtone]] (2024)
|-
| [[1701/1690|Kuhnausma]]
|
|
| 1701/1690
| {{Monzo| -1 5 -1 1 0 -2 }}
| 11.232
| [[Phlub]] (2026)
|-
| [[Dinos comma]]
| Lathuthuquinguma
| L3uu5gM
| 531441/528125
| {{Monzo| 0 12 -5 0 0 -2 }}
| 10.836
| [[Dummy Index]] (2024)
|-
| [[169/168|Buzurgisma, dhanvantarisma]]
| Thothoruma
| 3oorM
| 169/168
| {{Monzo| -3 -1 0 -1 0 2 }}
| 10.274
| [[Margo Schulter]] (2012) for ''buzurgisma''
|-
| [[3042/3025|Diagassormisma]]
| Bitholuguma
| 2(3o1ug)M
| 3042/3025
| {{Monzo| 1 2 -2 0 -2 2 }}
| 9.7020
| [[User:Xenllium|Xenllium]] (2022)
|-
| Greater nelindic comma
| Thothoquinru-ayoyoma
| 3oo5rayyM
| 16900/16807
| {{Monzo| 2 0 2 -5 0 2 }}
| 9.5532
| [[User:Kaiveran|Kaiveran]] (2019)
|-
| [[1287/1280|Catadictma]]
| Thologuma
| 3o1ogM
| 1287/1280
| {{Monzo| -8 2 -1 0 1 1 }}
| 9.4419
|
|-
| [[Glacier comma]]
| Quinthuma
| 5(3u)M
| 373248/371293
| {{Monzo| 9 6 0 0 0 -5 }}
| 9.0917
| [[ArrowHead294]] (2024)
|-
| [[196/195|Mynucuma]]
| Thuzozoguma
| 3uzzgM
| 196/195
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.8554
|
|-
| [[1625/1617|Sopreisma]]
| Tholururutriyoma
| 3urr3yM
| 1625/1617
| {{Monzo| 0 -1 3 -2 -1 1 }}
| 8.5440
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[640/637|Huntma]], lesser nelindic comma
| Thururuyoma
| 3urryM
| 640/637
| {{Monzo| 7 0 1 -2 0 -1 }}
| 8.1342
| [[User:Kaiveran|Kaiveran]] (2019) for ''lesser nelindic comma''
|-
|Sonoma
|Lala-tritritho-aquadyoma
|LL9(3o)4yM
|6627812108125/
6597069766656
|{{Monzo|-41 -1 4 0 0 9}}
|8.0488
|[https://x.com/vib_gen/status/2038852033244246443 Vib, Misohito Nakai] (2026)
|-
| [[2197/2187|Threedie]]
| Satrithoma
| s3(3o)M
| 2197/2187
| {{Monzo| 0 -7 0 0 0 3 }}
| 7.8980
|
|-
| [[Tridecimal nakaisma]]
| Quinsa-quadtritrithu-azoma
| 5s36(3u)zM
|
| {{Monzo| 132 -1 0 1 0 -36 }}
| 7.8751
| [[User:原井玉葱郎|Misohito Nakai]] (2025)
|-
| [[4394/4375|Hebrewsma]]
| Tritho-aruquadguma
| 3(3o)r4gM
| 4394/4375
| {{Monzo| 1 0 -4 -1 0 3 }}
| 7.5022
| [[Eliora]] (2022)
|-
| [[1188/1183|Kestrel comma]]
| Thuthuloruma
| 3uu1orM
| 1188/1183
| {{Monzo| 2 3 0 -1 1 -2 }}
| 7.3017
| [[Gene Ward Smith]] (2011)
|-
| [[30E comma|2D9 comma]]
| Thotriyoma
| 3o3yM
| 131625/131072
| {{Monzo|-17 4 3 0 0 1}}
| 7.2888
| [https://twitter.com/Regret_March/status/1709762093749252209 Figreflekt] (2023) but [https://twitter.com/Figreflekt/status/1710195052520337680 revised later]{{dead link}}
|-
| [[Brontesisma]]
| Trithu-azozoyoma
| 3(3u)zzM
| 2205/2197
| {{Monzo| 0 2 1 2 0 -3 }}
| 6.2925
| [[Francium]] (2025)
|-
| [[Praveensma]]
| Thoquadzoma
| 3o4zM
| 31213/31104
| {{Monzo| -7 -5 0 4 0 1 }}
| 6.0563
| [[Praveen Venkataramana]] (2022)
|-
| [[1573/1568|Lambeth comma]]
| Thobiloruma
| 3o2(1or)M
| 1573/1568
| {{Monzo| -5 0 0 -2 2 1 }}
| 5.5117
| [[Flora Canou]] (2022)
|-
| [[325/324|Marveltwin comma]]
| Thoyoyoma
| 3oyyM
| 325/324
| {{Monzo| -2 -4 2 0 0 1 }}
| 5.3351
|
|-
| [[Valerisma]], Hunt 13-cycle comma
| Laquinbithoma
| L10(3o)M
| 137858491849 / 137438953472
| {{Monzo| -37 0 0 0 0 10 }}
| 5.2766
| [[Mason Green]] (2016)
|-
| [[351/350|Ratwolfsma]]
| Thoruguguma
| 3orggM
| 351/350
| {{Monzo| -1 3 -2 -1 0 1 }}
| 4.9393
|
|-
| [[352/351|Major minthma, major gentle comma]], 11/13-kleisma
| Thuloma
| 3u1oM
| 352/351
| {{Monzo| 5 -3 0 0 1 -1 }}
| 4.9253
| See the page.
|-
| [[364/363|Minor minthma, minor gentle comma]]
| Tholuluzoma
| 3o1uuzM
| 364/363
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.7627
| See the page.
|-
| [[847/845|Cuthbert comma]]
| Bithulo-azoguma
| 2(3u1o)zgM
| 847/845
| {{Monzo| 0 0 -1 1 2 -2 }}
| 4.0928
|
|}

=== 17-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[2048/2023|Susurrisma, susurration comma]]
| Susuruma
| 17uurM
| 2048/2023
| {{Monzo| 11 0 0 -1 0 0 -2 }}
| 21.263
| [[User:Jerdle|Jerdle]] (2026)
|-
| [[85/84|Monk comma]]
| Soruyoma
| 17oryM
| 85/84
| {{Monzo| -2 -1 1 -1 0 0 1 }}
| 20.488
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[289/286|Lum comma]]
| Sosothuluma
| 17oo3u1uM
| 289/286
| {{Monzo| -1 0 0 0 -1 -1 2 }}
| 18.065
|
|-
| [[2197/2176|Mey comma]]
| Sutrithov
| 17u3(3o)M
| 2197/2176
| {{Monzo| -7 0 0 0 0 3 -1 }}
| 16.628
|
|-
| [[429/425|Middle semisixthma]]
| Suthologuguma
| 17u3o1oggM
| 429/425
| {{Monzo| 0 1 -2 0 1 1 -1 }}
| 16.218
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[4131/4096|Septendecimal comma]], Hunt flat 2 comma
| Lasoma
| L17oM
| 4131/4096
| {{Monzo| -12 5 0 0 0 0 1 }}
| 14.730
| [[Flora Canou]] (2020) for ''septendecimal comma''
|-
| [[120/119|Lynchisma]]
| Suruyoma
| 17uryM
| 120/119
| {{Monzo| 3 1 1 -1 0 0 -1 }}
| 14.487
| [[User:Xenllium|Xenllium]] (2022)
|-
| 23-17-comma, 23 semitone comma
| Trila-twethesoma
| 3L23(17o)M
|
| {{Monzo| -94 0 0 0 0 0 23 }}
| 13.974
|
|-
| [[136/135|Diatisma]], diatic comma, fiventeen comma, septendecimal semicomma
| Soguma
| 17ogM
| 136/135
| {{Monzo| 3 -3 -1 0 0 0 1 }}
| 12.777
| See the page.
|-
| [[154/153|Augustma]]
| Sulozoma
| 17u1ozM
| 154/153
| {{Monzo| 1 -2 0 1 1 0 -1 }}
| 11.278
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[170/169|Major naiadma]]
| Sothuthuyoma
| 17o3uuyM
| 170/169
| {{Monzo| 1 0 1 0 0 -2 1 }}
| 10.214
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2187/2176|Septendecimal schisma]]
| Lasuma
| L17uM
| 2187/2176
| {{Monzo| -7 7 0 0 0 0 -1 }}
| 8.7296
| [[Plainsound Music Edition]] (2020)
|-
| [[1452/1445|Small semisixthma]]
| Susulologuma
| 17uu1oogM
| 1452/1445
| {{Monzo| 2 1 -1 0 2 0 -2 }}
| 8.3664
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[Mean thirds comma]]
| Lasosoyoma
| L17ooyM
| 1053405/1048576
| {{Monzo|-20 6 1 0 0 0 2}}
| 7.9545
| [[User:Jerdle|Jerdle]] (2022)
|-
| [[221/220|Minor naiadma]]
| Sotholuguma
| 17o3o1ugM
| 221/220
| {{Monzo| -2 0 -1 0 -1 1 1 }}
| 7.8514
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2057/2048|Blume comma]]
| Sololoma
| 17o1ooM
| 2057/2048
| {{monzo| -11 0 0 0 2 0 1 }}
| 7.5913
| [[Douglas Blumeyer]]
|-
| [[256/255|Charisma]], charic comma, septendecimal kleisma
| Suguma
| 17ugM
| 256/255
| {{Monzo| 8 -1 -1 0 0 0 -1 }}
| 6.7759
| See the page.
|-
| [[273/272|Tannisma, prototannisma]]
| Suthozoma
| 17u3ozM
| 273/272
| {{Monzo| -4 1 0 1 0 1 -1}}
| 6.3532
| [[Scott Dakota]] (2017) for ''tannisma'' [[Flora Canou]] (2023) for ''prototannisma''
|-
| [[289/288|Semitonisma]], septendecimal semitones comma, septendecimal 6-cent comma
| Sosoma
| 17ooM
| 289/288
| {{Monzo| -5 -2 0 0 0 0 2 }}
| 6.0008
| [[Flora Canou]] (2023) ''for semitonisma''
|-
| [[375/374|Ursulisma]]
| Sulutriyoma
| 17u1u3yM
| 375/374
| {{Monzo| -1 1 3 0 -1 0 -1 }}
| 4.6228
| [[Dawson Berry]], [[User:VIxen|VIxen]] (2023)
|-
| [[442/441|Seminaiadma]]
| Sothoruruma
| 17o3orrM
| 442/441
| {{Monzo| 1 -2 0 -2 0 1 1 }}
| 3.9213
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[80-17-comma]], 17-ripple <strike>integer cents</strike> comma{{clarify}}
| Lesa-quinquadquadsuma
| 11s80(17u)M
|
| {{Monzo| 327 0 0 0 0 0 -80 }}
| 3.5672
| [[Eliora]] (2022) for ''80-17-comma''
|}

=== 19-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[135/133|Nuruyoma]]
| Nuruyoma
| 19uryM
| 135/133
| {{Monzo| 0 3 1 -1 0 0 0 -1 }}
| 25.84
|
|-
| [[76/75|Large undevicesimal 1/9-tone]]
| Noguguma
| 19oggM
| 76/75
| {{Monzo| 2 -1 -2 0 0 0 0 1 }}
| 22.931
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[77/76|Small undevicesimal 1/9-tone]]
| Nulozoma
| 19u1ozM
| 77/76
| {{Monzo| -2 0 0 1 1 0 0 -1 }}
| 22.631
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[96/95|19th-partial chroma]]
| Nuguma
| 19ugM
| 96/95
| {{Monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.128
| [[User:Flirora|Flirora]] (2021)
|-
| [[Ume comma]]
| Nutrisoma
| 19u3(17o)M
| 4913/4864
| {{Monzo| -8 0 0 0 0 0 3 -1 }}
| 17.353
|
|-
| [[729/722|Undevicesimal diaschisma]]
| Lanunuma
| L19uuM
| 729/722
| {{Monzo| -1 6 0 0 0 0 0 -2 }}
| 16.704
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[6912/6859|Deviaug comma]]
| Trinuma
| 3(19u)M
| 6912/6859
| {{Monzo| 8 3 0 0 0 0 0 -3 }}
| 13.326
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[133/132|Minithirdma]]
| Noluzoma
| 19o1uzM
| 133/132
| {{Monzo| -2 -1 0 1 -1 0 0 1 }}
| 13.066
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[153/152|Ganassisma]], Ganassi's comma
| Nusoma
| 19u17oM
| 153/152
| {{Monzo| -3 2 0 0 0 0 1 -1 }}
| 11.352
| [[User:Flirora|+merlan #flirora]] (2021)
|-
| [[171/170|Malcolmisma]]
| Nosuguma
| 19o17ugM
| 171/170
| {{Monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.154
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[131072/130321|Undevicesimal diminished comma]], Hunt 19-cycle comma
| Saquadnuma
| s4(19u)M
| 131072 / 130321
| {{Monzo| 17 0 0 0 0 0 0 -4 }}
| 9.9479
|
|-
| [[Eye comma]]
| Nubisoluma
| 19u2(17o1u)M
| 2312/2299
| {{Monzo| 3 0 0 0 -2 0 2 -1 }}
| 9.7619
|
|-
| [[363/361|Godzillisma]]
| Binuloma
| 2(19u1o)M
| 363/361
| {{Monzo| 0 1 0 0 2 0 0 -2 }}
| 9.5649
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[190/189|Cotylisma]]
| Noruyoma
| 19oryM
| 190/189
| {{Monzo| 1 -3 1 -1 0 0 0 1 }}
| 9.1358
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[209/208|Yama comma]]
| Nothuloma
| 19o3u1oM
| 209/208
| {{Monzo| -4 0 0 0 1 -1 0 1 }}
| 8.3033
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[210/209|Spleen comma]]
| Nuluzoyoma
| 19u1uzyM
| 210/209
| {{Monzo| 1 1 1 1 -1 0 0 -1 }}
| 8.2637
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1083/1078|Bihendrixma]]
| Nonolururuma
| 19oo1urrM
| 1083/1078
| {{Monzo| -1 1 0 -2 -1 0 0 2 }}
| 8.0113
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[286/285|Chthonisma]]
| Nuthologuma
| 19u3o1ogM
| 286/285
| {{Monzo| 1 -1 -1 0 1 1 0 -1 }}
| 6.0639
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[324/323|Photisma]]
| Nusuma
| 19u17uM
| 324/323
| {{Monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.3516
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[343/342|Nutrisma]]
| Nutrizoma
| 19u3zM
| 343/342
| {{Monzo| -1 -2 0 3 0 0 0 -1 }}
| 5.0547
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Triraptor comma]]
| Trinuso-azoguma
| 3(19u17o)zgM
| 34391/34295
| {{Monzo|0 0 -1 1 0 0 3 -3}}
| 4.8394
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[361/360|Go comma]], dudon comma
| Nonoguma
| 19oogM
| 361/360
| {{Monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.8023
| [[User:Xenwolf|Xenwolf]] (2013) for ''go comma''
|-
| [[400/399|Devichroma]]
| Nuruyoyoma
| 19uryyM
| 400/399
| {{Monzo| 4 -1 2 -1 0 0 0 -1 }}
| 4.3335
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[456/455|Abnobisma]]
| Nothuruguma
| 19o3urgM
| 456/455
| {{Monzo| 3 1 -1 -1 0 -1 0 1 }}
| 3.8007
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[476/475|Hedwigma]]
| Nusozoguguma
| 19u17ozggM
| 476/475
| {{Monzo| 2 0 -2 1 0 0 1 -1 }}
| 3.6409
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[495/494|Eulalisma]]
| Nuthuloyoma
| 19u3u1oyM
| 495/494
| {{Monzo| -1 2 1 0 1 -1 0 -1 }}
| 3.5010
| [[User:Xenllium|Xenllium]] (2023)
|}

=== 23-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[187/184|Twethusoloma]]
| Twethusoloma
| 23u17o1oM
| 187/184
| 2.11.17.23 {{monzo| -3 1 1 -1 }}
| 27.999
|
|-
| [[69/68|Large vicesimotertial 1/8-tone]]
| Twethosuma
| 23o17uM
| 69/68
| 2.3.17.23 {{monzo| -2 1 -1 1 }}
| 25.274
|[[User:Xenllium|Xenllium]] (2023)
|-
| [[70/69|Small vicesimotertial 1/8-tone]]
| Twethuzoyoma
| 23uzyM
| 70/69
| 2.3.5.7.23 {{monzo| 1 -1 1 1 -1 }}
| 24.910
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[92/91|Undinisma]]
| Twethothuruma
| 23o3urM
| 92/91
| 2.7.13.23 {{monzo| 2 -1 -1 1 }}
| 18.921
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[736/729|23-limit Tenney/Cage comma]]
| Satwethoma
| s23oM
| 736/729
| 2.3.23 {{monzo| 5 -6 1 }}
| 16.544
| [[Stephen Weigel]] (2023)
|-
| [[115/114|Yarmanisma]]
| Twethonuyoma
| 23o19uyM
| 115/114
| 2.3.5.19.23 {{monzo| -1 -1 1 -1 1 }}
| 15.120
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[161/160|Major kirnbergerisma]]
| Twethozoguma
| 23ozgM
| 161/160
| 2.5.7.23 {{monzo| -5 -1 1 1 }}
| 10.787
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[162/161|Minor kirnbergerisma]]
| Twethuruma
| 23urM
| 162/161
| 2.3.7.23 {{monzo| 1 4 -1 -1 }}
| 10.720
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[208/207|Vicetone comma]]
| Twethuthoma
| 23u3oM
| 208/207
| 2.3.13.23 {{monzo| 4 -2 1 -1 }}
| 8.3433
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[231/230|Major neutravicema]]
| Twethulozoguma
| 23u1ozgM
| 231/230
| {{monzo| -1 1 -1 1 1 0 0 0 -1 }}
| 7.5108
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Vicesimotertial schisma]]
| Lala-twethuma
| LL23uM
| 387420489 / 385875968
| 2.3.23 {{monzo| -24 18 -1 }}
| 6.9157
|
|-
| [[253/252|Middle neutravicema]]
| Twetholoruma
| 23o1orM
| 253/252
| 2.3.7.11.23 {{monzo| -2 -2 -1 1 1 }}
| 6.8564
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[276/275|Minor neutravicema]]
| Twetholuguguma
| 23o1uggM
| 276/275
| 2.3.5.11.23 {{monzo| 2 1 -2 -1 1 }}
| 6.2840
| [[User:Xenllium|Xenllium]] (2023)
|-
| 21-23-comma
| Trisa-septritwethuma
| 3s21(23u)M
| 281474976710656 / 280462473659039
| 2.23 {{monzo| 95 -21 }}
| 6.2387
|
|-
| [[300/299|Major naiadvicema]]
| Twethuthuyoyoma
| 23u3uyyM
| 300/299
| 2.3.5.13.23 {{monzo| 2 1 2 -1 -1 }}
| 5.7804
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[323/322|Major semivicema]]
| Twethunosoruma
| 23u19o17orM
| 323/322
| 2.7.17.19.23 {{monzo| -1 -1 1 1 -1 }}
| 5.3682
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[391/390|Minor naiadvicema]]
| Twethosothuguma
| 23o17o3ugM
| 391/390
| {{monzo| -1 -1 -1 0 0 -1 1 0 1 }}
| 4.4334
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[392/391|Minor semivicema]]
| Twethusuzozoma
| 23u17uzzM
| 392/391
| 2.7.17.23 {{monzo| 3 2 -1 -1 }}
| 4.4221
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[460/459|Scanisma, vicewolf comma]]
| Twethosuyoma
| 23o17uyM
| 460/459
| 2.3.5.17.23 {{monzo| 2 -3 1 -1 1 }}
| 3.7676
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[484/483|Pittsburghisma]]
| Twethuloloruma
| 23u1oorM
| 484/483
| 2.3.7.11.23 {{monzo| 2 -1 -1 2 -1 }}
| 3.5806
| [[User:Xenllium|Xenllium]] (2023)
|}

=== 29-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| Classical mediant of Didymus' and Archytas' commas
| Twenothuluyoma
| 29o3u1uyM
| 145/143
| 5.11.13.29 {{monzo| 1 -1 -1 1 }}
| 24.045
|
|-
| [[88/87|Farewell comma]]
| Twenuloma
| 29u1oM
| 88/87
| 2.3.11.29 {{monzo| 3 -1 1 -1 }}
| 19.786
|
|-
| [[116/115|Sironisma]]
| Twenotwethuguma
| 29o23ugM
| 116/115
| 2.5.23.29 {{monzo| 2 -1 -1 1 }}
| 14.989
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[117/116|Lomisma]]
| Twenuthoma
| 29u3oM
| 117/116
| 2.3.13.29 {{monzo| -2 2 1 -1 }}
| 14.860
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[145/144|29th-partial chroma]]
| Twenoyoma
| 29oyM
| 145/144
| 2.3.5.29 {{monzo| -4 -2 1 1 }}
| 11.981
| [[User:Flirora|Flirora]] (2023)
|-
| [[175/174|Major chthonovinema]]
| Twenuzoyoyoma
| 29uzyyM
| 175/174
| 2.3.5.7.29 {{monzo| -1 -1 2 1 -1 }}
| 9.9211
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[204/203|Kallistisma]]
| Twenusoruma
| 29u17orM
| 204/203
| 2.3.7.17.29 {{monzo| 2 1 -1 1 -1 }}
| 8.5073
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[232/231|Major paravinema]]
| Twenoluruma
| 29o1urM
| 232/231
| 2.3.7.11.29 {{monzo| 3 -1 -1 -1 1 }}
| 7.4783
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[Jackpot comma]]
| Laseptwenoma
| L7(29o)M
| 17249876309 / 17179869184
| 2.29 {{monzo| -34 7 }}
| 7.0404
| [[Eliora]] (2024)
|-
| [[261/260|Major vinetonema]]
| Twenothuguma
| 29o3ugM
| 261/260
| 2.3.5.13.29 {{monzo| -2 2 -1 -1 1 }}
| 6.6458
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[290/289|Brunisma]]
| Twenosusuyoma
| 29o17uuyM
| 290/289
| 2.5.17.29 {{monzo| 1 1 -2 1 }}
| 5.9801
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[320/319|Minor paravinema]]
| Twenuluyoma
| 29u1uyM
| 320/319
| 2.5.11.29 {{monzo| 6 1 -1 -1 }}
| 5.4186
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[378/377|Major semivinema]]
| Twenuthuzoma
| 29u3uzM
| 378/377
| 2.3.7.13.29 {{monzo| 1 3 1 -1 -1 }}
| 4.5861
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[406/405|Minor semivinema]]
| Twenozoguma
| 29ozgM
| 406/405
| 2.3.5.7.29 {{monzo| 1 -4 -1 1 1 }}
| 4.2694
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[494/493|Minor vinetonema]]
| Twenunosuthoma
| 29u19o17u3oM
| 494/493
| 2.13.17.19.29 {{monzo| 1 1 -1 1 -1 }}
| 3.5081
| [[User:Xenllium|Xenllium]] (2024)
|}

=== 31-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[63/62|Co-archytas comma]]
| Thiwuzoma
| 31uzM
| 63/62
| 2.3.7.31 {{monzo| -1 2 1 -1 }}
| 27.700
| [[Xenwolf]] (2020)
|-
| [[93/92|Tricema]]
| Thiwotwethuma
| 31o23uM
| 93/92
| 2.3.23.31 {{monzo| -2 1 -1 1 }}
| 18.716
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[125/124|Twizzler]]
| Thiwutriyoma
| 31u3yM
| 125/124
| 2.5.31 {{monzo| -2 3 -1 }}
| 13.906
| [[Xenwolf]] (2020)
|-
| [[155/154|Scyllisma]]
| Thiwoluruyoma
| 31o1uryM
| 155/154
| 2.5.7.11.31 {{monzo| -1 1 -1 -1 1 }}
| 11.205
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[156/155|Xanthippisma]]
| Thiwuthoguma
| 31u3ogM
| 156/155
| 2.3.5.13.31 {{monzo| 2 1 -1 1 -1 }}
| 11.133
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[187/186|Lambertisma]]
| Thiwusoloma
| 31u17o1oM
| 187/186
| 2.3.11.17.31 {{monzo| -1 -1 1 1 -1 }}
| 9.2828
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[217/216|Tricesimoprimal kleisma]]
| Thiwozoma
| 31ozM
| 217/216
| 2.3.7.31 {{monzo| -3 -3 1 1 }}
| 7.9965
|
|-
| [[Doctorsma]]
| Latrithiwu-athuquadzoma
| L3(31u)3u4zM
| 388962/387283
| 2.3.7.13.31 {{Monzo|1 4 4 -1 -3}}
| 7.4892
| [[User:Stavats|Stavats]] (2026)
|-
| [[248/247|Lameisma]]
| Thiwonuthuma
| 31o19u3uM
| 248/247
| 2.13.19.31 {{monzo| 3 -1 -1 1 }}
| 6.9949
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[280/279|Tricetone comma]]
| Thiwuzoyoma
| 31uzyM
| 280/279
| 2.3.5.7.31 {{monzo| 3 -2 1 1 -1 }}
| 6.1940
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Junebug comma]]
| Thiwutwenotwethunusotholuzoyoma
| 31u29o23u19u17o3o1uzyM
| 448630/447051
| {{monzo| 1 -1 1 1 -1 1 1 -1 -1 1 -1 }}
| 6.1040
| [[Tristan Bay]] (2024)
|-
| [[341/340|Californisma]]
| Thiwosuloguma
| 31o17u1ogM
| 341/340
| 2.5.11.17.31 {{monzo| -2 -1 1 -1 1 }}
| 5.0844
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[342/341|Endymisma]]
| Thiwunoluma
| 31u19o1uM
| 342/341
| 2.3.11.19.31 {{monzo| 1 2 -1 1 -1 }}
| 5.0695
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[435/434|Chinthisma]]
| Thiwutwenoruyoma
| 31u29oryM
| 435/434
| 2.3.5.7.29.31 {{monzo| -1 1 1 -1 1 -1 }}
| 3.9844
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[465/464|Alektisma]]
| Thiwotwenuyoma
| 31o29uyM
| 465/464
| 2.3.5.29.31 {{monzo| -4 1 1 -1 1 }}
| 3.7271
| [[User:Xenllium|Xenllium]] (2025)
|}

=== Higher-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[714984/704969|Lightyear comma]]
|
| 89u331o3M
| 714984/704969
| 2.3.31.89 {{monzo| 3 1 3 -3 }}
| 24.421
| [[Budjarn Lambeth]] (2023)
|-
| [[82/81|41-limit Johnston comma]]
| Fowoma
| 41oM
| 82/81
| 2.3.41 {{monzo| 1 -4 1 }}
| 21.242
| [[Stephen Weigel]] (2023)
|-
| [[2883/2848|Lilac comma]]
|
| 89u31ooM
| 2883/2848
| 2.3.31.89 {{monzo| -5 1 2 -1 }}
| 21.146
| [[Budjarn Lambeth]] (2023)
|-
| [[86/85|43-limit 10th-tone]], large quadracesimotertial 1/10-tone
| Fothosuguma
| 43o17ugM
| 86/85
| 2.5.17.43 {{monzo| 1 -1 -1 1 }}
| 20.249
| [[Mister Shaf]] (2025)
|-
| [[87/86|43-limit 10th-tone]], small quadracesimotertial 1/10-tone
| Fothutwenoma
| 43u29oM
| 87/86
| 2.3.29.43 {{monzo| -1 1 1 -1 }}
| 20.014
|
|-
| [[89/88|Tailwind comma]]
|
| 89o1uM
| 89/88
| 2.11.89 {{monzo| -3 -1 1 }}
| 19.562
| [[Budjarn Lambeth]] (2023)
|-
| [[389/385|Rebbe comma]]
|
| 389o1urgM
| 389/385
| 5.7.11.389 {{monzo| -1 -1 -1 1 }}
| 17.8794
| [[Mister Shaf]] (2025)
|-
| [[8277/8192|Lilly pilly comma]]
|
| 89o31oM
| 8277/8192
| 2.3.31.89 {{monzo| -13 1 1 1 }}
| 17.871
| [[Budjarn Lambeth]] (2023)
|-
| [[8000/7921|Incisor comma]]
|
| 89uu3yM
| 8000/7921
| 2.5.89 {{monzo| 6 3 -2 }}
| 17.181
| [[Budjarn Lambeth]] (2023)
|-
| [[129/128|43-limit Johnston comma]]
| Fothoma
| 43oM
| 129/128
| 2.3.43 {{monzo| -7 1 1 }}
| 13.473
| [[Stephen Weigel]] (2023)
|-
| [[979/972|Basement comma]]
|
| 89o1oM
| 979/972
| 2.3.11.89 {{monzo| -2 -5 1 1 }}
| 12.423
| [[Budjarn Lambeth]] (2023)
|-
| [[226/225|Reversed marvel comma]]
|
| 113oggM
| 226/225
| 2.3.5.113 {{monzo| 1 -2 -2 1 }}
| 7.6773
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Sidereal comma]]
|
| 73u61ogM
| 366/365
| 2.3.5.61.73 {{monzo| 1 1 -1 1 -1 }}
| 4.7366
| [[User:Frostburn|Frostburn]] (2024)
|-
| [[381/380|Five feet comma]]
|
| 127o19ugM
| 381/380
| 2.3.5.19.127 [-2 1 -1 -1 1⟩
| 4.5499
| [[Eliora]] (2022)
|-
| [[481/480|Semaphorisma]]
| Thisothoguma
| 37o3ogM
| 481/480
| 2.3.5.13.37 {{monzo| -5 -1 -1 1 1 }}
| 3.6030
| [[User:Kaiveran|Kaiveran]] (2023)
|}

== List of irrational commas ==
For intervals expressible as edosteps, see [[Interval size measure]]. We skip them here for brevity.

{| class="wikitable"
! Name(s)
! colspan="2" | [[Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[Caffeinterval]]
|
|
| 2((7/12) - (1/sqrt(3)))
|
| 7.1797
| [[User:R-4981|R-4981]] (2023)
|}

== See also ==
* [[Comma and diesis]]

== References ==

[[Category:Small commas| ]] 
[[Category:Lists of commas]]
{{Todo|complete table|add color name}}

343/342

2026-04-28T23:55:39Z

TallKite:

{{Infobox Interval
| Name = nutrisma
| Color name = 19uz32, nutrizo 2nd, 19u3zM, nutrizoma
| Comma = yes
}}
'''343/342''', the '''nutrisma''', is a [[19-limit]] [[small comma]]. It is the amount by which a stack of two [[7/6]]s exceeds [[19/14]]. By tempering it out is defined the '''nutrismic temperament''', which enables the [[nutrismic chords]].

== Etymology ==
The name ''nutrisma'' was named by [[User:Xenllium|Xenllium]] in 2023. It is a play of its [[Color notation|color name]], ''nutrizoma''.

== See also ==
* [[List of superparticular intervals]]

[[Category:Nutrismic]]
[[Category:Commas named after their color name]]

Unnoticeable comma

2026-04-28T21:41:37Z

TallKite: update comma color names, done up to twetha

'''Unnoticeable commas''' are very small intervals. These [[comma]]s are called "unnoticeable" because, being equal to or less than 3.5{{cent}}, they are smaller than the average peak [[just-noticeable difference]] (JND) of human pitch perception, as illustrated by the research of [[Aaron Andrew Hunt]]<ref>[http://musictheory.zentral.zone/huntsystem2.html#2 H-Pi Instruments | ''Hunt System Scale §The JND'']</ref>. It is improbable that even a trained listener would be able to notice these intervals, and as such they are a prime target for psychoacoustically informed [[microtempering]]. (However, a considerably larger comma can be unnoticeable in an [[adaptive just intonation|adaptive]] tuning context. Instead of one large pitch shift of the entire comma, there can be many small pitch shifts of a fraction of a comma, one per chord change. Given this, a noticeable 3-limit comma that arguably deserves inclusion is the [[mercator comma]], corresponding to using [[53edo]] for the circle of fifths.) In [[Sagittal notation]], intervals in the smaller part of this category are [[schismina]]s, and intervals in the larger part of this category are [[schisma (interval region)|schismas]]<ref>[https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>.

For commas over 100{{c}} in size, see [[Large comma]]; for commas in between 30–100{{c}} in size, see [[Medium comma]]; and for commas between 3.5–30{{c}} in size, see [[Small comma]].

Due to the wide range of sizes, cents values here and on the other commas pages are listed to 5 significant digits, instead of to a fixed number of decimal places as is the [[conventions|convention]] elsewhere on the wiki.

The commas might be numerous, but there is no need to memorize all the names. For pretty much all use cases, it is perfectly acceptable to just refer to a comma by the monzo or ratio, whichever is simpler.

== List of commas, by prime limit ==
=== 3-limit commas ===
{{See also| Table of 3-limit commas }}

{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Qian's large comma]]
| 359wama
| 359wM
| <abbr title="1934329451767021421190980423006270754962252499447372679942534652297789463068718331568475476554301659845788554312924531179306109686817232569946089263295619210341718686733067/1932268761508629172347675945465993672149463664853217499328617625725759571144780212268096883290961288981231808015751088588682539330521493827871454336733540374348490407411712">(344 digits)</abbr>
| {{Monzo| -569 359 }}
| 1.8453
| Chen Yingshi (2009)
|-
| [[Qian's small comma]], sasktel comma
| 306wama
| 306wM
| <abbr title="99895953610111751404211111353381321783955140565279076827493022708011895642232499843849795298031743077114461795885011932654335221737225129801285632/99793888233710926097676673961542382339552034110870991187709058567130998942396826836880350287497238272034603157195937657211050782186192219658614729">(292 digits)</abbr>
| {{Monzo| 485 -306 }}
| 1.7697
| Chen Yingshi (2009) for ''Qian's small comma''
|-
| [[Satanic comma]]
| 665wama
| 665wM
| <abbr title="193034257116813465350415306746516837350333763798962117430788985740786485136987943002905988649011085058426719117038711696606024631330152759176330399379617346789616335692978372064681236597226671488585092334981423081811727458166457361300251189808300631437024118571790058070714566731059066970852059271394655662607817543843/193025830561934107162947985381047541665608072055952185017491682078771915023799273387871154500424503798663213600460826789274033295999330021731389427128542432710187362934652673115221889249890533772697227171395058697282798274445240687006095271729621464100656563293799180557568945517759802372156455525060659659679134121984">(636 digits)</abbr>
| {{Monzo| -1054 665 }}
| 0.075575
| [[Marc Jones]] (1990)
|-
| 15601-comma
| 15601wama
| 15601wM
| (14888 digits)
| {{Monzo| 24727 -15601 }}
| 0.031499
|
|-
| 31867-comma
| 31867wama
| 31867wM
| (30410 digits)
| {{Monzo| -50508 31867 }}
| 0.012577
|
|-
| [[Archangelic comma]]
| 190537wama
| 190537wM
| (181820 digits)
| {{monzo| 301994 -190537 }}
| 0.00011162
| [[Dawson Berry]] (2022)
|}

=== 5-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Dodifo comma]]
| Quadla-sepquinyoma
| 4L35yM
| <abbr title="2910383045673370361328125 / 2904698108822600835661824">(50 digits)</abbr>
| {{Monzo| -67 -9 35 }}
| 3.3850
| [[Petr Pařízek]] (2011)
|-
| [[Vishnuzma]], semisuper comma
| Sasepbiguma
| s14gM
| 6115295232 / 6103515625
| {{Monzo| 23 6 -14 }}
| 3.3380
| [[Gene Ward Smith]] (2001), for ''semisuper comma''
|-
| [[Enneadeca]], 19-tone-comma
| Neyoma
| 19yM
| 19073486328125 / 19042491875328
| {{Monzo| -14 -19 19 }}
| 2.8155
| [[Gene Ward Smith]] (2001)
|-
| [[Vavoom comma]]
| Quinla-seyoma
| 5L17yM
| <abbr title="295578376007080078125 / 295147905179352825856">(42 digits)</abbr>
| {{Monzo| -68 18 17 }}
| 2.5232
| [[Paul Erlich]] (2002)
|-
| [[Alphatricot comma]]
| Quadsa-triyoma
| 4s3yM
| 68719476736000 / 68630377364883
| {{Monzo| 39 -29 3 }}
| 2.2461
| [[Paul Erlich]] (2002)
|-
| [[Schisma]]
| Layoma
| LyM
| 32805 / 32768
| {{Monzo| -15 8 1 }}
| 1.9537
| [[Hermann von Helmholtz]], [[Alexander Ellis]] (1875)
|-
| [[Aluminium comma]]
| Sepsa-theguma
| 7s13gM
| <abbr title="4951760157141521099596496896 / 4946966739525117513427734375">(56 digits)</abbr>
| {{Monzo| 92 -39 -13 }}
| 1.6767
| [[Eliora]] (2023)
|-
| [[Counterschisma]]
| Tribilaguma
| 6LgM
| <abbr title="2954312706550833698643 / 2951479051793528258560">(44 digits)</abbr>
| {{Monzo| -69 45 -1 }}
| 1.6613
| [[Gene Ward Smith]] (2003)
|-
| [[Neon comma]]
| Laquinquinbiguma
| L50gM
| <abbr title="88900702359186211632409599176343552 / 88817841970012523233890533447265625">(70 digits)</abbr>
| {{monzo| 21 60 -50 }}
| 1.6144
| [[Eliora]] (2024)
|-
| [[Septendecima]]
| Lala-sebiyoma
| LL34yM
| <abbr title="582076609134674072265625 / 581595589965365114830848">(48 digits)</abbr>
| {{Monzo| -52 -17 34 }}
| 1.4313
|
|-
| [[Luna comma]], hemithirds comma
| Sasa-quintriguma
| ss15gM
| 274877906944 / 274658203125
| {{Monzo| 38 -2 -15 }}
| 1.3843
| [[Gene Ward Smith]] (2002)
|-
| [[Minortone comma]], minortonma
| Trila-seguma
| 3L17gM
| <abbr title="50031545098999707 / 50000000000000000">(34 digits)</abbr>
| {{Monzo| -16 35 -17 }}
| 1.0919
| [[Gene Ward Smith]] (2002)
|-
| [[Ennealimma]]
| Satritribiyoma
| s18yM
| 7629394531250 / 7625597484987
| {{Monzo| 1 -27 18 }}
| 0.86183
| [[Paul Erlich]], [[Gene Ward Smith]] (2001)
|-
| Astro comma
| Tribisa-thiweguma
| 6s31gM
| <abbr title="2475880078570760549798248448 / 2474715001881122589111328125">(56 digits)</abbr>
| {{Monzo| 91 -12 -31 }}
| 0.81486
| [[Paul Erlich]] (2002)
|-
| [[Gaster comma]]
| Quadbila-neguma
| 8L19gM
| <abbr title="22528399544939174411840147874772641 / 22517998136852480000000000000000000">(70 digits)</abbr>
| {{Monzo| -70 72 -19 }}
| 0.79950
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Niobium comma]]
|
|
|
| {{Monzo| -875 492 41 }}
| 0.72269
| [[Eliora]] (2023)
|-
| [[Kwazy comma]]
| Quadla-quadquadyoma
| 4L16yM
| <abbr title="9010162353515625 / 9007199254740992">(32 digits)</abbr>
| {{Monzo| -53 10 16 }}
| 0.56943
| [[Paul Erlich]], [[Gene Ward Smith]] (2002)
|-
| Whoosh
| Saletriguma
| s33gM
| <abbr title="116450459770592056836096 / 116415321826934814453125">(48 digits)</abbr>
| {{Monzo| 37 25 -33 }}
| 0.52246
| [[Paul Erlich]] (2002)
|-
| Egads
| Setriyoma
| 51yM
| <abbr title="444089209850062616169452667236328125 / 444002166576103304796646509039845376">(72 digits)</abbr>
| {{Monzo| -36 -52 51 }}
| 0.33936
| [[Paul Erlich]] (2002)
|-
| [[Monzisma]]
| Quinsa-yoyoma
| 5syyM
| <abbr title="450359962737049600 / 450283905890997363">(36 digits)</abbr>
| {{Monzo| 54 -37 2 }}
| 0.29240
| [[Margo Schulter]] (2001)
|-
| Fortune
| Tritrila-sepbiyoma
| 9L14yM
| <abbr title="162285243890121480027996826171875 / 162259276829213363391578010288128">(66 digits)</abbr>
| {{Monzo| -107 47 14 }}
| 0.27703
| [[Paul Erlich]] (2002)
|-
| Gross
| Quinbisa-foseguma
| 10s47gM
| <abbr title="22300745198530623141535718272648361505980416 / 22297583945629639856633730232715606689453125">(88 digits)</abbr>
| {{Monzo| 144 -22 -47 }}
| 0.24543
| [[Paul Erlich]] (2002)
|-
| Senior
| Quadla-sepquinguma
| 4L35gM
| <abbr title="381520424476945831628649898809 / 381469726562500000000000000000">(60 digits)</abbr>
| {{Monzo| -17 62 -35 }}
| 0.23007
| [[Paul Erlich]] (2002)
|-
| [[Counterquectisma]], deltapion
| Quintritrilayoma
| 45LyM
|
| {{Monzo| -500 314 1 }}
| 0.18399
| [[Eliora]] (2023)
|-
| [[Septarium comma]]
| Sasepquadtriguma
| s84gM
| <abbr title="51704256436926231056548749215693807357721577836111615492096 / 51698788284564229679463043254372678347863256931304931640625">(118 digits)</abbr>
| {{Monzo| 73 77 -84 }}
| 0.18310
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Quectisma]]
|
|
|
| {{Monzo| 554 -351 1 }}
| 0.10841
| [[User:CompactStar|CompactStar]] (2023)
|-
| Raider
| Tritrisa-thiseyoma
| 9s37yM
| <abbr title="171798691840000000000000000000000000000000000000 / 171792506910670443678820376588540424234035840667">(96 digits)</abbr>
| {{Monzo| 71 -99 37 }}
| 0.062327
| [[Gene Ward Smith]] (2002)
|-
| Pirate
| Quinla-sepsepyoma
| 5L49yM
| <abbr title="17763568394002504646778106689453125 / 17763086495282268024161967871623168">(70 digits)</abbr>
| {{Monzo| -90 -15 49 }}
| 0.046966
| [[Paul Erlich]] (2002)
|-
| Viking
| Nela-siweyoma
| 19L61yM
|
| {{Monzo| -251 69 61 }}
| 0.031605
| [[Gene Ward Smith]] (2002)
|-
| [[Kirnberger's atom]]
| Sepbisa-quadtriguma
| 14s12gM
| <abbr title="2923003274661805836407369665432566039311865085952 / 2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
| {{Monzo| 161 -84 -12 }}
| 0.015361
|
|-
| Selenia
|
|
|
| {{Monzo| -433 -137 280 }}
| 0.0047636
|
|-
| Titania
|
|
|
| {{Monzo| 2746 -521 -827 }}
| 0.0031829
|
|-
| Quark
|
|
|
| {{Monzo| -573 237 85 }}
| 8.8361 × 10-4
|
|-
| Scamp
|
|
|
| {{Monzo| -5836 4293 -417 }}
| 3.3472 × 10-5
|
|-
| Rover
|
|
|
| {{Monzo| 1634 1502 -1729 }}
| 2.7513 × 10-5
|
|-
| Rascal
|
|
|
| {{Monzo| -7470 2791 1312 }}
| 5.9596 × 10-6
|
|}

=== 7-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Decovulture comma]]
| Sasa-biruguguma
| ss2rggM
| 67108864 / 66976875
| {{Monzo| 26 -7 -4 -2 }}
| 3.4083
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Rainy comma]]
| Laquinzo-atriyoma
| L5za3yM
| 2100875/2097152
| {{Monzo| -21 0 3 5 }}
| 3.0706
| [[User:Flirora|+merlan #flirora]] (2020)
|-
| [[Pontiqak comma]]
| Lazozotritriyoma
| Lzz9yM
| 95703125 / 95551488
| {{Monzo| -17 -6 9 2 }}
| 2.7452
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Pessoalisma]]
| Sasa-tribiru-aguguma
| ss6raggM
| 2147483648 / 2144153025
| {{Monzo| 31 -6 -2 -6 }}
| 2.6871
| [[Dawson Berry]] (2023)
|-
| [[Mitonisma]]
| Laquadzo-aguma
| L4zagM
| 5250987/5242880
| {{Monzo| -20 7 -1 4 }}
| 2.6749
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Horwell comma]]
| Lazoquinyoma
| Lz5yM
| 65625/65536
| {{Monzo| -16 1 5 1 }}
| 2.3495
| [[Gene Ward Smith]] (2005)
|-
| [[Forge comma]]
| Lala-trizo-aquinguma
| LL3za5gM
| 1640558367 / 1638400000
| {{Monzo| -19 14 -5 3 }}
| 2.2792
| [[Dawson Berry]] (2022)
|-
| [[109-7-comma]]
|
|
|
| {{Monzo| -306 0 0 109 }}
| 2.0238
|
|-
| [[Neptunisma]]
| Laruruleyoma
| Lrr11yM
| 48828125 / 48771072
| {{monzo| -12 -5 11 -2 }}
| 2.0240
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Slendroschisma]], slendric schisma
| Sasa-quinbiruma
| ss10rM
| 68719476736 / 68641485507
| {{monzo| 36 -5 0 -10 }}
| 1.9659
| [[User:Xenllium|Xenllium]] (2024), modified by [[Flora Canou]] (2025)
|-
| [[Septimal ennealimma]]
| Tritrizoma
| 9zM
| 40353607 / 40310784
| {{Monzo| -11 -9 0 9 }}
| 1.8382
| [[Eliora]] (2021)
|-
| [[Meter]]
| Latriru-asepyoma
| L3ra7yM
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.6283
|
|-
| [[Scheme comma]]
| Lala-rutriguma
| LLr3gM
| 14348907 / 14336000
| {{Monzo| -14 15 -3 -1}}
| 1.5580
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[Breeze comma]]
| Laquadru-atriyoma
| L4ra3yM
| 2460375 / 2458624
| {{Monzo| -10 9 3 -4 }}
| 1.2325
| [[Tristan Bay]] (2024)
|-
| [[Wizma]]
| Quinzo-ayoyoma
| 5zayyM
| 420175/419904
| {{Monzo| -6 -8 2 5 }}
| 1.1170
|
|-
| [[Supermatertisma]]
| Lasepru-atritriyoma
| L7ra9yM
| 52734375 / 52706752
| {{Monzo| -6 3 9 -7 }}
| 0.90708
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Trienstonisma]]
| Laquinru-aguma
| L5ragM
| 43046721 / 43025920
| {{monzo| -9 16 -1 -5 }}
| 0.83677
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[2401/2400|Breedsma]]
| Bizozoguma
| 2zzgM
| 2401/2400
| {{monzo| -5 -1 -2 4 }}
| 0.72120
| [[Gene Ward Smith]] (2004)
|-
| [[Gariatom]]
| Quintrila-tribizoma
| 15L6zM
|
| {{monzo| -169 96 0 6 }}
| 0.63552
| [[Tristan Bay]] (2023)
|-
| 171-9/7-comma
| Quadtribisa-netritrizoma
| 24s171zM
|
| {{monzo| 62 -342 0 171 }}
| 0.61971
|
|-
| [[Supermasesquartisma]]
| Laquadbiru-aquinyoma
| L8ra5yM
| 184528125 / 184473632
| {{monzo| -5 10 5 -8 }}
| 0.51133
| [[User:Xenllium|Xenllium]] (2021)
|-
| 571-7-comma
|
|
|
| {{monzo| 1603 0 0 -571 }}
| 0.40741
|
|-
| [[Ragisma]]
| Zoquadyoma
| z4yM
| 4375/4374
| {{Monzo| -1 -7 4 1 }}
| 0.39576
|
|-
| [[Septiruthenia]], septimal ruthenia
| Nela-lequadzoma
| 19L44zM
|
| {{Monzo| -263 88 0 44 }}
| 0.37996
| [[Eliora]] (2022)
|-
| [[Akjaysma]]
| Trisa-sepruguma
| 3s7rgM
| 140737488355328 / 140710042265625
| {{Monzo| 47 -7 -7 -7 }}
| 0.33765
| [[Aaron Krister Johnson]] (2011)
|-
| [[Landscape comma]]
| Trizoguguma
| 3zggM
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.32544
| [[Yahya Abdal-Aziz]] (2005)
|-
| [[Izar comma]], bapbo schismina
| Saquadtrizo-asepguma
| s12za7gM
| 13841287201 / 13839609375
| {{Monzo| 0 -11 -7 12 }}
| 0.20987
|
|-
| [[Nanisma]]
| Quinbisaruma
| 10srM
| <abbr title="649037107316853453566312041152512 / 648966242035284859600333477874109">(66 digits)</abbr>
| {{Monzo| 109 -67 0 -1 }}
| 0.18904
| [[Margo Schulter]] (2002)
|-
| laleruyo (171&1547&3125)
| Laleruyoma
| L11ryM
| 3955078125 / 3954653486
| {{Monzo| -1 4 11 -11 }}
| 0.18588
|
|-
| 87-fold starling comma
| Twenetrizotriguma
|
|
| {{Monzo| 86 174 -261 87 }}
| 0.14469
|
|-
| [[Revopentisma]]
| Sasa-neruma
| ss19rM
| <abbr title="11399736556781568 / 11398895185373143">(34 digits)</abbr>
| {{Monzo| 47 4 0 -19 }}
| 0.12778
| [[Eliora]] (2023)
|-
| [[Starscape comma]]
| Latritriru-ayoma
| L9rayM
| 645700815 / 645657712
| {{Monzo| -4 17 1 -9 }}
| 0.11557
|
|-
| [[Nommisma]]
| Quinla-zoyoyoma
| 5LzzyM
| <abbr title="36030948116563575 / 36028797018963968">(34 digits)</abbr>
| {{Monzo| -55 30 2 1 }}
| 0.10336
| [[Flora Canou]] (2023)
|-
| [[Euzenius comma]]
| Sabiruquinyoma
| s2r5yM
| 78125000 / 78121827
| {{Monzo| 3 -13 10 -2 }}
| 0.070314
|
|-
| [[Exodia comma]]
| Trila-quadbizo-aleyoma
| 3L8za11yM
| 281484423828125 / 281474976710656
| {{Monzo| -48 0 11 8 }}
| 0.058104
|
|-
| [[Septrongisma]]
| Lala-sepru-atritriguma
| LL7ra9gM
| 205891132094649 / 205885750000000
| {{Monzo| -7 30 -9 -7 }}
| 0.045256
| [[User:Xenllium|Xenllium]] (2026)
|-
| 171&1547&4973 comma
| Satwethezo-atritribiguma
| s23za18gM
| <abbr title="54737494680161832686 / 54736736297607421875">(40 digits)</abbr>
| {{Monzo| 1 -15 -18 23 }}
| 0.023986
|
|-
| [[Technologisma]]
| Trisa-quinbiru-aguma
| 3s10ragM
| <abbr title="2251799813685248 / 2251783932057135">(32 digits)</abbr>
| {{Monzo| 51 -13 -1 -10 }}
| 0.012210
| [[User:Godtone|Godtone]] (2024)
|-
| Termite
| Satritribiru-athiseyoma
| s18ra37yM
| <abbr title="37252902984619140625000000000 / 37252879910233655318543787489">(58 digits)</abbr>
| {{Monzo| 9 -28 37 -18 }}
| 0.0010723
|
|-
| Neutrino
|
|
|
| {{Monzo| 1889 -2145 138 424 }}
| 1.6361 × 10-10
|
|}

=== 11-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Lifthrasirsma]]
| Sasa-biluguma
| ss2(1ug)M
| 536870912 / 535869675
| {{Monzo| 29 -11 -2 0 -2 }}
| 3.2317
| [[Flora Canou]] (2023)
|-
| [[540/539|Swetisma]]
| Lururuyoma
| 1urryM
| 540/539
| {{Monzo| 2 3 1 -2 -1 }}
| 3.2090
| [[Manuel Op de Coul]] (2004)
|-
| [[Anthill comma]]
| Satrilo-ayoyoma
| s3(1o)yyM
| 532400/531441
| {{Monzo| 4 -12 2 0 3 }}
| 3.1212
| [[Budjarn Lambeth]] (2024)
|-
| [[4000/3993|Wizardharry comma]], pine comma
| Triluyoma
| 3(1uy)M
| 4000/3993
| {{Monzo| 5 -1 3 0 -3 }}
| 3.0323
| [[User:Godtone|Godtone]] (2023) for ''pine comma''
|-
| 23-11/7-comma
| Twetheluzoma
| 23(1uz)M
| <abbr title="896819112839771466727424 / 895430243255237372246531">(48 digits)</abbr>
| {{Monzo| 15 0 0 23 -23 }}
| 2.6832
|
|-
| [[Symbiotic comma]]
| Salozoma
| s1ozM
| 19712/19683
| {{Monzo| 8 -9 0 1 1 }}
| 2.5488
| [[Dawson Berry]] (2020)
|-
| [[5632/5625|Vishdel comma]]
| Saloquadguma
| s1o4gM
| 5632/5625
| {{Monzo| 9 -2 -4 0 1 }}
| 2.1531
|
|-
| [[Nexus comma]], nexisma
| Tribiloma
| 6(1o)M
| 1771561/1769472
| {{Monzo| -16 -3 0 0 6 }}
| 2.0427
| [[Dawson Berry]] (2020)
|-
| [[Reef comma]]
| Salubizoguma
| s1u2zgM
| 200704/200475
| {{Monzo| 12 -6 -2 2 -1 }}
| 1.9764
| [[Dawson Berry]] (2022)
|-
| [[41503/41472|Argyria]], tinge
| Lolotrizoma
| 1oo3zM
| 41503/41472
| {{Monzo| -9 -4 0 3 2 }}
| 1.2936
| [[Gayle Young]] (2018) and [[Todd Harrop]] (2020) for ''tinge'' [[Lériendil]] (2024) for ''argyria''
|-
| [[Alpharabian schisma]]
| Trisa-tritriloma
| 3s9(1o)M
| 618121839509504 / 617673396283947
| {{Monzo| 18 -31 0 0 9 }}
| 1.2565
| [[Dawson Berry]] (2024)
|-
| [[Olympia]]
| Salururuma
| s1urrM
| 131072/130977
| {{Monzo| 17 -5 0 -2 -1 }}
| 1.2552
| [[Flora Canou]] (2021)
|-
| [[37-11-comma]], 11-cycle schisma
| Quinsa-thiseluma
| 5s37(1u)M
| <abbr title="340282366920938463463374607431768211456 / 340039485861577398992406882305761986971">(78 digits)</abbr>
| {{Monzo| 128 0 0 0 -37 }}
| 1.2361
|
|-
| [[Seascape comma]], undecimal hemifourths comma
| Bilozoguguma
| 2(1ozgg)M
| 160083/160000
| {{Monzo| -8 3 -4 2 2 }}
| 0.89784
| [[User:Tristanbay|Tristan Bay]] (2024) for ''seascape comma''
|-
| [[Sesdecal comma]]
| Laquadlu-asepyoma
| L4(1u)7yM
| 234375/234256
| {{Monzo| -4 1 7 0 -4 }}
| 0.87923
|
|-
| [[Triagnoshenisma]]
| Trila-trilo-aguma
| 3L3(1o)gM
| 171885556953 / 171798691840
| {{Monzo| -35 17 -1 0 3 }}
| 0.87513
| [[Dawson Berry]], [[User:Frostburn|Frostburn]] (2024)
|-
| [[Frameshift comma]]
| Quadla-triluma
| 4L3(1u)M
| 22876792454961 / 22866405883904
| {{Monzo| -34 28 0 0 -3 }}
| 0.78620
| [[Dawson Berry]] (2021)
|-
| [[Chrysia]]
| Quadlo-atriruma
| 4(1o)3rM
| 43923/43904
| {{Monzo| -7 1 0 -3 4 }}
| 0.74905
| [[VIxen]] (2025)
|-
| [[Sossmarvel comma]]
| Trila-lusepruyoyoma
| 3L1u7ryyM
| <abbr title="9730975341796875 / 9726998192586752">(32 digits)</abbr>
| {{Monzo| -30 13 14 -7 -1 }}
| 0.70772
| [[Dawson Berry]] (2022)
|-
| [[3025/3024|Lehmerisma]]
| Loloruyoyoma
| 1ooryyM
| 3025/3024
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57240
| [[Gene Ward Smith]] (2004)
|-
| [[Ptolemi-nicema]]
| Quinbisa-twethetriluyoyoma
| 10s69(1uyy)M
|
| {{Monzo| 137 -138 138 0 -69 }}
| 0.56437
| [[User:Eliora|Eliora]] (2023)
|-
| [[Elysia]]
| Bilutrizoma
| 2(1u3z)M
| 117649/117612
| {{Monzo| -2 -5 0 6 -2 }}
| 0.54455
| [[Lériendil]] (2024)
|-
| [[Quartisma]]
| Saquinlu-azoma
| s5(1u)zM
| 117440512 / 117406179
| {{Monzo| 24 -6 0 1 -5 }}
| 0.50619
| [[Dawson Berry]] (2020)
|-
| [[9801/9800|Kalisma]], Gauss' comma
| Biloruguma
| 2(1org)M
| 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| 0.17665
| [[Margo Schulter]] (2000) [[Gene Ward Smith]] (2004) for ''Gauss' comma''
|-
| [[151263/151250|Odiheim comma]]
| Luluquinzo-aquadguma
| 1uu5za4gM
| 151263/151250
| {{Monzo| -1 2 -4 5 -2 }}
| 0.14879
|
|-
| [[Countercentisma]]
|
|
| (3776 digits)
| {{Monzo| -1 -3300 2700 0 -300 }}
| 0.14187
| [[User:CompactStar|CompactStar]] (2024)
|-
| [[Spoob]]
| Tribiluzozoguma
| 6(1uzzg)M
| 27682574402 / 27680640625
| {{Monzo| 1 0 -6 12 -6 }}
| 0.12094
| [[User:Eliora|Eliora]] (2023)
|-
| [[Luxma]]
| Saquinlu-aquadguma
| s5(1u)4gM
| 100663296/100656875
| {{Monzo|25 1 -4 0 -5}}
| 0.11043
| [[User:Tristanbay|Tristan Bay]] (2026)
|-
| [[Syntonoschisma]]
| Trisa-lusepyoma
| 3s1u7yM
| 83886080000000 / 83881572334857
| {{Monzo|30 -27 7 0 -1}}
| 0.093031
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Parimo]]
| Satribilo-aguma
| s6(1o)gM
| 1771561/1771470
| {{Monzo|-1 -11 -1 0 6}}
| 0.088931
|
|-
| Parisma
| Laquadlu-aruruguma
| L4(1u)rrgM
| 14348907 / 14348180
| {{Monzo|-2 15 -1 -2 -4}}
| 0.087717
|
|-
| [[Blare comma]]
| Laquadquadlo-aquadtrizoma
| L16(1o)12zM
| <abbr title="636003407850068828189211361 / 635974777627126753067532288">(54 digits)</abbr>
| {{Monzo|-51 -24 0 12 16}}
| 0.077935
| [[User:Akselai|Akselai]] (2024)
|-
| Ultimo
| Quadlo-asepru-ayoyoma
| 4(1o)7rayyM
| 3294225/3294172
| {{Monzo|-2 2 2 -7 4}}
| 0.027854
|
|-
| Parismina
| Sasa-quinbilo-azozoma
| ss10(1o)zzM
| 2541867610898 / 2541865828329
| {{Monzo|1 -26 0 2 10}}
| 0.0012141
|
|}

=== 13-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Wilschisma]]
| Sathoyoma
| s3oyM
| 532480/531441
| {{Monzo| 13 -12 1 0 0 1 }}
| 3.3814
| [[Flora Canou]] (2021)
|-
| Bean
| Sathuquinluma
| s3u5(1u)M
| 2097152/2093663
| {{Monzo| 21 0 0 0 -5 -1 }}
| 2.8826
|
|-
| [[625/624|Tunbarsma]]
| Thuquadyoma
| 3u4yM
| 625/624
| {{Monzo| -4 -1 4 0 0 -1 }}
| 2.7722
|
|-
| [[Fifthless mohohomma]]
| Thuthululuyoma
| 3uu1uuyM
| 20480/20449
| {{Monzo| 12 0 1 0 -2 -2 }}
| 2.6225
| [[User:原井玉葱郎|Misohito Nakai]] (2025)
|-
| [[676/675|Island comma]]
| Bithoguma
| 2(3og)M
| 676/675
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.5629
| [[Mike Battaglia]] (2011)
|-
| [[729/728|Squbema]]
| Lathuruma
| L3urM
| 729/728
| {{Monzo| -3 6 0 -1 0 -1 }}
| 2.3764
|
|-
| [[2200/2197|Petrma]]
| Trithu-aloyoyoma
| 3(3u)1oyyM
| 2200/2197
| {{Monzo| 3 0 2 0 1 -3 }}
| 2.3624
|
|-
| [[Tridecimal eighth-octave comma]]
| Thotrilo-aguma
| 3o3(1o)gM
| 17303/17280
| {{Monzo| -7 -3 -1 0 3 1 }}
| 2.3028
| [[Budjarn Lambeth]] (2024)
|-
| [[Sinarabian comma]]
| Lathotriluma
| L3o3(1u)M
| 85293/85184
| {{Monzo| -6 8 0 0 -3 1 }}
| 2.2138
| [[Dawson Berry]] (2025)
|-
| [[1575/1573|Nicola]]
| Thululuzoyoyoma
| 3u1uuzyyM
| 1575/1573
| {{Monzo| 0 2 2 1 -2 -1 }}
| 2.1998
|
|-
| [[Navicular comma]]
| Trithu-aluzoma
| 3(3u)1uzM
| 24192/24167
| {{Monzo| 7 3 0 1 -1 -3 }}
| 1.7900
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1001/1000|Sinbadma]]
| Tholozotriguma
| 3o1oz3gM
| 1001/1000
| {{Monzo| -3 0 -3 1 1 1 }}
| 1.7303
|
|-
| [[4459/4455|Tristanisma]]
| Tholutrizo-aguma
| 3o1u3zagM
| 4459/4455
| {{Monzo| 0 -4 -1 3 -1 1}}
| 1.5537
| [[User:Tristanbay|Tristan Bay]] (2024)
|-
| [[Tridecapyth comma]]
| Trisathoma
| 3s3oM
| 3489660928 / 3486784401
| {{Monzo| 28 -20 0 0 0 1 }}
| 1.4276
| [[Dawson Berry]] (2021)
|-
| [[Cantonisma]]
| Trithoru-ayoma
| 3(3or)yM
| 10985/10976
| {{Monzo| -5 0 1 -3 0 3 }}
| 1.4190
| [[Margo Schulter]] (2013)
|-
| [[Punctisma]]
| Sathutrizoguma
| s3u3zgM
| 43904/43875
| {{Monzo| 7 -3 -3 3 0 -1 }}
| 1.1439
| [[User:Jerdle|Jerdle]], [[Dawson Berry]] (2023)
|-
| [[Neguschisma]]
| Lala-thulozoma
| LL3u1ozM
| 13640319 / 13631488
| {{Monzo| -20 11 0 1 1 -1 }}
| 1.1212
| [[User:Tristanbay|Tristan Bay]] (2024)
|-
| [[1716/1715|Lummic comma]]
| Tholotriru-aguma
| 3o1o3ragM
| 1716/1715
| {{Monzo| 2 1 -1 -3 1 1 }}
| 1.0092
|
|-
| [[Pseudovishnuzma]]
| Sasa-thozosepbiguma
| ss3oz14gM
| 6106906624 / 6103515625
| {{Monzo| 26 0 -14 1 0 1 }}
| 0.96157
| [[User:Eliora|Eliora]] (2023)
|-
| [[Sercloreminisma]]
| Bithuthuzo-aguma
| 2(3uuz)gM
| 142884/142805
| {{Monzo| 2 6 -1 2 0 -4 }}
| 0.95746
| [[Francium]] (2024)
|-
| [[2080/2079|Ibnsinma, sinaisma]]
| Tholuruyoma
| 3o1uryM
| 2080/2079
| {{Monzo| 5 -3 1 -1 -1 1 }}
| 0.83252
| [[Margo Schulter]], [[Gene Ward Smith]] (2012) [[Flora Canou]] (2023)
|-
| [[Phaotic comma]], phaotisma
| Sathotriyoma
| s3u3yM
| 256000/255879
| {{Monzo| 11 -9 3 0 0 -1 }}
| 0.81847
| [[Dawson Berry]] (2021)
|-
| [[Kuragesma]]
| Tritho-aquadlu-ayoma
| 3(3o)4(1u)gM
| 43940/43923
| {{Monzo| 2 -1 1 0 -4 3 }}
| 0.66993
| [[User:原井玉葱郎|Misohito Nakai]] (2025)
|-
| [[Barbadisma]]
| Quadla-thuyoma
| 4L3uyM
| 114383962274805 / 114349209288704
| {{Monzo| -43 28 1 0 0 -1 }}
| 0.52608
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[4096/4095|Minisma]]
| Sathuruguma
| s3urgM
| 4096/4095
| {{Monzo| 12 -2 -1 -1 0 -1 }}
| 0.42272
| [[Flora Canou]] (2026)
|-
| [[4225/4224|Leprechaun comma]]
| Thotholuyoyoma
| 3oo1uyyM
| 4225/4224
| {{Monzo| -7 -1 2 0 -1 2 }}
| 0.40981
| [[Margo Schulter]] (2012)
|-
| [[6656/6655|Jacobin comma]]
| Thotrilu-aguma
| 3o3(1u)gM
| 6656/6655
| {{Monzo| 9 0 -1 0 -3 1 }}
| 0.26012
| [[Gene Ward Smith]] (2014)
|-
| [[Catasma]]
| Latrithuyoyoma
| L3(3uyy)M
| 140625/140608
| {{Monzo| -6 2 6 0 0 -3 }}
| 0.20930
| [[Tristan Bay]] (2023)
|-
| [[492128/492075|13^3⋅7/25 schismina]]
| Satritho-azoguguma
| s3(3o)zggM
| 492128/492075
| {{Monzo| 5 -9 -2 1 0 3 }}
| 0.18646
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[Harmonisma]]
| Thuthutrilo-aruma
| 3uu3(1o)rM
| 10648/10647
| {{Monzo| 3 -2 0 -1 3 -2 }}
| 0.16260
| [[Margo Schulter]] (2002)
|-
| [[Pentonisma]]
| Saquinthuzoguma
| s5(3uzg)M
| 281974669312 / 281950621875
| {{Monzo| 24 -5 -5 5 0 -5 }}
| 0.14765
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Pontigailimma]]
| Thururuquinguma
| 3urr5gM
| 1990656/1990625
| {{Monzo| 13 5 -5 -2 0 -1 }}
| 0.026960
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[Grossmisma]]
| septholo-azoguma
| 7(3o1o)zgM
| <abbr title="8559537565427849 / 8559456430325760">(32 digits)</abbr>
| {{Monzo| -30 -13 -1 1 7 7 }}
| 0.016410
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Chalmersia]]
| Lathotholuruguguma
| L3oo1urggM
| 123201/123200
| {{Monzo| -6 6 -2 -1 -1 2 }}
| 0.01405
| [[Gene Ward Smith]] (2003)
|-
| [[Lanasma]]
| Trila-septrithu-aquinquadbizoma
| 3L21(3u)40zM
| <abbr title="6366805760909027985741435139224001 / 6366804434232663711262864979263488">(68 digits)</abbr>
| {{Monzo| -33 -1 0 40 0 -21 }}
| 3.6074 × 10-4
| [[User:Angekallikoita|Ange]] (2025)
|}

=== 17-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Quinticular comma]]
| Saquinsoma
| s5(17o)M
| 1419857/1417176
| {{Monzo| -3 -11 0 0 0 0 5 }}
| 3.2720
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[561/560|Monardisma]]
| Soloruguma
| 17o1orgM
| 561/560
| {{Monzo| -4 1 -1 -1 1 0 1 }}
| 3.0887
| [[Scott Dakota]] (2023)
|-
| [[595/594|Dakotisma]]
| Soluzoyoma
| 17o1uzyM
| 595/594
| {{Monzo| -1 -3 1 1 -1 0 1 }}
| 2.9121
| [[Praveen Venkataramana]] (2022)
|-
| [[715/714|September comma]]
| Sutholoruyoma
| 17u3o1oryM
| 715/714
| {{Monzo| -1 -1 1 -1 1 1 -1 }}
| 2.4230
| [[Scott Dakota]] (2021)
|-
| [[833/832|Horizma, horizon comma]]
| Sothuzozoma
| 17o3uzzM
| 833/832
| {{Monzo| -6 0 0 2 0 -1 1 }}
| 2.0796
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[936/935|Ainisma, ainic comma]]
| Sutholuguma
| 17u3o1ugM
| 936/935
| {{Monzo| 3 2 -1 0 -1 1 -1 }}
| 1.8506
| [[Dawson Berry]] (2020)
|-
| [[2025/2023|Fidesma]]
| Susuruyoyoma
| 17uuryyM
| 2025/2023
| {{Monzo| 0 4 2 -1 0 0 -2 }}
| 1.7107
| [[Dawson Berry]] (2023)
|-
| [[1089/1088|Twosquare comma]]
| Suloloma
| 17u1ooM
| 1089/1088
| {{Monzo| -6 2 0 0 2 0 -1 }}
| 1.5905
| [[User:Kaiveran|Kaiveran]] (2018)
|-
| [[1156/1155|Quadrantonisma]]
| Sosoluruguma
| 17oo1urgM
| 1156/1155
| {{Monzo| 2 -1 -1 -1 -1 0 2 }}
| 1.4983
| [[Flora Canou]] (2023)
|-
| [[1225/1224|Noellisma]]
| Subizoyoma
| 17u2zyM
| 1225/1224
| {{Monzo| -3 -2 2 2 0 0 -1 }}
| 1.4138
| [[Flora Canou]] (2022)
|-
| [[1275/1274|Cimbrisma]]
| Sothubiruyoma
| 17o3u2ryM
| 1275/1274
| {{Monzo| -1 1 2 -2 0 -1 1 }}
| 1.3584
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1701/1700|Palingenetic comma, palingenesis]]
| Suzoguguma
| 17uzggM
| 1701/1700
| {{Monzo| -2 5 -2 1 0 0 -1 }}
| 1.0181
| [[Dawson Berry]] (2021)
|-
| [[Laser comma]]
| Lasorutriyoma
| L17or3yM
| 57375/57344
| {{Monzo| -13 3 3 -1 0 0 1 }}
| 0.93564
| [[User:Jerdle|Jerdle]] (2023)
|-
| [[2058/2057|Xenisma]]
| Sululutrizoma
| 17u1uu3zM
| 2058/2057
| {{Monzo| 1 1 0 3 -2 0 -1 }}
| 0.84143
| [[Margo Schulter]] (2000)
|-
| [[11016/11011|Cyclops comma]]
| Sothululuruma
| 17o3u1uurM
| 11016/11011
| {{Monzo| 3 4 0 -1 -2 -1 1 }}
| 0.78596
| [[Eliora]] (2022)
|-
| [[24576/24565|Mavka comma]], archagallisma
| Trisu-aguma
| 3(17u)gM
| 24576/24565
| {{Monzo| 13 1 -1 0 0 0 -3 }}
| 0.77506
| [[Eliora]] (2022) for ''mavka comma''
|-
| [[2431/2430|Heptacircle comma]]
| Sothologuma
| 17o3o1ogM
| 2431/2430
| {{Monzo| -1 -5 -1 0 1 1 1 }}
| 0.71230
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2500/2499|Sperasma]]
| Subiruyoyoma
| 17u2ryyM
| 2500/2499
| {{Monzo| 2 -1 4 -2 0 0 -1 }}
| 0.69263
| [[Dawson Berry]] (2023)
|-
| [[2601/2600|Sextantonisma]]
| Sosothuguguma
| 17oo3uggM
| 2601/2600
| {{Monzo| -3 2 -2 0 0 -1 2 }}
| 0.66573
| [[Flora Canou]] (2023)
|-
| [[Semisixthmisma]]
| Trisu-athutriloma
| 3(17u)3u3(1o)M
| 63888/63869
| {{Monzo| 4 1 0 0 3 -1 -3 }}
| 0.51494
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[4914/4913|Baladisma]]
| Trisu-athozoma
| 3(17u)3ozM
| 4914/4913
| {{Monzo| 1 3 0 1 0 1 -3 }}
| 0.35234
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[5832/5831|Chlorisma]]
| Sutriruma
| 17u3rM
| 5832/5831
| {{Monzo| 3 6 0 -3 0 0 -1 }}
| 0.29688
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[Galileisma]]
| Lalesu-aguma
| L11(17u)gM
| 171382426877952 / 171359481538165
| {{Monzo| 14 21 -1 0 0 0 -11 }}
| 0.23180
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Centisma]]
|
|
|
| 2.3.17 {{Monzo| -1001 -400 400 }}
| 0.16345
| [[CompactStar]] (2023)
|-
| [[Flashma]]
| Sotholuzotriguma
| 17o3o1uz3gM
| 12376/12375
| {{Monzo| 3 -2 -3 1 -1 1 1 }}
| 0.13989
| [[User:Kaiveran|Kaiveran]] (2016)
|-
| [[Sparkisma]]
| Sululuruyoyoma
| 17u1uuryyM
| 14400/14399
| {{Monzo| 6 2 2 -1 -2 0 -1 }}
| 0.12023
| [[User:Kaiveran|Kaiveran]] (2016)
|-
| [[Insanobromisma]]
| Sepquinsuyoyoma
| 35(17uyy)M
|
| {{Monzo| 36 -35 70 0 0 0 -35 }}
| 0.095608
| [[Eliora]] (2023)
|-
| [[1257795/1257728|Large triquarterisma]]
| Latrisulo-azoyoma
| L3(17u1o)zyM
| 1257795/1257728
| {{Monzo| -8 3 1 1 3 0 -3 }}
| 0.092222
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[471648/471625|Small triquarterisma]]
| Triso-alutriruguma
| 3(17o)1u3(rg)M
| 471648/471625
| {{Monzo| 5 1 -3 -3 -1 0 3 }}
| 0.084426
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[28561/28560|Pisanoisma]]
| Suquadtho-aruguma
| 17u4(3o)rgM
| 28561/28560
| {{Monzo| -4 -1 -1 -1 0 4 -1 }}
| 0.060616
| [[Budjarn Lambeth]] (2024)
|-
| [[E-shaped comma]]
| Susuthoquadzoma
| 17uu3o4zM
| 31213/31212
| {{Monzo| -2 -3 0 4 0 1 -2 }}
| 0.055466
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Lateral comma]]
| Sasuthotholoyoma
| s17u3oo1oyM
| 37180/37179
| {{Monzo| 2 -7 1 0 1 2 -1 }}
| 0.046564
| [[User:Akselai|Akselai]] (2024)
|-
| [[Clevelandisma]]
| Sotribizoguma
| 17o6(zg)M
| 2000033/2000000
| {{Monzo| -7 0 -6 6 0 0 1 }}
| 0.028565
| [[Lériendil]] (2024)
|-
| [[Scintillisma]]
| Lasuthuluquadzo-aguma
| L17u3u1u4zagM
| 194481/194480
| {{Monzo| -4 4 -1 4 -1 -1 -1 }}
| 0.0089018
| [[Margo Schulter]] (2012)
|-
| [[Aksial comma]]
| Sotritho-aquinru-aguma
| 17o3(3o)5ragM
| 336141/336140
| {{Monzo| -2 2 -1 -5 0 3 1 }}
| 0.0051503
| [[User:Akselai|Akselai]] (2024)
|}

=== 19-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[513/512|Undevicesimal schisma]], undevicesimal formal comma, Boethius' comma
| Lanoma
| L19oM
| 513/512
| 2.3.19 {{Monzo| -9 3 1 }}
| 3.3780
| Plainsound Music Edition (2020)<ref>[https://marsbat.space/pdfs/HEJI2legend+series.pdf The Helmholtz-Ellis JI Pitch Notation (HEJI)]</ref> for ''undevicesimal schisma''
|-
| [[6144/6137|Langwisma]]
| Nunusuma
| 19uu17uM
| 6144/6137
| {{Monzo| 11 1 0 0 0 0 -1 -2 }}
| 1.9736
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[969/968|Kingfisher comma]]
| Nosoluluma
| 19o17o1uuM
| 969/968
| {{Monzo| -3 1 0 0 -2 0 1 1 }}
| 1.7875
| [[Budjarn Lambeth]] (2025)
|-
| [[Mercurial comma]]
| Quinnosu-abiruyoma
| 5(19o17u)rryyM
| 557122275 / 556583944
| {{Monzo| -3 2 2 -2 0 0 -5 5 }}
| 1.6736
| [[User:Yourmusic Productions|Yourmusic Productions]] (2020)
|-
| [[1216/1215|Password, Eratosthenes' comma]]
| Sanoguma
| s19ogM
| 1216/1215
| {{Monzo| 6 -5 -1 0 0 0 0 1 }}
| 1.4243
|
|-
| [[1331/1330|Solvejgsma]]
| Nutrilo-aruguma
| 19u3(1o)rgM
| 1331/1330
| {{Monzo| -1 0 -1 -1 3 0 0 -1 }}
| 1.3012
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1445/1444|Aureusma]]
| Nunusosoyoma
| 19uu17ooyM
| 1445/1444
| {{Monzo| -2 0 1 0 0 0 2 -2 }}
| 1.1985
| [[Flora Canou]] (2021)
|-
| [[1521/1520|Pinkanberry]]
| Nuthothoguma
| 19u3oogM
| 1521/1520
| {{Monzo| -4 2 -1 0 0 2 0 -1 }}
| 1.1386
| [[User:Kaiveran|Kaiveran]] (2021)
|-
| [[1540/1539|Kevolisma]]
| Nulozoyoma
| 19u1ozyM
| 1540/1539
| {{Monzo| 2 -4 1 1 1 0 0 -1 }}
| 1.1245
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3213/3211|Cobaltomenisma]]
| Nusothuthuzoma
| 19u17o3uuzM
| 3213/3211
| {{Monzo| 0 3 0 1 0 -2 1 -1 }}
| 1.0780
| [[Francium]] (2025)
|-
| [[1729/1728|Ramanujanisma]]
| Nothozoma
| 19o3ozM
| 1729/1728
| {{Monzo| -6 -3 0 1 0 1 0 1 }}
| 1.0016
| [[Frédéric Gagné]] (2021)
|-
| [[3971/3969|Heartlandisma]]
| Nonoloruruma
| 19oo1orrM
| 3971/3969
| {{Monzo| 0 -4 0 -2 1 0 0 2 }}
| 0.87216
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[830297/829939|Minthtone schismina]]
| Trinuso-abitholuma
| 3(19u17o)2(3o1u)M
| 830297/829939
| {{Monzo| 0 0 0 0 -2 2 3 -3 }}
| 0.74662
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[2376/2375|Trichthonisma]]
| Nulotriguma
| 19u1o3gM
| 2376/2375
| {{Monzo| 3 3 -3 0 1 0 0 -1 }}
| 0.72879
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Crawma]]
| Nuquadso-atrithuma
| 19u4(17o)3(3u)M
| 83521/83486
| {{Monzo| -1 0 0 0 0 -3 4 -1 }}
| 0.72564
| [[groundfault]] (2023)
|-
| [[2432/2431|Blumeyer comma]]
| Nosuthuluma
| 19o17u3u1uM
| 2432/2431
| {{Monzo| 7 0 0 0 -1 -1 -1 1 }}
| 0.71200
| [[Douglas Blumeyer]] (2015)
|-
| [[93347/93312|Trilute comma]]
| Notrisoma
| 19o3(17o)M
| 93347/93312
| {{Monzo| -7 -6 0 0 0 0 3 1 }}
| 0.64924
| [[Xenllium]] (2025)
|-
| [[2926/2925|Neovulture comma, neovulturisma]]
| Nothulozoguguma
| 19o3u1ozggM
| 2926/2925
| {{Monzo| 1 -2 -2 1 1 -1 0 1 }}
| 0.59177
| [[Xenllium]] (2023)
|-
| [[3136/3135|Neomirkwai comma, neomirkwaisma]]
| Nuluzozoguma
| 19u1uzzgM
| 3136/3135
| {{Monzo| 6 -1 -1 2 -1 0 0 -1 }}
| 0.55214
| [[Xenllium]] (2023)
|-
| [[116640/116603|Large tridevisemma]]
| Trinu-asuyoma
| 3(19u)17uyM
| 116640/116603
| {{Monzo| 5 6 1 0 0 0 -1 -3 }}
| 0.54926
| [[Xenllium]] (2025)
|-
| [[3250/3249|Martebisma]]
| Nunuthotriyoma
| 19uu3o3yM
| 3250/3249
| {{Monzo| 1 -2 3 0 0 1 0 -2 }}
| 0.53277
| [[Xenllium]] (2023)
|-
| [[48013/48000|Small tridevisemma]]
| Trino-azotriguma
| 3(19o)z3gM
| 48013/48000
| {{Monzo| -7 -1 -3 1 0 0 0 3 }}
| 0.46881
| [[Xenllium]] (2025)
|-
| [[4200/4199|Neosatanisma]]
| Nusuthuzoyoyoma
| 19u17u3uzyyM
| 4200/4199
| {{Monzo| 3 1 2 1 0 -1 -1 -1 }}
| 0.41225
| [[Xenllium]] (2023)
|-
| [[176000/175959|Triseptichrome comma]]
| Nulotriruyoma
| 19u1o3(ry)M
| 176000/175959
| {{Monzo| 7 -3 3 -3 1 0 0 -1 }}
| 0.40335
| [[Xenllium]] (2025)
|-
| [[5776/5775|Neovish comma, neovishma]]
| Nonoluruguguma
| 19oo1urggM
| 5776/5775
| {{Monzo| 4 -1 -2 -1 -1 0 0 2 }}
| 0.29975
| [[Xenllium]] (2023)
|-
| [[5929/5928|Manzanisma]]
| Nuthubilozoma
| 19u3u2(1oz)M
| 5929/5928
| {{Monzo| -3 -1 0 2 2 -1 0 -1 }}
| 0.29202
| [[Xenllium]] (2025)
|-
| [[5985/5984|Neogrendel comma, neogrendelisma]]
| Nosuluzoyoma
| 19o17u1uzyM
| 5985/5984
| {{Monzo| -5 2 1 1 -1 0 -1 1 }}
| 0.28929
| [[Xenllium]] (2023)
|-
| BMO schismina
| Sabinothuma
| s2(19o3u)M
| 369664/369603
| {{Monzo| 10 -7 0 0 0 -2 0 2 }}
| 0.28570
|
|-
| [[6175/6174|Neonewtisma]]
| Nothotriru-ayoyoma
| 19o3o3rayyM
| 6175/6174
| {{Monzo| -1 -2 2 -3 0 1 0 1 }}
| 0.28038
| [[Xenllium]] (2023)
|-
| [[6860/6859|Devicubisma]]
| Trinuzo-ayoma
| 3(19uz)yM
| 6860/6859
| {{Monzo| 2 0 1 3 0 0 0 -3 }}
| 0.25238
| [[Xenllium]] (2025)
|-
| [[Undevicesimal counterschisma]]
| Seplanuma
| 7L19uM
| <abbr title=" 717897987691852588770249 / 717799705396186072481792">(48 digits)</abbr>
| 2.3.19 {{Monzo| -75 50 -1 }}
| 0.23703
| [[Xenllium]] (2025)
|-
| Frouggie comma
| Nusuquinthu-aquadloma
| 19u17u5(3u)4(1o)M
| 119939072 / 119927639
| {{Monzo| 13 0 0 0 4 -5 -1 -1 }}
| 0.16503
| [[Douglas Blumeyer]] (2020)
|-
| [[12636/12635|Padriellisma]]
| Nunuthoruguma
| 19uu3orgM
| 12636/12635
| {{Monzo| 2 5 -1 -1 0 1 0 -2 }}
| 0.13701
| [[Xenllium]] (2025)
|-
| [[Lakisma]]
| Saquadnoso-aguma
| s4(19o17o)gM
| 10884540241 / 10883911680
| {{Monzo| -12 -12 -1 0 0 0 4 4 }}
| 0.09998
| [[User:Flirora|+merlan #flirora]] (2021)
|-
| [[Aubertisma]]
| Nosothutrilu-arutriyoma
| 19o17o3u3(1u)r3yM
| 121125/121121
| {{monzo| 0 1 3 -1 -3 -1 1 1 }}
| 0.057173
| [[Francium]] (2025)
|-
| Pollar comma
| Nunusuquintho-aluluma
| 19uu17u5(3o)1uuM
| 742586/742577
| {{Monzo| 1 0 0 0 -2 5 -1 -2 }}
| 0.020982
| [[Douglas Blumeyer]] (2020)
|-
| [[Decimillisma]], 19-limit decimill
| Sanosorurutriguma
| s19o17orr3gM
| 165376/165375
| {{Monzo| 9 -3 -3 -2 0 0 1 1 }}
| 0.010469
| [[Flora Canou]] (2021), for ''decimillisma''
|-
| [[65/19 atom]]
| Sasa-nuthoyoma
| ss19u3oyM
| 272629760 / 272629233
| {{Monzo| 22 -15 1 0 0 1 0 -1 }}
| 0.0033465
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[Devicisma]]
| Nunusothutrilo-azoguma
| 19uu17o3u3(1o)zgM
| 633556/633555
| {{Monzo| 2 -3 -1 1 3 -1 1 -2 }}
| 0.0027326
| [[Xenllium]] (2023)
|-
| [[11859211/11859210|Tredekisma]]
| Quadno-athoquadlu-azoguma
| 19o43o1u4zgM
| 11859211/11859210
| {{Monzo| -1 -4 -1 1 -4 1 0 4 }}
| 0.000146
| [[Eufalesio]] (2026)
|}

=== 23-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[507/506|Laodicisma]]
| Twethuthotholuma
| 23u3oo1uM
| 507/506
| 2.3.11.13.23 {{Monzo| -1 1 -1 2 -1 }}
| 3.4180
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[529/528|Preziosisma]]
| Bitwetho-aluma
| 23oo1uM
| 529/528
| 2.3.11.23 {{Monzo| -4 -1 -1 2 }}
| 3.2758
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[576/575|Worcester comma]]
| Twethuguguma
| 23uggM
| 576/575
| 2.3.5.23 {{Monzo| 6 2 -2 -1 }}
| 3.0082
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[9765625/9750528]]
| Labitwethuquinyoma
| L23uu10yM
| 9765625/9750528
| 2.3.5.23 {{Monzo| -11 -2 10 -2 }}
| 2.6784
|
|-
| [[736/735|Harvardisma]]
| Twethoruruguma
| 23orrgM
| 736/735
| 2.3.5.7.23 {{Monzo| 5 -1 -1 -2 1 }}
| 2.3538
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[760/759|Squadronisma]]
| Twethunoluyoma
| 23u19o1uyM
| 760/759
| {{Monzo| 3 -1 1 0 -1 0 0 1 -1 }}
| 2.2794
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[875/874|Nymphisma]]
| Twethunuzotriyoma
| 23u19uz3yM
| 875/874
| 2.5.7.19.23 {{Monzo| -1 3 1 -1 -1 }}
| 1.9797
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[897/896|Lysistratisma]]
| Twethothoruma
| 23o3orM
| 897/896
| 2.3.7.13.23 {{Monzo| -7 1 -1 1 1 }}
| 1.9311
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3014656/3011499|23/17-schisma]]
| Sasa-twethosuma
| ss23o17uM
| 3014656/3011499
| 2.3.17.23 {{monzo| 17 -11 -1 1 }}
| 1.8139
|
|-
| [[2187/2185|Vashegyitisma]]
| Latwethunuguma
| L23u19ugM
| 2187/2185
| 3.5.19.23 {{monzo| 7 -1 -1 -1 }}
| 1.5839
| [[Francium]] (2025)
|-
| [[1105/1104|Fragarisma]]
| Twethusothoyoma
| 23u17o3oyM
| 1105/1104
| {{Monzo| -4 -1 1 0 0 1 1 0 -1 }}
| 1.5674
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[7942/7935|Brigade comma]]
| Bitwethuno-aloguma
| 23uu19oo1ogM
| 7942/7935
| {{Monzo| 1 -1 -1 0 1 0 0 2 -2 }}
| 1.5266
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1197/1196|Rodessisma]]
| Twethunothuzoma
| 23u19o3uzM
| 1197/1196
| {{Monzo| -2 2 0 1 0 -1 0 1 -1 }}
| 1.4469
| [[Lériendil]] (2025)
|-
| [[1288/1287|Triaphonisma]], santisma
| Twethothuluzoma
| 23o3u1uzM
| 1288/1287
| {{Monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 1.3446
| [[User:Xenllium|Xenllium]] (2023) for ''santisma''
|-
| [[1496/1495|Turkisma]]
| Twethusothuloguma
| 23u17o3u1ogM
| 1496/1495
| {{Monzo| 3 0 -1 0 1 -1 1 0 -1 }}
| 1.1576
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[7429/7425|Gordaitisma]]
| Twethonosoluguguma
| 23o19o17o1uggM
| 7429/7425
| {{monzo| 0 -3 -2 0 -1 0 1 1 1 }}
| 0.93240
| [[Francium]] (2025)
|-
| [[1863/1862|Antinousisma]]
| Twethonururuma
| 23o19urrM
| 1863/1862
| 2.3.7.19.23 {{Monzo| -1 4 -2 -1 1 }}
| 0.92952
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Kaifsma]]
| Twethunosutholuzozoguma
| 23u19o17u3o1uzzgM
| 193648/193545
| {{Monzo| 4 -2 -1 2 -1 1 -1 1 -1 }}
| 0.92108
| [[User:Kaiveran|Kaiveran]] (2022)
|-
| [[2024/2023|Artifisma]], insincere comma
| Twethosusuloruma
| 23o17uu1orM
| 2024/2023
| 2.7.11.17.23 {{Monzo| 3 -1 1 -2 1 }}
| 0.85556
| [[User:Xenllium|Xenllium]] (2023) for ''insincere comma''
|-
| [[2025/2024|Cupcake comma]], cupcakesma
| Latwethuluyoyoma
| L23u1uyyM
| 2025/2024
| 2.3.5.11.23 {{Monzo| -3 4 2 -1 -1 }}
| 0.85514
| See the page.
|-
| [[2185/2184|Guangdongisma]]
| Twethonothuruyoma
| 23o19o3uryM
| 2185/2184
| {{Monzo| -3 -1 1 -1 0 -1 0 1 1 }}
| 0.79251
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2300/2299|Travellisma]]
| Twethonubiluyoma
| 23o19u1uuyyM
| 2300/2299
| 2.5.11.19.23 {{Monzo| 2 2 -2 -1 1 }}
| 0.75287
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2646/2645|Biyativice comma]]
| Bitwethuzo-aguma
| 23uuzzgM
| 2646/2645
| 2.3.5.7.23 {{Monzo| 1 3 -1 2 -2 }}
| 0.65441
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2737/2736|Kotkisma]]
| Twethonusozoma
| 23o19u17ozM
| 2737/2736
| {{Monzo| -4 -2 0 1 0 0 1 -1 1 }}
| 0.63265
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Kwadransma]]
| Quadtwethuma
| 4(23u)M
| 279936/279841
| {{Monzo| 7 7 0 0 0 0 0 0 -4 }}
| 0.58762
| [[Lériendil]] (2024)
|-
| [[3060/3059|Vicious comma]], viciousma
| Twethunusoruyoma
| 23u19u17oryM
| 3060/3059
| {{Monzo| 2 2 1 -1 0 0 1 -1 -1 }}
| 0.56586
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3381/3380|Mikkolisma]], seminaiadvice comma
| Twethothuthuzozoguma
| 23o3uuzzgM
| 3381/3380
| {{Monzo| -2 1 -1 2 0 -2 0 0 1 }}
| 0.51212
| See the page.
|-
| [[6877/6875|Grossvice comma]]
| Bitwetho-atholuquadguma
| 23oo3o1u4gM
| 6877/6875
| {{Monzo| 0 0 -4 0 -1 1 0 0 2 }}
| 0.50356
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3520/3519|Vicedim comma]]
| Twethusuloyoma
| 23u17u1oyM
| 3520/3519
| {{Monzo| 6 -2 1 0 1 0 -1 0 -1 }}
| 0.49190
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3888/3887|Shoalma]], vicetride comma
| Twethuthuthuma
| 23u3uuM
| 3888/3887
| 2.3.13.23 {{Monzo| 4 5 -2 -1 }}
| 0.44533
| See the page.
|-
| [[8075/8073|Hagendorfisma]]
| Twethunosothuyoyoma
| 23u19o17o3uyyM
| 8075/8073
| {{monzo| 0 -3 2 0 0 -1 1 1 -1 }}
| 0.42884
| [[Francium]] (2025)
|-
| [[4693/4692|Viceaug comma]]
| Twethunonosuthoma
| 23u19oo17u3oM
| 4693/4692
| {{Monzo| -2 -1 0 0 0 1 -1 2 -1 }}
| 0.36894
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[4761/4760|Demiquartervice comma]]
| Bitwetho-asuruguma
| 23oo17urgM
| 4761/4760
| {{Monzo| -3 2 -1 -1 0 0 -1 0 2 }}
| 0.36367
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[5083/5082|Broadviewsma]]
| Twethosotholuluruma
| 23o17o3o1uurM
| 5083/5082
| {{Monzo| -1 -1 0 -1 -2 1 1 0 1 }}
| 0.34063
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[8625/8624|Beerglass comma]]
| Twetholururutriyoma
| 23o1urr3yM
| 8625/8624
| {{Monzo| -4 1 3 -2 -1 0 0 0 1 }}
| 0.20073
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Galeaclolusisma]]
| Twethususutholuquadyoma
| 23u17uu3o1u4yM
| 73125/73117
| {{Monzo| 0 2 4 0 -1 1 -2 0 -1 }}
| 0.18941
| [[Francium]] (2025)
|-
| [[10626/10625|Demiglace comma]]
| Twethosulozoquadguma
| 23o17u1oz4gM
| 10626/10625
| {{Monzo| 1 1 -4 1 1 0 -1 0 1 }}
| 0.16293
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Vicetertisma]]
| Tritwethu-athothoma
| 3(23u)3ooM
| 12168/12167
| 2.3.13.23 {{Monzo| 3 2 2 -3 }}
| 0.14228
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Joshuavoisma]]
| Twethusutholozoyoyoma
| 23u17u3o1ozyyM
| 25025/25024
| {{monzo| -6 0 2 1 1 1 -1 0 -1 }}
| 0.06918
| [[Francium]] (2024)
|-
| [[Diarithmedia]]
| Bitwethozo-aguma
| 23oozzgM
| 25921/25920
| 2.3.5.7.23 {{monzo| -6 -4 -1 2 2 }}
| 0.066790
| [[Flora Canou]] (2023), modified by [[Lériendil]] (2024)
|-
| [[Jeffbenisma]]
| Labitwethu-anutholuzoyoma
| L23uu19u3o1uzyM
| 110565/110561
| {{monzo| 0 5 1 1 -1 1 0 -1 -2 }}
| 0.062633
| [[Francium]] (2025)
|}

=== 29-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[551/550|Minor chthonovinema]]
| Twenonolugugu
| 29o19o1ugg2
| 551/550
| 2.5.11.19.29 {{monzo| -1 -2 -1 1 1 }}
| 3.1448
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[552/551|Sigelindisma]]
| Twenutwethonu
| 29u23o19u1
| 552/551
| 2.3.19.23.29 {{monzo| 3 1 -1 1 -1 }}
| 3.1391
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[609/608|Vineyard comma]]
| Twenonuzo
| 29o19uz1
| 609/608
| 2.3.7.19.29 {{monzo| -5 1 1 -1 1 }}
| 2.8451
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[638/637|Moirisma]]
| Twenothuloruru
| 29o3u1orr-2
| 638/637
| 2.7.11.13.29 {{monzo| 1 -2 1 -1 1 }}
| 2.7157
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[726/725|Joellisma]]
| Twenubilogu
| 29u1oogg1
| 726/725
| 2.3.5.11.29 {{monzo| 1 1 -2 2 -1 }}
| 2.3863
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[783/782|Norisma]]
| Twenotwethusu
| 29o23u17u-2
| 783/782
| 2.3.17.23.29 {{monzo| -1 3 -1 -1 1 }}
| 2.2124
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[784/783|Biminorisma]], spoogalactic comma
| Twenuzozo
| 29uzz2
| 784/783
| 2.3.7.29 {{monzo| 4 -3 2 -1 }}
| 2.2096
| [[Scott Dakota]] for ''biminorisma''
|-
| [[841/840|Arabellisma]]
| Bitweno-arugu
| 29oorg1
| 841/840
| 2.3.5.7.29 {{monzo| -3 -1 -1 -1 2 }}
| 2.0598
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1015/1014|Christisma]]
| Twenothuthuzoyo
| 29o3uuzy1
| 1015/1014
| {{monzo| -1 -1 1 1 0 -2 0 0 0 1 }}
| 1.7065
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1045/1044|Michelisma]]
| Twenunoloyo
| 29u19o1oy1
| 1045/1044
| {{monzo| -2 -2 1 0 1 0 0 1 0 -1 }}
| 1.6575
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1276/1275|Ucclisma]]
| Twenosulogugu
| 29o17u1ogg1
| 1276/1275
| {{monzo| 2 -1 -2 0 1 0 -1 0 0 1 }}
| 1.3573
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1450/1449|Raimondisma]]
| Twenotwethuruyoyo
| 29o23uryy-2
| 1450/1449
| {{monzo| 1 -2 2 -1 0 0 0 0 -1 1 }}
| 1.1944
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1596/1595|Itzigsohnisma]]
| Twenunoluzogu
| 29u19o1uzg2
| 1596/1595
| {{monzo| 2 1 -1 1 -1 0 0 1 0 -1 }}
| 1.0851
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1625/1624|Norcma]]
| Twenuthorutriyo
| 29u3or3y-2
| 1625/1624
| 2.5.7.13.29 {{monzo| -3 3 -1 1 -1 }}
| 1.0657
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1683/1682|Castafiorisma]]
| Bitwenu-asolo
| 29uu17o1o1
| 1683/1682
| 2.3.11.17.29 {{monzo| -1 2 1 1 -2 }}
| 1.0290
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2001/2000|Major discoverisma]]
| Twenotwethotrigu
| 29o23o3g2
| 2001/2000
| 2.3.5.23.29 {{monzo| -4 1 -3 1 1 }}
| 0.86540
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2002/2001|Minor discoverisma]]
| Twenutwethutholozo
| 29u23u3o1oz1
| 2002/2001
| {{monzo| 1 -1 0 1 1 1 0 0 -1 -1 }}
| 0.86497
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2176/2175|Donarisma]]
| Twenusogugu
| 29u17ogg2
| 2176/2175
| 2.3.5.17.29 {{monzo| 7 -1 -2 1 -1 }}
| 0.79579
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2205/2204|Glinkisma]]
| Twenunuzozoyo
| 29u19uzzy1
| 2205/2204
| {{monzo| -2 2 1 2 0 0 0 -1 0 -1 }}
| 0.78532
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2262/2261|Mitidikisma]]
| Twenonusuthoru
| 29o19u17u3or-2
| 2262/2261
| {{monzo| 1 1 0 -1 0 1 -1 -1 0 1 }}
| 0.76552
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2465/2464|Laservine comma]]
| Twenosoluruyo
| 29o17o1ury1
| 2465/2464
| {{monzo| -5 0 1 -1 -1 0 1 0 0 1 }}
| 0.70247
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2640/2639|Hällströmisma]]
| Twenuthuloruyo
| 29u3u1ory-2
| 2640/2639
| {{monzo| 4 1 1 -1 1 -1 0 0 0 -1 }}
| 0.65589
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2755/2754|Avicema]]
| Twenonosuyo
| 29o19o17uy1
| 2755/2754
| {{monzo| -1 -4 1 0 0 0 -1 1 0 1 }}
| 0.62851
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2784/2783|Domeykisma]]
| Twenotwethululu
| 29o23u1uu1
| 2784/2783
| 2.3.11.23.29 {{monzo| 5 1 -2 -1 1 }}
| 0.62196
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[9251/9248|Helevenisma]]
| Bitwenosu-alo
| 29oo17uu1o-2
| 9251/9248
| 2.11.17.29 {{monzo| -5 1 -2 2 }}
| 0.56151
| [[Zhea Erose]] (2023)
|-
| [[3249/3248|Musashinisma]]
| Twenunonoru
| 29u19oor1
| 3249/3248
| 2.3.7.19.29 {{monzo| -4 2 -1 2 -1 }}
| 0.53293
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[3451/3450|Mentorisma]]
| Twenotwethusozogugu
| 29o23u17ozgg2
| 3451/3450
| {{monzo| -1 -1 -2 1 0 0 1 0 -1 1 }}
| 0.50173
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Bronxisma]]
| Twenununusolozo
| 29u19uu17o1oz1
| 10472/10469
| {{monzo| 3 0 0 1 1 0 1 -2 0 -1 }}
| 0.49603
| [[Francium]] (2024)
|-
| [[3510/3509|Veederisma]]
| Twenutholuluyo
| 29u3o1uuy1
| 3510/3509
| {{monzo| 1 3 1 0 -2 1 0 0 0 -1 }}
| 0.49330
| [[Francium]] (2024)
|-
| [[4641/4640|Vinecute comma]]
| Twenusothozogu
| 29u17o3ozg2
| 4641/4640
| {{monzo| -5 1 -1 1 0 1 1 0 0 -1 }}
| 0.37307
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[4785/4784|Petrovsma]]
| Twenotwethuthuloyo
| 29o23u3u1oy-2
| 4785/4784
| {{monzo| -4 1 1 0 1 -1 0 0 -1 1 }}
| 0.36184
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[4901/4900|Large grapevine]]
| Twenothothobirugu
| 29o3oorrgg1
| 4901/4900
| 2.5.7.13.29 {{monzo| -2 -2 -2 2 1 }}
| 0.35328
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Mynucuvine comma]]
| Labitwenu-athuyo
| L29uu3uy-2
| 10935/10933
| 3.5.13.29 {{monzo| 7 1 -1 -2 }}
| 0.31667
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[5888/5887|Vinocular comma]]
| Bitwenu-atwethoru
| 29uu23or1
| 5888/5887
| 2.7.23.29 {{monzo| 8 -1 1 -2 }}
| 0.29405
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[5916/5915|Woudisma]]
| Twenosothuthurugu
| 29o17o3uurg1
| 5916/5915
| {{monzo| 2 1 -1 -1 0 -2 1 0 0 1 }}
| 0.29266
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[7425/7424|Small grapevine]]
| Latwenuloyoyo
| L29u1oyy-2
| 7425/7424
| 2.3.5.11.29 {{monzo| -8 3 2 1 -1 }}
| 0.23318
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[8671/8670|Vinous comma]], vinousma
| Twenotwethosusuthogu
| 29o23o17uu3og1
| 8671/8670
| {{monzo| -1 -1 -1 0 0 1 -2 0 1 1 }}
| 0.19967
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[9802/9801|Kakisma]]
|
|
| 9802/9801
| {{Monzo| 1 -4 0 0 -2 2 0 0 0 1 }}
| 0.17663
| [[Lériendil]] (2025)
|-
| [[10557/10556|Rowlandisma]]
| Twenutwethosothuru
| 29u23o17o3ur1
| 10557/10556
| {{monzo| -2 3 0 -1 0 -1 1 0 1 -1 }}
| 0.16400
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Uma]], umic comma
| Twenotwethoquadru-agu
| 29o23o4rg-2
| 12006/12005
| {{monzo| 1 2 -1 -4 0 0 0 0 1 1 }}
| 0.14420
|
|-
| [[Vuillafansisma]]
| Twenunosoluyo
| 29u19o17o1uy2
| 25840/25839
| {{monzo| 4 -4 1 0 -1 0 1 1 0 -1 }}
| 0.067000
| [[Francium]] (2024)
|-
| [[Odyssey comma]]
| Bitwenotwetho-athulurutrigu
| 29oo23oo3u1ur3g2
| 4004001/4004000
| {{monzo| -5 2 -3 -1 -1 -1 0 0 2 2 }}
| 0.00043238
| [[Tristan Bay]] (2024)
|}

=== 31-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[496/495|Navatonisma]]
| Thiwolugu
| 31o1ug1
| 496/495
| 2.3.5.11.31 {{monzo| 4 -2 -1 -1 1 }}
| 3.4939
| [[User:FilterNashi|FilterNashi]] (2025)
|-
| [[528/527|Rezisma]]
| Thiwusulo
| 31u17u1o1
| 528/527
| 2.3.11.17.31 {{monzo| 4 1 1 -1 -1 }}
| 3.2820
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[589/588|Croatisma]]
| Thiwonoruru
| 31o19orr-2
| 589/588
| 2.3.7.19.31 {{monzo| -2 -1 -2 1 1 }}
| 2.9418
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[621/620|Owowhatsthisma]]
| Thiwutwethogu
| 31u23og2
| 621/620
| 2.3.5.23.31 {{monzo| -2 3 -1 1 -1 }}
| 2.7901
| [[HEHEHE I AM A SUPAHSTAR SAGA]] (2021)
|-
| [[651/650|Antiklisma]]
| Thiwothuzogugu
| 31o3uzgg1
| 651/650
| 2.3.5.7.13.31 {{monzo| -1 1 -2 1 -1 1 }}
| 2.6614
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[714/713|Ululisma]]
| Thiwutwethusozo
| 31u23u17oz2
| 714/713
| 2.3.7.17.23.31 {{monzo| 1 1 1 1 -1 -1 }}
| 2.4264
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[961/960|Tricesimoprimal quartertones comma]]
| Bithiwo-agu
| 31oog-2
| 961/960
| 2.3.5.31 {{monzo| -6 -1 -1 2 }}
| 1.8024
| [[Godtone]] (2024)
|-
| [[1024/1023|Kibisma]]
| Thiwulu
| 31u1u2
| 1024/1023
| 2.3.11.31 {{Monzo| 10 -1 -1 -1 }}
| 1.6915
| See the page.
|-
| [[2233/2232|Kuznetsovisma]]
| Thiwotwenolozo
| 31u29o1oz2
| 2233/2232
| 2.3.7.11.29.31 {{monzo| -3 -2 1 1 1 -1 }}
| 0.77547
| [[Francium]] (2024)
|-
| [[3969/3968|Yunzee comma]]
| Lathiwuzozo
| L31uzz2
| 3969/3968
| 2.3.7.31 {{monzo| -7 4 2 -1 }}
| 0.43624
| [[User:Kaiveran|Kaiveran]] (2021)
|-
| [[4186/4185|Tamashimisma]]
| Thiwutwethothozogu
| 31u23o3ozg3
| 4186/4185
| 2.3.5.7.13.23.31 {{monzo| 1 -3 -1 1 1 1 -1 }}
| 0.41363
| [[Francium]] (2025)
|-
| [[4992/4991|Kalmanisma]]
| Thiwutwethuthoru
| 31u23u3or1
| 4992/4991
| 2.3.7.13.23.31 {{monzo| 7 1 -1 1 -1 -1 }}
| 0.34684
| [[Francium]] (2024)
|-
| [[5797/5796|Bivojisma]]
| Thiwotwethusoloru
| 31o23u17o1or-2
| 5797/5796
| 2.3.7.11.17.23.31 {{monzo| -2 -2 -1 1 1 -1 1 }}
| 0.29867
| [[Francium]] (2025)
|-
| [[6076/6075|Large ricegrain]]
| Sathiwobizogu
| s31ozzgg2
| 6076/6075
| 2.3.5.7.31 {{monzo| 2 -5 -2 2 1 }}
| 0.28495
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[6480/6479|Scarlattisma]]
| Thiwunuluyo
| 31u19u1uy1
| 6480/6479
| 2.3.5.11.19.31 {{monzo| 4 4 1 -1 -1 -1 }}
| 0.26719
| [[Francium]] (2024)
|-
| [[6728/6727|Sushi comma]]
| Bithiwutweno-aru
| 31uu29oor2
| 6728/6727
| 2.7.29.31 {{monzo| 3 -1 2 -2 }}
| 0.25734
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[16337/16335|Brown rice comma]]
| Bithiwo-asolulugu
| 31oo17o1uug1
| 16337/16335
| 3.5.11.17.31 {{monzo| -3 -1 -2 1 2 }}
| 0.21195
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[8464/8463|Polishookisma]]
| Thiwubitwetho-athuru
| 31u23oo3ur2
| 8464/8463
| 2.3.7.13.23.31 {{monzo| 4 -1 -1 -1 2 -1 }}
| 0.20455
| [[Francium]] (2024)
|-
| [[8960/8959|Small ricegrain]]
| Thiwususuzoyo
| 31u17uuzy1
| 8960/8959
| 2.5.7.17.31 {{monzo| 8 1 1 -2 -1 }}
| 0.19323
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Acronymisma]]
| Thiwotrithu-azo
| 31o3(3u)z-2
| 17577/17576
| 2.3.7.13.31 {{monzo| -3 4 1 -3 1 }}
| 0.098497
| [[User:Lériendil|Lériendil]] (2025)
|-
| [[Totziensisma]]
| Thiwotwetholurutrigu
| 31o23o1ur3g1
| 19251/19250
| 2.3.5.7.11.23.31 {{monzo| -1 3 -3 -1 -1 1 1 }}
| 0.089932
| [[Francium]] (2024)
|-
| [[Honeybrookisma]]
| Thiwobitwenu-atwethutho
| 31o29uu23u3o-2
| 19344/19343
| 2.3.13.23.29.31 {{monzo| 4 1 1 -1 -2 1 }}
| 0.089500
| [[Francium]] (2024)
|-
| [[Tricecubisma]]
| Trithiwu-anozozo
| 3(31u)19ozz4
| 29792/29791
| 2.7.19.31 {{monzo| 5 2 1 -3 }}
| 0.058112
| [[User:Xenllium|Xenllium]] (2025)
|}

=== 37-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[666/665|Beastisma]]
| Thisonurugu
| 37o19urg1
| 666/665
| 2.3.5.7.19.37 {{monzo| 1 2 -1 -1 -1 1 }}
| 2.6014
| [[Francium]] (2025)
|-
| [[667/666|Denisisma]]
| Thisutwenotwetho
| 37u29o23o1
| 667/666
| 2.3.23.29.37 {{monzo| -1 -2 1 1 -1 }}
| 2.5975
| [[Francium]] (2025)
|-
| [[703/702|Noemisma]]
| Thisonothu
| 37o19o3u2
| 703/702
| 2.3.13.19.37 {{monzo| -1 -3 -1 1 1 }}
| 2.4644
| [[Francium]] (2025)
|-
| [[704/703|Minimyna]]
| Thisunulo
| 37u19u1o-2
| 704/703
| 2.11.19.37 {{monzo| 6 1 -1 -1 }}
| 2.4609
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[741/740|Botolphisma]]
| Thisunothogu
| 37u19o3og1
| 741/740
| 2.3.5.13.19.37 {{monzo| -2 1 -1 1 1 -1 }}
| 2.3379
| [[Francium]] (2025)
|-
| [[851/850|Zeissisma]]
| Thisotwethosugugu
| 37o23o17ugg2
| 851/850
| 2.5.17.23.37 {{monzo| -1 -2 -1 1 1 }}
| 2.0355
| [[Francium]] (2025)
|-
| [[925/924|Alphonsinisma]]
| Thisoluruyoyo
| 37o1uryy1
| 925/924
| 2.3.5.7.11.37 {{monzo| -2 -1 2 -1 -1 1 }}
| 1.8726
| [[Francium]] (2025)
|-
| [[1036/1035|Ganymedisma]]
| Thisotwethuzogu
| 37o23uzg2
| 1036/1035
| 2.3.5.7.23.37 {{monzo| 2 -2 -1 1 -1 1 }}
| 1.6719
| [[Francium]] (2025)
|-
| [[1184/1183|Gaeisma]]
| Thisothuthuru
| 37o3uur1
| 1184/1183
| 2.7.13.37 {{monzo| 5 -1 -2 1}}
| 1.4628
| [[Francium]] (2025)
|-
| [[1332/1331|Marconisma]]
| Thisotrilu
| 37o3(1u)2
| 1332/1331
| 2.3.11.37 {{monzo| 2 2 -3 1 }}
| 1.3002
| [[Francium]] (2024)
|-
| [[1480/1479|Aunusisma]]
| Thisotwenusuyo
| 37o29u17uy1
| 1480/1479
| 2.3.5.17.29.37 {{monzo| 3 -1 1 -1 -1 1 }}
| 1.1702
| [[Francium]] (2024)
|-
| [[1665/1664|Gabisma]]
| Thisothuyo
| 37o3uy1
| 1665/1664
| 2.3.5.13.37 {{monzo| -7 2 1 -1 1 }}
| 1.0401
| [[Francium]] (2025)
|-
| [[1666/1665|Gentisma]]
| Thisusozozogu
| 37u17ozzg2
| 1666/1665
| 2.3.5.7.17.37 {{monzo| 1 -2 -1 2 1 -1 }}
| 1.0395
| [[Francium]] (2025)
|-
| [[1702/1701|Kalaharisma]]
| Sathisotwethoru
| S37o23or2
| 1702/1701
| 2.3.7.23.37 {{monzo| 1 -5 -1 1 1 }}
| 1.0175
| [[Francium]] (2025)
|-
| [[1925/1924|Misericorde]]
| Thisuthulozoyoyo
| 37u3u1ozyy-2
| 1925/1924
| 2.5.7.11.13.37 {{monzo| -2 2 1 1 -1 -1 }}
| 0.89958
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[2109/2108|Dhotelisma]]
| Thisothiwunosu
| 37o31u19o17u2
| 2109/2108
| 2.3.17.19.31.37 {{monzo| -2 1 -1 1 -1 1 }}
| 0.82107
| [[Francium]] (2025)
|-
| [[2146/2145|Stentorisma]]
| Thisotwenothulugu
| 37o29o3u1ug2
| 2146/2145
| 2.3.5.11.13.29.37 {{monzo| 1 -1 -1 -1 -1 1 1 }}
| 0.80691
| [[Francium]] (2024)
|-
| [[2553/2552|Viljevisma]]
| Thisotwenutwetholu
| 37o29u23o1u2
| 2553/2552
| 2.3.11.23.29.37 {{monzo| -3 1 -1 1 -1 1 }}
| 0.67825
| [[Francium]] (2024)
|-
| [[2850/2849|Mozhaiskisma]]
| Thisunoluruyoyo
| 37u19o1uryy-2
| 2850/2849
| 2.3.5.7.11.19.37 {{monzo| 1 1 2 -1 -1 1 -1 }}
| 0.60756
| [[Francium]] (2025)
|-
| [[3146/3145|Datonisma]]
| Thisusutholologu
| 37u17u3o1oog-2
| 3146/3145
| 2.5.11.13.17.37 {{monzo| 1 -1 2 1 -1 -1 }}
| 0.55038
| [[Francium]] (2025)
|-
| [[3220/3219|Murayamisma]]
| Thisutwenutwethozoyo
| 37u29u23ozy1
| 3220/3219
| 2.3.5.7.23.29.37 {{monzo| 2 -1 1 1 1 -1 -1 }}
| 0.53773
| [[Francium]] (2025)
|-
| [[3256/3255|Daguerrisma]]
| Thisothiwulorugu
| 37o31u1org2
| 3256/3255
| 2.3.5.7.11.31.37 {{monzo| 3 -1 -1 -1 1 -1 1 }}
| 0.53179
| [[Francium]] (2025)
|-
| [[3367/3366|Alexisma]]
| Thisosutholuzo
| 37o17u3o1uz2
| 3367/3366
| 2.3.7.11.13.17.37 {{monzo| -1 -2 1 -1 1 -1 1 }}
| 0.51425
| [[Francium]] (2025)
|-
| [[3553/3552|Meranisma]]
| Thisunosolo
| 37u19o17o1o1
| 3553/3552
| 2.3.11.17.19.37 {{monzo| -5 -1 1 1 1 -1 }}
| 0.48733
| [[Francium]] (2024)
|-
| [[3626/3625|Ohsakisma]]
| Thisotwenuzozotrigu
| 37o29uzz3g3
| 3626/3625
| 2.5.7.29.37 {{monzo| 1 -3 2 -1 1 }}
| 0.47752
| [[Francium]] (2025)
|-
| [[3627/3626|Sayersisma]]
| Thisuthiwothoruru
| 37u31o3orr-3
| 3627/3626
| 2.3.7.13.31.37 {{monzo| -1 2 -2 1 1 -1 }}
| 0.47738
| [[Francium]] (2025)
|-
| [[3774/3773|Megumisma]]
| Thisosolutriru
| 37o17o1u3r1
| 3774/3773
| 2.3.7.11.17.37 {{monzo| 1 1 -3 -1 1 1 }}
| 0.45879
| [[Francium]] (2025)
|-
| [[4625/4624|Shchedrinisma]]
| Thisosusutriyo
| 37o17uu3y-2
| 4625/4624
| 2.5.17.37 {{monzo| -4 3 -2 1 }}
| 0.37436
| [[Francium]] (2024)
|-
| [[5292/5291|Bullionisma]]
| Thisuthuluzozo
| 37u3u1uzz1
| 5292/5291
| 2.3.7.11.13.37 {{monzo| 2 3 2 -1 -1 -1 }}
| 0.32717
| [[Eliora]] (2023)
|-
| [[5440/5439|Teraosma]]
| Thisusoruruyo
| 37u17orry-2
| 5440/5439
| 2.3.5.7.17.37 {{monzo| 6 -1 1 -2 1 -1 }}
| 0.31827
| [[Francium]] (2025)
|-
| [[7105/7104|Yousyozanisma]]
| Thisutwenozozoyo
| 37u29ozzy1
| 7105/7104
| 2.3.5.7.29.37 {{monzo| -6 -1 1 2 1 -1 }}
| 0.24368
| [[Francium]] (2025)
|-
| [[7696/7695|Liebisma]]
| Sathisonuthogu
| S37o19u3og2
| 7696/7695
| 2.3.5.13.19.37 {{monzo| 4 -4 -1 1 -1 1 }}
| 0.22497
| [[Francium]] (2025)
|-
| [[8991/8990|Solidarity comma]]
| Thisothiwutwenugu
| 37o31u29ug2
| 8991/8990
| 2.3.5.29.31.37 {{monzo| -1 5 -1 -1 -1 1 }}
| 0.19256
| [[Francium]] (2025)
|-
| [[9177/9176|Donsaarisma]]
| Thisuthiwutwethonozo
| 37u31u23o19oz2
| 9177/9176
| 2.3.7.19.23.31.37 {{monzo| -3 1 1 1 1 -1 -1 }}
| 0.18866
| [[Francium]] (2025)
|-
| [[9251/9250|Harchisma]]
| Thisubitweno-alotrigu
| 37u29oo1o3g1
| 9251/9250
| 2.5.11.29.37 {{monzo| -1 -3 1 2 -1 }}
| 0.18715
| [[Francium]] (2025)
|-
| [[9361/9360|Friesachisma]]
| Thisotwethothulogu
| 37o23o3u1og2
| 9361/9360
| 2.3.5.11.13.23.37 {{monzo| -4 -2 -1 1 -1 1 1}}
| 0.18495
| [[Francium]] (2024)
|-
| [[Zangarisma]]
| Thisososolu
| 37o17oo1u3
| 10693/10692
| 2.3.11.17.37 {{monzo| -2 -5 -1 2 1 }}
| 0.16191
| [[Francium]] (2024)
|-
| [[23275/23273|Sunrise comma]]
| Bithisu-anosubizoyo
| 37uu19o17uzzyy-2
| 23275/23273
| 5.7.17.19.37 {{monzo| 2 2 -1 1 -2 }}
| 0.14877
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Sinachopoulosisma]]
| Thisotwethonunuluzo
| 37o23o19uu1uz2
| 11914/11913
| 2.3.7.11.19.23.37 {{monzo| 1 -1 1 -1 -2 1 1 }}
| 0.14532
| [[Francium]] (2024)
|-
| [[12321/12320|Zurakowskisma]]
| Bithiso-alurugu
| 37oo1urg2
| 12321/12320
| 2.3.5.7.11.37 {{monzo| -5 2 -1 -1 -1 2 }}
| 0.14052
| [[Francium]] (2025)
|-
| [[13690/13689|Lesleymartin comma]]
| Bithisothu-ayo
| 37oo3uuy2
| 13690/13689
| 2.3.5.13.37 {{monzo| 1 -4 1 -2 2 }}
| 0.12646
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Berylisma]]
| Quadthisolu
| 4(37o1u)4
| 1874161/1874048
| 2.11.37 {{monzo| -7 -4 4 }}
| 0.10439
| [[User:Jerdle|Jerdle]] (2023)
|-
| [[Genzelisma]]
| Thisotwenonusogu
| 37o29o19u17og2
| 18241/18240
| 2.3.5.17.19.29.37 {{monzo| -6 -1 -1 1 -1 1 1 }}
| 0.094912
| [[Francium]] (2024)
|-
| [[33264/33263|Maryrogersisma]]
| Thisuthiwutwenulozo
| 37u31u29u1oz1
| 33264/33263
| 2.3.7.11.29.31.37 {{monzo| 4 3 1 1 -1 -1 -1 }}
| 0.052046
| [[Francium]] (2025)
|-
| [[Jouvisma]]
| Thisotwethotholuluzogu
| 37o23o3o1uuzg3
| 77441/77440
| 2.5.7.11.13.23.37 {{monzo| -7 -1 1 -2 1 1 1 }}
| 0.022356
| [[Francium]] (2024)
|}

=== 41-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[575/574|Renatisma]]
| Fowutwethoruyoyo
| 41u23oryy1
| 575/574
| 2.5.7.23.41 {{monzo| -1 2 -1 1 -1 }}
| 3.0135
| [[Francium]] (2025)
|-
| [[616/615|Ellisma]]
| Fowulozogu
| 41u1ozg2
| 616/615
| 2.3.5.7.11.41 {{monzo| 3 -1 -1 1 1 -1 }}
| 2.8127
| [[Francium]] (2024)
|-
| [[780/779|Wiesentisma]]
| Fowunuthoyo
| 41u19u3oy1
| 780/779
| 2.3.5.13.19.41 {{monzo| 2 1 1 1 -1 -1 }}
| 2.2210
| [[Francium]] (2025)
|-
| [[1025/1024|Kilobytisma]]
| Fowoyoyo
| 41oyy-2
| 1025/1024
| 2.5.41 {{Monzo| -10 2 1 }}
| 1.6898
| [[CompactStar]] (2024)
|-
| [[1026/1025|Ingridisma]]
| Fowunogugu
| 41u19ogg2
| 1026/1025
| 2.3.5.19.41 {{monzo| 1 3 -2 1 -1 }}
| 1.6882
| [[Francium]] (2025)
|-
| [[1190/1189|Pelagisma]]
| Fowutwenusozoyo
| 41u29u17ozy2
| 1190/1189
| 2.5.7.17.29.41 {{monzo| 1 1 1 1 -1 -1 }}
| 1.4554
| [[Francium]] (2025)
|-
| [[1518/1517|Rovaniemisma]]
| Fowuthisutwetholo
| 41u37u23o1o1
| 1518/1517
| 2.3.11.23.37.41 {{monzo|1 1 1 1 -1 -1 }}
| 1.1408
| [[Francium]] (2025)
|-
| [[1682/1681|Shaftesburisma]]
| Bifowutweno
| 41uu29oo2
| 1682/1681
| 2.29.41 {{monzo| 1 2 -2 }}
| 1.0296
| [[Budjarn Lambeth]] (2023)
|-
| [[2255/2254|Qinghaisma]]
| Fowotwethuloruruyo
| 41o23u1orry-3
| 2255/2254
| 2.5.7.11.23.41 {{monzo| -1 1 -2 1 -1 1 }}
| 0.76790
| [[Francium]] (2025)
|-
| [[2871/2870|Schoberisma]]
| Fowutwenolorugu
| 41u29o1org1
| 2871/2870
| 2.3.5.7.11.29.41 {{monzo| -1 2 -1 -1 1 1 -1 }}
| 0.60311
| [[Francium]] (2025)
|-
| [[3773/3772|Smithsonianisma]]
| Fowutwethulotrizo
| 41u23u1o3z2
| 3773/3772
| 2.7.11.23.41 {{monzo| -2 3 1 -1 -1 }}
| 0.45891
| [[Francium]] (2025)
|-
| [[4060/4059|Deipylosisma]]
| Fowutwenoluzoyo
| 41u29o1uzy2
| 4060/4059
| 2.3.5.7.11.29.41 {{monzo| 2 -2 1 1 -1 1 -1 }}
| 0.42646
| [[Francium]] (2025)
|-
| [[4675/4674|Ohbokisma]]
| Fowunusoloyoyo
| 41u19u17o1oyy1
| 4675/4674
| 2.3.5.11.17.19.41 {{monzo| -1 -1 2 1 1 -1 -1 }}
| 0.37036
| [[Francium]] (2025)
|-
| [[4921/4920|Volontisma]]
| Fowuthisonozogu
| 41u37o19ozg3
| 4921/4920
| 2.3.5.7.19.37.41 {{monzo| -3 -1 -1 1 1 1 -1 }}
| 0.35184
| [[Francium]] (2025)
|-
| [[5577/5576|Priestlisma]]
| Fowusuthotholo
| 41u17u3oo1o1
| 5577/5576
| 2.3.11.13.17.41 {{monzo| -3 1 1 2 -1 -1 }}
| 0.31045
| [[Francium]] (2025)
|-
| [[6069/6068|Cevolanisma]]
| Fowuthisusosozo
| 41u37u17ooz2
| 6069/6068
| 2.3.7.17.37.41 {{monzo| -2 1 1 2 -1 -1 }}
| 0.28528
| [[Francium]] (2025)
|-
| [[6930/6929|Bedanisma]]
| Fowuthuthulozoyo
| 41u3uu1ozy1
| 6930/6929
| 2.3.5.7.11.13.41 {{monzo| 1 2 1 1 1 -2 -1 }}
| 0.24984
| [[Francium]] (2025)
|-
| [[7176/7175|Kunijisma]]
| Fowutwethothorugugu
| 41u23o3orgg2
| 7176/7175
| 2.3.5.7.13.23.41 {{monzo| 3 1 -2 -1 1 1 -1 }}
| 0.24127
| [[Francium]] (2025)
|-
| [[8569/8568|Mamelisma]]
| Fowonosuloru
| 41o19o17u1or-2
| 8569/8568
| 2.3.7.11.17.19.41 {{monzo| -3 -2 -1 1 -1 1 1 }}
| 0.20205
| [[Francium]] (2025)
|-
| [[9472/9471|Brugesisma]]
| Fowuthisoluru
| 41u37o1ur2
| 9472/9471
| 2.3.7.11.37.41 {{monzo| 8 -1 -1 -1 1 -1 }}
| 0.18278
| [[Francium]] (2025)
|-
| [[Etampesisma]]
| Fowutwethunotholuzo
| 41u23u19o3o1uz2
| 10374/10373
| 2.3.7.11.13.19.23.41 {{monzo| 1 1 1 -1 1 1 -1 -1 }}
| 0.16689
| [[Francium]] (2024)
|-
| [[11440/11439|Massironisma]]
| Fowuthiwutholoyo
| 41u31u3o1oy2
| 11440/11439
| 2.3.5.11.13.31.41 {{monzo| 4 -2 1 1 1 -1 -1 }}
| 0.15134
| [[Francium]] (2025)
|-
| [[15376/15375|Martakisma]]
| Fowubithiwo-atrigu
| 41u31oo3g1
| 15376/15375
| 2.3.5.31.41 {{monzo| 4 -1 -3 2 -1 }}
| 0.11260
| [[Francium]] (2025)
|-
| [[Canupisma]]
| Fowutwenuthotrizo-agu
| 41u29u3o3zag3
| 17836/17835
| 2.3.5.7.13.29.41 {{monzo| 2 -1 -1 3 1 -1 -1 }}
| 0.097067
| [[Francium]] (2024)
|-
| [[76384/76383|Vernonisma]]
| Fowuthiwotwethulozo
| 41u31o23u1oz1
| 76384/76383
| 2.3.7.11.23.31.41 {{monzo| 5 -4 1 1 -1 1 -1 }}
| 0.022665
| [[Francium]] (2025)
|-
| [[Mebisma]]
| Safowuthiwulugugu
| s41u31u1ugg3
| 1048576/1048575
| 2.3.5.11.31.41 {{Monzo| 20 -1 -2 -1 -1 -1 }}
| 0.0016510
| See the page.
|}

=== 43-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[646/645|Kastalisma]]
| Fothunosogu
| 43u19o17og2
| 646/645
| 2.3.5.17.19.43 {{monzo| 1 -1 -1 1 1 -1 }}
| 2.6820
| [[Francium]] (2025)
|-
| [[990/989|Yerkesisma]]
| Fothutwethuloyo
| 43u23u1oy-2
| 990/989
| 2.3.5.11.23.43 {{monzo| 1 2 1 1 -1 -1 }}
| 1.7496
| [[Francium]] (2025)
|-
| [[1333/1332|Cevelonisma]]
| Fothothisuthiwo
| 43o37u31o-2
| 1333/1332
| 2.3.31.37.43 {{monzo| -2 -2 1 -1 1 }}
| 1.2992
| [[Francium]] (2025)
|-
| [[1377/1376|Roberbauxisma]]
| Lafothuso
| L43u17o1
| 1377/1376
| 2.3.17.43 {{monzo| -5 4 1 -1}}
| 1.2577
| [[Francium]] (2025)
|-
| [[1463/1462|Nordenmarkisma]]
| Fothunosulozo
| 43u19o17uoz1
| 1463/1462
| 2.7.11.17.19.43 {{monzo| -1 1 1 -1 1 -1 }}
| 1.1838
| [[Francium]] (2025)
|-
| [[Magnetisma]]
| Tritrila-quinquadtrifo thutweno
| 9L60(43u29o)-8
|
| 2.3.29.43 {{monzo| -61 60 60 -60 }}
| 0.86936
| [[User:Eliora|Eliora]] (2022)
|-
| [[2925/2924|Beattisma]]
| Fothusuthoyoyo
| 43u17u3oyy-2
| 2925/2924
| 2.3.5.13.17.43 {{monzo| -2 2 2 1 -1 -1 }}
| 0.59198
| [[Francium]] (2025)
|-
| [[3312/3311|Pedersenisma]]
| Fothutwetholuru
| 43u23o1ur1
| 3312/3311
| 2.3.7.11.23.43 {{monzo| 4 2 -1 -1 1 -1 }}
| 0.52279
| [[Francium]] (2025)
|-
| [[4000/3999|Hipparchusisma]]
| Fothuthiwutriyo
| 43u31u3y1
| 4000/3999
| 2.3.5.31.43 {{monzo| 5 -1 3 -1 -1 }}
| 0.43286
| [[Francium]] (2025)
|-
| [[4301/4300|Boydenisma]]
| Fothutwethosologugu
| 43u23o17o1ogg2
| 4301/4300
| 2.5.11.17.23.43 {{monzo| -2 -2 1 1 1 -1 }}
| 0.40257
| [[Francium]] (2025)
|-
| [[4774/4773|Hobetsisma]]
| Fothuthisuthiwolozo
| 43u37u31o1oz-2
| 4774/4773
| 2.3.7.11.31.37.43 {{monzo| 1 -1 1 1 1 -1 -1 }}
| 0.36268
| [[Francium]] (2025)
|-
| [[5720/5719|Halweaverisma]]
| Fothunutholoruyo
| 43u19u3o1ory-2
| 5720/5719
| 2.5.7.11.13.19.43 {{monzo| 3 1 -1 1 1 -1 -1 }}
| 0.30269
| [[Francium]] (2025)
|-
| [[7225/7224|Huntressisma]]
| Fothusosoruyoyo
| 43u17ooryy1
| 7225/7224
| 2.3.5.7.17.43 {{monzo| -3 -1 2 -1 2 -1 }}
| 0.23963
| [[Francium]] (2025)
|-
| [[7956/7955|Yajinisma]]
| Fothuthisusothogu
| 43u37u17o3og1
| 7956/7955
| 2.3.5.13.17.37.43 {{monzo| 2 2 -1 1 1 -1 -1 }}
| 0.21761
| [[Francium]] (2025)
|-
| [[9504/9503|Lionelisma]]
| Fothusuthulo
| 43u17u3u1o-2
| 9504/9503
| 2.3.11.13.17.43 {{monzo| 5 3 1 -1 -1 -1 }}
| 0.18217
| [[Francium]] (2025)
|-
| [[9633/9632|Coturisma]]
| Fothunothothoru
| 43u19o3oor1
| 9633/9632
| 2.3.7.13.19.43 {{monzo| -5 1 -1 2 1 -1 }}
| 0.17973
| [[Francium]] (2025)
|-
| [[Girardisma]]
| Fothunoloyoyo
| 43u19o1oyy1
| 10450/10449
| 2.3.5.11.19.43 {{monzo| 1 -5 2 1 1 -1 }}
| 0.16567
| [[Francium]] (2024)
|-
| [[Kaguyisma]]
| Fothutwethusoluyo
| 43u23u17o1uy1
| 10880/10879
| 2.5.11.17.23.43 {{monzo| 7 1 -1 1 -1 -1 }}
| 0.15913
| [[Francium]] (2024)
|-
| [[Manheimisma]]
| Fothutwenosuloloyo
| 43u29o17u1ooy-2
| 17545/17544
| 2.3.5.11.17.29.43 {{monzo| -3 -1 1 2 -1 1 -1 }}
| 0.098677
| [[Francium]] (2024)
|-
| [[Dimigenes comma]]
| Fothosepyo
| 43o7y-2
| 3359375/3359232
| 2.3.5.43 {{monzo| -9 -8 7 1 }}
| 0.073696
| [[User:Akselai|Akselai]] (2024)
|-
| [[27048/27047|Jangongisma]]
| Fothuthisutwethosuzozo
| 43u37u23o17uzz1
| 27048/27047
| 2.3.7.17.23.37.43 {{monzo| 3 1 2 -1 1 -1 -1 }}
| 0.064007
| [[Francium]] (2025)
|-
| [[29241/29240|Locquirecisma]]
| Fothunonosugu
| 43u19oo17ug1
| 29241/29240
| 2.3.5.17.19.43 {{monzo| -3 4 -1 -1 2 -1 }}
| 0.059207
| [[Francium]] (2025)
|}

=== Higher-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[7936/7921|Lily comma]]
|
| 89uu31o1
| 7936/7921
| 2.31.89 {{monzo| 8 1 -2 }}
| 3.2753
| [[Budjarn Lambeth]] (2023)
|-
| [[Molar comma]]
|
| 3(89o)1u3g1
| 704969/704000
| 2.5.11.89 {{monzo| -9 -3 -1 3 }}
| 2.3813
| [[Budjarn Lambeth]] (2023)
|-
| [[750/749|Ancient Chinese tempering comma]]{{Clarify}}
|
| 107ur3y-2
| 750/749
| 2.3.5.7.107 {{monzo| 1 1 3 -1 -1 }}
| 2.3099
|
|-
| [[1176/1175|Lucidorisma]]
| Fosubizogu
| 47uzzgg2
| 1176/1175
| 2.3.5.7.47 {{monzo| 3 1 -2 2 -1 }}
| 1.4728
| [[Francium]] (2025)
|-
| [[2520/2519|Platonisma]]
|
| 229u1uzy1
| 2520/2519
| 2.3.5.7.11.229 {{monzo| 3 2 1 1 -1 -1 }}
| 0.68713
| [[User:Eliora|Eliora]] (2022)
|-
| [[5041/5040|Third brown pair comma]], 19th highly compositema
|
| 71oorg1
| 5041/5040
| 2.3.5.7.71 {{monzo| -4 -2 -1 -1 2 }}
| 0.34347
| [[Eliora]] (2022)
|-
| [[7777/7776|Pulsar comma]]
|
| 101o1oz2
| 7777/7776
| 2.3.7.11.101 {{monzo| -5 -5 1 1 1 }}
| 0.22262
| [[Eliora]] (2022)
|-
| Palimilli
|
| 1003001o23o1u1
| 23069023 / 23068672
| 2.11.23.1003001 {{monzo| -21 -1 1 1 }}
| 0.026341
| [[User:Kaiveran|Kaiveran]] (2021)
|-
| [[Sulbasutrisma]]
|
| 577oo17uu-2
| 332929/332928
| 2.3.17.577 {{monzo| -7 -2 -2 2 }}
| 0.0052000
| [[User:2^67-1|Cole]] (2024)
|-
| [[Zudilisma]]
|
| 4L397u23ur-5
| 68630377364883 / 68630356164608
| 2.3.7.23.397 {{monzo| -30 29 -1 -1 -1 }}
| 0.00053479
| [[Budjarn Lambeth]] (2023)
|-
| [[Borcherdsma]]
|
| 71u3(59u)47o31o 29o19o3u1uur5y-2
| 160561400000 / 160561399999
| 2.5.7.11.13.19.29.31.47.59.71 {{monzo| 6 5 -1 -2 -1 1 1 1 1 -3 -1 }}
| 1.0783 × 10-8
| [[User:Akselai|Akselai]] (2024)
|}

== Notes ==
<references />

== See also ==
* [[Comma and diesis]]

[[Category:Unnoticeable commas| ]] 
[[Category:Lists of commas]]
{{Todo|complete table|add color name}}

Unnoticeable comma

2026-04-28T21:41:26Z

TallKite: /* Higher-limit commas */

'''Unnoticeable commas''' are very small intervals. These [[comma]]s are called "unnoticeable" because, being equal to or less than 3.5{{cent}}, they are smaller than the average peak [[just-noticeable difference]] (JND) of human pitch perception, as illustrated by the research of [[Aaron Andrew Hunt]]<ref>[http://musictheory.zentral.zone/huntsystem2.html#2 H-Pi Instruments | ''Hunt System Scale §The JND'']</ref>. It is improbable that even a trained listener would be able to notice these intervals, and as such they are a prime target for psychoacoustically informed [[microtempering]]. (However, a considerably larger comma can be unnoticeable in an [[adaptive just intonation|adaptive]] tuning context. Instead of one large pitch shift of the entire comma, there can be many small pitch shifts of a fraction of a comma, one per chord change. Given this, a noticeable 3-limit comma that arguably deserves inclusion is the [[mercator comma]], corresponding to using [[53edo]] for the circle of fifths.) In [[Sagittal notation]], intervals in the smaller part of this category are [[schismina]]s, and intervals in the larger part of this category are [[schisma (interval region)|schismas]]<ref>[https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>.

For commas over 100{{c}} in size, see [[Large comma]]; for commas in between 30–100{{c}} in size, see [[Medium comma]]; and for commas between 3.5–30{{c}} in size, see [[Small comma]].

Due to the wide range of sizes, cents values here and on the other commas pages are listed to 5 significant digits, instead of to a fixed number of decimal places as is the [[conventions|convention]] elsewhere on the wiki. Except for the 3-limit commas, the [[color notation|color name]] refers to both the comma and the temperament created when it is tempered out.

The commas might be numerous, but there is no need to memorise all the names. For pretty much all use cases, it is perfectly acceptable – preferred, even – to just refer to a comma by the monzo or ratio, whichever is simpler.

== List of commas, by prime limit ==
=== 3-limit commas ===
{{See also| Table of 3-limit commas }}

{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Qian's large comma]]
| Wa-359
| w-359
| <abbr title="1934329451767021421190980423006270754962252499447372679942534652297789463068718331568475476554301659845788554312924531179306109686817232569946089263295619210341718686733067/1932268761508629172347675945465993672149463664853217499328617625725759571144780212268096883290961288981231808015751088588682539330521493827871454336733540374348490407411712">(344 digits)</abbr>
| {{Monzo| -569 359 }}
| 1.8453
| Chen Yingshi (2009)
|-
| [[Qian's small comma]], sasktel comma
| Wa-306
| w-306
| <abbr title="99895953610111751404211111353381321783955140565279076827493022708011895642232499843849795298031743077114461795885011932654335221737225129801285632/99793888233710926097676673961542382339552034110870991187709058567130998942396826836880350287497238272034603157195937657211050782186192219658614729">(292 digits)</abbr>
| {{Monzo| 485 -306 }}
| 1.7697
| Chen Yingshi (2009) for ''Qian's small comma''
|-
| [[Satanic comma]]
| Wa-665
| w-665
| <abbr title="193034257116813465350415306746516837350333763798962117430788985740786485136987943002905988649011085058426719117038711696606024631330152759176330399379617346789616335692978372064681236597226671488585092334981423081811727458166457361300251189808300631437024118571790058070714566731059066970852059271394655662607817543843/193025830561934107162947985381047541665608072055952185017491682078771915023799273387871154500424503798663213600460826789274033295999330021731389427128542432710187362934652673115221889249890533772697227171395058697282798274445240687006095271729621464100656563293799180557568945517759802372156455525060659659679134121984">(636 digits)</abbr>
| {{Monzo| -1054 665 }}
| 0.075575
| [[Marc Jones]] (1990)
|-
| 15601-comma
| Wa-15601
| w-15601
| (14888 digits)
| {{Monzo| 24727 -15601 }}
| 0.031499
|
|-
| 31867-comma
| Wa-31867
| w-31867
| (30410 digits)
| {{Monzo| -50508 31867 }}
| 0.012577
|
|-
| [[Archangelic comma]]
| Wa-190537
| w-190537
| (181820 digits)
| {{monzo| 301994 -190537 }}
| 0.00011162
| [[Dawson Berry]] (2022)
|}

=== 5-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Dodifo comma]]
| Quadla-sepquinyo
| 4L35y-9
| <abbr title="2910383045673370361328125 / 2904698108822600835661824">(50 digits)</abbr>
| {{Monzo| -67 -9 35 }}
| 3.3850
| [[Petr Pařízek]] (2011)
|-
| [[Vishnuzma]], semisuper comma
| Sasepbigu
| s14g4
| 6115295232 / 6103515625
| {{Monzo| 23 6 -14 }}
| 3.3380
| [[Gene Ward Smith]] (2001), for ''semisuper comma''
|-
| [[Enneadeca]], 19-tone-comma
| Neyo
| 19y-4
| 19073486328125 / 19042491875328
| {{Monzo| -14 -19 19 }}
| 2.8155
| [[Gene Ward Smith]] (2001)
|-
| [[Vavoom comma]]
| Quinla-seyo
| 5L17y-7
| <abbr title="295578376007080078125 / 295147905179352825856">(42 digits)</abbr>
| {{Monzo| -68 18 17 }}
| 2.5232
| [[Paul Erlich]] (2002)
|-
| [[Alphatricot comma]]
| Quadsa-triyo
| 4s3y3
| 68719476736000 / 68630377364883
| {{Monzo| 39 -29 3 }}
| 2.2461
| [[Paul Erlich]] (2002)
|-
| [[Schisma]]
| Layo
| Ly-2
| 32805 / 32768
| {{Monzo| -15 8 1 }}
| 1.9537
| [[Hermann von Helmholtz]], [[Alexander Ellis]] (1875)
|-
| [[Aluminium comma]]
| Sepsa-thegu
| 7s13g8
| <abbr title="4951760157141521099596496896 / 4946966739525117513427734375">(56 digits)</abbr>
| {{Monzo| 92 -39 -13 }}
| 1.6767
| [[Eliora]] (2023)
|-
| [[Counterschisma]]
| Tribilagu
| 6Lg-5
| <abbr title="2954312706550833698643 / 2951479051793528258560">(44 digits)</abbr>
| {{Monzo| -69 45 -1 }}
| 1.6613
| [[Gene Ward Smith]] (2003)
|-
| [[Neon comma]]
| Laquinquinbigu
| L50g8
| <abbr title="88900702359186211632409599176343552 / 88817841970012523233890533447265625">(70 digits)</abbr>
| {{monzo| 21 60 -50 }}
| 1.6144
| [[Eliora]] (2024)
|-
| [[Septendecima]]
| Lala-sebiyo
| LL34y-8
| <abbr title="582076609134674072265625 / 581595589965365114830848">(48 digits)</abbr>
| {{Monzo| -52 -17 34 }}
| 1.4313
|
|-
| [[Luna comma]], hemithirds comma
| Sasa-quintrigu
| ss15g5
| 274877906944 / 274658203125
| {{Monzo| 38 -2 -15 }}
| 1.3843
| [[Gene Ward Smith]] (2002)
|-
| [[Minortone comma]], minortonma
| Trila-segu
| 3L17g2
| <abbr title="50031545098999707 / 50000000000000000">(34 digits)</abbr>
| {{Monzo| -16 35 -17 }}
| 1.0919
| [[Gene Ward Smith]] (2002)
|-
| [[Ennealimma]]
| Satritribiyo
| s18y-3
| 7629394531250 / 7625597484987
| {{Monzo| 1 -27 18 }}
| 0.86183
| [[Paul Erlich]], [[Gene Ward Smith]] (2001)
|-
| Astro comma
| Tribisa-thiwegu
| 6s31g10
| <abbr title="2475880078570760549798248448 / 2474715001881122589111328125">(56 digits)</abbr>
| {{Monzo| 91 -12 -31 }}
| 0.81486
| [[Paul Erlich]] (2002)
|-
| [[Gaster comma]]
| Quadbila-negu
| 8L19g-3
| <abbr title="22528399544939174411840147874772641 / 22517998136852480000000000000000000">(70 digits)</abbr>
| {{Monzo| -70 72 -19 }}
| 0.79950
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Niobium comma]]
|
|
|
| {{Monzo| -875 492 41 }}
| 0.72269
| [[Eliora]] (2023)
|-
| [[Kwazy comma]]
| Quadla-quadquadyo
| 4L16y-6
| <abbr title="9010162353515625 / 9007199254740992">(32 digits)</abbr>
| {{Monzo| -53 10 16 }}
| 0.56943
| [[Paul Erlich]], [[Gene Ward Smith]] (2002)
|-
| Whoosh
| Saletrigu
| s33g7
| <abbr title="116450459770592056836096 / 116415321826934814453125">(48 digits)</abbr>
| {{Monzo| 37 25 -33 }}
| 0.52246
| [[Paul Erlich]] (2002)
|-
| Egads
| Setriyo
| 51y-9
| <abbr title="444089209850062616169452667236328125 / 444002166576103304796646509039845376">(72 digits)</abbr>
| {{Monzo| -36 -52 51 }}
| 0.33936
| [[Paul Erlich]] (2002)
|-
| [[Monzisma]]
| Quinsa-yoyo
| 5syy4
| <abbr title="450359962737049600 / 450283905890997363">(36 digits)</abbr>
| {{Monzo| 54 -37 2 }}
| 0.29240
| [[Margo Schulter]] (2001)
|-
| Fortune
| Tritrila-sepbiyo
| 9L14y-9
| <abbr title="162285243890121480027996826171875 / 162259276829213363391578010288128">(66 digits)</abbr>
| {{Monzo| -107 47 14 }}
| 0.27703
| [[Paul Erlich]] (2002)
|-
| Gross
| Quinbisa-fosegu
| 10s47g15
| <abbr title="22300745198530623141535718272648361505980416 / 22297583945629639856633730232715606689453125">(88 digits)</abbr>
| {{Monzo| 144 -22 -47 }}
| 0.24543
| [[Paul Erlich]] (2002)
|-
| Senior
| Quadla-sepquingu
| 4L35g4
| <abbr title="381520424476945831628649898809 / 381469726562500000000000000000">(60 digits)</abbr>
| {{Monzo| -17 62 -35 }}
| 0.23007
| [[Paul Erlich]] (2002)
|-
| [[Counterquectisma]], deltapion
| Quintritrilayo
| 45Ly-31
|
| {{Monzo| -500 314 1 }}
| 0.18399
| [[Eliora]] (2023)
|-
| [[Septarium comma]]
| Sasepquadtrigu
| s84g15
| <abbr title="51704256436926231056548749215693807357721577836111615492096 / 51698788284564229679463043254372678347863256931304931640625">(118 digits)</abbr>
| {{Monzo| 73 77 -84 }}
| 0.18310
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Quectisma]]
|
|
|
| {{Monzo| 554 -351 1 }}
| 0.10841
| [[User:CompactStar|CompactStar]] (2023)
|-
| Raider
| Tritrisa-thiseyo
| 9s37y1
| <abbr title="171798691840000000000000000000000000000000000000 / 171792506910670443678820376588540424234035840667">(96 digits)</abbr>
| {{Monzo| 71 -99 37 }}
| 0.062327
| [[Gene Ward Smith]] (2002)
|-
| Pirate
| Quinla-sepsepyo
| 5L49y-12
| <abbr title="17763568394002504646778106689453125 / 17763086495282268024161967871623168">(70 digits)</abbr>
| {{Monzo| -90 -15 49 }}
| 0.046966
| [[Paul Erlich]] (2002)
|-
| Viking
| Nela-siweyo
| 19L61y-23
|
| {{Monzo| -251 69 61 }}
| 0.031605
| [[Gene Ward Smith]] (2002)
|-
| [[Kirnberger's atom]]
| Sepbisa-quadtrigu
| 14s12g12
| <abbr title="2923003274661805836407369665432566039311865085952 / 2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
| {{Monzo| 161 -84 -12 }}
| 0.015361
|
|-
| Selenia
|
|
|
| {{Monzo| -433 -137 280 }}
| 0.0047636
|
|-
| Titania
|
|
|
| {{Monzo| 2746 -521 -827 }}
| 0.0031829
|
|-
| Quark
| Twethebila-sequinyo
|
|
| {{Monzo| -573 237 85 }}
| 8.8361 × 10-4
|
|-
| Scamp
|
|
|
| {{Monzo| -5836 4293 -417 }}
| 3.3472 × 10-5
|
|-
| Rover
|
|
|
| {{Monzo| 1634 1502 -1729 }}
| 2.7513 × 10-5
|
|-
| Rascal
|
|
|
| {{Monzo| -7470 2791 1312 }}
| 5.9596 × 10-6
|
|}

=== 7-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Decovulture comma]]
| Sasa-birugugu
| ss2rgg2
| 67108864 / 66976875
| {{Monzo| 26 -7 -4 -2 }}
| 3.4083
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Rainy comma]]
| Laquinzo-atriyo
| L5za3y2
| 2100875/2097152
| {{Monzo| -21 0 3 5 }}
| 3.0706
| [[User:Flirora|+merlan #flirora]] (2020)
|-
| [[Pontiqak comma]]
| Lazozotritriyo
| Lzz9y-2
| 95703125 / 95551488
| {{Monzo| -17 -6 9 2 }}
| 2.7452
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Pessoalisma]]
| Sasa-tribiru-agugu
| ss6ragg-2
| 2147483648 / 2144153025
| {{Monzo| 31 -6 -2 -6 }}
| 2.6871
| [[Dawson Berry]] (2023)
|-
| [[Mitonisma]]
| Laquadzo-agu
| L4zag2
| 5250987/5242880
| {{Monzo| -20 7 -1 4 }}
| 2.6749
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Horwell comma]]
| Lazoquinyo
| Lz5y-2
| 65625/65536
| {{Monzo| -16 1 5 1 }}
| 2.3495
| [[Gene Ward Smith]] (2005)
|-
| [[Forge comma]]
| Lala-trizo-aquingu
| LL3za5g2
| 1640558367 / 1638400000
| {{Monzo| -19 14 -5 3 }}
| 2.2792
| [[Dawson Berry]] (2022)
|-
| [[109-7-comma]]
|
|
|
| {{Monzo| -306 0 0 109 }}
| 2.0238
|
|-
| [[Neptunisma]]
| Laruruleyo
| Lrr11y-4
| 48828125 / 48771072
| {{monzo| -12 -5 11 -2 }}
| 2.0240
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Slendroschisma]], slendric schisma
| Sasa-quinbiru
| ss10r-4
| 68719476736 / 68641485507
| {{monzo| 36 -5 0 -10 }}
| 1.9659
| [[User:Xenllium|Xenllium]] (2024), modified by [[Flora Canou]] (2025)
|-
| [[Septimal ennealimma]]
| Tritrizo
| 9z5
| 40353607 / 40310784
| {{Monzo| -11 -9 0 9 }}
| 1.8382
| [[Eliora]] (2021)
|-
| [[Meter]]
| Latriru-asepyo
| L3ra7y-4
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.6283
|
|-
| [[Scheme comma]]
| Lala-rutrigu
| LLr3g-2
| 14348907 / 14336000
| {{Monzo| -14 15 -3 -1}}
| 1.5580
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[Breeze comma]]
| Laquadru-atriyo
| L4ra3y-4
| 2460375 / 2458624
| {{Monzo| -10 9 3 -4 }}
| 1.2325
| [[Tristan Bay]] (2024)
|-
| [[Wizma]]
| Quinzo-ayoyo
| 5zayy3
| 420175/419904
| {{Monzo| -6 -8 2 5 }}
| 1.1170
|
|-
| [[Supermatertisma]]
| Lasepru-atritriyo
| L7ra9y-6
| 52734375 / 52706752
| {{Monzo| -6 3 9 -7 }}
| 0.90708
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Trienstonisma]]
| Laquinru-agu
| L5rag-4
| 43046721 / 43025920
| {{monzo| -9 16 -1 -5 }}
| 0.83677
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[2401/2400|Breedsma]]
| Bizozogu
| 2zzg3
| 2401/2400
| {{monzo| -5 -1 -2 4 }}
| 0.72120
| [[Gene Ward Smith]] (2004)
|-
| [[Gariatom]]
| Quintrila-tribizo
| 15L6z-8
|
| {{monzo| -169 96 0 6 }}
| 0.63552
| [[Tristan Bay]] (2023)
|-
| 171-9/7-comma
| Quadtribisa-netritrizo
|
|
| {{monzo| 62 -342 0 171 }}
| 0.61971
|
|-
| [[Supermasesquartisma]]
| Laquadbiru-aquinyo
| L8ra5y-6
| 184528125 / 184473632
| {{monzo| -5 10 5 -8 }}
| 0.51133
| [[User:Xenllium|Xenllium]] (2021)
|-
| 571-7-comma
|
|
|
| {{monzo| 1603 0 0 -571 }}
| 0.40741
|
|-
| [[Ragisma]]
| Zoquadyo
| z4y1
| 4375/4374
| {{Monzo| -1 -7 4 1 }}
| 0.39576
|
|-
| [[Septiruthenia]], septimal ruthenia
| Nela-lequadzo
| 19L44z8
|
| {{Monzo| -263 88 0 44 }}
| 0.37996
| [[Eliora]] (2022)
|-
| [[Akjaysma]]
| Trisa-seprugu
| 3s7rg1
| 140737488355328 / 140710042265625
| {{Monzo| 47 -7 -7 -7 }}
| 0.33765
| [[Aaron Krister Johnson]] (2011)
|-
| [[Landscape comma]]
| Trizogugu
| 3zgg3
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.32544
| [[Yahya Abdal-Aziz]] (2005)
|-
| [[Izar comma]], bapbo schismina
| Saquadtrizo-asepgu
| s12za7g8
| 13841287201 / 13839609375
| {{Monzo| 0 -11 -7 12 }}
| 0.20987
|
|-
| [[Nanisma]]
| Quinbisaru
| 10sr7
| <abbr title="649037107316853453566312041152512 / 648966242035284859600333477874109">(66 digits)</abbr>
| {{Monzo| 109 -67 0 -1 }}
| 0.18904
| [[Margo Schulter]] (2002)
|-
| laleruyo (171&1547&3125)
| Laleruyo
| L11ry-8
| 3955078125 / 3954653486
| {{Monzo| -1 4 11 -11 }}
| 0.18588
|
|-
| 87-fold starling comma
| Twenetrizotrigu
|
|
| {{Monzo| 86 174 -261 87 }}
| 0.14469
|
|-
| [[Revopentisma]]
| Sasa-neru
| ss19r-8
| <abbr title="11399736556781568 / 11398895185373143">(34 digits)</abbr>
| {{Monzo| 47 4 0 -19 }}
| 0.12778
| [[Eliora]] (2023)
|-
| [[Starscape comma]]
| Latritriru-ayo
| L9ray-6
| 645700815 / 645657712
| {{Monzo| -4 17 1 -9 }}
| 0.11557
|
|-
| [[Nommisma]]
| Quinla-zoyoyo
| 5Lzzy-4
| <abbr title="36030948116563575 / 36028797018963968">(34 digits)</abbr>
| {{Monzo| -55 30 2 1 }}
| 0.10336
| [[Flora Canou]] (2023)
|-
| [[Euzenius comma]]
| Sabiruquinyo
| s2r5y-3
| 78125000 / 78121827
| {{Monzo| 3 -13 10 -2 }}
| 0.070314
|
|-
| [[Exodia comma]]
| Trila-quadbizo-aleyo
| 3L8za11y1
| 281484423828125 / 281474976710656
| {{Monzo| -48 0 11 8 }}
| 0.058104
|
|-
| [[Septrongisma]]
| Lala-sepru-atritrigu
| LL7ra9g-4
| 205891132094649 / 205885750000000
| {{Monzo| -7 30 -9 -7 }}
| 0.045256
| [[User:Xenllium|Xenllium]] (2026)
|-
| 171&1547&4973 comma
| Satwethezo-atritribigu
| s23za18g15
| <abbr title="54737494680161832686 / 54736736297607421875">(40 digits)</abbr>
| {{Monzo| 1 -15 -18 23 }}
| 0.023986
|
|-
| [[Technologisma]]
| Trisa-quinbiru-agu
| 3s10rag-3
| <abbr title="2251799813685248 / 2251783932057135">(32 digits)</abbr>
| {{Monzo| 51 -13 -1 -10 }}
| 0.012210
| [[User:Godtone|Godtone]] (2024)
|-
| Termite
| Satritribiru-athiseyo
| s18ra37y-14
| <abbr title="37252902984619140625000000000 / 37252879910233655318543787489">(58 digits)</abbr>
| {{Monzo| 9 -28 37 -18 }}
| 0.0010723
|
|-
| Neutrino
|
|
|
| {{Monzo| 1889 -2145 138 424 }}
| 1.6361 × 10-10
|
|}

=== 11-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Lifthrasirsma]]
| Sasa-bilugu
| ss2(1ug)3
| 536870912 / 535869675
| {{Monzo| 29 -11 -2 0 -2 }}
| 3.2317
| [[Flora Canou]] (2023)
|-
| [[540/539|Swetisma]]
| Lururuyo
| 1urry-2
| 540/539
| {{Monzo| 2 3 1 -2 -1 }}
| 3.2090
| [[Manuel Op de Coul]] (2004)
|-
| [[Anthill comma]]
| Satrilo-ayoyo
| s3(1o)yy1
| 532400/531441
| {{Monzo| 4 -12 2 0 3 }}
| 3.1212
| [[Budjarn Lambeth]] (2024)
|-
| [[4000/3993|Wizardharry comma]], pine comma
| Triluyo
| 3(1uy)1
| 4000/3993
| {{Monzo| 5 -1 3 0 -3 }}
| 3.0323
| [[User:Godtone|Godtone]] (2023) for ''pine comma''
|-
| 23-11/7-comma
| Twetheluzo
| 23(1uz)14
| <abbr title="896819112839771466727424 / 895430243255237372246531">(48 digits)</abbr>
| {{Monzo| 15 0 0 23 -23 }}
| 2.6832
|
|-
| [[Symbiotic comma]]
| Salozo
| s1oz2
| 19712/19683
| {{Monzo| 8 -9 0 1 1 }}
| 2.5488
| [[Dawson Berry]] (2020)
|-
| [[5632/5625|Vishdel comma]]
| Saloquadgu
| s1o4g2
| 5632/5625
| {{Monzo| 9 -2 -4 0 1 }}
| 2.1531
|
|-
| [[Nexus comma]], nexisma
| Tribilo
| 6(1o)-2
| 1771561/1769472
| {{Monzo| -16 -3 0 0 6 }}
| 2.0427
| [[Dawson Berry]] (2020)
|-
| [[Reef comma]]
| Salubizogu
| s1u2zg3
| 200704/200475
| {{Monzo| 12 -6 -2 2 -1 }}
| 1.9764
| [[Dawson Berry]] (2022)
|-
| [[41503/41472|Argyria]], tinge
| Lolotrizo
| 1oo3z2
| 41503/41472
| {{Monzo| -9 -4 0 3 2 }}
| 1.2936
| [[Gayle Young]] (2018) and [[Todd Harrop]] (2020) for ''tinge'' [[Lériendil]] (2024) for ''argyria''
|-
| [[Alpharabian schisma]]
| Trisa-tritrilo
| 3s9(1o)2
| 618121839509504 / 617673396283947
| {{Monzo| 18 -31 0 0 9 }}
| 1.2565
| [[Dawson Berry]] (2024)
|-
| [[Olympia]]
| Salururu
| s1urr1
| 131072/130977
| {{Monzo| 17 -5 0 -2 -1 }}
| 1.2552
| [[Flora Canou]] (2021)
|-
| [[37-11-comma]], 11-cycle schisma
| Quinsa-thiselu
| 5s37(1u)9
| <abbr title="340282366920938463463374607431768211456 / 340039485861577398992406882305761986971">(78 digits)</abbr>
| {{Monzo| 128 0 0 0 -37 }}
| 1.2361
|
|-
| [[Seascape comma]], undecimal hemifourths comma
| Bilozogugu
| 2(1ozgg)2
| 160083/160000
| {{Monzo| -8 3 -4 2 2 }}
| 0.89784
| [[User:Tristanbay|Tristan Bay]] (2024) for ''seascape comma''
|-
| [[Sesdecal comma]]
| Laquadlu-asepyo
| L4(1u)7y-2
| 234375/234256
| {{Monzo| -4 1 7 0 -4 }}
| 0.87923
|
|-
| [[Triagnoshenisma]]
| Trila-trilo-agu
| 3L3(1o)g-3
| 171885556953 / 171798691840
| {{Monzo| -35 17 -1 0 3 }}
| 0.87513
| [[Dawson Berry]], [[User:Frostburn|Frostburn]] (2024)
|-
| [[Frameshift comma]]
| Quadla-trilu
| 4L3(1u)-3
| 22876792454961 / 22866405883904
| {{Monzo| -34 28 0 0 -3 }}
| 0.78620
| [[Dawson Berry]] (2021)
|-
| [[Chrysia]]
| Quadlo-atriru
| 4(1o)3r-3
| 43923/43904
| {{Monzo| -7 1 0 -3 4 }}
| 0.74905
| [[VIxen]] (2025)
|-
| [[Sossmarvel comma]]
| Trila-lusepruyoyo
| 3L1u7ryy-8
| <abbr title="9730975341796875 / 9726998192586752">(32 digits)</abbr>
| {{Monzo| -30 13 14 -7 -1 }}
| 0.70772
| [[Dawson Berry]] (2022)
|-
| [[3025/3024|Lehmerisma]]
| Loloruyoyo
| 1ooryy-2
| 3025/3024
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57240
| [[Gene Ward Smith]] (2004)
|-
| [[Ptolemi-nicema]]
| Quinbisa-twethetriluyoyo
| 10s69(1uyy)-8
|
| {{Monzo| 137 -138 138 0 -69 }}
| 0.56437
| [[User:Eliora|Eliora]] (2023)
|-
| [[Elysia]]
| Bilutrizo
| 2(1u3z)4
| 117649/117612
| {{Monzo| -2 -5 0 6 -2 }}
| 0.54455
| [[Lériendil]] (2024)
|-
| [[Quartisma]]
| Saquinlu-azo
| s5(1u)z3
| 117440512 / 117406179
| {{Monzo| 24 -6 0 1 -5 }}
| 0.50619
| [[Dawson Berry]] (2020)
|-
| [[9801/9800|Kalisma]], Gauss' comma
| Bilorugu
| 2(1org)-2
| 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| 0.17665
| [[Margo Schulter]] (2000) [[Gene Ward Smith]] (2004) for ''Gauss' comma''
|-
| [[151263/151250|Odiheim comma]]
| Luluquinzo-aquadgu
| 1uu5z4g4
| 151263/151250
| {{Monzo| -1 2 -4 5 -2 }}
| 0.14879
|
|-
| [[Countercentisma]]
|
|
| (3776 digits)
| {{Monzo| -1 -3300 2700 0 -300 }}
| 0.14187
| [[User:CompactStar|CompactStar]] (2024)
|-
| [[Spoob]]
| Tribiluzozogu
| 6(1uzzg)8
| 27682574402 / 27680640625
| {{Monzo| 1 0 -6 12 -6 }}
| 0.12094
| [[User:Eliora|Eliora]] (2023)
|-
| [[Luxma]]
| Saquinlu-aquadgu
| s5(1u)4g3
| 100663296/100656875
| {{Monzo|25 1 -4 0 -5}}
| 0.11043
| [[User:Tristanbay|Tristan Bay]] (2026)
|-
| [[Syntonoschisma]]
| Trisa-lusepyo
| 3s1u7y2
| 83886080000000 / 83881572334857
| {{Monzo|30 -27 7 0 -1}}
| 0.093031
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Parimo]]
| Satribilo-agu
| s6(1o)g1
| 1771561/1771470
| {{Monzo|-1 -11 -1 0 6}}
| 0.088931
|
|-
| Parisma
| Laquadlu-arurugu
| L4(1u)rrg-2
| 14348907 / 14348180
| {{Monzo|-2 15 -1 -2 -4}}
| 0.087717
|
|-
| [[Blare comma]]
| Laquadquadlo-aquadtrizo
| L16(1o)12z4
| <abbr title="636003407850068828189211361 / 635974777627126753067532288">(54 digits)</abbr>
| {{Monzo|-51 -24 0 12 16}}
| 0.077935
| [[User:Akselai|Akselai]] (2024)
|-
| Ultimo
| Quadlo-asepru-ayoyo
| 4(1o)7ryy-5
| 3294225/3294172
| {{Monzo|-2 2 2 -7 4}}
| 0.027854
|
|-
| Parismina
| Sasa-quinbilo-azozo
| ss10(1o)zz2
| 2541867610898 / 2541865828329
| {{Monzo|1 -26 0 2 10}}
| 0.0012141
|
|}

=== 13-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Wilschisma]]
| Sathoyo
| s3oy2
| 532480/531441
| {{Monzo| 13 -12 1 0 0 1 }}
| 3.3814
| [[Flora Canou]] (2021)
|-
| Bean
| Sathuquinlu
| s3u5(1u)2
| 2097152/2093663
| {{Monzo| 21 0 0 0 -5 -1 }}
| 2.8826
|
|-
| [[625/624|Tunbarsma]]
| Thuquadyo
| 3u4y-2
| 625/624
| {{Monzo| -4 -1 4 0 0 -1 }}
| 2.7722
|
|-
| [[Fifthless mohohomma]]
| Thuthululuyo
| 3uu1uuy1
| 20480/20449
| {{Monzo| 12 0 1 0 -2 -2 }}
| 2.6225
| [[User:原井玉葱郎|Misohito Nakai]] (2025)
|-
| [[676/675|Island comma]]
| Bithogu
| 2(3og)2
| 676/675
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.5629
| [[Mike Battaglia]] (2011)
|-
| [[729/728|Squbema]]
| Lathuru
| L3ur-2
| 729/728
| {{Monzo| -3 6 0 -1 0 -1 }}
| 2.3764
|
|-
| [[2200/2197|Petrma]]
| Trithu-aloyoyo
| 3(3u)1oyy-2
| 2200/2197
| {{Monzo| 3 0 2 0 1 -3 }}
| 2.3624
|
|-
| [[Tridecimal eighth-octave comma]]
| Thotrilo-agu
| 3o3(1o)g1
| 17303/17280
| {{Monzo| -7 -3 -1 0 3 1 }}
| 2.3028
| [[Budjarn Lambeth]] (2024)
|-
| [[Sinarabian comma]]
| Lathotrilu
| L3o3(1u)
| 85293/85184
| {{Monzo| -6 8 0 0 -3 1 }}
| 2.2138
| [[Dawson Berry]] (2025)
|-
| [[1575/1573|Nicola]]
| Thululuzoyoyo
| 3u1uuzyy1
| 1575/1573
| {{Monzo| 0 2 2 1 -2 -1 }}
| 2.1998
|
|-
| [[Navicular comma]]
| Trithu-aluzo
| 3(3u)1uz1
| 24192/24167
| {{Monzo| 7 3 0 1 -1 -3 }}
| 1.7900
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1001/1000|Sinbadma]]
| Tholozotrigu
| 3o1oz3g2
| 1001/1000
| {{Monzo| -3 0 -3 1 1 1 }}
| 1.7303
|
|-
| [[4459/4455|Tristanisma]]
| Tholutrizo-agu
| 3o1u3zag3
| 4459/4455
| {{Monzo| 0 -4 -1 3 -1 1}}
| 1.5537
| [[User:Tristanbay|Tristan Bay]] (2024)
|-
| [[Tridecapyth comma]]
| Trisatho
| 3s3o3
| 3489660928 / 3486784401
| {{Monzo| 28 -20 0 0 0 1 }}
| 1.4276
| [[Dawson Berry]] (2021)
|-
| [[Cantonisma]]
| Trithoru-ayo
| 3(3or)y-2
| 10985/10976
| {{Monzo| -5 0 1 -3 0 3 }}
| 1.4190
| [[Margo Schulter]] (2013)
|-
| [[Punctisma]]
| Sathutrizogu
| s3u3zg3
| 43904/43875
| {{Monzo| 7 -3 -3 3 0 -1 }}
| 1.1439
| [[User:Jerdle|Jerdle]], [[Dawson Berry]] (2023)
|-
| [[Neguschisma]]
| Lala-thulozo
| LL3u1oz-2
| 13640319 / 13631488
| {{Monzo| -20 11 0 1 1 -1 }}
| 1.1212
| [[User:Tristanbay|Tristan Bay]] (2024)
|-
| [[1716/1715|Lummic comma]]
| Tholotriru-agu
| 3o1o3rag-2
| 1716/1715
| {{Monzo| 2 1 -1 -3 1 1 }}
| 1.0092
|
|-
| [[Pseudovishnuzma]]
| Sasa-thozosepbigu
| ss3oz14g5
| 6106906624 / 6103515625
| {{Monzo| 26 0 -14 1 0 1 }}
| 0.96157
| [[User:Eliora|Eliora]] (2023)
|-
| [[Sercloreminisma]]
| Bithuthuzo-agu
| 2(3uuz)g1
| 142884/142805
| {{Monzo| 2 6 -1 2 0 -4 }}
| 0.95746
| [[Francium]] (2024)
|-
| [[2080/2079|Ibnsinma, sinaisma]]
| Tholuruyo
| 3o1ury1
| 2080/2079
| {{Monzo| 5 -3 1 -1 -1 1 }}
| 0.83252
| [[Margo Schulter]], [[Gene Ward Smith]] (2012) [[Flora Canou]] (2023)
|-
| [[Phaotic comma]], phaotisma
| Sathotriyo
| s3u3y1
| 256000/255879
| {{Monzo| 11 -9 3 0 0 -1 }}
| 0.81847
| [[Dawson Berry]] (2021)
|-
| [[Kuragesma]]
| Tritho-aquadlu-ayo
| 3(3o)4(1u)g2
| 43940/43923
| {{Monzo| 2 -1 1 0 -4 3 }}
| 0.66993
| [[User:原井玉葱郎|Misohito Nakai]] (2025)
|-
| [[Barbadisma]]
| Quadla-thuyo
| 4L3uy-4
| 114383962274805 / 114349209288704
| {{Monzo| -43 28 1 0 0 -1 }}
| 0.52608
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[4096/4095|Minisma]]
| Sathurugu
| s3urg1
| 4096/4095
| {{Monzo| 12 -2 -1 -1 0 -1 }}
| 0.42272
| [[Flora Canou]] (2026)
|-
| [[4225/4224|Leprechaun comma]]
| Thotholuyoyo
| 3oo1uyy1
| 4225/4224
| {{Monzo| -7 -1 2 0 -1 2 }}
| 0.40981
| [[Margo Schulter]] (2012)
|-
| [[6656/6655|Jacobin comma]]
| Thotrilu-agu
| 3o3(1u)g2
| 6656/6655
| {{Monzo| 9 0 -1 0 -3 1 }}
| 0.26012
| [[Gene Ward Smith]] (2014)
|-
| [[Catasma]]
| Latrithuyoyo
| L3(3uyy)-3
| 140625/140608
| {{Monzo| -6 2 6 0 0 -3 }}
| 0.20930
| [[Tristan Bay]] (2023)
|-
| [[492128/492075|13^3⋅7/25 schismina]]
| Satritho-azogugu
| s3(3o)zgg3
| 492128/492075
| {{Monzo| 5 -9 -2 1 0 3 }}
| 0.18646
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[Harmonisma]]
| Thuthutrilo-aru
| 3uu3(1o)r-2
| 10648/10647
| {{Monzo| 3 -2 0 -1 3 -2 }}
| 0.16260
| [[Margo Schulter]] (2002)
|-
| [[Pentonisma]]
| Saquinthuzogu
| s5(3uzg)4
| 281974669312 / 281950621875
| {{Monzo| 24 -5 -5 5 0 -5 }}
| 0.14765
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Pontigailimma]]
| Thururuquingu
| 3urr5g1
| 1990656/1990625
| {{Monzo| 13 5 -5 -2 0 -1 }}
| 0.026960
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[Grossmisma]]
| septholo-azogu
| 7(3o1o)zg2
| <abbr title="8559537565427849 / 8559456430325760">(32 digits)</abbr>
| {{Monzo| -30 -13 -1 1 7 7 }}
| 0.016410
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Chalmersia]]
| Lathotholurugugu
| L3oo1urgg1
| 123201/123200
| {{Monzo| -6 6 -2 -1 -1 2 }}
| 0.01405
| [[Gene Ward Smith]] (2003)
|-
| [[Lanasma]]
| Trila-septrithu-aquinquadbizo
| 3L21(3u)40z13
| <abbr title="6366805760909027985741435139224001 / 6366804434232663711262864979263488">(68 digits)</abbr>
| {{Monzo| -33 -1 0 40 0 -21 }}
| 3.6074 × 10-4
| [[User:Angekallikoita|Ange]] (2025)
|}

=== 17-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Quinticular comma]]
| Saquinso
| s5(17o)4
| 1419857/1417176
| {{Monzo| -3 -11 0 0 0 0 5 }}
| 3.2720
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[561/560|Monardisma]]
| Solorugu
| 17o1org1
| 561/560
| {{Monzo| -4 1 -1 -1 1 0 1 }}
| 3.0887
| [[Scott Dakota]] (2023)
|-
| [[595/594|Dakotisma]]
| Soluzoyo
| 17o1uzy2
| 595/594
| {{Monzo| -1 -3 1 1 -1 0 1 }}
| 2.9121
| [[Praveen Venkataramana]] (2022)
|-
| [[715/714|September comma]]
| Sutholoruyo
| 17u3o1ory-2
| 715/714
| {{Monzo| -1 -1 1 -1 1 1 -1 }}
| 2.4230
| [[Scott Dakota]] (2021)
|-
| [[833/832|Horizma, horizon comma]]
| Sothuzozo
| 17o3uzz2
| 833/832
| {{Monzo| -6 0 0 2 0 -1 1 }}
| 2.0796
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[936/935|Ainisma, ainic comma]]
| Sutholugu
| 17u3o1ug1
| 936/935
| {{Monzo| 3 2 -1 0 -1 1 -1 }}
| 1.8506
| [[Dawson Berry]] (2020)
|-
| [[2025/2023|Fidesma]]
| Susuruyoyo
| 17uuryy-3
| 2025/2023
| {{Monzo| 0 4 2 -1 0 0 -2 }}
| 1.7107
| [[Dawson Berry]] (2023)
|-
| [[1089/1088|Twosquare comma]]
| Sulolo
| 17u1oo-2
| 1089/1088
| {{Monzo| -6 2 0 0 2 0 -1 }}
| 1.5905
| [[User:Kaiveran|Kaiveran]] (2018)
|-
| [[1156/1155|Quadrantonisma]]
| Sosolurugu
| 17oo1urg2
| 1156/1155
| {{Monzo| 2 -1 -1 -1 -1 0 2 }}
| 1.4983
| [[Flora Canou]] (2023)
|-
| [[1225/1224|Noellisma]]
| Subizoyo
| 17u2zy1
| 1225/1224
| {{Monzo| -3 -2 2 2 0 0 -1 }}
| 1.4138
| [[Flora Canou]] (2022)
|-
| [[1275/1274|Cimbrisma]]
| Sothubiruyo
| 17o3u2ry-2
| 1275/1274
| {{Monzo| -1 1 2 -2 0 -1 1 }}
| 1.3584
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1701/1700|Palingenetic comma, palingenesis]]
| Suzogugu
| 17uzgg1
| 1701/1700
| {{Monzo| -2 5 -2 1 0 0 -1 }}
| 1.0181
| [[Dawson Berry]] (2021)
|-
| [[Laser comma]]
| Lasorutriyo
| L17or3y-2
| 57375/57344
| {{Monzo| -13 3 3 -1 0 0 1 }}
| 0.93564
| [[User:Jerdle|Jerdle]] (2023)
|-
| [[2058/2057|Xenisma]]
| Sululutrizo
| 17u1uu3z2
| 2058/2057
| {{Monzo| 1 1 0 3 -2 0 -1 }}
| 0.84143
| [[Margo Schulter]] (2000)
|-
| [[11016/11011|Cyclops comma]]
| Sothululuru
| 17o3u1uur1
| 11016/11011
| {{Monzo| 3 4 0 -1 -2 -1 1 }}
| 0.78596
| [[Eliora]] (2022)
|-
| [[24576/24565|Mavka comma]], archagallisma
| Trisu-agu
| 3(17u)g-2
| 24576/24565
| {{Monzo| 13 1 -1 0 0 0 -3 }}
| 0.77506
| [[Eliora]] (2022) for ''mavka comma''
|-
| [[2431/2430|Heptacircle comma]]
| Sothologu
| 17o3o1og2
| 2431/2430
| {{Monzo| -1 -5 -1 0 1 1 1 }}
| 0.71230
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2500/2499|Sperasma]]
| Subiruyoyo
| 17u2ryy-3
| 2500/2499
| {{Monzo| 2 -1 4 -2 0 0 -1 }}
| 0.69263
| [[Dawson Berry]] (2023)
|-
| [[2601/2600|Sextantonisma]]
| Sosothugugu
| 17oo3ugg2
| 2601/2600
| {{Monzo| -3 2 -2 0 0 -1 2 }}
| 0.66573
| [[Flora Canou]] (2023)
|-
| [[Semisixthmisma]]
| Trisu-athutrilo
| 3(17u)3u3(1o)-3
| 63888/63869
| {{Monzo| 4 1 0 0 3 -1 -3 }}
| 0.51494
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[4914/4913|Baladisma]]
| Trisu-athozo
| 3(17u)3oz-2
| 4914/4913
| {{Monzo| 1 3 0 1 0 1 -3 }}
| 0.35234
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[5832/5831|Chlorisma]]
| Sutriru
| 17u3r-3
| 5832/5831
| {{Monzo| 3 6 0 -3 0 0 -1 }}
| 0.29688
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[Galileisma]]
| Lalesu-agu
| L11(17u)g-7
| 171382426877952 / 171359481538165
| {{Monzo| 14 21 -1 0 0 0 -11 }}
| 0.23180
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Centisma]]
|
|
|
| 2.3.17 {{Monzo| -1001 -400 400 }}
| 0.16345
| [[CompactStar]] (2023)
|-
| [[Flashma]]
| Sotholuzotrigu
| 17o3o1uz3g3
| 12376/12375
| {{Monzo| 3 -2 -3 1 -1 1 1 }}
| 0.13989
| [[User:Kaiveran|Kaiveran]] (2016)
|-
| [[Sparkisma]]
| Sululuruyoyo
| 17u1uuryy-2
| 14400/14399
| {{Monzo| 6 2 2 -1 -2 0 -1 }}
| 0.12023
| [[User:Kaiveran|Kaiveran]] (2016)
|-
| [[Insanobromisma]]
| Sepquinsuyoyo
| 35(17uyy)-29
|
| {{Monzo| 36 -35 70 0 0 0 -35 }}
| 0.095608
| [[Eliora]] (2023)
|-
| [[1257795/1257728|Large triquarterisma]]
| Latrisulo-azoyo
| L3(17u1o)azy-3
| 1257795/1257728
| {{Monzo| -8 3 1 1 3 0 -3 }}
| 0.092222
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[471648/471625|Small triquarterisma]]
| triso-alutrirugu
| 3(17o)a1u3(rg)2
| 471648/471625
| {{Monzo| 5 1 -3 -3 -1 0 3 }}
| 0.084426
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[28561/28560|Pisanoisma]]
| Suquadtho-arugu
| 17u4(3o)rg1
| 28561/28560
| {{Monzo| -4 -1 -1 -1 0 4 -1 }}
| 0.060616
| [[Budjarn Lambeth]] (2024)
|-
| [[E-shaped comma]]
| Susuthoquadzo
| 17uu3o4z2
| 31213/31212
| {{Monzo| -2 -3 0 4 0 1 -2 }}
| 0.055466
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Lateral comma]]
| Sasuthotholoyo
| s17u3oo1oy1
| 37180/37179
| {{Monzo| 2 -7 1 0 1 2 -1 }}
| 0.046564
| [[User:Akselai|Akselai]] (2024)
|-
| [[Clevelandisma]]
| Sotribizogu
| 17o6(zg)5
| 2000033/2000000
| {{Monzo| -7 0 -6 6 0 0 1 }}
| 0.028565
| [[Lériendil]] (2024)
|-
| [[Scintillisma]]
| Lasuthuluquadzo-agu
| L17u3u1u4zag2
| 194481/194480
| {{Monzo| -4 4 -1 4 -1 -1 -1 }}
| 0.0089018
| [[Margo Schulter]] (2012)
|-
| [[Aksial comma]]
| Sotritho-aquinru-agu
| 17o3(3o)5rag-2
| 336141/336140
| {{Monzo| -2 2 -1 -5 0 3 1 }}
| 0.0051503
| [[User:Akselai|Akselai]] (2024)
|}

=== 19-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[513/512|Undevicesimal schisma]], undevicesimal formal comma, Boethius' comma
| Lano
| L19o1
| 513/512
| 2.3.19 {{Monzo| -9 3 1 }}
| 3.3780
| Plainsound Music Edition (2020)<ref>[https://marsbat.space/pdfs/HEJI2legend+series.pdf The Helmholtz-Ellis JI Pitch Notation (HEJI)]</ref> for ''undevicesimal schisma''
|-
| [[6144/6137|Langwisma]]
| Nunusu
| 19uu17u-2
| 6144/6137
| {{Monzo| 11 1 0 0 0 0 -1 -2 }}
| 1.9736
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[969/968|Kingfisher comma]]
| Nosolulu
| 19o17o1uu2
| 969/968
| {{Monzo| -3 1 0 0 -2 0 1 1 }}
| 1.7875
| [[Budjarn Lambeth]] (2025)
|-
| [[Mercurial comma]]
| Quinnosu-abiruyo
| 5(19o17u)2ry-3
| 557122275 / 556583944
| {{Monzo| -3 2 2 -2 0 0 -5 5 }}
| 1.6736
| [[User:Yourmusic Productions|Yourmusic Productions]] (2020)
|-
| [[1216/1215|Password, Eratosthenes' comma]]
| Sanogu
| s19og2
| 1216/1215
| {{Monzo| 6 -5 -1 0 0 0 0 1 }}
| 1.4243
|
|-
| [[1331/1330|Solvejgsma]]
| Nutrilo-arugu
| 19u3(1o)rg-2
| 1331/1330
| {{Monzo| -1 0 -1 -1 3 0 0 -1 }}
| 1.3012
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1445/1444|Aureusma]]
| Nunusosoyo
| 19uu17ooy1
| 1445/1444
| {{Monzo| -2 0 1 0 0 0 2 -2 }}
| 1.1985
| [[Flora Canou]] (2021)
|-
| [[1521/1520|Pinkanberry]]
| Nuthothogu
| 19u3oog1
| 1521/1520
| {{Monzo| -4 2 -1 0 0 2 0 -1 }}
| 1.1386
| [[User:Kaiveran|Kaiveran]] (2021)
|-
| [[1540/1539|Kevolisma]]
| Nulozoyo
| 19u1ozy1
| 1540/1539
| {{Monzo| 2 -4 1 1 1 0 0 -1 }}
| 1.1245
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3213/3211|Cobaltomenisma]]
| Nusothuthuzo
| 19u17o3uuz1
| 3213/3211
| {{Monzo| 0 3 0 1 0 -2 1 -1 }}
| 1.0780
| [[Francium]] (2025)
|-
| [[1729/1728|Ramanujanisma]]
| Nothozo
| 19o3oz2
| 1729/1728
| {{Monzo| -6 -3 0 1 0 1 0 1 }}
| 1.0016
| [[Frédéric Gagné]] (2021)
|-
| [[3971/3969|Heartlandisma]]
| Nonoloruru
| 19oo1orr1
| 3971/3969
| {{Monzo| 0 -4 0 -2 1 0 0 2 }}
| 0.87216
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[830297/829939|Minthtone schismina]]
| Trinuso-abitholu
| 3(19u17o)2(3o1u)2
| 830297/829939
| {{Monzo| 0 0 0 0 -2 2 3 -3 }}
| 0.74662
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[2376/2375|Trichthonisma]]
| Nulotrigu
| 19u1o3g1
| 2376/2375
| {{Monzo| 3 3 -3 0 1 0 0 -1 }}
| 0.72879
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Crawma]]
| Nuquadso-atrithu
| 19u4(17o)3(3u)2
| 83521/83486
| {{Monzo| -1 0 0 0 0 -3 4 -1 }}
| 0.72564
| [[groundfault]] (2023)
|-
| [[2432/2431|Blumeyer comma]]
| Nosuthulu
| 19o17u3u1u1
| 2432/2431
| {{Monzo| 7 0 0 0 -1 -1 -1 1 }}
| 0.71200
| [[Douglas Blumeyer]] (2015)
|-
| [[93347/93312|Trilute comma]]
| Notriso
| 19o3(17o)3
| 93347/93312
| {{Monzo| -7 -6 0 0 0 0 3 1 }}
| 0.64924
| [[Xenllium]] (2025)
|-
| [[2926/2925|Neovulture comma, neovulturisma]]
| Nothulozogugu
| 19o3u1ozgg2
| 2926/2925
| {{Monzo| 1 -2 -2 1 1 -1 0 1 }}
| 0.59177
| [[Xenllium]] (2023)
|-
| [[3136/3135|Neomirkwai comma, neomirkwaisma]]
| Nuluzozogu
| 19u1uzzg2
| 3136/3135
| {{Monzo| 6 -1 -1 2 -1 0 0 -1 }}
| 0.55214
| [[Xenllium]] (2023)
|-
| [[116640/116603|Large tridevisemma]]
| Trinu-asuyo
| 3(19u)17uy-3
| 116640/116603
| {{Monzo| 5 6 1 0 0 0 -1 -3 }}
| 0.54926
| [[Xenllium]] (2025)
|-
| [[3250/3249|Martebisma]]
| Nunuthotriyo
| 19uu3o3y-2
| 3250/3249
| {{Monzo| 1 -2 3 0 0 1 0 -2 }}
| 0.53277
| [[Xenllium]] (2023)
|-
| [[48013/48000|Small tridevisemma]]
| Trino-azotrigu
| 3(19o)z3g3
| 48013/48000
| {{Monzo| -7 -1 -3 1 0 0 0 3 }}
| 0.46881
| [[Xenllium]] (2025)
|-
| [[4200/4199|Neosatanisma]]
| Nusuthuzoyoyo
| 19u17u3uzyy-2
| 4200/4199
| {{Monzo| 3 1 2 1 0 -1 -1 -1 }}
| 0.41225
| [[Xenllium]] (2023)
|-
| [[176000/175959|Triseptichrome comma]]
| Nulotriruyo
| 19u1o3(ry)-3
| 176000/175959
| {{Monzo| 7 -3 3 -3 1 0 0 -1 }}
| 0.40335
| [[Xenllium]] (2025)
|-
| [[5776/5775|Neovish comma, neovishma]]
| Nonolurugugu
| 19oo1urgg2
| 5776/5775
| {{Monzo| 4 -1 -2 -1 -1 0 0 2 }}
| 0.29975
| [[Xenllium]] (2023)
|-
| [[5929/5928|Manzanisma]]
| Nuthubilozo
| 19u3u2(1oz)1
| 5929/5928
| {{Monzo| -3 -1 0 2 2 -1 0 -1 }}
| 0.29202
| [[Xenllium]] (2025)
|-
| [[5985/5984|Neogrendel comma, neogrendelisma]]
| Nosuluzoyo
| 19o17u1uzy1
| 5985/5984
| {{Monzo| -5 2 1 1 -1 0 -1 1 }}
| 0.28929
| [[Xenllium]] (2023)
|-
| BMO schismina
| Sabinothu
| s2(19o3u)2
| 369664/369603
| {{Monzo| 10 -7 0 0 0 -2 0 2 }}
| 0.28570
|
|-
| [[6175/6174|Neonewtisma]]
| Nothotriru-ayoyo
| 19o3o3rayy-2
| 6175/6174
| {{Monzo| -1 -2 2 -3 0 1 0 1 }}
| 0.28038
| [[Xenllium]] (2023)
|-
| [[6860/6859|Devicubisma]]
| Trinuzo-ayo
| 3(19uz)y1
| 6860/6859
| {{Monzo| 2 0 1 3 0 0 0 -3 }}
| 0.25238
| [[Xenllium]] (2025)
|-
| [[Undevicesimal counterschisma]]
| Seplanu
| 7L19u-6
| <abbr title=" 717897987691852588770249 / 717799705396186072481792">(48 digits)</abbr>
| 2.3.19 {{Monzo| -75 50 -1 }}
| 0.23703
| [[Xenllium]] (2025)
|-
| Frouggie comma
| Nusuquinthu-aquadlo
| 19u17u5(3u)4(1o)-3
| 119939072 / 119927639
| {{Monzo| 13 0 0 0 4 -5 -1 -1 }}
| 0.16503
| [[Douglas Blumeyer]] (2020)
|-
| [[12636/12635|Padriellisma]]
| Nunuthorugu
| 19uu3org-2
| 12636/12635
| {{Monzo| 2 5 -1 -1 0 1 0 -2 }}
| 0.13701
| [[Xenllium]] (2025)
|-
| [[Lakisma]]
| Saquadnoso-agu
| s4(19o17o)g5
| 10884540241 / 10883911680
| {{Monzo| -12 -12 -1 0 0 0 4 4 }}
| 0.09998
| [[User:Flirora|+merlan #flirora]] (2021)
|-
| [[Aubertisma]]
| Nosothutrilu-arutriyo
| 19o17o3u3(1u)r3y1
| 121125/121121
| {{monzo| 0 1 3 -1 -3 -1 1 1 }}
| 0.057173
| [[Francium]] (2025)
|-
| Pollar comma
| Nunusuquintho-alulu
| 19uu17u5(3o)1uu1
| 742586/742577
| {{Monzo| 1 0 0 0 -2 5 -1 -2 }}
| 0.020982
| [[Douglas Blumeyer]] (2020)
|-
| [[Decimillisma]], 19-limit decimill
| Sanosorurutrigu
| s19o17orr3g2
| 165376/165375
| {{Monzo| 9 -3 -3 -2 0 0 1 1 }}
| 0.010469
| [[Flora Canou]] (2021), for ''decimillisma''
|-
| [[65/19 atom]]
| Sasa-nuthoyo
| ss19u3oy2
| 272629760 / 272629233
| {{Monzo| 22 -15 1 0 0 1 0 -1 }}
| 0.0033465
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[Devicisma]]
| Nunusothutrilo-azogu
| 19uu17o3u3(1o)zg1
| 633556/633555
| {{Monzo| 2 -3 -1 1 3 -1 1 -2 }}
| 0.0027326
| [[Xenllium]] (2023)
|-
| [[11859211/11859210|Tredekisma]]
| Quadno-athoquadlu-azogu
| 19o43o1u4zg4
| 11859211/11859210
| {{Monzo| -1 -4 -1 1 -4 1 0 4 }}
| 0.000146
| [[Eufalesio]] (2026)
|}

=== 23-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[507/506|Laodicisma]]
| Twethuthotholu
| 23u3oo1u1
| 507/506
| 2.3.11.13.23 {{Monzo| -1 1 -1 2 -1 }}
| 3.4180
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[529/528|Preziosisma]]
| Bitwetho-alu
| 23oo1u2
| 529/528
| 2.3.11.23 {{Monzo| -4 -1 -1 2 }}
| 3.2758
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[576/575|Worcester comma]]
| Twethugugu
| 23ugg1
| 576/575
| 2.3.5.23 {{Monzo| 6 2 -2 -1 }}
| 3.0082
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[9765625/9750528]]
| Labitwethuquinyo
| L23uu10y-4
| 9765625/9750528
| 2.3.5.23 {{Monzo| -11 -2 10 -2 }}
| 2.6784
|
|-
| [[736/735|Harvardisma]]
| Twethorurugu
| 23orrg1
| 736/735
| 2.3.5.7.23 {{Monzo| 5 -1 -1 -2 1 }}
| 2.3538
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[760/759|Squadronisma]]
| Twethunoluyo
| 23u19o1uy1
| 760/759
| {{Monzo| 3 -1 1 0 -1 0 0 1 -1 }}
| 2.2794
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[875/874|Nymphisma]]
| Twethunuzotriyo
| 23u19uz3y-2
| 875/874
| 2.5.7.19.23 {{Monzo| -1 3 1 -1 -1 }}
| 1.9797
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[897/896|Lysistratisma]]
| Twethothoru
| 23o3or1
| 897/896
| 2.3.7.13.23 {{Monzo| -7 1 -1 1 1 }}
| 1.9311
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3014656/3011499|23/17-schisma]]
| Sasa-twethosu
| ss23o17u2
| 3014656/3011499
| 2.3.17.23 {{monzo| 17 -11 -1 1 }}
| 1.8139
|
|-
| [[2187/2185|Vashegyitisma]]
| Latwethunugu
| L23u19ug-2
| 2187/2185
| 3.5.19.23 {{monzo| 7 -1 -1 -1 }}
| 1.5839
| [[Francium]] (2025)
|-
| [[1105/1104|Fragarisma]]
| Twethusothoyo
| 23u17o3oy1
| 1105/1104
| {{Monzo| -4 -1 1 0 0 1 1 0 -1 }}
| 1.5674
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[7942/7935|Brigade comma]]
| Bitwethuno-alogu
| 23uu19oo1og1
| 7942/7935
| {{Monzo| 1 -1 -1 0 1 0 0 2 -2 }}
| 1.5266
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1197/1196|Rodessisma]]
| Twethunothuzo
| 23u19o3uz1
| 1197/1196
| {{Monzo| -2 2 0 1 0 -1 0 1 -1 }}
| 1.4469
| [[Lériendil]] (2025)
|-
| [[1288/1287|Triaphonisma]], santisma
| Twethothuluzo
| 23o3u1uz2
| 1288/1287
| {{Monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 1.3446
| [[User:Xenllium|Xenllium]] (2023) for ''santisma''
|-
| [[1496/1495|Turkisma]]
| Twethusothulogu
| 23u17o3u1og1
| 1496/1495
| {{Monzo| 3 0 -1 0 1 -1 1 0 -1 }}
| 1.1576
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[7429/7425|Gordaitisma]]
| Twethonosolugugu
| 23o19o17o1ugg3
| 7429/7425
| {{monzo| 0 -3 -2 0 -1 0 1 1 1 }}
| 0.93240
| [[Francium]] (2025)
|-
| [[1863/1862|Antinousisma]]
| Twethonururu
| 23o19urr-2
| 1863/1862
| 2.3.7.19.23 {{Monzo| -1 4 -2 -1 1 }}
| 0.92952
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Kaifsma]]
| Twethunosutholuzozogu
| 23u19o17u3o1uzzg2
| 193648/193545
| {{Monzo| 4 -2 -1 2 -1 1 -1 1 -1 }}
| 0.92108
| [[User:Kaiveran|Kaiveran]] (2022)
|-
| [[2024/2023|Artifisma]], insincere comma
| Twethosusuloru
| 23o17uu1or-2
| 2024/2023
| 2.7.11.17.23 {{Monzo| 3 -1 1 -2 1 }}
| 0.85556
| [[User:Xenllium|Xenllium]] (2023) for ''insincere comma''
|-
| [[2025/2024|Cupcake comma]], cupcakesma
| Latwethuluyoyo
| L23u1uyy-2
| 2025/2024
| 2.3.5.11.23 {{Monzo| -3 4 2 -1 -1 }}
| 0.85514
| See the page.
|-
| [[2185/2184|Guangdongisma]]
| Twethonothuruyo
| 23o19o3ury1
| 2185/2184
| {{Monzo| -3 -1 1 -1 0 -1 0 1 1 }}
| 0.79251
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2300/2299|Travellisma]]
| Twethonubiluyo
| 23o19u1uuyy1
| 2300/2299
| 2.5.11.19.23 {{Monzo| 2 2 -2 -1 1 }}
| 0.75287
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2646/2645|Biyativice comma]]
| Bitwethuzo-agu
| 23uuzzg1
| 2646/2645
| 2.3.5.7.23 {{Monzo| 1 3 -1 2 -2 }}
| 0.65441
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2737/2736|Kotkisma]]
| Twethonusozo
| 23o19u17oz2
| 2737/2736
| {{Monzo| -4 -2 0 1 0 0 1 -1 1 }}
| 0.63265
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Kwadransma]]
| Quadtwethu
| 4(23u)-3
| 279936/279841
| {{Monzo| 7 7 0 0 0 0 0 0 -4 }}
| 0.58762
| [[Lériendil]] (2024)
|-
| [[3060/3059|Vicious comma]], viciousma
| Twethunusoruyo
| 23u19u17ory-2
| 3060/3059
| {{Monzo| 2 2 1 -1 0 0 1 -1 -1 }}
| 0.56586
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3381/3380|Mikkolisma]], seminaiadvice comma
| Twethothuthuzozogu
| 23o3uuzzg2
| 3381/3380
| {{Monzo| -2 1 -1 2 0 -2 0 0 1 }}
| 0.51212
| See the page.
|-
| [[6877/6875|Grossvice comma]]
| Bitwetho-atholuquadgu
| 23oo3o1u4g3
| 6877/6875
| {{Monzo| 0 0 -4 0 -1 1 0 0 2 }}
| 0.50356
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3520/3519|Vicedim comma]]
| Twethusuloyo
| 23u17u1oy-2
| 3520/3519
| {{Monzo| 6 -2 1 0 1 0 -1 0 -1 }}
| 0.49190
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3888/3887|Shoalma]], vicetride comma
| Twethuthuthu
| 23u3uu-2
| 3888/3887
| 2.3.13.23 {{Monzo| 4 5 -2 -1 }}
| 0.44533
| See the page.
|-
| [[8075/8073|Hagendorfisma]]
| Twethunosothuyoyo
| 23u19o17o3uyy1
| 8075/8073
| {{monzo| 0 -3 2 0 0 -1 1 1 -1 }}
| 0.42884
| [[Francium]] (2025)
|-
| [[4693/4692|Viceaug comma]]
| Twethunonosutho
| 23u19oo17u3o1
| 4693/4692
| {{Monzo| -2 -1 0 0 0 1 -1 2 -1 }}
| 0.36894
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[4761/4760|Demiquartervice comma]]
| Bitwetho-asurugu
| 23oo17urg1
| 4761/4760
| {{Monzo| -3 2 -1 -1 0 0 -1 0 2 }}
| 0.36367
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[5083/5082|Broadviewsma]]
| Twethosotholuluru
| 23o17o3o1uur2
| 5083/5082
| {{Monzo| -1 -1 0 -1 -2 1 1 0 1 }}
| 0.34063
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[8625/8624|Beerglass comma]]
| Twetholururutriyo
| 23o1urr3y-2
| 8625/8624
| {{Monzo| -4 1 3 -2 -1 0 0 0 1 }}
| 0.20073
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Galeaclolusisma]]
| Twethususutholuquadyo
| 23u17uu3o1u4y-3
| 73125/73117
| {{Monzo| 0 2 4 0 -1 1 -2 0 -1 }}
| 0.18941
| [[Francium]] (2025)
|-
| [[10626/10625|Demiglace comma]]
| Twethosulozoquadgu
| 23o17u1oz4g2
| 10626/10625
| {{Monzo| 1 1 -4 1 1 0 -1 0 1 }}
| 0.16293
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Vicetertisma]]
| Tritwethu-athotho
| 3(23u)3oo-2
| 12168/12167
| 2.3.13.23 {{Monzo| 3 2 2 -3 }}
| 0.14228
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Joshuavoisma]]
| Twethusutholozoyoyo
| 23u17u3o1ozyy-2
| 25025/25024
| {{monzo| -6 0 2 1 1 1 -1 0 -1 }}
| 0.06918
| [[Francium]] (2024)
|-
| [[Diarithmedia]]
| Bitwethozo-agu
| 23oozzg3
| 25921/25920
| 2.3.5.7.23 {{monzo| -6 -4 -1 2 2 }}
| 0.066790
| [[Flora Canou]] (2023), modified by [[Lériendil]] (2024)
|-
| [[Jeffbenisma]]
| Labitwethu-anutholuzoyo
| L23uu19u3o1uzy-2
| 110565/110561
| {{monzo| 0 5 1 1 -1 1 0 -1 -2 }}
| 0.062633
| [[Francium]] (2025)
|}

=== 29-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[551/550|Minor chthonovinema]]
| Twenonolugugu
| 29o19o1ugg2
| 551/550
| 2.5.11.19.29 {{monzo| -1 -2 -1 1 1 }}
| 3.1448
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[552/551|Sigelindisma]]
| Twenutwethonu
| 29u23o19u1
| 552/551
| 2.3.19.23.29 {{monzo| 3 1 -1 1 -1 }}
| 3.1391
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[609/608|Vineyard comma]]
| Twenonuzo
| 29o19uz1
| 609/608
| 2.3.7.19.29 {{monzo| -5 1 1 -1 1 }}
| 2.8451
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[638/637|Moirisma]]
| Twenothuloruru
| 29o3u1orr-2
| 638/637
| 2.7.11.13.29 {{monzo| 1 -2 1 -1 1 }}
| 2.7157
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[726/725|Joellisma]]
| Twenubilogu
| 29u1oogg1
| 726/725
| 2.3.5.11.29 {{monzo| 1 1 -2 2 -1 }}
| 2.3863
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[783/782|Norisma]]
| Twenotwethusu
| 29o23u17u-2
| 783/782
| 2.3.17.23.29 {{monzo| -1 3 -1 -1 1 }}
| 2.2124
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[784/783|Biminorisma]], spoogalactic comma
| Twenuzozo
| 29uzz2
| 784/783
| 2.3.7.29 {{monzo| 4 -3 2 -1 }}
| 2.2096
| [[Scott Dakota]] for ''biminorisma''
|-
| [[841/840|Arabellisma]]
| Bitweno-arugu
| 29oorg1
| 841/840
| 2.3.5.7.29 {{monzo| -3 -1 -1 -1 2 }}
| 2.0598
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1015/1014|Christisma]]
| Twenothuthuzoyo
| 29o3uuzy1
| 1015/1014
| {{monzo| -1 -1 1 1 0 -2 0 0 0 1 }}
| 1.7065
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1045/1044|Michelisma]]
| Twenunoloyo
| 29u19o1oy1
| 1045/1044
| {{monzo| -2 -2 1 0 1 0 0 1 0 -1 }}
| 1.6575
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1276/1275|Ucclisma]]
| Twenosulogugu
| 29o17u1ogg1
| 1276/1275
| {{monzo| 2 -1 -2 0 1 0 -1 0 0 1 }}
| 1.3573
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1450/1449|Raimondisma]]
| Twenotwethuruyoyo
| 29o23uryy-2
| 1450/1449
| {{monzo| 1 -2 2 -1 0 0 0 0 -1 1 }}
| 1.1944
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1596/1595|Itzigsohnisma]]
| Twenunoluzogu
| 29u19o1uzg2
| 1596/1595
| {{monzo| 2 1 -1 1 -1 0 0 1 0 -1 }}
| 1.0851
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1625/1624|Norcma]]
| Twenuthorutriyo
| 29u3or3y-2
| 1625/1624
| 2.5.7.13.29 {{monzo| -3 3 -1 1 -1 }}
| 1.0657
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[1683/1682|Castafiorisma]]
| Bitwenu-asolo
| 29uu17o1o1
| 1683/1682
| 2.3.11.17.29 {{monzo| -1 2 1 1 -2 }}
| 1.0290
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2001/2000|Major discoverisma]]
| Twenotwethotrigu
| 29o23o3g2
| 2001/2000
| 2.3.5.23.29 {{monzo| -4 1 -3 1 1 }}
| 0.86540
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2002/2001|Minor discoverisma]]
| Twenutwethutholozo
| 29u23u3o1oz1
| 2002/2001
| {{monzo| 1 -1 0 1 1 1 0 0 -1 -1 }}
| 0.86497
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2176/2175|Donarisma]]
| Twenusogugu
| 29u17ogg2
| 2176/2175
| 2.3.5.17.29 {{monzo| 7 -1 -2 1 -1 }}
| 0.79579
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2205/2204|Glinkisma]]
| Twenunuzozoyo
| 29u19uzzy1
| 2205/2204
| {{monzo| -2 2 1 2 0 0 0 -1 0 -1 }}
| 0.78532
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2262/2261|Mitidikisma]]
| Twenonusuthoru
| 29o19u17u3or-2
| 2262/2261
| {{monzo| 1 1 0 -1 0 1 -1 -1 0 1 }}
| 0.76552
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2465/2464|Laservine comma]]
| Twenosoluruyo
| 29o17o1ury1
| 2465/2464
| {{monzo| -5 0 1 -1 -1 0 1 0 0 1 }}
| 0.70247
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2640/2639|Hällströmisma]]
| Twenuthuloruyo
| 29u3u1ory-2
| 2640/2639
| {{monzo| 4 1 1 -1 1 -1 0 0 0 -1 }}
| 0.65589
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2755/2754|Avicema]]
| Twenonosuyo
| 29o19o17uy1
| 2755/2754
| {{monzo| -1 -4 1 0 0 0 -1 1 0 1 }}
| 0.62851
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[2784/2783|Domeykisma]]
| Twenotwethululu
| 29o23u1uu1
| 2784/2783
| 2.3.11.23.29 {{monzo| 5 1 -2 -1 1 }}
| 0.62196
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[9251/9248|Helevenisma]]
| Bitwenosu-alo
| 29oo17uu1o-2
| 9251/9248
| 2.11.17.29 {{monzo| -5 1 -2 2 }}
| 0.56151
| [[Zhea Erose]] (2023)
|-
| [[3249/3248|Musashinisma]]
| Twenunonoru
| 29u19oor1
| 3249/3248
| 2.3.7.19.29 {{monzo| -4 2 -1 2 -1 }}
| 0.53293
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[3451/3450|Mentorisma]]
| Twenotwethusozogugu
| 29o23u17ozgg2
| 3451/3450
| {{monzo| -1 -1 -2 1 0 0 1 0 -1 1 }}
| 0.50173
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Bronxisma]]
| Twenununusolozo
| 29u19uu17o1oz1
| 10472/10469
| {{monzo| 3 0 0 1 1 0 1 -2 0 -1 }}
| 0.49603
| [[Francium]] (2024)
|-
| [[3510/3509|Veederisma]]
| Twenutholuluyo
| 29u3o1uuy1
| 3510/3509
| {{monzo| 1 3 1 0 -2 1 0 0 0 -1 }}
| 0.49330
| [[Francium]] (2024)
|-
| [[4641/4640|Vinecute comma]]
| Twenusothozogu
| 29u17o3ozg2
| 4641/4640
| {{monzo| -5 1 -1 1 0 1 1 0 0 -1 }}
| 0.37307
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[4785/4784|Petrovsma]]
| Twenotwethuthuloyo
| 29o23u3u1oy-2
| 4785/4784
| {{monzo| -4 1 1 0 1 -1 0 0 -1 1 }}
| 0.36184
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[4901/4900|Large grapevine]]
| Twenothothobirugu
| 29o3oorrgg1
| 4901/4900
| 2.5.7.13.29 {{monzo| -2 -2 -2 2 1 }}
| 0.35328
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Mynucuvine comma]]
| Labitwenu-athuyo
| L29uu3uy-2
| 10935/10933
| 3.5.13.29 {{monzo| 7 1 -1 -2 }}
| 0.31667
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[5888/5887|Vinocular comma]]
| Bitwenu-atwethoru
| 29uu23or1
| 5888/5887
| 2.7.23.29 {{monzo| 8 -1 1 -2 }}
| 0.29405
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[5916/5915|Woudisma]]
| Twenosothuthurugu
| 29o17o3uurg1
| 5916/5915
| {{monzo| 2 1 -1 -1 0 -2 1 0 0 1 }}
| 0.29266
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[7425/7424|Small grapevine]]
| Latwenuloyoyo
| L29u1oyy-2
| 7425/7424
| 2.3.5.11.29 {{monzo| -8 3 2 1 -1 }}
| 0.23318
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[8671/8670|Vinous comma]], vinousma
| Twenotwethosusuthogu
| 29o23o17uu3og1
| 8671/8670
| {{monzo| -1 -1 -1 0 0 1 -2 0 1 1 }}
| 0.19967
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[9802/9801|Kakisma]]
|
|
| 9802/9801
| {{Monzo| 1 -4 0 0 -2 2 0 0 0 1 }}
| 0.17663
| [[Lériendil]] (2025)
|-
| [[10557/10556|Rowlandisma]]
| Twenutwethosothuru
| 29u23o17o3ur1
| 10557/10556
| {{monzo| -2 3 0 -1 0 -1 1 0 1 -1 }}
| 0.16400
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Uma]], umic comma
| Twenotwethoquadru-agu
| 29o23o4rg-2
| 12006/12005
| {{monzo| 1 2 -1 -4 0 0 0 0 1 1 }}
| 0.14420
|
|-
| [[Vuillafansisma]]
| Twenunosoluyo
| 29u19o17o1uy2
| 25840/25839
| {{monzo| 4 -4 1 0 -1 0 1 1 0 -1 }}
| 0.067000
| [[Francium]] (2024)
|-
| [[Odyssey comma]]
| Bitwenotwetho-athulurutrigu
| 29oo23oo3u1ur3g2
| 4004001/4004000
| {{monzo| -5 2 -3 -1 -1 -1 0 0 2 2 }}
| 0.00043238
| [[Tristan Bay]] (2024)
|}

=== 31-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[496/495|Navatonisma]]
| Thiwolugu
| 31o1ug1
| 496/495
| 2.3.5.11.31 {{monzo| 4 -2 -1 -1 1 }}
| 3.4939
| [[User:FilterNashi|FilterNashi]] (2025)
|-
| [[528/527|Rezisma]]
| Thiwusulo
| 31u17u1o1
| 528/527
| 2.3.11.17.31 {{monzo| 4 1 1 -1 -1 }}
| 3.2820
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[589/588|Croatisma]]
| Thiwonoruru
| 31o19orr-2
| 589/588
| 2.3.7.19.31 {{monzo| -2 -1 -2 1 1 }}
| 2.9418
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[621/620|Owowhatsthisma]]
| Thiwutwethogu
| 31u23og2
| 621/620
| 2.3.5.23.31 {{monzo| -2 3 -1 1 -1 }}
| 2.7901
| [[HEHEHE I AM A SUPAHSTAR SAGA]] (2021)
|-
| [[651/650|Antiklisma]]
| Thiwothuzogugu
| 31o3uzgg1
| 651/650
| 2.3.5.7.13.31 {{monzo| -1 1 -2 1 -1 1 }}
| 2.6614
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[714/713|Ululisma]]
| Thiwutwethusozo
| 31u23u17oz2
| 714/713
| 2.3.7.17.23.31 {{monzo| 1 1 1 1 -1 -1 }}
| 2.4264
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[961/960|Tricesimoprimal quartertones comma]]
| Bithiwo-agu
| 31oog-2
| 961/960
| 2.3.5.31 {{monzo| -6 -1 -1 2 }}
| 1.8024
| [[Godtone]] (2024)
|-
| [[1024/1023|Kibisma]]
| Thiwulu
| 31u1u2
| 1024/1023
| 2.3.11.31 {{Monzo| 10 -1 -1 -1 }}
| 1.6915
| See the page.
|-
| [[2233/2232|Kuznetsovisma]]
| Thiwotwenolozo
| 31u29o1oz2
| 2233/2232
| 2.3.7.11.29.31 {{monzo| -3 -2 1 1 1 -1 }}
| 0.77547
| [[Francium]] (2024)
|-
| [[3969/3968|Yunzee comma]]
| Lathiwuzozo
| L31uzz2
| 3969/3968
| 2.3.7.31 {{monzo| -7 4 2 -1 }}
| 0.43624
| [[User:Kaiveran|Kaiveran]] (2021)
|-
| [[4186/4185|Tamashimisma]]
| Thiwutwethothozogu
| 31u23o3ozg3
| 4186/4185
| 2.3.5.7.13.23.31 {{monzo| 1 -3 -1 1 1 1 -1 }}
| 0.41363
| [[Francium]] (2025)
|-
| [[4992/4991|Kalmanisma]]
| Thiwutwethuthoru
| 31u23u3or1
| 4992/4991
| 2.3.7.13.23.31 {{monzo| 7 1 -1 1 -1 -1 }}
| 0.34684
| [[Francium]] (2024)
|-
| [[5797/5796|Bivojisma]]
| Thiwotwethusoloru
| 31o23u17o1or-2
| 5797/5796
| 2.3.7.11.17.23.31 {{monzo| -2 -2 -1 1 1 -1 1 }}
| 0.29867
| [[Francium]] (2025)
|-
| [[6076/6075|Large ricegrain]]
| Sathiwobizogu
| s31ozzgg2
| 6076/6075
| 2.3.5.7.31 {{monzo| 2 -5 -2 2 1 }}
| 0.28495
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[6480/6479|Scarlattisma]]
| Thiwunuluyo
| 31u19u1uy1
| 6480/6479
| 2.3.5.11.19.31 {{monzo| 4 4 1 -1 -1 -1 }}
| 0.26719
| [[Francium]] (2024)
|-
| [[6728/6727|Sushi comma]]
| Bithiwutweno-aru
| 31uu29oor2
| 6728/6727
| 2.7.29.31 {{monzo| 3 -1 2 -2 }}
| 0.25734
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[16337/16335|Brown rice comma]]
| Bithiwo-asolulugu
| 31oo17o1uug1
| 16337/16335
| 3.5.11.17.31 {{monzo| -3 -1 -2 1 2 }}
| 0.21195
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[8464/8463|Polishookisma]]
| Thiwubitwetho-athuru
| 31u23oo3ur2
| 8464/8463
| 2.3.7.13.23.31 {{monzo| 4 -1 -1 -1 2 -1 }}
| 0.20455
| [[Francium]] (2024)
|-
| [[8960/8959|Small ricegrain]]
| Thiwususuzoyo
| 31u17uuzy1
| 8960/8959
| 2.5.7.17.31 {{monzo| 8 1 1 -2 -1 }}
| 0.19323
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Acronymisma]]
| Thiwotrithu-azo
| 31o3(3u)z-2
| 17577/17576
| 2.3.7.13.31 {{monzo| -3 4 1 -3 1 }}
| 0.098497
| [[User:Lériendil|Lériendil]] (2025)
|-
| [[Totziensisma]]
| Thiwotwetholurutrigu
| 31o23o1ur3g1
| 19251/19250
| 2.3.5.7.11.23.31 {{monzo| -1 3 -3 -1 -1 1 1 }}
| 0.089932
| [[Francium]] (2024)
|-
| [[Honeybrookisma]]
| Thiwobitwenu-atwethutho
| 31o29uu23u3o-2
| 19344/19343
| 2.3.13.23.29.31 {{monzo| 4 1 1 -1 -2 1 }}
| 0.089500
| [[Francium]] (2024)
|-
| [[Tricecubisma]]
| Trithiwu-anozozo
| 3(31u)19ozz4
| 29792/29791
| 2.7.19.31 {{monzo| 5 2 1 -3 }}
| 0.058112
| [[User:Xenllium|Xenllium]] (2025)
|}

=== 37-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[666/665|Beastisma]]
| Thisonurugu
| 37o19urg1
| 666/665
| 2.3.5.7.19.37 {{monzo| 1 2 -1 -1 -1 1 }}
| 2.6014
| [[Francium]] (2025)
|-
| [[667/666|Denisisma]]
| Thisutwenotwetho
| 37u29o23o1
| 667/666
| 2.3.23.29.37 {{monzo| -1 -2 1 1 -1 }}
| 2.5975
| [[Francium]] (2025)
|-
| [[703/702|Noemisma]]
| Thisonothu
| 37o19o3u2
| 703/702
| 2.3.13.19.37 {{monzo| -1 -3 -1 1 1 }}
| 2.4644
| [[Francium]] (2025)
|-
| [[704/703|Minimyna]]
| Thisunulo
| 37u19u1o-2
| 704/703
| 2.11.19.37 {{monzo| 6 1 -1 -1 }}
| 2.4609
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[741/740|Botolphisma]]
| Thisunothogu
| 37u19o3og1
| 741/740
| 2.3.5.13.19.37 {{monzo| -2 1 -1 1 1 -1 }}
| 2.3379
| [[Francium]] (2025)
|-
| [[851/850|Zeissisma]]
| Thisotwethosugugu
| 37o23o17ugg2
| 851/850
| 2.5.17.23.37 {{monzo| -1 -2 -1 1 1 }}
| 2.0355
| [[Francium]] (2025)
|-
| [[925/924|Alphonsinisma]]
| Thisoluruyoyo
| 37o1uryy1
| 925/924
| 2.3.5.7.11.37 {{monzo| -2 -1 2 -1 -1 1 }}
| 1.8726
| [[Francium]] (2025)
|-
| [[1036/1035|Ganymedisma]]
| Thisotwethuzogu
| 37o23uzg2
| 1036/1035
| 2.3.5.7.23.37 {{monzo| 2 -2 -1 1 -1 1 }}
| 1.6719
| [[Francium]] (2025)
|-
| [[1184/1183|Gaeisma]]
| Thisothuthuru
| 37o3uur1
| 1184/1183
| 2.7.13.37 {{monzo| 5 -1 -2 1}}
| 1.4628
| [[Francium]] (2025)
|-
| [[1332/1331|Marconisma]]
| Thisotrilu
| 37o3(1u)2
| 1332/1331
| 2.3.11.37 {{monzo| 2 2 -3 1 }}
| 1.3002
| [[Francium]] (2024)
|-
| [[1480/1479|Aunusisma]]
| Thisotwenusuyo
| 37o29u17uy1
| 1480/1479
| 2.3.5.17.29.37 {{monzo| 3 -1 1 -1 -1 1 }}
| 1.1702
| [[Francium]] (2024)
|-
| [[1665/1664|Gabisma]]
| Thisothuyo
| 37o3uy1
| 1665/1664
| 2.3.5.13.37 {{monzo| -7 2 1 -1 1 }}
| 1.0401
| [[Francium]] (2025)
|-
| [[1666/1665|Gentisma]]
| Thisusozozogu
| 37u17ozzg2
| 1666/1665
| 2.3.5.7.17.37 {{monzo| 1 -2 -1 2 1 -1 }}
| 1.0395
| [[Francium]] (2025)
|-
| [[1702/1701|Kalaharisma]]
| Sathisotwethoru
| S37o23or2
| 1702/1701
| 2.3.7.23.37 {{monzo| 1 -5 -1 1 1 }}
| 1.0175
| [[Francium]] (2025)
|-
| [[1925/1924|Misericorde]]
| Thisuthulozoyoyo
| 37u3u1ozyy-2
| 1925/1924
| 2.5.7.11.13.37 {{monzo| -2 2 1 1 -1 -1 }}
| 0.89958
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[2109/2108|Dhotelisma]]
| Thisothiwunosu
| 37o31u19o17u2
| 2109/2108
| 2.3.17.19.31.37 {{monzo| -2 1 -1 1 -1 1 }}
| 0.82107
| [[Francium]] (2025)
|-
| [[2146/2145|Stentorisma]]
| Thisotwenothulugu
| 37o29o3u1ug2
| 2146/2145
| 2.3.5.11.13.29.37 {{monzo| 1 -1 -1 -1 -1 1 1 }}
| 0.80691
| [[Francium]] (2024)
|-
| [[2553/2552|Viljevisma]]
| Thisotwenutwetholu
| 37o29u23o1u2
| 2553/2552
| 2.3.11.23.29.37 {{monzo| -3 1 -1 1 -1 1 }}
| 0.67825
| [[Francium]] (2024)
|-
| [[2850/2849|Mozhaiskisma]]
| Thisunoluruyoyo
| 37u19o1uryy-2
| 2850/2849
| 2.3.5.7.11.19.37 {{monzo| 1 1 2 -1 -1 1 -1 }}
| 0.60756
| [[Francium]] (2025)
|-
| [[3146/3145|Datonisma]]
| Thisusutholologu
| 37u17u3o1oog-2
| 3146/3145
| 2.5.11.13.17.37 {{monzo| 1 -1 2 1 -1 -1 }}
| 0.55038
| [[Francium]] (2025)
|-
| [[3220/3219|Murayamisma]]
| Thisutwenutwethozoyo
| 37u29u23ozy1
| 3220/3219
| 2.3.5.7.23.29.37 {{monzo| 2 -1 1 1 1 -1 -1 }}
| 0.53773
| [[Francium]] (2025)
|-
| [[3256/3255|Daguerrisma]]
| Thisothiwulorugu
| 37o31u1org2
| 3256/3255
| 2.3.5.7.11.31.37 {{monzo| 3 -1 -1 -1 1 -1 1 }}
| 0.53179
| [[Francium]] (2025)
|-
| [[3367/3366|Alexisma]]
| Thisosutholuzo
| 37o17u3o1uz2
| 3367/3366
| 2.3.7.11.13.17.37 {{monzo| -1 -2 1 -1 1 -1 1 }}
| 0.51425
| [[Francium]] (2025)
|-
| [[3553/3552|Meranisma]]
| Thisunosolo
| 37u19o17o1o1
| 3553/3552
| 2.3.11.17.19.37 {{monzo| -5 -1 1 1 1 -1 }}
| 0.48733
| [[Francium]] (2024)
|-
| [[3626/3625|Ohsakisma]]
| Thisotwenuzozotrigu
| 37o29uzz3g3
| 3626/3625
| 2.5.7.29.37 {{monzo| 1 -3 2 -1 1 }}
| 0.47752
| [[Francium]] (2025)
|-
| [[3627/3626|Sayersisma]]
| Thisuthiwothoruru
| 37u31o3orr-3
| 3627/3626
| 2.3.7.13.31.37 {{monzo| -1 2 -2 1 1 -1 }}
| 0.47738
| [[Francium]] (2025)
|-
| [[3774/3773|Megumisma]]
| Thisosolutriru
| 37o17o1u3r1
| 3774/3773
| 2.3.7.11.17.37 {{monzo| 1 1 -3 -1 1 1 }}
| 0.45879
| [[Francium]] (2025)
|-
| [[4625/4624|Shchedrinisma]]
| Thisosusutriyo
| 37o17uu3y-2
| 4625/4624
| 2.5.17.37 {{monzo| -4 3 -2 1 }}
| 0.37436
| [[Francium]] (2024)
|-
| [[5292/5291|Bullionisma]]
| Thisuthuluzozo
| 37u3u1uzz1
| 5292/5291
| 2.3.7.11.13.37 {{monzo| 2 3 2 -1 -1 -1 }}
| 0.32717
| [[Eliora]] (2023)
|-
| [[5440/5439|Teraosma]]
| Thisusoruruyo
| 37u17orry-2
| 5440/5439
| 2.3.5.7.17.37 {{monzo| 6 -1 1 -2 1 -1 }}
| 0.31827
| [[Francium]] (2025)
|-
| [[7105/7104|Yousyozanisma]]
| Thisutwenozozoyo
| 37u29ozzy1
| 7105/7104
| 2.3.5.7.29.37 {{monzo| -6 -1 1 2 1 -1 }}
| 0.24368
| [[Francium]] (2025)
|-
| [[7696/7695|Liebisma]]
| Sathisonuthogu
| S37o19u3og2
| 7696/7695
| 2.3.5.13.19.37 {{monzo| 4 -4 -1 1 -1 1 }}
| 0.22497
| [[Francium]] (2025)
|-
| [[8991/8990|Solidarity comma]]
| Thisothiwutwenugu
| 37o31u29ug2
| 8991/8990
| 2.3.5.29.31.37 {{monzo| -1 5 -1 -1 -1 1 }}
| 0.19256
| [[Francium]] (2025)
|-
| [[9177/9176|Donsaarisma]]
| Thisuthiwutwethonozo
| 37u31u23o19oz2
| 9177/9176
| 2.3.7.19.23.31.37 {{monzo| -3 1 1 1 1 -1 -1 }}
| 0.18866
| [[Francium]] (2025)
|-
| [[9251/9250|Harchisma]]
| Thisubitweno-alotrigu
| 37u29oo1o3g1
| 9251/9250
| 2.5.11.29.37 {{monzo| -1 -3 1 2 -1 }}
| 0.18715
| [[Francium]] (2025)
|-
| [[9361/9360|Friesachisma]]
| Thisotwethothulogu
| 37o23o3u1og2
| 9361/9360
| 2.3.5.11.13.23.37 {{monzo| -4 -2 -1 1 -1 1 1}}
| 0.18495
| [[Francium]] (2024)
|-
| [[Zangarisma]]
| Thisososolu
| 37o17oo1u3
| 10693/10692
| 2.3.11.17.37 {{monzo| -2 -5 -1 2 1 }}
| 0.16191
| [[Francium]] (2024)
|-
| [[23275/23273|Sunrise comma]]
| Bithisu-anosubizoyo
| 37uu19o17uzzyy-2
| 23275/23273
| 5.7.17.19.37 {{monzo| 2 2 -1 1 -2 }}
| 0.14877
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Sinachopoulosisma]]
| Thisotwethonunuluzo
| 37o23o19uu1uz2
| 11914/11913
| 2.3.7.11.19.23.37 {{monzo| 1 -1 1 -1 -2 1 1 }}
| 0.14532
| [[Francium]] (2024)
|-
| [[12321/12320|Zurakowskisma]]
| Bithiso-alurugu
| 37oo1urg2
| 12321/12320
| 2.3.5.7.11.37 {{monzo| -5 2 -1 -1 -1 2 }}
| 0.14052
| [[Francium]] (2025)
|-
| [[13690/13689|Lesleymartin comma]]
| Bithisothu-ayo
| 37oo3uuy2
| 13690/13689
| 2.3.5.13.37 {{monzo| 1 -4 1 -2 2 }}
| 0.12646
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Berylisma]]
| Quadthisolu
| 4(37o1u)4
| 1874161/1874048
| 2.11.37 {{monzo| -7 -4 4 }}
| 0.10439
| [[User:Jerdle|Jerdle]] (2023)
|-
| [[Genzelisma]]
| Thisotwenonusogu
| 37o29o19u17og2
| 18241/18240
| 2.3.5.17.19.29.37 {{monzo| -6 -1 -1 1 -1 1 1 }}
| 0.094912
| [[Francium]] (2024)
|-
| [[33264/33263|Maryrogersisma]]
| Thisuthiwutwenulozo
| 37u31u29u1oz1
| 33264/33263
| 2.3.7.11.29.31.37 {{monzo| 4 3 1 1 -1 -1 -1 }}
| 0.052046
| [[Francium]] (2025)
|-
| [[Jouvisma]]
| Thisotwethotholuluzogu
| 37o23o3o1uuzg3
| 77441/77440
| 2.5.7.11.13.23.37 {{monzo| -7 -1 1 -2 1 1 1 }}
| 0.022356
| [[Francium]] (2024)
|}

=== 41-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[575/574|Renatisma]]
| Fowutwethoruyoyo
| 41u23oryy1
| 575/574
| 2.5.7.23.41 {{monzo| -1 2 -1 1 -1 }}
| 3.0135
| [[Francium]] (2025)
|-
| [[616/615|Ellisma]]
| Fowulozogu
| 41u1ozg2
| 616/615
| 2.3.5.7.11.41 {{monzo| 3 -1 -1 1 1 -1 }}
| 2.8127
| [[Francium]] (2024)
|-
| [[780/779|Wiesentisma]]
| Fowunuthoyo
| 41u19u3oy1
| 780/779
| 2.3.5.13.19.41 {{monzo| 2 1 1 1 -1 -1 }}
| 2.2210
| [[Francium]] (2025)
|-
| [[1025/1024|Kilobytisma]]
| Fowoyoyo
| 41oyy-2
| 1025/1024
| 2.5.41 {{Monzo| -10 2 1 }}
| 1.6898
| [[CompactStar]] (2024)
|-
| [[1026/1025|Ingridisma]]
| Fowunogugu
| 41u19ogg2
| 1026/1025
| 2.3.5.19.41 {{monzo| 1 3 -2 1 -1 }}
| 1.6882
| [[Francium]] (2025)
|-
| [[1190/1189|Pelagisma]]
| Fowutwenusozoyo
| 41u29u17ozy2
| 1190/1189
| 2.5.7.17.29.41 {{monzo| 1 1 1 1 -1 -1 }}
| 1.4554
| [[Francium]] (2025)
|-
| [[1518/1517|Rovaniemisma]]
| Fowuthisutwetholo
| 41u37u23o1o1
| 1518/1517
| 2.3.11.23.37.41 {{monzo|1 1 1 1 -1 -1 }}
| 1.1408
| [[Francium]] (2025)
|-
| [[1682/1681|Shaftesburisma]]
| Bifowutweno
| 41uu29oo2
| 1682/1681
| 2.29.41 {{monzo| 1 2 -2 }}
| 1.0296
| [[Budjarn Lambeth]] (2023)
|-
| [[2255/2254|Qinghaisma]]
| Fowotwethuloruruyo
| 41o23u1orry-3
| 2255/2254
| 2.5.7.11.23.41 {{monzo| -1 1 -2 1 -1 1 }}
| 0.76790
| [[Francium]] (2025)
|-
| [[2871/2870|Schoberisma]]
| Fowutwenolorugu
| 41u29o1org1
| 2871/2870
| 2.3.5.7.11.29.41 {{monzo| -1 2 -1 -1 1 1 -1 }}
| 0.60311
| [[Francium]] (2025)
|-
| [[3773/3772|Smithsonianisma]]
| Fowutwethulotrizo
| 41u23u1o3z2
| 3773/3772
| 2.7.11.23.41 {{monzo| -2 3 1 -1 -1 }}
| 0.45891
| [[Francium]] (2025)
|-
| [[4060/4059|Deipylosisma]]
| Fowutwenoluzoyo
| 41u29o1uzy2
| 4060/4059
| 2.3.5.7.11.29.41 {{monzo| 2 -2 1 1 -1 1 -1 }}
| 0.42646
| [[Francium]] (2025)
|-
| [[4675/4674|Ohbokisma]]
| Fowunusoloyoyo
| 41u19u17o1oyy1
| 4675/4674
| 2.3.5.11.17.19.41 {{monzo| -1 -1 2 1 1 -1 -1 }}
| 0.37036
| [[Francium]] (2025)
|-
| [[4921/4920|Volontisma]]
| Fowuthisonozogu
| 41u37o19ozg3
| 4921/4920
| 2.3.5.7.19.37.41 {{monzo| -3 -1 -1 1 1 1 -1 }}
| 0.35184
| [[Francium]] (2025)
|-
| [[5577/5576|Priestlisma]]
| Fowusuthotholo
| 41u17u3oo1o1
| 5577/5576
| 2.3.11.13.17.41 {{monzo| -3 1 1 2 -1 -1 }}
| 0.31045
| [[Francium]] (2025)
|-
| [[6069/6068|Cevolanisma]]
| Fowuthisusosozo
| 41u37u17ooz2
| 6069/6068
| 2.3.7.17.37.41 {{monzo| -2 1 1 2 -1 -1 }}
| 0.28528
| [[Francium]] (2025)
|-
| [[6930/6929|Bedanisma]]
| Fowuthuthulozoyo
| 41u3uu1ozy1
| 6930/6929
| 2.3.5.7.11.13.41 {{monzo| 1 2 1 1 1 -2 -1 }}
| 0.24984
| [[Francium]] (2025)
|-
| [[7176/7175|Kunijisma]]
| Fowutwethothorugugu
| 41u23o3orgg2
| 7176/7175
| 2.3.5.7.13.23.41 {{monzo| 3 1 -2 -1 1 1 -1 }}
| 0.24127
| [[Francium]] (2025)
|-
| [[8569/8568|Mamelisma]]
| Fowonosuloru
| 41o19o17u1or-2
| 8569/8568
| 2.3.7.11.17.19.41 {{monzo| -3 -2 -1 1 -1 1 1 }}
| 0.20205
| [[Francium]] (2025)
|-
| [[9472/9471|Brugesisma]]
| Fowuthisoluru
| 41u37o1ur2
| 9472/9471
| 2.3.7.11.37.41 {{monzo| 8 -1 -1 -1 1 -1 }}
| 0.18278
| [[Francium]] (2025)
|-
| [[Etampesisma]]
| Fowutwethunotholuzo
| 41u23u19o3o1uz2
| 10374/10373
| 2.3.7.11.13.19.23.41 {{monzo| 1 1 1 -1 1 1 -1 -1 }}
| 0.16689
| [[Francium]] (2024)
|-
| [[11440/11439|Massironisma]]
| Fowuthiwutholoyo
| 41u31u3o1oy2
| 11440/11439
| 2.3.5.11.13.31.41 {{monzo| 4 -2 1 1 1 -1 -1 }}
| 0.15134
| [[Francium]] (2025)
|-
| [[15376/15375|Martakisma]]
| Fowubithiwo-atrigu
| 41u31oo3g1
| 15376/15375
| 2.3.5.31.41 {{monzo| 4 -1 -3 2 -1 }}
| 0.11260
| [[Francium]] (2025)
|-
| [[Canupisma]]
| Fowutwenuthotrizo-agu
| 41u29u3o3zag3
| 17836/17835
| 2.3.5.7.13.29.41 {{monzo| 2 -1 -1 3 1 -1 -1 }}
| 0.097067
| [[Francium]] (2024)
|-
| [[76384/76383|Vernonisma]]
| Fowuthiwotwethulozo
| 41u31o23u1oz1
| 76384/76383
| 2.3.7.11.23.31.41 {{monzo| 5 -4 1 1 -1 1 -1 }}
| 0.022665
| [[Francium]] (2025)
|-
| [[Mebisma]]
| Safowuthiwulugugu
| s41u31u1ugg3
| 1048576/1048575
| 2.3.5.11.31.41 {{Monzo| 20 -1 -2 -1 -1 -1 }}
| 0.0016510
| See the page.
|}

=== 43-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[646/645|Kastalisma]]
| Fothunosogu
| 43u19o17og2
| 646/645
| 2.3.5.17.19.43 {{monzo| 1 -1 -1 1 1 -1 }}
| 2.6820
| [[Francium]] (2025)
|-
| [[990/989|Yerkesisma]]
| Fothutwethuloyo
| 43u23u1oy-2
| 990/989
| 2.3.5.11.23.43 {{monzo| 1 2 1 1 -1 -1 }}
| 1.7496
| [[Francium]] (2025)
|-
| [[1333/1332|Cevelonisma]]
| Fothothisuthiwo
| 43o37u31o-2
| 1333/1332
| 2.3.31.37.43 {{monzo| -2 -2 1 -1 1 }}
| 1.2992
| [[Francium]] (2025)
|-
| [[1377/1376|Roberbauxisma]]
| Lafothuso
| L43u17o1
| 1377/1376
| 2.3.17.43 {{monzo| -5 4 1 -1}}
| 1.2577
| [[Francium]] (2025)
|-
| [[1463/1462|Nordenmarkisma]]
| Fothunosulozo
| 43u19o17uoz1
| 1463/1462
| 2.7.11.17.19.43 {{monzo| -1 1 1 -1 1 -1 }}
| 1.1838
| [[Francium]] (2025)
|-
| [[Magnetisma]]
| Tritrila-quinquadtrifo thutweno
| 9L60(43u29o)-8
|
| 2.3.29.43 {{monzo| -61 60 60 -60 }}
| 0.86936
| [[User:Eliora|Eliora]] (2022)
|-
| [[2925/2924|Beattisma]]
| Fothusuthoyoyo
| 43u17u3oyy-2
| 2925/2924
| 2.3.5.13.17.43 {{monzo| -2 2 2 1 -1 -1 }}
| 0.59198
| [[Francium]] (2025)
|-
| [[3312/3311|Pedersenisma]]
| Fothutwetholuru
| 43u23o1ur1
| 3312/3311
| 2.3.7.11.23.43 {{monzo| 4 2 -1 -1 1 -1 }}
| 0.52279
| [[Francium]] (2025)
|-
| [[4000/3999|Hipparchusisma]]
| Fothuthiwutriyo
| 43u31u3y1
| 4000/3999
| 2.3.5.31.43 {{monzo| 5 -1 3 -1 -1 }}
| 0.43286
| [[Francium]] (2025)
|-
| [[4301/4300|Boydenisma]]
| Fothutwethosologugu
| 43u23o17o1ogg2
| 4301/4300
| 2.5.11.17.23.43 {{monzo| -2 -2 1 1 1 -1 }}
| 0.40257
| [[Francium]] (2025)
|-
| [[4774/4773|Hobetsisma]]
| Fothuthisuthiwolozo
| 43u37u31o1oz-2
| 4774/4773
| 2.3.7.11.31.37.43 {{monzo| 1 -1 1 1 1 -1 -1 }}
| 0.36268
| [[Francium]] (2025)
|-
| [[5720/5719|Halweaverisma]]
| Fothunutholoruyo
| 43u19u3o1ory-2
| 5720/5719
| 2.5.7.11.13.19.43 {{monzo| 3 1 -1 1 1 -1 -1 }}
| 0.30269
| [[Francium]] (2025)
|-
| [[7225/7224|Huntressisma]]
| Fothusosoruyoyo
| 43u17ooryy1
| 7225/7224
| 2.3.5.7.17.43 {{monzo| -3 -1 2 -1 2 -1 }}
| 0.23963
| [[Francium]] (2025)
|-
| [[7956/7955|Yajinisma]]
| Fothuthisusothogu
| 43u37u17o3og1
| 7956/7955
| 2.3.5.13.17.37.43 {{monzo| 2 2 -1 1 1 -1 -1 }}
| 0.21761
| [[Francium]] (2025)
|-
| [[9504/9503|Lionelisma]]
| Fothusuthulo
| 43u17u3u1o-2
| 9504/9503
| 2.3.11.13.17.43 {{monzo| 5 3 1 -1 -1 -1 }}
| 0.18217
| [[Francium]] (2025)
|-
| [[9633/9632|Coturisma]]
| Fothunothothoru
| 43u19o3oor1
| 9633/9632
| 2.3.7.13.19.43 {{monzo| -5 1 -1 2 1 -1 }}
| 0.17973
| [[Francium]] (2025)
|-
| [[Girardisma]]
| Fothunoloyoyo
| 43u19o1oyy1
| 10450/10449
| 2.3.5.11.19.43 {{monzo| 1 -5 2 1 1 -1 }}
| 0.16567
| [[Francium]] (2024)
|-
| [[Kaguyisma]]
| Fothutwethusoluyo
| 43u23u17o1uy1
| 10880/10879
| 2.5.11.17.23.43 {{monzo| 7 1 -1 1 -1 -1 }}
| 0.15913
| [[Francium]] (2024)
|-
| [[Manheimisma]]
| Fothutwenosuloloyo
| 43u29o17u1ooy-2
| 17545/17544
| 2.3.5.11.17.29.43 {{monzo| -3 -1 1 2 -1 1 -1 }}
| 0.098677
| [[Francium]] (2024)
|-
| [[Dimigenes comma]]
| Fothosepyo
| 43o7y-2
| 3359375/3359232
| 2.3.5.43 {{monzo| -9 -8 7 1 }}
| 0.073696
| [[User:Akselai|Akselai]] (2024)
|-
| [[27048/27047|Jangongisma]]
| Fothuthisutwethosuzozo
| 43u37u23o17uzz1
| 27048/27047
| 2.3.7.17.23.37.43 {{monzo| 3 1 2 -1 1 -1 -1 }}
| 0.064007
| [[Francium]] (2025)
|-
| [[29241/29240|Locquirecisma]]
| Fothunonosugu
| 43u19oo17ug1
| 29241/29240
| 2.3.5.17.19.43 {{monzo| -3 4 -1 -1 2 -1 }}
| 0.059207
| [[Francium]] (2025)
|}

=== Higher-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[7936/7921|Lily comma]]
|
| 89uu31o1
| 7936/7921
| 2.31.89 {{monzo| 8 1 -2 }}
| 3.2753
| [[Budjarn Lambeth]] (2023)
|-
| [[Molar comma]]
|
| 3(89o)1u3g1
| 704969/704000
| 2.5.11.89 {{monzo| -9 -3 -1 3 }}
| 2.3813
| [[Budjarn Lambeth]] (2023)
|-
| [[750/749|Ancient Chinese tempering comma]]{{Clarify}}
|
| 107ur3y-2
| 750/749
| 2.3.5.7.107 {{monzo| 1 1 3 -1 -1 }}
| 2.3099
|
|-
| [[1176/1175|Lucidorisma]]
| Fosubizoguma
| 47uzzgg2
| 1176/1175
| 2.3.5.7.47 {{monzo| 3 1 -2 2 -1 }}
| 1.4728
| [[Francium]] (2025)
|-
| [[2520/2519|Platonisma]]
|
| 229u1uzy1
| 2520/2519
| 2.3.5.7.11.229 {{monzo| 3 2 1 1 -1 -1 }}
| 0.68713
| [[User:Eliora|Eliora]] (2022)
|-
| [[5041/5040|Third brown pair comma]], 19th highly compositema
|
| 71oorg1
| 5041/5040
| 2.3.5.7.71 {{monzo| -4 -2 -1 -1 2 }}
| 0.34347
| [[Eliora]] (2022)
|-
| [[7777/7776|Pulsar comma]]
|
| 101o1oz2
| 7777/7776
| 2.3.7.11.101 {{monzo| -5 -5 1 1 1 }}
| 0.22262
| [[Eliora]] (2022)
|-
| Palimilli
|
| 1003001o23o1u1
| 23069023 / 23068672
| 2.11.23.1003001 {{monzo| -21 -1 1 1 }}
| 0.026341
| [[User:Kaiveran|Kaiveran]] (2021)
|-
| [[Sulbasutrisma]]
|
| 577oo17uu-2
| 332929/332928
| 2.3.17.577 {{monzo| -7 -2 -2 2 }}
| 0.0052000
| [[User:2^67-1|Cole]] (2024)
|-
| [[Zudilisma]]
|
| 4L397u23ur-5
| 68630377364883 / 68630356164608
| 2.3.7.23.397 {{monzo| -30 29 -1 -1 -1 }}
| 0.00053479
| [[Budjarn Lambeth]] (2023)
|-
| [[Borcherdsma]]
|
| 71u3(59u)47o31o 29o19o3u1uur5y-2
| 160561400000 / 160561399999
| 2.5.7.11.13.19.29.31.47.59.71 {{monzo| 6 5 -1 -2 -1 1 1 1 1 -3 -1 }}
| 1.0783 × 10-8
| [[User:Akselai|Akselai]] (2024)
|}

== Notes ==
<references />

== See also ==
* [[Comma and diesis]]

[[Category:Unnoticeable commas| ]] 
[[Category:Lists of commas]]
{{Todo|complete table|add color name}}

Small comma

2026-04-28T09:10:59Z

TallKite: typos

A '''small comma''' is a [[comma]] whose size is approximately between 3.5 and 30 cents. These intervals are in the range from just noticeable up to usable as melodic steps. The actual perception of course varies. In [[Sagittal notation]], intervals in the smaller part of this category are [[kleisma (interval region)|kleismas]], and intervals in the larger part of this category are [[comma (interval region)|commas]]<ref>[https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>.

For commas over 100 cents in size, see [[Large comma]]; for commas in between 30 and 100 cents in size, see [[Medium comma]]; and for commas under 3.5 cents in size, see [[Unnoticeable comma]].

Due to the wide range of sizes, cents values here and on the other commas pages are listed to 5 significant digits, instead of to a fixed number of decimal places as is the [[Xenharmonic Wiki: Conventions|convention]] elsewhere on the wiki.

The commas might be numerous, but there is no need to memorise all the names. For pretty much all use cases, it is perfectly acceptable to just refer to a comma by the monzo or ratio, whichever is simpler.

== List of commas, by prime limit ==
=== 3-limit commas ===
{{See also| Table of 3-limit commas }}

{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cents]]
! Named by
|-
| 241-comma
| 241wama
| 241wM
|
| {{Monzo| 382 -241 }}
| 28.845
|
|-
| 65-comma, Pythagorean septimal comma
| 65wama
| 65wM
|
| {{Monzo| -103 65 }}
| 27.075
|
|-
| [[Pythagorean comma]]
| Lalawama
| LLwM
| 531441 / 524288
| {{Monzo| -19 12 }}
| 23.460
|
|-
| [[41-comma]], Pythagorean countercomma, countercomp comma
| 41wama
| 41wM
| <abbr title="36893488147419103232 / 36472996377170786403">(40 digits)</abbr>
| {{Monzo| 65 -41 }}
| 19.845
| [[Flora Canou]] (2022)
|-
| [[94-comma]], garistearn comma
| 94wama
| 94wM
|
| {{Monzo| 149 -94 }}
| 16.230
| [[User:Xenllium|Xenllium]] (2021)
|-
| 147-comma
| 147wama
| 147wM
|
| {{Monzo| 233 -147 }}
| 12.615
|
|-
| 200-comma, Pythagorean integer-cent ET comma
| 200wama
| 200wM
|
| {{Monzo| 317 -200 }}
| 8.9998
|
|-
| 253-comma
| 253wama
| 253wM
|
| {{Monzo| 401 -253 }}
| 5.3848
|
|-
| [[Mercator's comma]], 53-comma
| 53wama
| 53wM
| <abbr title="19383245667680019896796723 / 19342813113834066795298816">(52 digits)</abbr>
| {{Monzo| -84 53 }}
| 3.6150
|
|}

=== 5-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cents]]
! Named by
|-
| [[Magic comma]], small diesis
| Laquinyoma
| L5yM
| 3125 / 3072
| {{Monzo| -10 -1 5 }}
| 29.614
|
|-
| [[Triscordial comma]]
| Tribila-triyoma
| 6L3yM
| <abbr title="18761829412124890125 / 18446744073709551616">(40 digits)</abbr>
| {{Monzo| -64 36 3 }}
| 29.321
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Hendecatonic comma]]
| Trisa-leguma
| 3s11gM
| 8796093022208 / 8649755859375
| {{Monzo| 43 -11 -11 }}
| 29.044
|
|-
| [[Devil's tridecalimma]]
| Lala-theguma
| LL13gM
| 2541865828329 / 2500000000000
| {{Monzo| -11 26 -13 }}
| 28.752
|
|-
| [[Anthoine comma]]
| Trila-quinquadyoma
| 3L20yM
| 286102294921875 / 281474976710656
| {{Monzo| -48 1 20 }}
| 28.229
|
|-
| [[Tetracot comma]], minimal diesis
| Saquadyoma
| s4yM
| 20000 / 19683
| {{Monzo| 5 -9 4 }}
| 27.660
|
|-
| [[Biscordial comma]]
| Quadla-yoyoma
| 4LyyM
| 571919811374025 / 562949953421312
| {{Monzo| -49 28 2 }}
| 27.367
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Semaja comma]]
| Lala-neyoma
| LL19yM
| 19073486328125 / 18786186952704
| {{Monzo| -33 -7 19 }}
| 26.276
| [[Petr Pařízek]] (2011)
|-
| [[Quanic comma]]
| Sepsa-quinyoma
| 7s5yM
|
| {{Monzo| 74 -54 5 }}
| 25.999
|
|-
| [[Roda]], rodan comma
| Sasa-triyoma
| ss3yM
| 131072000 / 129140163
| {{Monzo| 20 -17 3 }}
| 25.706
|
|-
| [[Gracecordial comma]]
| Trilayoma
| 3LyM
| 17433922005 / 17179869184
| {{Monzo| -34 20 1 }}
| 25.414
|
|-
| [[Trisedodge comma]]
| Saquintriguma
| s15gM
| 30958682112 / 30517578125
| {{Monzo| 19 10 -15 }}
| 24.844
| [[Petr Pařízek]] (2011)
|-
| [[Birds comma]]
| Quadsa-thiweguma
| 4s31gM
|
| {{Monzo| 72 0 -31 }}
| 24.275
|
|-
| [[Neuk comma]]
| Trisa-yoyoma
| 3syyM
| 858993459200 / 847288609443
| {{Monzo| 35 -25 2 }}
| 23.752
| [[User:CompactStar|CompactStar]] (2023)
|-
| [[Maja comma]]
| Saseyoma
| s17yM
| 762939453125 / 753145430616
| {{Monzo| -3 -23 17 }}
| 22.368
| [[Petr Pařízek]] (2011)
|-
| [[Satin comma]]
| Quinbisa-triyoma
| 10s3yM
|
| {{Monzo| 104 -70 3 }}
| 22.091
|
|-
| [[Misneb comma]]
| Quadla-quintriyoma
| 4L15yM
| <abbr title="145964630126953125 / 144115188075855872">(36 digits)</abbr>
| {{Monzo| -57 14 15 }}
| 22.076
| [[Petr Pařízek]] (2011)
|-
| [[Syntonic comma]], Didymus comma, meantone comma
| Guma
| gM
| 81 / 80
| {{Monzo| -4 4 -1 }}
| 21.506
|
|-
| [[Maquila comma]]
| Trisa-seguma
| 3s17gM
| 562949953421312 / 556182861328125
| {{Monzo| 49 -6 -17 }}
| 20.937
| [[Petr Pařízek]] (2011)
|-
| [[Sfourth comma]]
| Lala-neguma
| LL19gM
| 617673396283947 / 610351562500000
| {{Monzo| -5 31 -19 }}
| 20.644
| [[Petr Pařízek]] (2011)
|-
| [[Diaschisma]]
| Saguguma
| sggM
| 2048 / 2025
| {{Monzo| 11 -4 -2 }}
| 19.553
| Hermann von Helmholtz, Alexander Ellis (1875)
|-
| [[Countermeantone comma]]
| Quinquadguma
| 20gM
| 96402615118848 / 95367431640625
| {{Monzo| 10 23 -20 }}
| 18.691
| [[Petr Pařízek]] (2012)
|-
| [[Ditonma]]
| Lala-theyoma
| LL13yM
| 1220703125 / 1207959552
| {{Monzo| -27 -2 13 }}
| 18.168
| [[Petr Pařízek]] (2011)
|-
| [[Misty comma]]
| Sasa-triguma
| ss3gM
| 67108864 / 66430125
| {{Monzo| 26 -12 -3 }}
| 17.599
| [[Paul Erlich]] (2002)
|-
| [[Quintile comma]]
| Trila-quinguma
| 3L5gM
| 847288609443 / 838860800000
| {{Monzo| -28 25 -5 }}
| 17.306
| See the dedicated page.
|-
| [[Enneadecacot comma]]
| Tribisa-neguma
| 6s19gM
| <abbr title="604462909807314587353088 / 598546211414337158203125">(48 digits)</abbr>
| {{monzo| 79 -22 -19 }}
| 17.029
| [[User:CompactStar|CompactStar]] (2024)
|-
| [[Oquatonic comma]]
| Quadla-sepquadyoma
| 4L28yM
| <abbr title="37252902984619140625 / 36893488147419103232">(40 digits)</abbr>
| {{Monzo| -65 0 28 }}
| 16.784
| [[Petr Pařízek]] (2011)
|-
| [[Undim comma]]
| Trisa-quadguma
| 3s4gM
| 2199023255552 / 2179240250625
| {{Monzo| 41 -20 -4 }}
| 15.645
| [[Petr Pařízek]] (2011)
|-
| [[Graviton]], gravity comma
| Lala-tribiguma
| LL6gM
| 129140163 / 128000000
| {{Monzo| -13 17 -6 }}
| 15.353
| [[Mike Battaglia]] (2011)
|-
| [[Majvam comma]]
| Sasa-lebiguma
| ss22gM
| <abbr title="2404631929946112 / 2384185791015625">(32 digits)</abbr>
| {{Monzo| 40 7 -22 }}
| 14.783
| [[Petr Pařízek]] (2011)
|-
| [[Quartonic comma]]
| Saleyoma
| s11yM
| 390625000 / 387420489
| {{Monzo| 3 -18 11 }}
| 14.261
|
|-
| [[Untritonic comma]]
| Quadla-tritriyoma
| 4L9yM
| <abbr title="2270041927734375 / 2251799813685248">(32 digits)</abbr>
| {{Monzo| -51 19 9 }}
| 13.968
| [[Petr Pařízek]] (2011)
|-
| [[Quindromeda comma]]
| Quinsa-quinguma
| 5s5gM
| <abbr title="72057594037927936 / 71489976421753125">(34 digits)</abbr>
| {{Monzo| 56 -28 -5 }}
| 13.691
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Sensipent comma]], medium semicomma
| Sepguma
| 7gM
| 78732 / 78125
| {{Monzo| 2 9 -7 }}
| 13.399
|
|-
| [[Copper comma]]
| Theneyoma
| 41L29yM
|
| {{Monzo| -481 261 29 }}
| 13.353
| [[Eliora]] (2023)
|-
| [[Counterwürschmidt comma]]
| Trisa-twetheguma
| 3s23gM
| <abbr title="36028797018963968 / 35762786865234375">(34 digits)</abbr>
| {{Monzo| 55 -1 -23 }}
| 12.830
| [[Flora Canou]] (2023)
|-
| [[Tertiosec comma]]
| Laquadtribiyoma
| 6L24yM
|
| {{Monzo| -89 21 24 }}
| 12.584
| [[Petr Pařízek]] (2012)
|-
| [[Submajor comma]]
| Trila-quadbiyoma
| 3L8yM
| 69198046875 / 68719476736
| {{Monzo| -36 11 8 }}
| 12.015
|
|-
| [[Würschmidt comma]]
| Saquadbiguma
| s8gM
| 393216 / 390625
| {{Monzo| 17 1 -8 }}
| 11.445
|
|-
| [[Bicommatic comma]]
| Quadla-quinbiguma
| 4L10gM
| <abbr title="1350851717672992089 / 1342177280000000000">(38 digits)</abbr>
| {{Monzo| -37 38 -10 }}
| 11.153
|
|-
| [[Counterhanson comma]]
| Quinquinyoma
| 25yM
| <abbr title="298023223876953125 / 296148833645101056">(36 digits)</abbr>
| {{Monzo| -20 -24 25 }}
| 10.923
| [[Petr Pařízek]] (2012)
|-
| [[Countritonic comma]]
| Quadsa-tritriyoma
| 4s9yM
| <abbr title="16777216000000000 / 16677181699666569">(34 digits)</abbr>
| {{Monzo| 33 -34 9 }}
| 10.353
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Semicomma]], Fokker's comma
| Lasepyoma
| L7yM
| 2109375 / 2097152
| {{Monzo| -21 3 7 }}
| 10.061
|
|-
| [[Heptacot comma]]
| Sepsa-sepguma
| 7s7gM
|
| {{Monzo| 86 -44 -7 }}
| 9.7840
| [[Tristan Bay]] (2024)
|-
| [[Escapade comma]]
| Sasa-tritriguma
| ss9gM
| 4294967296 / 4271484375
| {{Monzo| 32 -7 -9 }}
| 9.4916
| [[Paul Erlich]] (2002)
|-
| [[Undetritisma]], twentcufo comma
| Trila-leguma
| 3L11gM
| 205891132094649 / 204800000000000
| {{Monzo| -22 30 -11 }}
| 9.1992
|
|-
| [[15625/15552|Kleisma]], semicomma majeur
| Tribiyoma
| 6yM
| 15625 / 15552
| {{Monzo| -6 -5 6 }}
| 8.1073
| {{W|Shohé Tanaka}} (1890)
|-
| [[Quintosec comma]]
| Quadsa-quinbiguma
| 4s10gM
| 140737488355328 / 140126044921875
| {{Monzo| 47 -15 -10 }}
| 7.5378
| [[Petr Pařízek]] (2012)
|-
| 59-5-comma
| Quadbisa-fineguma
| 8s59gM
|
| {{Monzo| 137 0 -59 }}
| 7.4909
|
|-
| [[Unidecma]]
| Laquadtriguma
| L12gM
| 31381059609 / 31250000000
| {{Monzo| -7 22 -12 }}
| 7.2455
|
|-
| [[Mutt comma]]
| Trila-septriyoma
| 3L21yM
| 476837158203125 / 474989023199232
| {{Monzo| -44 -3 21 }}
| 6.7230
| [[Gene Ward Smith]] (2004)
|-
| [[Sulfur comma]]
| Lela-quadquadguma
| 11L16gM
|
| {{Monzo| -115 96 -16 }}
| 6.6607
|
|-
| [[Amity comma]]
| Saquinyoma
| s5yM
| 1600000 / 1594323
| {{Monzo| 9 -13 5 }}
| 6.1536
| [[Gene Ward Smith]] (2001)
|-
| [[Parakleisma]]
| Theguma
| 13gM
| 1224440064 / 1220703125
| {{Monzo| 8 14 -13 }}
| 5.2917
| [[Gene Ward Smith]] (2001)
|-
| [[Gammic comma]]
| Laquinquadyoma
| L20yM
| 95367431640625 / 95105071448064
| {{Monzo| -29 -11 20 }}
| 4.7693
|
|-
| [[Squarschmidt comma]]
| Quadsa-tweneguma
| 4s29gM
| <abbr title="186773283746309210112 / 186264514923095703125">(42 digits)</abbr>
| {{Monzo| 61 4 -29 }}
| 4.7223
| [[Petr Pařízek]] (2012)
|-
| Huntian 15-cycle comma
| Quadtrisa-fotheguma
| 12s43gM
|
| {{Monzo| 168 -43 -43 }}
| 4.4453
|
|-
| [[Barium comma]]
| Quadtribila-sepquadbiguma
| 24L56gM
|
| {{Monzo| -225 224 -56 }}
| 4.3522
| [[Eliora]] (2023)
|-
| [[Vulture comma]]
| Sasa-quadyoma
| ss4yM
| 10485760000 / 10460353203
| {{Monzo| 24 -21 4 }}
| 4.1998
| [[Paul Erlich]] (2002)
|-
| [[Dipromethia]]
| Thebila-siweyoma
| 26L61yM
|
| {{Monzo| -335 122 61 }}
| 3.6467
| [[Eliora]] (2023)
|-
| [[Lafa comma]]
| Tribisa-quadtriguma
| 6s12gM
|
| {{Monzo| 77 -31 -12 }}
| 3.6304
| [[Petr Pařízek]] (2011)
|}

=== 7-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cents]]
! Named by
|-
| [[3645/3584|Schismean comma]]
| Laruyoma
| LryM
| 3645 / 3584
| {{Monzo| -9 6 1 -1 }}
| 29.218
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Doublehearted comma]]
| Quadbizoma
| 8zM
| 5764801 / 5668704
| {{Monzo| -5 -11 0 8 }}
| 29.102
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Frostburn comma]]
| Quadru-asepyoma
| 4ra7yM
| 78125 / 76832
| {{Monzo| -5 0 7 -4 }}
| 28.892
|
|-
| [[686/675|Senga]]
| Trizo-aguguma
| 3zaggM
| 686 / 675
| {{Monzo| 1 -3 -2 3 }}
| 27.985
| [[Gene Ward Smith]] (2005)
|-
| 23-21-comma
| Sepla-twethezoma
| 7L23zM
|
| {{Monzo| -101 23 0 23 }}
| 27.961
|
|-
| [[64/63|Septimal comma]], Archytas' comma, Leipziger Komma
| Ruma
| rM
| 64 / 63
| {{Monzo| 6 -2 0 -1 }}
| 27.264
|
|-
| [[Mandos comma]]
| Biruguguma
| 2rggM
| 31104 / 30625
| {{Monzo| 7 5 -4 -2 }}
| 26.868
|
|-
| [[Slither comma]]
| Satritriru-aquadyoma
| s9ra4yM
| 40960000 / 40353607
| {{Monzo| 16 0 4 -9 }}
| 25.822
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Bastille comma]]
|
|
|
| {{Monzo| 1426 0 -596 -15 }}
| 24.638
| [[Eliora]] (2023)
|-
| 33-7/5-comma
| Letrizoguma
| 33zgM
|
| {{Monzo| -16 0 -33 33 }}
| 22.902
|
|-
| Huntian 35-cycle comma
| Quintrisa-tritritribiruguma
| 15s54rgM
|
| {{Monzo| 277 0 -54 -54 }}
| 22.461
|
|-
| [[Blackjackisma]]
| Lasepru-aquadbiyoma
| L7ra8yM
| 854296875 / 843308032
| {{Monzo| -10 7 8 -7 }}
| 22.413
| [[Gene Ward Smith]] (2011)
|-
| [[Squalentine comma]]
| Laquadzo-atriguma
| L4za3gM
| 64827 / 64000
| {{Monzo| -9 3 -3 4 }}
| 22.227
|
|-
| [[875/864|Keema]]
| Zotriyoma
| z3yM
| 875 / 864
| {{Monzo| -5 -3 3 1 }}
| 21.902
| [[Gene Ward Smith]] (2005)
|-
| [[Betelgeuse comma]]
| Satritrizo-aguguma
| s9zaggM
| 40353607 / 39858075
| {{Monzo| 0 -13 -2 9 }}
| 21.391
|
|-
| [[3125/3087|Gariboh comma]]
| Triru-aquinyoma
| 3ra5yM
| 3125 / 3087
| {{Monzo| 0 -2 5 -3 }}
| 21.181
| [[Gene Ward Smith]] (2005)
|-
| [[Secanticornisma]]
| Laruquinguma
| Lr5gM
| 177147 / 175000
| {{Monzo| -3 11 -5 -1 }}
| 21.111
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[2430/2401|Nuwell comma]]
| Quadru-ayoma
| 4rayM
| 2430 / 2401
| {{Monzo| 1 5 1 -4 }}
| 20.785
| [[Gene Ward Smith]] (2005)
|-
| [[Septimagic comma]]
| Saquinzoma
| s5zM
| 537824 / 531441
| {{Monzo| 5 -12 0 5 }}
| 20.670
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Mermisma]]
| Sepruyoma
| 7ryM
| 2500000 / 2470629
| {{Monzo| 5 -1 7 -7 }}
| 20.460
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Negricorn comma]], small quadruple bluish
| Saquadzoguma
| s4zgM
| 153664 / 151875
| {{monzo| 6 -5 -4 4 }}
| 20.274
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Tolerant comma]]
| Sazoyoyoma
| szyyM
| 179200 / 177147
| {{Monzo| 10 -11 2 1 }}
| 19.948
|
|-
| [[Icosipentatonic comma]], 25-36/35-comma
| Quinquinruguma
| 25rgM
|
| {{Monzo| 49 50 -25 -25 }}
| 19.260
|
|-
| [[Valenwuer comma]]
| Sarutribiguma
| sr6gM
| 110592 / 109375
| {{Monzo| 12 3 -6 -1 }}
| 19.157
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Buzzardsma]], buzzard comma
| Saquadruma
| s4rM
| 65536 / 64827
| {{Monzo| 16 -3 0 -4 }}
| 18.831
| See the page.
|-
| Huntian 21-cycle comma
| Quadbisa-sepquadruma
| 8s28rM
|
| {{Monzo| 123 -28 0 -28 }}
| 18.135
|
|-
| [[Mirwomo comma]]
| Labizoyoma
| L2zyM
| 33075 / 32768
| {{Monzo| -15 3 2 2 }}
| 16.144
|
|-
| [[Catasyc comma]]
| Laruquadbiyoma
| Lr8yM
| 390625 / 387072
| {{Monzo| -11 -3 8 -1 }}
| 15.819
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Compass comma]]
| Quinruyoyoma
| 5ryyM
| 9765625 / 9680832
| {{monzo| -6 -2 10 -5 }}
| 15.098
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Sensibeta comma]]
| Satrizo-aquinyoma
| s3za5yM
| 1071875 / 1062882
| {{monzo| -1 -12 5 3 }}
| 14.586
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Trimyna comma]]
| Quinzoguma
| 5zgM
| 50421 / 50000
| {{monzo| -4 1 -5 5 }}
| 14.516
|
|-
| [[245/243|Sensamagic comma]]
| Zozoyoma
| zzyM
| 245 / 243
| {{monzo| 0 -5 1 2 }}
| 14.191
| [[Gene Ward Smith]] (2010)
|-
| [[126/125|Starling comma]], septimal semicomma
| Zotriguma
| z3gM
| 126 / 125
| {{Monzo| 1 2 -3 1 }}
| 13.795
|
|-
| [[Vermeil comma]], 34-49/48-comma
| Quinla-sequadzoma
| 5L68zM
|
| {{Monzo| -137 -34 0 68 }}
| 13.692
| [[User:Perry.k|Perry.k]] (2026)
|-
| [[4000/3969|Octagar comma]]
| Rurutriyoma
| rr3yM
| 4000 / 3969
| {{Monzo| 5 -4 3 -2 }}
| 13.469
| [[Gene Ward Smith]] (2005)
|-
| [[1728/1715|Orwellisma]]
| Triru-aguma
| 3ragM
| 1728 / 1715
| {{Monzo| 6 3 -1 -3 }}
| 13.074
|
|-
| [[Mynaslender comma]]
| Sepru-ayoma
| 7rayM
| 829440 / 823543
| {{Monzo| 11 4 1 -7 }}
| 12.352
| [[User:Xenllium|Xenllium]] (2021)
|-
| 35-7/5-comma
| Sepquinruyoma
| 35ryM
|
| {{monzo| 17 0 35 -35 }}
| 12.073
|
|-
| [[Chromatisma]]
| Trisa-triru-aquadquadyoma
| 3s3ra16yM
| <abbr title="640000000000000000 / 635585924776181463">(36 digits)</abbr>
| {{Monzo| 22 -32 16 -3 }}
| 11.982
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Mistisma]]
| Sazoquadguma
| sz4gM
| 458752 / 455625
| {{Monzo| 16 -6 -4 1 }}
| 11.841
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Bronzisma]]
| Satriru-aguguma
| s3raggM
| 2097152 / 2083725
| {{Monzo| 21 -5 -2 -3 }}
| 11.120
| [[User:Xenllium|Xenllium]] (2021)
|-
| 34-jubilismic comma
| Sequadzoguma
| 68zgM
|
| {{Monzo| -33 0 -68 68 }}
| 10.829
|
|-
| [[Fynn's comma]], Hunt 7-cycle comma
| Quadsa-thebiruma
| 4s26rM
| <abbr title="9444732965739290427392 / 9387480337647754305649">(44 digits)</abbr>
| {{Monzo| 73 0 0 -26 }}
| 10.526
| [[Fynn Cerulean]] (2026) for ''Fynn's comma''
|-
| [[Septiness comma]]
| Sasasepruma
| ss7rM
| 67108864 / 66706983
| {{Monzo| 26 -4 0 -7 }}
| 10.399
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[31-comma temperaments|31-35-comma]]
| Tritrila-thiwezoyoma
| 9L31zyM
|
| {{Monzo| -159 0 31 31 }}
| 9.3282
|
|-
| [[Quince comma]]
| Lasepzo-aguguma
| L7zaggM
| 823543 / 819200
| {{Monzo| -15 0 -2 7 }}
| 9.1539
|
|-
| [[Uniwiz comma]]
| Quadzoyoma
| 4zyM
| 1500625 / 1492992
| {{Monzo| -11 -6 4 4 }}
| 8.8285
|
|-
| [[Historisma]]
| Latribizoguma
| L6zgM
| 257298363 / 256000000
| {{Monzo| -14 7 -6 6 }}
| 8.7582
| [[ArrowHead294]] (2024)
|-
| [[1029/1024|Gamelisma]]
| Latrizoma
| L3zM
| 1029 / 1024
| {{Monzo| -10 1 0 3 }}
| 8.4327
|
|-
| [[Varunisma]]
| Labizoguguma
| L2zggM
| 321489 / 320000
| {{Monzo| -9 8 -4 2 }}
| 8.0370
|
|-
| [[225/224|Marvel comma]], septimal kleisma
| Ruyoyoma
| ryyM
| 225 / 224
| {{Monzo| -5 2 2 -1 }}
| 7.7115
| [[Gene Ward Smith]] (2003)
|-
| [[Dimcomp comma]]
| Quadruyoyoma
| 4ryyM
| 390625 / 388962
| {{Monzo| -1 -4 8 -4 }}
| 7.3861
|
|-
| [[Cataharry comma]]
| Labiruguma
| L2rgM
| 19683 / 19600
| {{Monzo| -4 9 -2 -2 }}
| 7.3158
| [[Gene Ward Smith]] (2005)
|-
| [[Procyon comma]]
| Sasepzo-atriguma
| s7za3gM
| 823543 / 820125
| {{Monzo| 0 -8 -3 7 }}
| 7.2002
|
|-
| [[Qiqi comma]]
| Sepruyoyoma
| 7ryyM
| 48828125000 / 48629390607
| {{Monzo| 3 -10 14 -7 }}
| 7.0606
| [[User:Angekallikoita|Ange]] (2025)
|-
| [[Mirkwai comma]]
| Quinru-aquadyoma
| 5ra4yM
| 16875 / 16807
| {{Monzo| 0 3 4 -5 }}
| 6.9903
| [[Gene Ward Smith]] (2005)
|-
| [[Canousma]]
| Saquadzo-atriyoma
| s4za3yM
| 4802000 / 4782969
| {{Monzo| 4 -14 3 4 }}
| 6.8748
| [[Flora Canou]] (2020)
|-
| [[Triwellisma]]
| Tribizo-asepguma
| 6za7gM
| 235298 / 234375
| {{Monzo| 1 -1 -7 6 }}
| 6.8044
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Stearnsma]]
| Latribiruma
| L6rM
| 118098 / 117649
| {{Monzo| 1 10 0 -6 }}
| 6.5946
|
|-
| [[10976/10935|Hemimage comma]]
| Satrizo-aguma
| s3zagM
| 10976 / 10935
| {{Monzo| 5 -7 -1 3 }}
| 6.4790
| [[Gene Ward Smith]] (2005)
|-
| [[3136/3125|Hemimean comma]]
| Zozoquinguma
| zz5gM
| 3136 / 3125
| {{Monzo| 6 0 -5 2 }}
| 6.0832
| [[Gene Ward Smith]] (2005)
|-
| [[5120/5103|Hemifamity comma]], 5/7-kleisma
| Saruyoma
| sryM
| 5120 / 5103
| {{Monzo| 10 -6 1 -1 }}
| 5.7578
| [[Gene Ward Smith]] (2005)
|-
| [[Parkleiness comma]]
| Zotritriguma
| z9gM
| 1959552 / 1953125
| {{Monzo| 7 7 -9 1 }}
| 5.6875
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Octaphore comma]], enneagari comma
| Sasa-quadbizoma
| ss8zM
| 94450499584 / 94143178827
| {{Monzo| 14 -23 0 8 }}
| 5.6422
| [[User:Unque|Unque]] (2024)
|-
| [[Linus comma]]
| Saquinbizoguma
| s10zgM
| 578509309952 / 576650390625
| {{Monzo| 11 -10 -10 10 }}
| 5.5719
| [[Gene Ward Smith]] (2011)
|-
| [[Reiwa comma]]
| Saquadru-asepyoma
| s4ra7yM
| 1280000000 / 1275989841
| {{monzo| 14 -12 7 -4 }}
| 5.4324
| [[Flora Canou]] (2022)
|-
| [[6144/6125|Porwell comma]]
| Sarurutriguma
| srr3gM
| 6144 / 6125
| {{Monzo| 11 1 -3 -2 }}
| 5.3620
| [[Gene Ward Smith]], [[Petr Pařízek]] (2010)
|-
| [[Acromagic comma]]
| Sasa-sepzo-aquadguma
| ss7za4gM
| 26985857024 / 26904200625
| {{Monzo| 15 -16 -4 7 }}
| 5.2466
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Cartoonisma]]
| Satritrizo-asepbiguma
| s9za14gM
| 165288374272 / 164794921875
| {{Monzo| 12 -3 -14 9 }}
| 5.1762
| [[Eliora]] (2023)
|-
| [[Hemfiness comma]]
| Saquadru-atriyoma
| s4ra3yM
| 4096000 / 4084101
| {{Monzo| 15 -5 3 -5 }}
| 5.0366
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Acrodec comma]]
| Sasa-tribizo-aquadbiguma
| ss6za8gM
| 7710244864 / 7688671875
| {{Monzo| 16 -9 -8 6 }}
| 4.8507
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Hewuermera comma]]
| Satribiru-aguma
| s6ragM
| 589824 / 588245
| {{Monzo| 16 2 -1 -6 }}
| 4.6408
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Hearts comma]]
| Trila-quadzoma
| 3L4zM
| 34451725707 / 34359738368
| {{Monzo| -35 15 0 4 }}
| 4.6286
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Lokisma]], loki comma
| Sasa-bizotriguma
| ss2z3gM
| 102760448 / 102515625
| {{Monzo| 21 -8 -6 2 }}
| 4.1295
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Garischisma]], septimal schisma
| Sasaruma
| ssrM
| 33554432 / 33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.8041
|
|-
| [[Wadisma]]
| Latritrizo-ayoma
| L9zayM
| 201768035 / 201326592
| {{Monzo| -26 -1 1 9 }}
| 3.7919
|
|-
| [[Septimal enneadeca]]
| Quinla-neruma
| 5L19rM
| <abbr title="1570042899082081611640534563 / 1566652225014704215735402496">(56 digits)</abbr>
| {{Monzo| -37 57 0 -19 }}
| 3.7428
| [[Godtone]] (2024)
|-
| [[Quasiorwellisma]]
| Sazoquinbiguma
| sz10gM
| 29360128 / 29296875
| {{Monzo| 22 -1 -10 1 }}
| 3.7338
|
|}

=== 11-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[Dew comma]]
| Saloma
| s1oM
| 180224 / 177147
| {{Monzo| 14 -11 0 0 1 }}
| 29.812
| [[User:CompactStar|CompactStar]] (2023)
|-
| [[Thuja comma]]
| Saquinlu-ayoma
| s5(1u)ayM
| 163840 / 161051
| {{Monzo| 15 0 1 0 -5 }}
| 29.724
|
|-
| [[625/616|Quadrikite comma]]
| Luruquadyoma
| 1ur4yM
| 625 / 616
| {{Monzo| -3 0 4 -1 -1 }}
| 25.111
| [[Praveen Venkataramana]], [[Lumi Pakkanen]] (2022)
|-
| [[1350/1331|Large tetracot diesis]]
| Trilu-ayoyoma
| 3(1u)ayyM
| 1350 / 1331
| {{Monzo| 1 3 2 0 -3 }}
| 24.539
| [[User:ArrowHead294|ArrowHead294]] (2024)
|-
| [[Sensmus comma]]
| Salozoguma
| s1ozgM
| 1232 / 1215
| {{Monzo| 4 -5 -1 1 1 }}
| 24.055
|
|-
| [[Sevnothrush comma]]
| Loquinguma
| 1o5gM
| 3168 / 3125
| {{Monzo| 5 2 -5 0 1 }}
| 23.659
|
|-
| [[245/242|Frostma]]
| Biluzo-ayoma
| 2(1uz)ayM
| 245 / 242
| {{Monzo| -1 0 1 2 -2 }}
| 21.330
| [[Praveen Venkataramana]] (2022)
|-
| [[Distarma]]
| Trilozoma
| 3(1o)zM
| 9317 / 9216
| {{Monzo|-10 -2 0 1 3}}
| 18.869
| [https://twitter.com/Lilly__Flores Lilly Flores] (2023)
|-
| [[1617/1600|Antimisma]]
| Lobizoguma
| 1o2zgM
| 1617 / 1600
| {{Monzo| -6 1 -2 2 1 }}
| 18.297
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[99/98|Mothwellsma]]
| Loruruma
| 1orrM
| 99 / 98
| {{Monzo| -1 2 0 -2 1 }}
| 17.576
|
|-
| [[1610510/1594323|Fifthchromisma]]
| Saquinlo-ayoma
| s5(1o)ayM
| 1610510 / 1594323
| {{Monzo| 1 -13 1 0 5 }}
| 17.488
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[100/99|Ptolemisma]]
| Luyoyoma
| 1uyyM
| 100 / 99
| {{Monzo| 2 -2 2 0 -1 }}
| 17.399
|
|-
| [[Hemimin comma]]
| Trilu-azoma
| 3(1u)azM
| 1344 / 1331
| {{Monzo| 6 1 0 1 -3 }}
| 16.827
|
|-
| [[Betarabian comma]]
| Laloloma
| L1ooM
| 264627 / 262144
| {{Monzo| -18 7 0 0 2 }}
| 16.321
| [[Dawson Berry]] (2020)
|-
| [[Biyatisma]]
| Lologuma
| 1oogM
| 121 / 120
| {{Monzo| -3 -1 -1 0 2 }}
| 14.367
| [[Gene Ward Smith]] (2010)
|-
| [[Absinthma]]
| Luluruyoma
| 1uuryM
| 2560 / 2541
| {{Monzo| 9 -1 1 -1 -2 }}
| 12.897
| [[User:Jerdle|Jerdle]] (2023)
|-
| [[2835/2816|35/11 kleisma]]
| Laluzoyoma
| L1uzyM
| 2835 / 2816
| {{Monzo| -8 4 1 1 -1 }}
| 11.642
|
|-
| [[Aphrowe comma]]
| Trilo-aruruma
| 3(1o)arrM
| 1331 / 1323
| {{Monzo| 0 -3 0 -2 3 }}
| 10.437
| [[Lériendil]] (2024)
|-
| [[2200/2187|Small tetracot diesis]]
| Saloyoyoma
| s1oyyM
| 2200 / 2187
| {{Monzo| 3 -7 2 0 1 }}
| 10.260
| [[User:ArrowHead294|ArrowHead294]]
|-
| [[Valinorsma]]
| Loruguguma
| 1orggM
| 176 / 175
| {{Monzo| 4 0 -2 -1 1 }}
| 9.8646
|
|-
| [[Pentacircle comma]]
| Saluzoma
| s1uzM
| 896 / 891
| {{Monzo| 7 -4 0 1 -1 }}
| 9.6880
|
|-
| [[Alpharabian comma]]
| Laquadloma
| L4(1o)M
| 131769 / 131072
| {{Monzo| -17 2 0 0 4 }}
| 9.1818
| [[Dawson Berry]] (2020)
|-
| [[Orgonisma]]
| Satrilu-aruruma
| s3(1u)arrM
| 65536 / 65219
| {{Monzo| 16 0 0 -2 -3 }}
| 8.3944
|
|-
| [[Quindecic comma]]
| Sasa-quintriloruma
| ss15(1or)M
|
| {{Monzo| 14 -15 0 -15 15 }}
| 8.0555
|
|-
| [[117649/117128]]
| Bilulutrizoma
| 2(1uu3z)M
| 117649 / 117128
| {{Monzo| -3 0 0 6 -4 }}
| 7.6837
|
|-
| [[Topsy comma]]
| Quadlo-atrizo-asepguma
| 4(1o)a3za7gM
| 5021863 / 5000000
| {{Monzo| -6 0 -7 3 4 }}
| 7.5535
|
|-
| [[4375/4356|Fantares comma]]
| Luluzoquadyoma
| 1uuz4yM
| 4375 / 4356
| {{Monzo| -2 -2 4 1 -2 }}
| 7.5349
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Semicanousma]]
| Quadlo-aguma
| 4(1o)agM
| 14641 / 14580
| {{Monzo| -2 -6 -1 0 4 }}
| 7.2281
| [[Flora Canou]] (2020)
|-
| [[243/242|Rastma]]
| Luluma
| 1uuM
| 243 / 242
| {{Monzo| -1 5 0 0 -2 }}
| 7.1391
|
|-
| [[3388/3375|Myhemiwell comma]]
| Lolozotriguma
| 1ooz3gM
| 3388 / 3375
| {{Monzo| 2 -3 -3 1 2 }}
| 6.6556
|
|-
| [[Pythrabian comma]]
| Trisaloma
| 3s1oM
| 94489280512 / 94143178827
| {{Monzo| 33 -23 0 0 1 }}
| 6.3529
| [[Dawson Berry]] (2021)
|-
| [[Semiporwellisma]]
| Saluluguma
| s1uugM
| 16384 / 16335
| {{Monzo| 14 -3 -1 0 -2 }}
| 5.1854
| [[Flora Canou]] (2021)
|-
| [[Octatonic comma]], undecimal octatonic comma
| Quadbiluma
| 8(1u)M
| 214990848 / 214358881
| {{Monzo| 15 8 0 0 -8 }}
| 5.0965
| [[Eliora]] (2023)
|-
| [[385/384|Keenanisma]]
| Lozoyoma
| 1ozyM
| 385 / 384
| {{Monzo| -7 -1 1 1 1 }}
| 4.5026
| [[Paul Erlich]] (2001)
|-
| [[Trimitone comma]]
| Lalotriguma
| L1o3gM
| 8019 / 8000
| {{Monzo| -6 6 -3 0 1 }}
| 4.1068
| [[User:Godtone|Godtone]] (2021)
|-
| [[4-cent comma]]
| Lutritryoma
| 1u9yM
| 1953125 / 1948617
| {{Monzo| 0 -11 9 0 -1 }}
| 4.0004
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[441/440|Werckisma]]
| Luzozoguma
| 1uzzgM
| 441 / 440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.9302
|
|-
| [[1375/1372|Moctdel comma]]
| Lotriruyo
| 1o3ryM
| 1375 / 1372
| {{Monzo| -2 0 3 -3 1 }}
| 3.7814
|
|-
| [[Unisquarisma]], unisquary comma
| Trilu-aquadzo-ayoma
| 3(1u)a4zayM
| 12005 / 11979
| {{Monzo| 0 -2 1 4 -3 }}
| 3.7535
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[6250/6237|Liganellus comma]], liganellisma
| Luruquinyoma
| 1ur5yM
| 6250 / 6237
| {{Monzo| 1 -4 5 -1 -1 }}
| 3.6047
| [[Dawson Berry]] (2020)
|}

=== 13-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color Name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[1600/1573|Cameratasma]]
|
|
| 1600/1573
| {{Monzo| 6 0 2 0 -2 -1 }}
| 29.464
| [[Phlub]] (2026)
|-
| [[Lovecraft comma]]
| Thothotriluma
| 3oo3(1u)M
| 1352/1331
| {{Monzo| 3 0 0 0 -3 2 }}
| 27.101
|
|-
| [[65/64|Wilsorma]]
| Thoyoma
| 3oyM
| 65/64
| {{Monzo| -6 0 1 0 0 1 }}
| 26.841
|
|-
| [[Hyperpyth comma]]
| Quadtho-aquinguma
| 4(3o)a5gM
| 28561/28125
| {{Monzo| 0 -2 -5 0 0 4 }}
| 26.632
|
|-
| [[66/65|Winmeanma]]
| Thuloguma
| 3u1ogM
| 66/65
| {{Monzo| 1 1 -1 0 1 -1 }}
| 26.432
|
|-
| [[343/338|Sooty fox comma]]
| Thuthutrizoma
| 3uu3zM
| 343/338
| {{Monzo| -1 0 0 3 0 -2 }}
| 25.422
| [[Budjarn Lambeth]] (2024)
|-
| [[Tetris comma]]
| Sathoma
| s3oM
| 6656/6561
| {{Monzo| 9 -8 0 0 0 1 }}
| 24.888
| [[Godtone]] (2024)
|-
| [[507/500|Large semisixthma]]
| Thothotriguma
| 3oo3gM
| 507/500
| {{Monzo| -2 1 -3 0 0 2 }}
| 24.069
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[78/77|Negustma]]
| Tholuruma
| 3o1urM
| 78/77
| {{Monzo| 1 1 0 -1 -1 1 }}
| 22.339
|
|-
| [[Greater tendoneutralisma]]
| Laquadbithoma
| L8(3o)M
| 815730721 / 805306368
| {{Monzo| -28 -1 0 0 0 8 }}
| 22.266
| [[Godtone]] (2024)
|-
| [[2025/2002|Beyoncisma]]
|
|
| 2025/2002
| {{Monzo| -1 4 2 -1 -1 -1 }}
| 19.776
| [[Phlub]] (2026)
|-
| [[91/90|Biome comma, superleap comma]]
| Thozoguma
| 3ozgM
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.130
|
|-
| [[8281/8192|Diahuntmisma]]
| Labithozoma
| L2(3oz)M
| 8281/8192
| {{Monzo| -13 0 0 2 0 2 }}
| 18.707
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[512/507|Tridecimal neutral thirds comma]]
| Thuthuma
| 3uuM
| 512/507
| {{Monzo| 9 -1 0 0 0 -2 }}
| 16.990
|
|-
| [[105/104|Animist comma]]
| Thuzoyoma
| 3uzyM
| 105/104
| {{Monzo| -3 1 1 1 0 -1 }}
| 16.567
|
|-
| [[28812/28561|Tesseract comma]]
| Quadthuzoma
| 4(3uz)M
| 28812/28561
| {{Monzo| 2 1 0 4 0 -4 }}
| 15.148
| [[User:Unque|Unque]] (2025)

|-
| [[Secorian comma]]
| Sathuzoma
| s3uzM
| 28672 / 28431
| {{Monzo| 12 -7 0 1 0 -1 }}
| 14.613
|
|-
| [[3159/3136|Mosaic comma]]
| Lathoruruma
| L3orrM
| 3159/3136
| {{Monzo| -6 5 0 -2 0 1}}
| 12.651
|
|-
| [[275/273|Gassorma]]
| Thuloruyoyoma
| 3u1oryyM
| 275/273
| {{Monzo| 0 -1 2 -1 1 -1 }}
| 12.637
|
|-
| [[144/143|Grossma]]
| Thuluma
| 3u1uM
| 144/143
| {{Monzo| 4 2 0 0 -1 -1 }}
| 12.064
|

|-
| [[Lesser tendoneutralisma]]
| Sasa-quadtrithuma
| ss12(3u)M
| 70368744177664 / 69894255367443
| {{Monzo| 46 -1 0 0 0 -12 }}
| 11.713
| [[Godtone]] (2024)
|-
| [[1701/1690|Kuhnausma]]
|
|
| 1701/1690
| {{Monzo| -1 5 -1 1 0 -2 }}
| 11.232
| [[Phlub]] (2026)
|-
| [[Dinos comma]]
| Lathuthuquinguma
| L3uu5gM
| 531441/528125
| {{Monzo| 0 12 -5 0 0 -2 }}
| 10.836
| [[Dummy Index]] (2024)
|-
| [[169/168|Buzurgisma, dhanvantarisma]]
| Thothoruma
| 3oorM
| 169/168
| {{Monzo| -3 -1 0 -1 0 2 }}
| 10.274
| [[Margo Schulter]] (2012) for ''buzurgisma''
|-
| [[3042/3025|Diagassormisma]]
| Bitholuguma
| 2(3o1ug)M
| 3042/3025
| {{Monzo| 1 2 -2 0 -2 2 }}
| 9.7020
| [[User:Xenllium|Xenllium]] (2022)
|-
| Greater nelindic comma
| Thothoquinru-ayoyoma
| 3oo5rayyM
| 16900/16807
| {{Monzo| 2 0 2 -5 0 2 }}
| 9.5532
| [[User:Kaiveran|Kaiveran]] (2019)
|-
| [[1287/1280|Catadictma]]
| Thologuma
| 3o1ogM
| 1287/1280
| {{Monzo| -8 2 -1 0 1 1 }}
| 9.4419
|
|-
| [[Glacier comma]]
| Quinthuma
| 5(3u)M
| 373248/371293
| {{Monzo| 9 6 0 0 0 -5 }}
| 9.0917
| [[ArrowHead294]] (2024)
|-
| [[196/195|Mynucuma]]
| Thuzozoguma
| 3uzzgM
| 196/195
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.8554
|
|-
| [[1625/1617|Sopreisma]]
| Tholururutriyoma
| 3urr3yM
| 1625/1617
| {{Monzo| 0 -1 3 -2 -1 1 }}
| 8.5440
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[640/637|Huntma]], lesser nelindic comma
| Thururuyoma
| 3urryM
| 640/637
| {{Monzo| 7 0 1 -2 0 -1 }}
| 8.1342
| [[User:Kaiveran|Kaiveran]] (2019) for ''lesser nelindic comma''
|-
|Sonoma
|Lala-tritritho-aquadyoma
|LL9(3o)a4yM
|6627812108125/
6597069766656
|{{Monzo|-41 -1 4 0 0 9}}
|8.0488
|[https://x.com/vib_gen/status/2038852033244246443 Vib, Misohito Nakai] (2026)
|-
| [[2197/2187|Threedie]]
| Satrithoma
| s3(3o)M
| 2197/2187
| {{Monzo| 0 -7 0 0 0 3 }}
| 7.8980
|
|-
| [[Tridecimal nakaisma]]
| Quinsa-quadtritrithu-azoma
| 5s36(3u)azM
|
| {{Monzo| 132 -1 0 1 0 -36 }}
| 7.8751
| [[User:原井玉葱郎|Misohito Nakai]] (2025)
|-
| [[4394/4375|Hebrewsma]]
| Tritho-aruquadguma
| 3(3o)ar4gM
| 4394/4375
| {{Monzo| 1 0 -4 -1 0 3 }}
| 7.5022
| [[Eliora]] (2022)
|-
| [[1188/1183|Kestrel comma]]
| Thuthuloruma
| 3uu1orM
| 1188/1183
| {{Monzo| 2 3 0 -1 1 -2 }}
| 7.3017
| [[Gene Ward Smith]] (2011)
|-
| [[30E comma|2D9 comma]]
| Thotriyoma
| 3o3yM
| 131625/131072
| {{Monzo|-17 4 3 0 0 1}}
| 7.2888
| [https://twitter.com/Regret_March/status/1709762093749252209 Figreflekt] (2023) but [https://twitter.com/Figreflekt/status/1710195052520337680 revised later]{{dead link}}
|-
| [[Brontesisma]]
| Trithu-azozoyoma
| 3(3u)azzM
| 2205/2197
| {{Monzo| 0 2 1 2 0 -3 }}
| 6.2925
| [[Francium]] (2025)
|-
| [[Praveensma]]
| Thoquadzoma
| 3o4zM
| 31213/31104
| {{Monzo| -7 -5 0 4 0 1 }}
| 6.0563
| [[Praveen Venkataramana]] (2022)
|-
| [[1573/1568|Lambeth comma]]
| Thobiloruma
| 3o2(1or)M
| 1573/1568
| {{Monzo| -5 0 0 -2 2 1 }}
| 5.5117
| [[Flora Canou]] (2022)
|-
| [[325/324|Marveltwin comma]]
| Thoyoyoma
| 3oyyM
| 325/324
| {{Monzo| -2 -4 2 0 0 1 }}
| 5.3351
|
|-
| [[Valerisma]], Hunt 13-cycle comma
| Laquinbithoma
| L10(3o)M
| 137858491849 / 137438953472
| {{Monzo| -37 0 0 0 0 10 }}
| 5.2766
| [[Mason Green]] (2016)
|-
| [[351/350|Ratwolfsma]]
| Thoruguguma
| 3orggM
| 351/350
| {{Monzo| -1 3 -2 -1 0 1 }}
| 4.9393
|
|-
| [[352/351|Major minthma, major gentle comma]], 11/13-kleisma
| Thuloma
| 3u1oM
| 352/351
| {{Monzo| 5 -3 0 0 1 -1 }}
| 4.9253
| See the page.
|-
| [[364/363|Minor minthma, minor gentle comma]]
| Tholuluzoma
| 3o1uuzM
| 364/363
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.7627
| See the page.
|-
| [[847/845|Cuthbert comma]]
| Bithulo-azoguma
| 2(3u1o)azgM
| 847/845
| {{Monzo| 0 0 -1 1 2 -2 }}
| 4.0928
|
|}

=== 17-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[2048/2023|Susurrisma, susurration comma]]
| Susuruma
| 17uurM
| 2048/2023
| {{Monzo| 11 0 0 -1 0 0 -2 }}
| 21.263
| [[User:Jerdle|Jerdle]] (2026)
|-
| [[85/84|Monk comma]]
| Soruyoma
| 17oryM
| 85/84
| {{Monzo| -2 -1 1 -1 0 0 1 }}
| 20.488
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[289/286|Lum comma]]
| Sosothuluma
| 17oo3u1uM
| 289/286
| {{Monzo| -1 0 0 0 -1 -1 2 }}
| 18.065
|
|-
| [[2197/2176|Mey comma]]
| Sutrithov
| 17u3(3o)M
| 2197/2176
| {{Monzo| -7 0 0 0 0 3 -1 }}
| 16.628
|
|-
| [[429/425|Middle semisixthma]]
| Suthologuguma
| 17u3o1oggM
| 429/425
| {{Monzo| 0 1 -2 0 1 1 -1 }}
| 16.218
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[4131/4096|Septendecimal comma]], Hunt flat 2 comma
| Lasoma
| L17oM
| 4131/4096
| {{Monzo| -12 5 0 0 0 0 1 }}
| 14.730
| [[Flora Canou]] (2020) for ''septendecimal comma''
|-
| [[120/119|Lynchisma]]
| Suruyoma
| 17uryM
| 120/119
| {{Monzo| 3 1 1 -1 0 0 -1 }}
| 14.487
| [[User:Xenllium|Xenllium]] (2022)
|-
| 23-17-comma, 23 semitone comma
| Trila-twethesoma
| 3L23(17o)M
|
| {{Monzo| -94 0 0 0 0 0 23 }}
| 13.974
|
|-
| [[136/135|Diatisma]], diatic comma, fiventeen comma, septendecimal semicomma
| Soguma
| 17ogM
| 136/135
| {{Monzo| 3 -3 -1 0 0 0 1 }}
| 12.777
| See the page.
|-
| [[154/153|Augustma]]
| Sulozoma
| 17u1ozM
| 154/153
| {{Monzo| 1 -2 0 1 1 0 -1 }}
| 11.278
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[170/169|Major naiadma]]
| Sothuthuyoma
| 17o3uuyM
| 170/169
| {{Monzo| 1 0 1 0 0 -2 1 }}
| 10.214
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2187/2176|Septendecimal schisma]]
| Lasuma
| L17uM
| 2187/2176
| {{Monzo| -7 7 0 0 0 0 -1 }}
| 8.7296
| [[Plainsound Music Edition]] (2020)
|-
| [[1452/1445|Small semisixthma]]
| Susulologuma
| 17uu1oogM
| 1452/1445
| {{Monzo| 2 1 -1 0 2 0 -2 }}
| 8.3664
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[Mean thirds comma]]
| Lasosoyoma
| L17ooyM
| 1053405/1048576
| {{Monzo|-20 6 1 0 0 0 2}}
| 7.9545
| [[User:Jerdle|Jerdle]] (2022)
|-
| [[221/220|Minor naiadma]]
| Sotholuguma
| 17o3o1ugM
| 221/220
| {{Monzo| -2 0 -1 0 -1 1 1 }}
| 7.8514
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2057/2048|Blume comma]]
| Sololoma
| 17o1ooM
| 2057/2048
| {{monzo| -11 0 0 0 2 0 1 }}
| 7.5913
| [[Douglas Blumeyer]]
|-
| [[256/255|Charisma]], charic comma, septendecimal kleisma
| Suguma
| 17ugM
| 256/255
| {{Monzo| 8 -1 -1 0 0 0 -1 }}
| 6.7759
| See the page.
|-
| [[273/272|Tannisma, prototannisma]]
| Suthozoma
| 17u3ozM
| 273/272
| {{Monzo| -4 1 0 1 0 1 -1}}
| 6.3532
| [[Scott Dakota]] (2017) for ''tannisma'' [[Flora Canou]] (2023) for ''prototannisma''
|-
| [[289/288|Semitonisma]], septendecimal semitones comma, septendecimal 6-cent comma
| Sosoma
| 17ooM
| 289/288
| {{Monzo| -5 -2 0 0 0 0 2 }}
| 6.0008
| [[Flora Canou]] (2023) ''for semitonisma''
|-
| [[375/374|Ursulisma]]
| Sulutriyoma
| 17u1u3yM
| 375/374
| {{Monzo| -1 1 3 0 -1 0 -1 }}
| 4.6228
| [[Dawson Berry]], [[User:VIxen|VIxen]] (2023)
|-
| [[442/441|Seminaiadma]]
| Sothoruruma
| 17o3orrM
| 442/441
| {{Monzo| 1 -2 0 -2 0 1 1 }}
| 3.9213
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[80-17-comma]], 17-ripple <strike>integer cents</strike> comma{{clarify}}
| Lesa-quinquadquadsuma
| 11s80(17u)M
|
| {{Monzo| 327 0 0 0 0 0 -80 }}
| 3.5672
| [[Eliora]] (2022) for ''80-17-comma''
|}

=== 19-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by

|-
| [[76/75|Large undevicesimal 1/9-tone]]
| Noguguma
| 19oggM
| 76/75
| {{Monzo| 2 -1 -2 0 0 0 0 1 }}
| 22.931
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[77/76|Small undevicesimal 1/9-tone]]
| Nulozoma
| 19u1ozM
| 77/76
| {{Monzo| -2 0 0 1 1 0 0 -1 }}
| 22.631
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[96/95|19th-partial chroma]]
| Nuguma
| 19ugM
| 96/95
| {{Monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.128
| [[User:Flirora|Flirora]] (2021)
|-
| [[Ume comma]]
| Nutrisoma
| 19u3(17o)M
| 4913/4864
| {{Monzo| -8 0 0 0 0 0 3 -1 }}
| 17.353
|
|-
| [[729/722|Undevicesimal diaschisma]]
| Lanunuma
| L19uuM
| 729/722
| {{Monzo| -1 6 0 0 0 0 0 -2 }}
| 16.704
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[6912/6859|Deviaug comma]]
| Trinuma
| 3(19u)M
| 6912/6859
| {{Monzo| 8 3 0 0 0 0 0 -3 }}
| 13.326
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[133/132|Minithirdma]]
| Noluzoma
| 19o1uzM
| 133/132
| {{Monzo| -2 -1 0 1 -1 0 0 1 }}
| 13.066
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[153/152|Ganassisma]], Ganassi's comma
| Nusoma
| 19u17oM
| 153/152
| {{Monzo| -3 2 0 0 0 0 1 -1 }}
| 11.352
| [[User:Flirora|+merlan #flirora]] (2021)
|-
| [[171/170|Malcolmisma]]
| Nosuguma
| 19o17ugM
| 171/170
| {{Monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.154
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[131072/130321|Undevicesimal diminished comma]], Hunt 19-cycle comma
| Saquadnuma
| s4(19u)M
| 131072 / 130321
| {{Monzo| 17 0 0 0 0 0 0 -4 }}
| 9.9479
|
|-
| [[Eye comma]]
| Nubisoluma
| 19u2(17o1u)M
| 2312/2299
| {{Monzo| 3 0 0 0 -2 0 2 -1 }}
| 9.7619
|
|-
| [[363/361|Godzillisma]]
| Binuloma
| 2(19u1o)M
| 363/361
| {{Monzo| 0 1 0 0 2 0 0 -2 }}
| 9.5649
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[190/189|Cotylisma]]
| Noruyoma
| 19oryM
| 190/189
| {{Monzo| 1 -3 1 -1 0 0 0 1 }}
| 9.1358
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[209/208|Yama comma]]
| Nothuloma
| 19o3u1oM
| 209/208
| {{Monzo| -4 0 0 0 1 -1 0 1 }}
| 8.3033
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[210/209|Spleen comma]]
| Nuluzoyoma
| 19u1uzyM
| 210/209
| {{Monzo| 1 1 1 1 -1 0 0 -1 }}
| 8.2637
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1083/1078|Bihendrixma]]
| Nonolururuma
| 19oo1urrM
| 1083/1078
| {{Monzo| -1 1 0 -2 -1 0 0 2 }}
| 8.0113
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[286/285|Chthonisma]]
| Nuthologuma
| 19u3o1ogM
| 286/285
| {{Monzo| 1 -1 -1 0 1 1 0 -1 }}
| 6.0639
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[324/323|Photisma]]
| Nusuma
| 19u17uM
| 324/323
| {{Monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.3516
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[343/342|Nutrisma]]
| Nutrizoma
| 19u3zM
| 343/342
| {{Monzo| -1 -2 0 3 0 0 0 -1 }}
| 5.0547
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Triraptor comma]]
| Trinuso-azoguma
| 3(19u17o)azgM
| 34391/34295
| {{Monzo|0 0 -1 1 0 0 3 -3}}
| 4.8394
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[361/360|Go comma]], dudon comma
| Nonoguma
| 19oogM
| 361/360
| {{Monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.8023
| [[User:Xenwolf|Xenwolf]] (2013) for ''go comma''
|-
| [[400/399|Devichroma]]
| Nuruyoyoma
| 19uryyM
| 400/399
| {{Monzo| 4 -1 2 -1 0 0 0 -1 }}
| 4.3335
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[456/455|Abnobisma]]
| Nothuruguma
| 19o3urgM
| 456/455
| {{Monzo| 3 1 -1 -1 0 -1 0 1 }}
| 3.8007
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[476/475|Hedwigma]]
| Nusozoguguma
| 19u17ozggM
| 476/475
| {{Monzo| 2 0 -2 1 0 0 1 -1 }}
| 3.6409
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[495/494|Eulalisma]]
| Nuthuloyoma
| 19u3u1oyM
| 495/494
| {{Monzo| -1 2 1 0 1 -1 0 -1 }}
| 3.5010
| [[User:Xenllium|Xenllium]] (2023)
|}

=== 23-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by

|-
| [[69/68|Large vicesimotertial 1/8-tone]]
| Twethosuma
| 23o17uM
| 69/68
| 2.3.17.23 {{monzo| -2 1 -1 1 }}
| 25.274
|[[User:Xenllium|Xenllium]] (2023)
|-
| [[70/69|Small vicesimotertial 1/8-tone]]
| Twethuzoyoma
| 23uzyM
| 70/69
| 2.3.5.7.23 {{monzo| 1 -1 1 1 -1 }}
| 24.910
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[92/91|Undinisma]]
| Twethothuruma
| 23o3urM
| 92/91
| 2.7.13.23 {{monzo| 2 -1 -1 1 }}
| 18.921
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[736/729|23-limit Tenney/Cage comma]]
| Satwethoma
| s23oM
| 736/729
| 2.3.23 {{monzo| 5 -6 1 }}
| 16.544
| [[Stephen Weigel]] (2023)
|-
| [[115/114|Yarmanisma]]
| Twethonuyoma
| 23o19uyM
| 115/114
| 2.3.5.19.23 {{monzo| -1 -1 1 -1 1 }}
| 15.120
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[161/160|Major kirnbergerisma]]
| Twethozoguma
| 23ozgM
| 161/160
| 2.5.7.23 {{monzo| -5 -1 1 1 }}
| 10.787
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[162/161|Minor kirnbergerisma]]
| Twethuruma
| 23urM
| 162/161
| 2.3.7.23 {{monzo| 1 4 -1 -1 }}
| 10.720
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[208/207|Vicetone comma]]
| Twethuthoma
| 23u3oM
| 208/207
| 2.3.13.23 {{monzo| 4 -2 1 -1 }}
| 8.3433
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[231/230|Major neutravicema]]
| Twethulozoguma
| 23u1ozgM
| 231/230
| {{monzo| -1 1 -1 1 1 0 0 0 -1 }}
| 7.5108
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Vicesimotertial schisma]]
| Lala-twethuma
| LL23uM
| 387420489 / 385875968
| 2.3.23 {{monzo| -24 18 -1 }}
| 6.9157
|
|-
| [[253/252|Middle neutravicema]]
| Twetholoruma
| 23o1orM
| 253/252
| 2.3.7.11.23 {{monzo| -2 -2 -1 1 1 }}
| 6.8564
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[276/275|Minor neutravicema]]
| Twetholuguguma
| 23o1uggM
| 276/275
| 2.3.5.11.23 {{monzo| 2 1 -2 -1 1 }}
| 6.2840
| [[User:Xenllium|Xenllium]] (2023)
|-
| 21-23-comma
| Trisa-septritwethuma
| 3s21(23u)M
| 281474976710656 / 280462473659039
| 2.23 {{monzo| 95 -21 }}
| 6.2387
|
|-
| [[300/299|Major naiadvicema]]
| Twethuthuyoyoma
| 23u3uyyM
| 300/299
| 2.3.5.13.23 {{monzo| 2 1 2 -1 -1 }}
| 5.7804
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[323/322|Major semivicema]]
| Twethunosoruma
| 23u19o17orM
| 323/322
| 2.7.17.19.23 {{monzo| -1 -1 1 1 -1 }}
| 5.3682
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[391/390|Minor naiadvicema]]
| Twethosothuguma
| 23o17o3ugM
| 391/390
| {{monzo| -1 -1 -1 0 0 -1 1 0 1 }}
| 4.4334
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[392/391|Minor semivicema]]
| Twethusuzozoma
| 23u17uzzM
| 392/391
| 2.7.17.23 {{monzo| 3 2 -1 -1 }}
| 4.4221
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[460/459|Scanisma, vicewolf comma]]
| Twethosuyoma
| 23o17uyM
| 460/459
| 2.3.5.17.23 {{monzo| 2 -3 1 -1 1 }}
| 3.7676
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[484/483|Pittsburghisma]]
| Twethuloloruma
| 23u1oorM
| 484/483
| 2.3.7.11.23 {{monzo| 2 -1 -1 2 -1 }}
| 3.5806
| [[User:Xenllium|Xenllium]] (2023)
|}

=== 29-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| Classical mediant of Didymus' and Archytas' commas
| Twenothuluyoma
| 29o3u1uyM
| 145/143
| 5.11.13.29 {{monzo| 1 -1 -1 1 }}
| 24.045
|
|-
| [[88/87|Farewell comma]]
| Twenuloma
| 29u1oM
| 88/87
| 2.3.11.29 {{monzo| 3 -1 1 -1 }}
| 19.786
|
|-
| [[116/115|Sironisma]]
| Twenotwethuguma
| 29o23ugM
| 116/115
| 2.5.23.29 {{monzo| 2 -1 -1 1 }}
| 14.989
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[117/116|Lomisma]]
| Twenuthoma
| 29u3oM
| 117/116
| 2.3.13.29 {{monzo| -2 2 1 -1 }}
| 14.860
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[145/144|29th-partial chroma]]
| Twenoyoma
| 29oyM
| 145/144
| 2.3.5.29 {{monzo| -4 -2 1 1 }}
| 11.981
| [[User:Flirora|Flirora]] (2023)
|-
| [[175/174|Major chthonovinema]]
| Twenuzoyoyoma
| 29uzyyM
| 175/174
| 2.3.5.7.29 {{monzo| -1 -1 2 1 -1 }}
| 9.9211
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[204/203|Kallistisma]]
| Twenusoruma
| 29u17orM
| 204/203
| 2.3.7.17.29 {{monzo| 2 1 -1 1 -1 }}
| 8.5073
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[232/231|Major paravinema]]
| Twenoluruma
| 29o1urM
| 232/231
| 2.3.7.11.29 {{monzo| 3 -1 -1 -1 1 }}
| 7.4783
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[Jackpot comma]]
| Laseptwenoma
| L7(29o)M
| 17249876309 / 17179869184
| 2.29 {{monzo| -34 7 }}
| 7.0404
| [[Eliora]] (2024)
|-
| [[261/260|Major vinetonema]]
| Twenothuguma
| 29o3ugM
| 261/260
| 2.3.5.13.29 {{monzo| -2 2 -1 -1 1 }}
| 6.6458
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[290/289|Brunisma]]
| Twenosusuyoma
| 29o17uuyM
| 290/289
| 2.5.17.29 {{monzo| 1 1 -2 1 }}
| 5.9801
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[320/319|Minor paravinema]]
| Twenuluyoma
| 29u1uyM
| 320/319
| 2.5.11.29 {{monzo| 6 1 -1 -1 }}
| 5.4186
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[378/377|Major semivinema]]
| Twenuthuzoma
| 29u3uzM
| 378/377
| 2.3.7.13.29 {{monzo| 1 3 1 -1 -1 }}
| 4.5861
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[406/405|Minor semivinema]]
| Twenozoguma
| 29ozgM
| 406/405
| 2.3.5.7.29 {{monzo| 1 -4 -1 1 1 }}
| 4.2694
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[494/493|Minor vinetonema]]
| Twenunosuthoma
| 29u19o17u3oM
| 494/493
| 2.13.17.19.29 {{monzo| 1 1 -1 1 -1 }}
| 3.5081
| [[User:Xenllium|Xenllium]] (2024)
|}

=== 31-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[63/62|Co-archytas comma]]
| Thiwuzoma
| 31uzM
| 63/62
| 2.3.7.31 {{monzo| -1 2 1 -1 }}
| 27.700
| [[Xenwolf]] (2020)
|-
| [[93/92|Tricema]]
| Thiwotwethuma
| 31o23uM
| 93/92
| 2.3.23.31 {{monzo| -2 1 -1 1 }}
| 18.716
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[125/124|Twizzler]]
| Thiwutriyoma
| 31u3yM
| 125/124
| 2.5.31 {{monzo| -2 3 -1 }}
| 13.906
| [[Xenwolf]] (2020)
|-
| [[155/154|Scyllisma]]
| Thiwoluruyoma
| 31o1uryM
| 155/154
| 2.5.7.11.31 {{monzo| -1 1 -1 -1 1 }}
| 11.205
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[156/155|Xanthippisma]]
| Thiwuthoguma
| 31u3ogM
| 156/155
| 2.3.5.13.31 {{monzo| 2 1 -1 1 -1 }}
| 11.133
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[187/186|Lambertisma]]
| Thiwusoloma
| 31u17o1oM
| 187/186
| 2.3.11.17.31 {{monzo| -1 -1 1 1 -1 }}
| 9.2828
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[217/216|Tricesimoprimal kleisma]]
| Thiwozoma
| 31ozM
| 217/216
| 2.3.7.31 {{monzo| -3 -3 1 1 }}
| 7.9965
|
|-
| [[Doctorsma]]
| Latrithiwu-athuquadzoma
| L3(31u)a3u4zM
| 388962/387283
| 2.3.7.13.31 {{Monzo|1 4 4 -1 -3}}
| 7.4892
| [[User:Stavats|Stavats]] (2026)
|-
| [[248/247|Lameisma]]
| Thiwonuthuma
| 31o19u3uM
| 248/247
| 2.13.19.31 {{monzo| 3 -1 -1 1 }}
| 6.9949
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[280/279|Tricetone comma]]
| Thiwuzoyoma
| 31uzyM
| 280/279
| 2.3.5.7.31 {{monzo| 3 -2 1 1 -1 }}
| 6.1940
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Junebug comma]]
| Thiwutwenotwethunusotholuzoyoma
| 31u29o23u19u17o3o1uzyM
| 448630/447051
| {{monzo| 1 -1 1 1 -1 1 1 -1 -1 1 -1 }}
| 6.1040
| [[Tristan Bay]] (2024)
|-
| [[341/340|Californisma]]
| Thiwosuloguma
| 31o17u1ogM
| 341/340
| 2.5.11.17.31 {{monzo| -2 -1 1 -1 1 }}
| 5.0844
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[342/341|Endymisma]]
| Thiwunoluma
| 31u19o1uM
| 342/341
| 2.3.11.19.31 {{monzo| 1 2 -1 1 -1 }}
| 5.0695
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[435/434|Chinthisma]]
| Thiwutwenoruyoma
| 31u29oryM
| 435/434
| 2.3.5.7.29.31 {{monzo| -1 1 1 -1 1 -1 }}
| 3.9844
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[465/464|Alektisma]]
| Thiwotwenuyoma
| 31o29uyM
| 465/464
| 2.3.5.29.31 {{monzo| -4 1 1 -1 1 }}
| 3.7271
| [[User:Xenllium|Xenllium]] (2025)
|}

=== Higher-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[714984/704969|Lightyear comma]]
|
| 89u331o3M
| 714984/704969
| 2.3.31.89 {{monzo| 3 1 3 -3 }}
| 24.421
| [[Budjarn Lambeth]] (2023)
|-
| [[82/81|41-limit Johnston comma]]
| Fowoma
| 41oM
| 82/81
| 2.3.41 {{monzo| 1 -4 1 }}
| 21.242
| [[Stephen Weigel]] (2023)
|-
| [[2883/2848|Lilac comma]]
|
| 89u31ooM
| 2883/2848
| 2.3.31.89 {{monzo| -5 1 2 -1 }}
| 21.146
| [[Budjarn Lambeth]] (2023)
|-
| [[86/85|43-limit 10th-tone]], large quadracesimotertial 1/10-tone
| Fothosuguma
| 43o17ugM
| 86/85
| 2.5.17.43 {{monzo| 1 -1 -1 1 }}
| 20.249
| [[Mister Shaf]] (2025)
|-
| [[87/86|43-limit 10th-tone]], small quadracesimotertial 1/10-tone
| Fothutwenoma
| 43u29oM
| 87/86
| 2.3.29.43 {{monzo| -1 1 1 -1 }}
| 20.014
|
|-
| [[89/88|Tailwind comma]]
|
| 89o1uM
| 89/88
| 2.11.89 {{monzo| -3 -1 1 }}
| 19.562
| [[Budjarn Lambeth]] (2023)
|-
| [[389/385|Rebbe comma]]
|
| 389o1urgM
| 389/385
| 5.7.11.389 {{monzo| -1 -1 -1 1 }}
| 17.8794
| [[Mister Shaf]] (2025)
|-
| [[8277/8192|Lilly pilly comma]]
|
| 89o31oM
| 8277/8192
| 2.3.31.89 {{monzo| -13 1 1 1 }}
| 17.871
| [[Budjarn Lambeth]] (2023)
|-
| [[8000/7921|Incisor comma]]
|
| 89uu3yM
| 8000/7921
| 2.5.89 {{monzo| 6 3 -2 }}
| 17.181
| [[Budjarn Lambeth]] (2023)
|-
| [[129/128|43-limit Johnston comma]]
| Fothoma
| 43oM
| 129/128
| 2.3.43 {{monzo| -7 1 1 }}
| 13.473
| [[Stephen Weigel]] (2023)
|-
| [[979/972|Basement comma]]
|
| 89o1oM
| 979/972
| 2.3.11.89 {{monzo| -2 -5 1 1 }}
| 12.423
| [[Budjarn Lambeth]] (2023)
|-
| [[226/225|Reversed marvel comma]]
|
| 113oggM
| 226/225
| 2.3.5.113 {{monzo| 1 -2 -2 1 }}
| 7.6773
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Sidereal comma]]
|
| 73u61ogM
| 366/365
| 2.3.5.61.73 {{monzo| 1 1 -1 1 -1 }}
| 4.7366
| [[User:Frostburn|Frostburn]] (2024)
|-
| [[381/380|Five feet comma]]
|
| 127o19ugM
| 381/380
| 2.3.5.19.127 [-2 1 -1 -1 1⟩
| 4.5499
| [[Eliora]] (2022)
|-
| [[481/480|Semaphorisma]]
| Thisothoguma
| 37o3ogM
| 481/480
| 2.3.5.13.37 {{monzo| -5 -1 -1 1 1 }}
| 3.6030
| [[User:Kaiveran|Kaiveran]] (2023)
|}

== List of irrational commas ==
For intervals expressible as edosteps, see [[Interval size measure]]. We skip them here for brevity.

{| class="wikitable"
! Name(s)
! colspan="2" | [[Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[Caffeinterval]]
|
|
| 2((7/12) - (1/sqrt(3)))
|
| 7.1797
| [[User:R-4981|R-4981]] (2023)
|}

== See also ==
* [[Comma and diesis]]

== References ==

[[Category:Small commas| ]] 
[[Category:Lists of commas]]
{{Todo|complete table|add color name}}

Small comma

2026-04-28T09:06:22Z

TallKite: updated comma color names

A '''small comma''' is a [[comma]] whose size is approximately between 3.5 and 30 cents. These intervals are in the range from just noticeable up to usable as melodic steps. The actual perception of course varies. In [[Sagittal notation]], intervals in the smaller part of this category are [[kleisma (interval region)|kleismas]], and intervals in the larger part of this category are [[comma (interval region)|commas]]<ref>[https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>.

For commas over 100 cents in size, see [[Large comma]]; for commas in between 30 and 100 cents in size, see [[Medium comma]]; and for commas under 3.5 cents in size, see [[Unnoticeable comma]].

Due to the wide range of sizes, cents values here and on the other commas pages are listed to 5 significant digits, instead of to a fixed number of decimal places as is the [[Xenharmonic Wiki: Conventions|convention]] elsewhere on the wiki.

The commas might be numerous, but there is no need to memorise all the names. For pretty much all use cases, it is perfectly acceptable to just refer to a comma by the monzo or ratio, whichever is simpler.

== List of commas, by prime limit ==
=== 3-limit commas ===
{{See also| Table of 3-limit commas }}

{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cents]]
! Named by
|-
| 241-comma
| 241wama
| 241wM
|
| {{Monzo| 382 -241 }}
| 28.845
|
|-
| 65-comma, Pythagorean septimal comma
| 65wama
| 65wM
|
| {{Monzo| -103 65 }}
| 27.075
|
|-
| [[Pythagorean comma]]
| Lalawama
| LLwM
| 531441 / 524288
| {{Monzo| -19 12 }}
| 23.460
|
|-
| [[41-comma]], Pythagorean countercomma, countercomp comma
| 41wama
| 41wM
| <abbr title="36893488147419103232 / 36472996377170786403">(40 digits)</abbr>
| {{Monzo| 65 -41 }}
| 19.845
| [[Flora Canou]] (2022)
|-
| [[94-comma]], garistearn comma
| 94wama
| 94wM
|
| {{Monzo| 149 -94 }}
| 16.230
| [[User:Xenllium|Xenllium]] (2021)
|-
| 147-comma
| 147wama
| 147wM
|
| {{Monzo| 233 -147 }}
| 12.615
|
|-
| 200-comma, Pythagorean integer-cent ET comma
| 200wama
| 200wM
|
| {{Monzo| 317 -200 }}
| 8.9998
|
|-
| 253-comma
| 253wama
| 253wM
|
| {{Monzo| 401 -253 }}
| 5.3848
|
|-
| [[Mercator's comma]], 53-comma
| 53wama
| 53wM
| <abbr title="19383245667680019896796723 / 19342813113834066795298816">(52 digits)</abbr>
| {{Monzo| -84 53 }}
| 3.6150
|
|}

=== 5-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cents]]
! Named by
|-
| [[Magic comma]], small diesis
| Laquinyoma
| L5yM
| 3125 / 3072
| {{Monzo| -10 -1 5 }}
| 29.614
|
|-
| [[Triscordial comma]]
| Tribila-triyoma
| 6L3yM
| <abbr title="18761829412124890125 / 18446744073709551616">(40 digits)</abbr>
| {{Monzo| -64 36 3 }}
| 29.321
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Hendecatonic comma]]
| Trisa-leguma
| 3s11gM
| 8796093022208 / 8649755859375
| {{Monzo| 43 -11 -11 }}
| 29.044
|
|-
| [[Devil's tridecalimma]]
| Lala-theguma
| LL13gM
| 2541865828329 / 2500000000000
| {{Monzo| -11 26 -13 }}
| 28.752
|
|-
| [[Anthoine comma]]
| Trila-quinquadyoma
| 3L20yM
| 286102294921875 / 281474976710656
| {{Monzo| -48 1 20 }}
| 28.229
|
|-
| [[Tetracot comma]], minimal diesis
| Saquadyoma
| s4yM
| 20000 / 19683
| {{Monzo| 5 -9 4 }}
| 27.660
|
|-
| [[Biscordial comma]]
| Quadla-yoyoma
| 4LyyM
| 571919811374025 / 562949953421312
| {{Monzo| -49 28 2 }}
| 27.367
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Semaja comma]]
| Lala-neyoma
| LL19yM
| 19073486328125 / 18786186952704
| {{Monzo| -33 -7 19 }}
| 26.276
| [[Petr Pařízek]] (2011)
|-
| [[Quanic comma]]
| Sepsa-quinyoma
| 7s5yM
|
| {{Monzo| 74 -54 5 }}
| 25.999
|
|-
| [[Roda]], rodan comma
| Sasa-triyoma
| ss3yM
| 131072000 / 129140163
| {{Monzo| 20 -17 3 }}
| 25.706
|
|-
| [[Gracecordial comma]]
| Trilayoma
| 3LyM
| 17433922005 / 17179869184
| {{Monzo| -34 20 1 }}
| 25.414
|
|-
| [[Trisedodge comma]]
| Saquintriguma
| s15gM
| 30958682112 / 30517578125
| {{Monzo| 19 10 -15 }}
| 24.844
| [[Petr Pařízek]] (2011)
|-
| [[Birds comma]]
| Quadsa-thiweguma
| 4s31gM
|
| {{Monzo| 72 0 -31 }}
| 24.275
|
|-
| [[Neuk comma]]
| Trisa-yoyoma
| 3syyM
| 858993459200 / 847288609443
| {{Monzo| 35 -25 2 }}
| 23.752
| [[User:CompactStar|CompactStar]] (2023)
|-
| [[Maja comma]]
| Saseyoma
| s17yM
| 762939453125 / 753145430616
| {{Monzo| -3 -23 17 }}
| 22.368
| [[Petr Pařízek]] (2011)
|-
| [[Satin comma]]
| Quinbisa-triyoma
| 10s3yM
|
| {{Monzo| 104 -70 3 }}
| 22.091
|
|-
| [[Misneb comma]]
| Quadla-quintriyoma
| 4L15yM
| <abbr title="145964630126953125 / 144115188075855872">(36 digits)</abbr>
| {{Monzo| -57 14 15 }}
| 22.076
| [[Petr Pařízek]] (2011)
|-
| [[Syntonic comma]], Didymus comma, meantone comma
| Guma
| gM
| 81 / 80
| {{Monzo| -4 4 -1 }}
| 21.506
|
|-
| [[Maquila comma]]
| Trisa-seguma
| 3s17gM
| 562949953421312 / 556182861328125
| {{Monzo| 49 -6 -17 }}
| 20.937
| [[Petr Pařízek]] (2011)
|-
| [[Sfourth comma]]
| Lala-neguma
| LL19gM
| 617673396283947 / 610351562500000
| {{Monzo| -5 31 -19 }}
| 20.644
| [[Petr Pařízek]] (2011)
|-
| [[Diaschisma]]
| Saguguma
| sggM
| 2048 / 2025
| {{Monzo| 11 -4 -2 }}
| 19.553
| Hermann von Helmholtz, Alexander Ellis (1875)
|-
| [[Countermeantone comma]]
| Quinquadguma
| 20gM
| 96402615118848 / 95367431640625
| {{Monzo| 10 23 -20 }}
| 18.691
| [[Petr Pařízek]] (2012)
|-
| [[Ditonma]]
| Lala-theyoma
| LL13yM
| 1220703125 / 1207959552
| {{Monzo| -27 -2 13 }}
| 18.168
| [[Petr Pařízek]] (2011)
|-
| [[Misty comma]]
| Sasa-triguma
| ss3gM
| 67108864 / 66430125
| {{Monzo| 26 -12 -3 }}
| 17.599
| [[Paul Erlich]] (2002)
|-
| [[Quintile comma]]
| Trila-quinguma
| 3L5gM
| 847288609443 / 838860800000
| {{Monzo| -28 25 -5 }}
| 17.306
| See the dedicated page.
|-
| [[Enneadecacot comma]]
| Tribisa-neguma
| 6s19gM
| <abbr title="604462909807314587353088 / 598546211414337158203125">(48 digits)</abbr>
| {{monzo| 79 -22 -19 }}
| 17.029
| [[User:CompactStar|CompactStar]] (2024)
|-
| [[Oquatonic comma]]
| Quadla-sepquadyoma
| 4L28yM
| <abbr title="37252902984619140625 / 36893488147419103232">(40 digits)</abbr>
| {{Monzo| -65 0 28 }}
| 16.784
| [[Petr Pařízek]] (2011)
|-
| [[Undim comma]]
| Trisa-quadguma
| 3s4gM
| 2199023255552 / 2179240250625
| {{Monzo| 41 -20 -4 }}
| 15.645
| [[Petr Pařízek]] (2011)
|-
| [[Graviton]], gravity comma
| Lala-tribiguma
| LL6gM
| 129140163 / 128000000
| {{Monzo| -13 17 -6 }}
| 15.353
| [[Mike Battaglia]] (2011)
|-
| [[Majvam comma]]
| Sasa-lebiguma
| ss22gM
| <abbr title="2404631929946112 / 2384185791015625">(32 digits)</abbr>
| {{Monzo| 40 7 -22 }}
| 14.783
| [[Petr Pařízek]] (2011)
|-
| [[Quartonic comma]]
| Saleyoma
| s11yM
| 390625000 / 387420489
| {{Monzo| 3 -18 11 }}
| 14.261
|
|-
| [[Untritonic comma]]
| Quadla-tritriyoma
| 4L9yM
| <abbr title="2270041927734375 / 2251799813685248">(32 digits)</abbr>
| {{Monzo| -51 19 9 }}
| 13.968
| [[Petr Pařízek]] (2011)
|-
| [[Quindromeda comma]]
| Quinsa-quinguma
| 5s5gM
| <abbr title="72057594037927936 / 71489976421753125">(34 digits)</abbr>
| {{Monzo| 56 -28 -5 }}
| 13.691
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Sensipent comma]], medium semicomma
| Sepguma
| 7gM
| 78732 / 78125
| {{Monzo| 2 9 -7 }}
| 13.399
|
|-
| [[Copper comma]]
| Theneyoma
| 41L29yM
|
| {{Monzo| -481 261 29 }}
| 13.353
| [[Eliora]] (2023)
|-
| [[Counterwürschmidt comma]]
| Trisa-twetheguma
| 3s23gM
| <abbr title="36028797018963968 / 35762786865234375">(34 digits)</abbr>
| {{Monzo| 55 -1 -23 }}
| 12.830
| [[Flora Canou]] (2023)
|-
| [[Tertiosec comma]]
| Laquadtribiyoma
| 6L24yM
|
| {{Monzo| -89 21 24 }}
| 12.584
| [[Petr Pařízek]] (2012)
|-
| [[Submajor comma]]
| Trila-quadbiyoma
| 3L8yM
| 69198046875 / 68719476736
| {{Monzo| -36 11 8 }}
| 12.015
|
|-
| [[Würschmidt comma]]
| Saquadbiguma
| s8gM
| 393216 / 390625
| {{Monzo| 17 1 -8 }}
| 11.445
|
|-
| [[Bicommatic comma]]
| Quadla-quinbiguma
| 4L10gM
| <abbr title="1350851717672992089 / 1342177280000000000">(38 digits)</abbr>
| {{Monzo| -37 38 -10 }}
| 11.153
|
|-
| [[Counterhanson comma]]
| Quinquinyoma
| 25yM
| <abbr title="298023223876953125 / 296148833645101056">(36 digits)</abbr>
| {{Monzo| -20 -24 25 }}
| 10.923
| [[Petr Pařízek]] (2012)
|-
| [[Countritonic comma]]
| Quadsa-tritriyoma
| 4s9yM
| <abbr title="16777216000000000 / 16677181699666569">(34 digits)</abbr>
| {{Monzo| 33 -34 9 }}
| 10.353
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Semicomma]], Fokker's comma
| Lasepyoma
| L7yM
| 2109375 / 2097152
| {{Monzo| -21 3 7 }}
| 10.061
|
|-
| [[Heptacot comma]]
| Sepsa-sepguma
| 7s7gM
|
| {{Monzo| 86 -44 -7 }}
| 9.7840
| [[Tristan Bay]] (2024)
|-
| [[Escapade comma]]
| Sasa-tritriguma
| ss9gM
| 4294967296 / 4271484375
| {{Monzo| 32 -7 -9 }}
| 9.4916
| [[Paul Erlich]] (2002)
|-
| [[Undetritisma]], twentcufo comma
| Trila-leguma
| 3L11gM
| 205891132094649 / 204800000000000
| {{Monzo| -22 30 -11 }}
| 9.1992
|
|-
| [[15625/15552|Kleisma]], semicomma majeur
| Tribiyoma
| 6yM
| 15625 / 15552
| {{Monzo| -6 -5 6 }}
| 8.1073
| {{W|Shohé Tanaka}} (1890)
|-
| [[Quintosec comma]]
| Quadsa-quinbiguma
| 4s10gM
| 140737488355328 / 140126044921875
| {{Monzo| 47 -15 -10 }}
| 7.5378
| [[Petr Pařízek]] (2012)
|-
| 59-5-comma
| Quadbisa-fineguma
| 8s59gM
|
| {{Monzo| 137 0 -59 }}
| 7.4909
|
|-
| [[Unidecma]]
| Laquadtriguma
| L12gM
| 31381059609 / 31250000000
| {{Monzo| -7 22 -12 }}
| 7.2455
|
|-
| [[Mutt comma]]
| Trila-septriyoma
| 3L21yM
| 476837158203125 / 474989023199232
| {{Monzo| -44 -3 21 }}
| 6.7230
| [[Gene Ward Smith]] (2004)
|-
| [[Sulfur comma]]
| Lela-quadquadguma
| 11L16gM
|
| {{Monzo| -115 96 -16 }}
| 6.6607
|
|-
| [[Amity comma]]
| Saquinyoma
| s5yM
| 1600000 / 1594323
| {{Monzo| 9 -13 5 }}
| 6.1536
| [[Gene Ward Smith]] (2001)
|-
| [[Parakleisma]]
| Theguma
| 13gM
| 1224440064 / 1220703125
| {{Monzo| 8 14 -13 }}
| 5.2917
| [[Gene Ward Smith]] (2001)
|-
| [[Gammic comma]]
| Laquinquadyoma
| L20yM
| 95367431640625 / 95105071448064
| {{Monzo| -29 -11 20 }}
| 4.7693
|
|-
| [[Squarschmidt comma]]
| Quadsa-tweneguma
| 4s29gM
| <abbr title="186773283746309210112 / 186264514923095703125">(42 digits)</abbr>
| {{Monzo| 61 4 -29 }}
| 4.7223
| [[Petr Pařízek]] (2012)
|-
| Huntian 15-cycle comma
| Quadtrisa-fotheguma
| 12s43gM
|
| {{Monzo| 168 -43 -43 }}
| 4.4453
|
|-
| [[Barium comma]]
| Quadtribila-sepquadbiguma
| 24L56gM
|
| {{Monzo| -225 224 -56 }}
| 4.3522
| [[Eliora]] (2023)
|-
| [[Vulture comma]]
| Sasa-quadyoma
| ss4yM
| 10485760000 / 10460353203
| {{Monzo| 24 -21 4 }}
| 4.1998
| [[Paul Erlich]] (2002)
|-
| [[Dipromethia]]
| Thebila-siweyoma
| 26L61yM
|
| {{Monzo| -335 122 61 }}
| 3.6467
| [[Eliora]] (2023)
|-
| [[Lafa comma]]
| Tribisa-quadtriguma
| 6s12gM
|
| {{Monzo| 77 -31 -12 }}
| 3.6304
| [[Petr Pařízek]] (2011)
|}

=== 7-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cents]]
! Named by
|-
| [[3645/3584|Schismean comma]]
| Laruyoma
| LryM
| 3645 / 3584
| {{Monzo| -9 6 1 -1 }}
| 29.218
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Doublehearted comma]]
| Quadbizoma
| 8zM
| 5764801 / 5668704
| {{Monzo| -5 -11 0 8 }}
| 29.102
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Frostburn comma]]
| Quadru-asepyoma
| 4ra7yM
| 78125 / 76832
| {{Monzo| -5 0 7 -4 }}
| 28.892
|
|-
| [[686/675|Senga]]
| Trizo-aguguma
| 3zaggM
| 686 / 675
| {{Monzo| 1 -3 -2 3 }}
| 27.985
| [[Gene Ward Smith]] (2005)
|-
| 23-21-comma
| Sepla-twethezoma
| 7L23zM
|
| {{Monzo| -101 23 0 23 }}
| 27.961
|
|-
| [[64/63|Septimal comma]], Archytas' comma, Leipziger Komma
| Ruma
| rM
| 64 / 63
| {{Monzo| 6 -2 0 -1 }}
| 27.264
|
|-
| [[Mandos comma]]
| Biruguguma
| 2rggM
| 31104 / 30625
| {{Monzo| 7 5 -4 -2 }}
| 26.868
|
|-
| [[Slither comma]]
| Satritriru-aquadyoma
| s9ra4yM
| 40960000 / 40353607
| {{Monzo| 16 0 4 -9 }}
| 25.822
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Bastille comma]]
|
|
|
| {{Monzo| 1426 0 -596 -15 }}
| 24.638
| [[Eliora]] (2023)
|-
| 33-7/5-comma
| Letrizoguma
| 33zgM
|
| {{Monzo| -16 0 -33 33 }}
| 22.902
|
|-
| Huntian 35-cycle comma
| Quintrisa-tritritribiruguma
| 15s54rgM
|
| {{Monzo| 277 0 -54 -54 }}
| 22.461
|
|-
| [[Blackjackisma]]
| Lasepru-aquadbiyoma
| L7ra8yM
| 854296875 / 843308032
| {{Monzo| -10 7 8 -7 }}
| 22.413
| [[Gene Ward Smith]] (2011)
|-
| [[Squalentine comma]]
| Laquadzo-atriguma
| L4za3gM
| 64827 / 64000
| {{Monzo| -9 3 -3 4 }}
| 22.227
|
|-
| [[875/864|Keema]]
| Zotriyoma
| z3yM
| 875 / 864
| {{Monzo| -5 -3 3 1 }}
| 21.902
| [[Gene Ward Smith]] (2005)
|-
| [[Betelgeuse comma]]
| Satritrizo-aguguma
| s9zaggM
| 40353607 / 39858075
| {{Monzo| 0 -13 -2 9 }}
| 21.391
|
|-
| [[3125/3087|Gariboh comma]]
| Triru-aquinyoma
| 3ra5yM
| 3125 / 3087
| {{Monzo| 0 -2 5 -3 }}
| 21.181
| [[Gene Ward Smith]] (2005)
|-
| [[Secanticornisma]]
| Laruquinguma
| Lr5gM
| 177147 / 175000
| {{Monzo| -3 11 -5 -1 }}
| 21.111
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[2430/2401|Nuwell comma]]
| Quadru-ayoma
| 4rayM
| 2430 / 2401
| {{Monzo| 1 5 1 -4 }}
| 20.785
| [[Gene Ward Smith]] (2005)
|-
| [[Septimagic comma]]
| Saquinzoma
| s5zM
| 537824 / 531441
| {{Monzo| 5 -12 0 5 }}
| 20.670
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Mermisma]]
| Sepruyoma
| 7ryM
| 2500000 / 2470629
| {{Monzo| 5 -1 7 -7 }}
| 20.460
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Negricorn comma]], small quadruple bluish
| Saquadzoguma
| s4zgM
| 153664 / 151875
| {{monzo| 6 -5 -4 4 }}
| 20.274
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Tolerant comma]]
| Sazoyoyoma
| szyyM
| 179200 / 177147
| {{Monzo| 10 -11 2 1 }}
| 19.948
|
|-
| [[Icosipentatonic comma]], 25-36/35-comma
| Quinquinruguma
| 25rgM
|
| {{Monzo| 49 50 -25 -25 }}
| 19.260
|
|-
| [[Valenwuer comma]]
| Sarutribiguma
| sr6gM
| 110592 / 109375
| {{Monzo| 12 3 -6 -1 }}
| 19.157
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Buzzardsma]], buzzard comma
| Saquadruma
| s4rM
| 65536 / 64827
| {{Monzo| 16 -3 0 -4 }}
| 18.831
| See the page.
|-
| Huntian 21-cycle comma
| Quadbisa-sepquadruma
| 8s28rM
|
| {{Monzo| 123 -28 0 -28 }}
| 18.135
|
|-
| [[Mirwomo comma]]
| Labizoyoma
| L2zyM
| 33075 / 32768
| {{Monzo| -15 3 2 2 }}
| 16.144
|
|-
| [[Catasyc comma]]
| Laruquadbiyoma
| Lr8yM
| 390625 / 387072
| {{Monzo| -11 -3 8 -1 }}
| 15.819
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Compass comma]]
| Quinruyoyoma
| 5ryyM
| 9765625 / 9680832
| {{monzo| -6 -2 10 -5 }}
| 15.098
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Sensibeta comma]]
| Satrizo-aquinyoma
| s3za5yM
| 1071875 / 1062882
| {{monzo| -1 -12 5 3 }}
| 14.586
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Trimyna comma]]
| Quinzoguma
| 5zgM
| 50421 / 50000
| {{monzo| -4 1 -5 5 }}
| 14.516
|
|-
| [[245/243|Sensamagic comma]]
| Zozoyoma
| zzyM
| 245 / 243
| {{monzo| 0 -5 1 2 }}
| 14.191
| [[Gene Ward Smith]] (2010)
|-
| [[126/125|Starling comma]], septimal semicomma
| Zotriguma
| z3gM
| 126 / 125
| {{Monzo| 1 2 -3 1 }}
| 13.795
|
|-
| [[Vermeil comma]], 34-49/48-comma
| Quinla-sequadzoma
| 5L68zM
|
| {{Monzo| -137 -34 0 68 }}
| 13.692
| [[User:Perry.k|Perry.k]] (2026)
|-
| [[4000/3969|Octagar comma]]
| Rurutriyoma
| rr3yM
| 4000 / 3969
| {{Monzo| 5 -4 3 -2 }}
| 13.469
| [[Gene Ward Smith]] (2005)
|-
| [[1728/1715|Orwellisma]]
| Triru-aguma
| 3ragM
| 1728 / 1715
| {{Monzo| 6 3 -1 -3 }}
| 13.074
|
|-
| [[Mynaslender comma]]
| Sepru-ayoma
| 7rayM
| 829440 / 823543
| {{Monzo| 11 4 1 -7 }}
| 12.352
| [[User:Xenllium|Xenllium]] (2021)
|-
| 35-7/5-comma
| Sepquinruyoma
| 35ryM
|
| {{monzo| 17 0 35 -35 }}
| 12.073
|
|-
| [[Chromatisma]]
| Trisa-triru-aquadquadyoma
| 3s3ra16yM
| <abbr title="640000000000000000 / 635585924776181463">(36 digits)</abbr>
| {{Monzo| 22 -32 16 -3 }}
| 11.982
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Mistisma]]
| Sazoquadguma
| sz4gM
| 458752 / 455625
| {{Monzo| 16 -6 -4 1 }}
| 11.841
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Bronzisma]]
| Satriru-aguguma
| s3raggM
| 2097152 / 2083725
| {{Monzo| 21 -5 -2 -3 }}
| 11.120
| [[User:Xenllium|Xenllium]] (2021)
|-
| 34-jubilismic comma
| Sequadzoguma
| 68zgM
|
| {{Monzo| -33 0 -68 68 }}
| 10.829
|
|-
| [[Fynn's comma]], Hunt 7-cycle comma
| Quadsa-thebiruma
| 4s26rM
| <abbr title="9444732965739290427392 / 9387480337647754305649">(44 digits)</abbr>
| {{Monzo| 73 0 0 -26 }}
| 10.526
| [[Fynn Cerulean]] (2026) for ''Fynn's comma''
|-
| [[Septiness comma]]
| Sasasepruma
| ss7rM
| 67108864 / 66706983
| {{Monzo| 26 -4 0 -7 }}
| 10.399
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[31-comma temperaments|31-35-comma]]
| Tritrila-thiwezoyoma
| 9L31zyM
|
| {{Monzo| -159 0 31 31 }}
| 9.3282
|
|-
| [[Quince comma]]
| Lasepzo-aguguma
| L7zaggM
| 823543 / 819200
| {{Monzo| -15 0 -2 7 }}
| 9.1539
|
|-
| [[Uniwiz comma]]
| Quadzoyoma
| 4zyM
| 1500625 / 1492992
| {{Monzo| -11 -6 4 4 }}
| 8.8285
|
|-
| [[Historisma]]
| Latribizoguma
| L6zgM
| 257298363 / 256000000
| {{Monzo| -14 7 -6 6 }}
| 8.7582
| [[ArrowHead294]] (2024)
|-
| [[1029/1024|Gamelisma]]
| Latrizoma
| L3zM
| 1029 / 1024
| {{Monzo| -10 1 0 3 }}
| 8.4327
|
|-
| [[Varunisma]]
| Labizoguguma
| L2zggM
| 321489 / 320000
| {{Monzo| -9 8 -4 2 }}
| 8.0370
|
|-
| [[225/224|Marvel comma]], septimal kleisma
| Ruyoyoma
| ryyM
| 225 / 224
| {{Monzo| -5 2 2 -1 }}
| 7.7115
| [[Gene Ward Smith]] (2003)
|-
| [[Dimcomp comma]]
| Quadruyoyoma
| 4ryyM
| 390625 / 388962
| {{Monzo| -1 -4 8 -4 }}
| 7.3861
|
|-
| [[Cataharry comma]]
| Labiruguma
| L2rgM
| 19683 / 19600
| {{Monzo| -4 9 -2 -2 }}
| 7.3158
| [[Gene Ward Smith]] (2005)
|-
| [[Procyon comma]]
| Sasepzo-atriguma
| s7za3gM
| 823543 / 820125
| {{Monzo| 0 -8 -3 7 }}
| 7.2002
|
|-
| [[Qiqi comma]]
| Sepruyoyoma
| 7ryyM
| 48828125000 / 48629390607
| {{Monzo| 3 -10 14 -7 }}
| 7.0606
| [[User:Angekallikoita|Ange]] (2025)
|-
| [[Mirkwai comma]]
| Quinru-aquadyoma
| 5ra4yM
| 16875 / 16807
| {{Monzo| 0 3 4 -5 }}
| 6.9903
| [[Gene Ward Smith]] (2005)
|-
| [[Canousma]]
| Saquadzo-atriyoma
| s4za3yM
| 4802000 / 4782969
| {{Monzo| 4 -14 3 4 }}
| 6.8748
| [[Flora Canou]] (2020)
|-
| [[Triwellisma]]
| Tribizo-asepguma
| 6za7gM
| 235298 / 234375
| {{Monzo| 1 -1 -7 6 }}
| 6.8044
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Stearnsma]]
| Latribiruma
| L6rM
| 118098 / 117649
| {{Monzo| 1 10 0 -6 }}
| 6.5946
|
|-
| [[10976/10935|Hemimage comma]]
| Satrizo-aguma
| s3zagM
| 10976 / 10935
| {{Monzo| 5 -7 -1 3 }}
| 6.4790
| [[Gene Ward Smith]] (2005)
|-
| [[3136/3125|Hemimean comma]]
| Zozoquinguma
| zz5gM
| 3136 / 3125
| {{Monzo| 6 0 -5 2 }}
| 6.0832
| [[Gene Ward Smith]] (2005)
|-
| [[5120/5103|Hemifamity comma]], 5/7-kleisma
| Saruyoma
| sryM
| 5120 / 5103
| {{Monzo| 10 -6 1 -1 }}
| 5.7578
| [[Gene Ward Smith]] (2005)
|-
| [[Parkleiness comma]]
| Zotritriguma
| z9gM
| 1959552 / 1953125
| {{Monzo| 7 7 -9 1 }}
| 5.6875
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Octaphore comma]], enneagari comma
| Sasa-quadbizoma
| ss8zM
| 94450499584 / 94143178827
| {{Monzo| 14 -23 0 8 }}
| 5.6422
| [[User:Unque|Unque]] (2024)
|-
| [[Linus comma]]
| Saquinbizoguma
| s10zgM
| 578509309952 / 576650390625
| {{Monzo| 11 -10 -10 10 }}
| 5.5719
| [[Gene Ward Smith]] (2011)
|-
| [[Reiwa comma]]
| Saquadru-asepyoma
| s4ra7yM
| 1280000000 / 1275989841
| {{monzo| 14 -12 7 -4 }}
| 5.4324
| [[Flora Canou]] (2022)
|-
| [[6144/6125|Porwell comma]]
| Sarurutriguma
| srr3gM
| 6144 / 6125
| {{Monzo| 11 1 -3 -2 }}
| 5.3620
| [[Gene Ward Smith]], [[Petr Pařízek]] (2010)
|-
| [[Acromagic comma]]
| Sasa-sepzo-aquadguma
| ss7za4gM
| 26985857024 / 26904200625
| {{Monzo| 15 -16 -4 7 }}
| 5.2466
| [[User:Xenllium|Xenllium]] (2026)
|-
| [[Cartoonisma]]
| Satritrizo-asepbiguma
| s9za14gM
| 165288374272 / 164794921875
| {{Monzo| 12 -3 -14 9 }}
| 5.1762
| [[Eliora]] (2023)
|-
| [[Hemfiness comma]]
| Saquadru-atriyoma
| s4ra3yM
| 4096000 / 4084101
| {{Monzo| 15 -5 3 -5 }}
| 5.0366
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Acrodec comma]]
| Sasa-tribizo-aquadbiguma
| ss6za8gM
| 7710244864 / 7688671875
| {{Monzo| 16 -9 -8 6 }}
| 4.8507
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Hewuermera comma]]
| Satribiru-aguma
| s6ragM
| 589824 / 588245
| {{Monzo| 16 2 -1 -6 }}
| 4.6408
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Hearts comma]]
| Trila-quadzoma
| 3L4zM
| 34451725707 / 34359738368
| {{Monzo| -35 15 0 4 }}
| 4.6286
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Lokisma]], loki comma
| Sasa-bizotriguma
| ss2z3gM
| 102760448 / 102515625
| {{Monzo| 21 -8 -6 2 }}
| 4.1295
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Garischisma]], septimal schisma
| Sasaruma
| ssrM
| 33554432 / 33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.8041
|
|-
| [[Wadisma]]
| Latritrizo-ayoma
| L9zayM
| 201768035 / 201326592
| {{Monzo| -26 -1 1 9 }}
| 3.7919
|
|-
| [[Septimal enneadeca]]
| Quinla-neruma
| 5L19rM
| <abbr title="1570042899082081611640534563 / 1566652225014704215735402496">(56 digits)</abbr>
| {{Monzo| -37 57 0 -19 }}
| 3.7428
| [[Godtone]] (2024)
|-
| [[Quasiorwellisma]]
| Sazoquinbiguma
| sz10gM
| 29360128 / 29296875
| {{Monzo| 22 -1 -10 1 }}
| 3.7338
|
|}

=== 11-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[Dew comma]]
| Saloma
| s1oM
| 180224 / 177147
| {{Monzo| 14 -11 0 0 1 }}
| 29.812
| [[User:CompactStar|CompactStar]] (2023)
|-
| [[Thuja comma]]
| Saquinlu-ayoma
| s5(1u)ayM
| 163840 / 161051
| {{Monzo| 15 0 1 0 -5 }}
| 29.724
|
|-
| [[625/616|Quadrikite comma]]
| Luruquadyoma
| 1ur4yM
| 625 / 616
| {{Monzo| -3 0 4 -1 -1 }}
| 25.111
| [[Praveen Venkataramana]], [[Lumi Pakkanen]] (2022)
|-
| [[1350/1331|Large tetracot diesis]]
| Trilu-ayoyoma
| 3(1u)ayyM
| 1350 / 1331
| {{Monzo| 1 3 2 0 -3 }}
| 24.539
| [[User:ArrowHead294|ArrowHead294]] (2024)
|-
| [[Sensmus comma]]
| Salozoguma
| s1ozgM
| 1232 / 1215
| {{Monzo| 4 -5 -1 1 1 }}
| 24.055
|
|-
| [[Sevnothrush comma]]
| Loquinguma
| 1o5gM
| 3168 / 3125
| {{Monzo| 5 2 -5 0 1 }}
| 23.659
|
|-
| [[245/242|Frostma]]
| Biluzo-ayoma
| 2(1uz)ayM
| 245 / 242
| {{Monzo| -1 0 1 2 -2 }}
| 21.330
| [[Praveen Venkataramana]] (2022)
|-
| [[Distarma]]
| Trilozoma
| 3(1o)zM
| 9317 / 9216
| {{Monzo|-10 -2 0 1 3}}
| 18.869
| [https://twitter.com/Lilly__Flores Lilly Flores] (2023)
|-
| [[1617/1600|Antimisma]]
| Lobizoguma
| 1o2zgM
| 1617 / 1600
| {{Monzo| -6 1 -2 2 1 }}
| 18.297
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[99/98|Mothwellsma]]
| Loruruma
| 1orrM
| 99 / 98
| {{Monzo| -1 2 0 -2 1 }}
| 17.576
|
|-
| [[1610510/1594323|Fifthchromisma]]
| Saquinlo-ayoma
| s5(1o)ayM
| 1610510 / 1594323
| {{Monzo| 1 -13 1 0 5 }}
| 17.488
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[100/99|Ptolemisma]]
| Luyoyoma
| 1uyyM
| 100 / 99
| {{Monzo| 2 -2 2 0 -1 }}
| 17.399
|
|-
| [[Hemimin comma]]
| Trilu-azoma
| 3(1u)zM
| 1344 / 1331
| {{Monzo| 6 1 0 1 -3 }}
| 16.827
|
|-
| [[Betarabian comma]]
| Laloloma
| L1ooM
| 264627 / 262144
| {{Monzo| -18 7 0 0 2 }}
| 16.321
| [[Dawson Berry]] (2020)
|-
| [[Biyatisma]]
| Lologuma
| 1oogM
| 121 / 120
| {{Monzo| -3 -1 -1 0 2 }}
| 14.367
| [[Gene Ward Smith]] (2010)
|-
| [[Absinthma]]
| Luluruyoma
| 1uuryM
| 2560 / 2541
| {{Monzo| 9 -1 1 -1 -2 }}
| 12.897
| [[User:Jerdle|Jerdle]] (2023)
|-
| [[2835/2816|35/11 kleisma]]
| Laluzoyoma
| L1uzyM
| 2835 / 2816
| {{Monzo| -8 4 1 1 -1 }}
| 11.642
|
|-
| [[Aphrowe comma]]
| Trilo-aruruma
| 3(1o)arrM
| 1331 / 1323
| {{Monzo| 0 -3 0 -2 3 }}
| 10.437
| [[Lériendil]] (2024)
|-
| [[2200/2187|Small tetracot diesis]]
| Saloyoyoma
| s1oyyM
| 2200 / 2187
| {{Monzo| 3 -7 2 0 1 }}
| 10.260
| [[User:ArrowHead294|ArrowHead294]]
|-
| [[Valinorsma]]
| Loruguguma
| 1orggM
| 176 / 175
| {{Monzo| 4 0 -2 -1 1 }}
| 9.8646
|
|-
| [[Pentacircle comma]]
| Saluzoma
| s1uzM
| 896 / 891
| {{Monzo| 7 -4 0 1 -1 }}
| 9.6880
|
|-
| [[Alpharabian comma]]
| Laquadloma
| L4(1o)M
| 131769 / 131072
| {{Monzo| -17 2 0 0 4 }}
| 9.1818
| [[Dawson Berry]] (2020)
|-
| [[Orgonisma]]
| Satrilu-aruruma
| s3(1u)arrM
| 65536 / 65219
| {{Monzo| 16 0 0 -2 -3 }}
| 8.3944
|
|-
| [[Quindecic comma]]
| Sasa-quintriloruma
| ss15(1or)M
|
| {{Monzo| 14 -15 0 -15 15 }}
| 8.0555
|
|-
| [[117649/117128]]
| Bilulutrizoma
| 2(1uu3z)M
| 117649 / 117128
| {{Monzo| -3 0 0 6 -4 }}
| 7.6837
|
|-
| [[Topsy comma]]
| Quadlo-atrizo-asepguma
| 4(1o)a3za7gM
| 5021863 / 5000000
| {{Monzo| -6 0 -7 3 4 }}
| 7.5535
|
|-
| [[4375/4356|Fantares comma]]
| Luluzoquadyoma
| 1uuz4yM
| 4375 / 4356
| {{Monzo| -2 -2 4 1 -2 }}
| 7.5349
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Semicanousma]]
| Quadlo-aguma
| 4(1o)agM
| 14641 / 14580
| {{Monzo| -2 -6 -1 0 4 }}
| 7.2281
| [[Flora Canou]] (2020)
|-
| [[243/242|Rastma]]
| Luluma
| 1uuM
| 243 / 242
| {{Monzo| -1 5 0 0 -2 }}
| 7.1391
|
|-
| [[3388/3375|Myhemiwell comma]]
| Lolozotriguma
| 1ooz3gM
| 3388 / 3375
| {{Monzo| 2 -3 -3 1 2 }}
| 6.6556
|
|-
| [[Pythrabian comma]]
| Trisaloma
| 3s1oM
| 94489280512 / 94143178827
| {{Monzo| 33 -23 0 0 1 }}
| 6.3529
| [[Dawson Berry]] (2021)
|-
| [[Semiporwellisma]]
| Saluluguma
| s1uugM
| 16384 / 16335
| {{Monzo| 14 -3 -1 0 -2 }}
| 5.1854
| [[Flora Canou]] (2021)
|-
| [[Octatonic comma]], undecimal octatonic comma
| Quadbiluma
| 8(1u)M
| 214990848 / 214358881
| {{Monzo| 15 8 0 0 -8 }}
| 5.0965
| [[Eliora]] (2023)
|-
| [[385/384|Keenanisma]]
| Lozoyoma
| 1ozyM
| 385 / 384
| {{Monzo| -7 -1 1 1 1 }}
| 4.5026
| [[Paul Erlich]] (2001)
|-
| [[Trimitone comma]]
| Lalotriguma
| L1o3gM
| 8019 / 8000
| {{Monzo| -6 6 -3 0 1 }}
| 4.1068
| [[User:Godtone|Godtone]] (2021)
|-
| [[4-cent comma]]
| Lutritryoma
| 1u9yM
| 1953125 / 1948617
| {{Monzo| 0 -11 9 0 -1 }}
| 4.0004
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[441/440|Werckisma]]
| Luzozoguma
| 1uzzgM
| 441 / 440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.9302
|
|-
| [[1375/1372|Moctdel comma]]
| Lotriruyo
| 1o3ryM
| 1375 / 1372
| {{Monzo| -2 0 3 -3 1 }}
| 3.7814
|
|-
| [[Unisquarisma]], unisquary comma
| Trilu-aquadzo-ayoma
| 3(1u)a4zayM
| 12005 / 11979
| {{Monzo| 0 -2 1 4 -3 }}
| 3.7535
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[6250/6237|Liganellus comma]], liganellisma
| Luruquinyoma
| 1ur5yM
| 6250 / 6237
| {{Monzo| 1 -4 5 -1 -1 }}
| 3.6047
| [[Dawson Berry]] (2020)
|}

=== 13-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color Name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[1600/1573|Cameratasma]]
|
|
| 1600/1573
| {{Monzo| 6 0 2 0 -2 -1 }}
| 29.464
| [[Phlub]] (2026)
|-
| [[Lovecraft comma]]
| Thothotriluma
| 3oo3(1u)M
| 1352/1331
| {{Monzo| 3 0 0 0 -3 2 }}
| 27.101
|
|-
| [[65/64|Wilsorma]]
| Thoyoma
| 3oyM
| 65/64
| {{Monzo| -6 0 1 0 0 1 }}
| 26.841
|
|-
| [[Hyperpyth comma]]
| Quadtho-aquinguma
| 4(3o)a5gM
| 28561/28125
| {{Monzo| 0 -2 -5 0 0 4 }}
| 26.632
|
|-
| [[66/65|Winmeanma]]
| Thuloguma
| 3u1ogM
| 66/65
| {{Monzo| 1 1 -1 0 1 -1 }}
| 26.432
|
|-
| [[343/338|Sooty fox comma]]
| Thuthutrizoma
| 3uu3zM
| 343/338
| {{Monzo| -1 0 0 3 0 -2 }}
| 25.422
| [[Budjarn Lambeth]] (2024)
|-
| [[Tetris comma]]
| Sathoma
| s3oM
| 6656/6561
| {{Monzo| 9 -8 0 0 0 1 }}
| 24.888
| [[Godtone]] (2024)
|-
| [[507/500|Large semisixthma]]
| Thothotriguma
| 3oo3gM
| 507/500
| {{Monzo| -2 1 -3 0 0 2 }}
| 24.069
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[78/77|Negustma]]
| Tholuruma
| 3o1urM
| 78/77
| {{Monzo| 1 1 0 -1 -1 1 }}
| 22.339
|
|-
| [[Greater tendoneutralisma]]
| Laquadbithoma
| L8(3o)M
| 815730721 / 805306368
| {{Monzo| -28 -1 0 0 0 8 }}
| 22.266
| [[Godtone]] (2024)
|-
| [[2025/2002|Beyoncisma]]
|
|
| 2025/2002
| {{Monzo| -1 4 2 -1 -1 -1 }}
| 19.776
| [[Phlub]] (2026)
|-
| [[91/90|Biome comma, superleap comma]]
| Thozoguma
| 3ozgM
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.130
|
|-
| [[8281/8192|Diahuntmisma]]
| Labithozoma
| L2(3oz)M
| 8281/8192
| {{Monzo| -13 0 0 2 0 2 }}
| 18.707
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[512/507|Tridecimal neutral thirds comma]]
| Thuthuma
| 3uuM
| 512/507
| {{Monzo| 9 -1 0 0 0 -2 }}
| 16.990
|
|-
| [[105/104|Animist comma]]
| Thuzoyoma
| 3uzyM
| 105/104
| {{Monzo| -3 1 1 1 0 -1 }}
| 16.567
|
|-
| [[28812/28561|Tesseract comma]]
| Quadthuzoma
| 4(3uz)M
| 28812/28561
| {{Monzo| 2 1 0 4 0 -4 }}
| 15.148
| [[User:Unque|Unque]] (2025)

|-
| [[Secorian comma]]
| Sathuzoma
| s3uzM
| 28672 / 28431
| {{Monzo| 12 -7 0 1 0 -1 }}
| 14.613
|
|-
| [[3159/3136|Mosaic comma]]
| Lathoruruma
| L3orrM
| 3159/3136
| {{Monzo| -6 5 0 -2 0 1}}
| 12.651
|
|-
| [[275/273|Gassorma]]
| Thuloruyoyoma
| 3u1oryyM
| 275/273
| {{Monzo| 0 -1 2 -1 1 -1 }}
| 12.637
|
|-
| [[144/143|Grossma]]
| Thuluma
| 3u1uM
| 144/143
| {{Monzo| 4 2 0 0 -1 -1 }}
| 12.064
|

|-
| [[Lesser tendoneutralisma]]
| Sasa-quadtrithuma
| ss12(3u)M
| 70368744177664 / 69894255367443
| {{Monzo| 46 -1 0 0 0 -12 }}
| 11.713
| [[Godtone]] (2024)
|-
| [[1701/1690|Kuhnausma]]
|
|
| 1701/1690
| {{Monzo| -1 5 -1 1 0 -2 }}
| 11.232
| [[Phlub]] (2026)
|-
| [[Dinos comma]]
| Lathuthuquinguma
| L3uu5gM
| 531441/528125
| {{Monzo| 0 12 -5 0 0 -2 }}
| 10.836
| [[Dummy Index]] (2024)
|-
| [[169/168|Buzurgisma, dhanvantarisma]]
| Thothoruma
| 3oorM
| 169/168
| {{Monzo| -3 -1 0 -1 0 2 }}
| 10.274
| [[Margo Schulter]] (2012) for ''buzurgisma''
|-
| [[3042/3025|Diagassormisma]]
| Bitholuguma
| 2(3o1ug)M
| 3042/3025
| {{Monzo| 1 2 -2 0 -2 2 }}
| 9.7020
| [[User:Xenllium|Xenllium]] (2022)
|-
| Greater nelindic comma
| Thothoquinru-ayoyoma
| 3oo5rayyM
| 16900/16807
| {{Monzo| 2 0 2 -5 0 2 }}
| 9.5532
| [[User:Kaiveran|Kaiveran]] (2019)
|-
| [[1287/1280|Catadictma]]
| Thologuma
| 3o1ogM
| 1287/1280
| {{Monzo| -8 2 -1 0 1 1 }}
| 9.4419
|
|-
| [[Glacier comma]]
| Quinthuma
| 5(3u)M
| 373248/371293
| {{Monzo| 9 6 0 0 0 -5 }}
| 9.0917
| [[ArrowHead294]] (2024)
|-
| [[196/195|Mynucuma]]
| Thuzozoguma
| 3uzzgM
| 196/195
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.8554
|
|-
| [[1625/1617|Sopreisma]]
| Tholururutriyoma
| 3urr3yM
| 1625/1617
| {{Monzo| 0 -1 3 -2 -1 1 }}
| 8.5440
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[640/637|Huntma]], lesser nelindic comma
| Thururuyoma
| 3urryM
| 640/637
| {{Monzo| 7 0 1 -2 0 -1 }}
| 8.1342
| [[User:Kaiveran|Kaiveran]] (2019) for ''lesser nelindic comma''
|-
|Sonoma
|Lala-tritritho-aquadyoma
|LL9(3o)a4yM
|6627812108125/
6597069766656
|{{Monzo|-41 -1 4 0 0 9}}
|8.0488
|[https://x.com/vib_gen/status/2038852033244246443 Vib, Misohito Nakai] (2026)
|-
| [[2197/2187|Threedie]]
| Satrithoma
| s3(3o)M
| 2197/2187
| {{Monzo| 0 -7 0 0 0 3 }}
| 7.8980
|
|-
| [[Tridecimal nakaisma]]
| Quinsa-quadtritrithu-azoma
| 5s36(3u)azM
|
| {{Monzo| 132 -1 0 1 0 -36 }}
| 7.8751
| [[User:原井玉葱郎|Misohito Nakai]] (2025)
|-
| [[4394/4375|Hebrewsma]]
| Tritho-aruquadguma
| 3(3o)ar4gM
| 4394/4375
| {{Monzo| 1 0 -4 -1 0 3 }}
| 7.5022
| [[Eliora]] (2022)
|-
| [[1188/1183|Kestrel comma]]
| Thuthuloruma
| 3uu1orM
| 1188/1183
| {{Monzo| 2 3 0 -1 1 -2 }}
| 7.3017
| [[Gene Ward Smith]] (2011)
|-
| [[30E comma|2D9 comma]]
| Thotriyoma
| 3o3yM
| 131625/131072
| {{Monzo|-17 4 3 0 0 1}}
| 7.2888
| [https://twitter.com/Regret_March/status/1709762093749252209 Figreflekt] (2023) but [https://twitter.com/Figreflekt/status/1710195052520337680 revised later]{{dead link}}
|-
| [[Brontesisma]]
| Trithu-azozoyoma
| 3(3u)azzM
| 2205/2197
| {{Monzo| 0 2 1 2 0 -3 }}
| 6.2925
| [[Francium]] (2025)
|-
| [[Praveensma]]
| Thoquadzoma
| 3o4zM
| 31213/31104
| {{Monzo| -7 -5 0 4 0 1 }}
| 6.0563
| [[Praveen Venkataramana]] (2022)
|-
| [[1573/1568|Lambeth comma]]
| Thobiloruma
| 3o2(1or)M
| 1573/1568
| {{Monzo| -5 0 0 -2 2 1 }}
| 5.5117
| [[Flora Canou]] (2022)
|-
| [[325/324|Marveltwin comma]]
| Thoyoyoma
| 3oyyM
| 325/324
| {{Monzo| -2 -4 2 0 0 1 }}
| 5.3351
|
|-
| [[Valerisma]], Hunt 13-cycle comma
| Laquinbithoma
| L10(3o)M
| 137858491849 / 137438953472
| {{Monzo| -37 0 0 0 0 10 }}
| 5.2766
| [[Mason Green]] (2016)
|-
| [[351/350|Ratwolfsma]]
| Thoruguguma
| 3orggM
| 351/350
| {{Monzo| -1 3 -2 -1 0 1 }}
| 4.9393
|
|-
| [[352/351|Major minthma, major gentle comma]], 11/13-kleisma
| Thuloma
| 3u1oM
| 352/351
| {{Monzo| 5 -3 0 0 1 -1 }}
| 4.9253
| See the page.
|-
| [[364/363|Minor minthma, minor gentle comma]]
| Tholuluzoma
| 3o1uuzM
| 364/363
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.7627
| See the page.
|-
| [[847/845|Cuthbert comma]]
| Bithulo-azoguma
| 2(3u1o)azgM
| 847/845
| {{Monzo| 0 0 -1 1 2 -2 }}
| 4.0928
|
|}

=== 17-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[2048/2023|Susurrisma, susurration comma]]
| Susuruma
| 17uurM
| 2048/2023
| {{Monzo| 11 0 0 -1 0 0 -2 }}
| 21.263
| [[User:Jerdle|Jerdle]] (2026)
|-
| [[85/84|Monk comma]]
| Soruyoma
| 17oryM
| 85/84
| {{Monzo| -2 -1 1 -1 0 0 1 }}
| 20.488
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[289/286|Lum comma]]
| Sosothuluma
| 17oo3u1uM
| 289/286
| {{Monzo| -1 0 0 0 -1 -1 2 }}
| 18.065
|
|-
| [[2197/2176|Mey comma]]
| Sutrithov
| 17u3(3o)M
| 2197/2176
| {{Monzo| -7 0 0 0 0 3 -1 }}
| 16.628
|
|-
| [[429/425|Middle semisixthma]]
| Suthologuguma
| 17u3o1oggM
| 429/425
| {{Monzo| 0 1 -2 0 1 1 -1 }}
| 16.218
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[4131/4096|Septendecimal comma]], Hunt flat 2 comma
| Lasoma
| L17oM
| 4131/4096
| {{Monzo| -12 5 0 0 0 0 1 }}
| 14.730
| [[Flora Canou]] (2020) for ''septendecimal comma''
|-
| [[120/119|Lynchisma]]
| Suruyoma
| 17uryM
| 120/119
| {{Monzo| 3 1 1 -1 0 0 -1 }}
| 14.487
| [[User:Xenllium|Xenllium]] (2022)
|-
| 23-17-comma, 23 semitone comma
| Trila-twethesoma
| 3L23(17o)M
|
| {{Monzo| -94 0 0 0 0 0 23 }}
| 13.974
|
|-
| [[136/135|Diatisma]], diatic comma, fiventeen comma, septendecimal semicomma
| Soguma
| 17ogM
| 136/135
| {{Monzo| 3 -3 -1 0 0 0 1 }}
| 12.777
| See the page.
|-
| [[154/153|Augustma]]
| Sulozoma
| 17u1ozM
| 154/153
| {{Monzo| 1 -2 0 1 1 0 -1 }}
| 11.278
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[170/169|Major naiadma]]
| Sothuthuyoma
| 17o3uuyM
| 170/169
| {{Monzo| 1 0 1 0 0 -2 1 }}
| 10.214
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2187/2176|Septendecimal schisma]]
| Lasuma
| L17uM
| 2187/2176
| {{Monzo| -7 7 0 0 0 0 -1 }}
| 8.7296
| [[Plainsound Music Edition]] (2020)
|-
| [[1452/1445|Small semisixthma]]
| Susulologuma
| 17uu1oogM
| 1452/1445
| {{Monzo| 2 1 -1 0 2 0 -2 }}
| 8.3664
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[Mean thirds comma]]
| Lasosoyoma
| L17ooyM
| 1053405/1048576
| {{Monzo|-20 6 1 0 0 0 2}}
| 7.9545
| [[User:Jerdle|Jerdle]] (2022)
|-
| [[221/220|Minor naiadma]]
| Sotholuguma
| 17o3o1ugM
| 221/220
| {{Monzo| -2 0 -1 0 -1 1 1 }}
| 7.8514
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2057/2048|Blume comma]]
| Sololoma
| 17o1ooM
| 2057/2048
| {{monzo| -11 0 0 0 2 0 1 }}
| 7.5913
| [[Douglas Blumeyer]]
|-
| [[256/255|Charisma]], charic comma, septendecimal kleisma
| Suguma
| 17ugM
| 256/255
| {{Monzo| 8 -1 -1 0 0 0 -1 }}
| 6.7759
| See the page.
|-
| [[273/272|Tannisma, prototannisma]]
| Suthozoma
| 17u3ozM
| 273/272
| {{Monzo| -4 1 0 1 0 1 -1}}
| 6.3532
| [[Scott Dakota]] (2017) for ''tannisma'' [[Flora Canou]] (2023) for ''prototannisma''
|-
| [[289/288|Semitonisma]], septendecimal semitones comma, septendecimal 6-cent comma
| Sosoma
| 17ooM
| 289/288
| {{Monzo| -5 -2 0 0 0 0 2 }}
| 6.0008
| [[Flora Canou]] (2023) ''for semitonisma''
|-
| [[375/374|Ursulisma]]
| Sulutriyoma
| 17u1u3yM
| 375/374
| {{Monzo| -1 1 3 0 -1 0 -1 }}
| 4.6228
| [[Dawson Berry]], [[User:VIxen|VIxen]] (2023)
|-
| [[442/441|Seminaiadma]]
| Sothoruruma
| 17o3orrM
| 442/441
| {{Monzo| 1 -2 0 -2 0 1 1 }}
| 3.9213
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[80-17-comma]], 17-ripple <strike>integer cents</strike> comma{{clarify}}
| Lesa-quinquadquadsuma
| 11s80(17u)M
|
| {{Monzo| 327 0 0 0 0 0 -80 }}
| 3.5672
| [[Eliora]] (2022) for ''80-17-comma''
|}

=== 19-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by

|-
| [[76/75|Large undevicesimal 1/9-tone]]
| Noguguma
| 19oggM
| 76/75
| {{Monzo| 2 -1 -2 0 0 0 0 1 }}
| 22.931
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[77/76|Small undevicesimal 1/9-tone]]
| Nulozoma
| 19u1ozM
| 77/76
| {{Monzo| -2 0 0 1 1 0 0 -1 }}
| 22.631
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[96/95|19th-partial chroma]]
| Nuguma
| 19ugM
| 96/95
| {{Monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.128
| [[User:Flirora|Flirora]] (2021)
|-
| [[Ume comma]]
| Nutrisoma
| 19u3(17o)M
| 4913/4864
| {{Monzo| -8 0 0 0 0 0 3 -1 }}
| 17.353
|
|-
| [[729/722|Undevicesimal diaschisma]]
| Lanunuma
| L19uuM
| 729/722
| {{Monzo| -1 6 0 0 0 0 0 -2 }}
| 16.704
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[6912/6859|Deviaug comma]]
| Trinuma
| 3(19u)M
| 6912/6859
| {{Monzo| 8 3 0 0 0 0 0 -3 }}
| 13.326
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[133/132|Minithirdma]]
| Noluzoma
| 19o1uzM
| 133/132
| {{Monzo| -2 -1 0 1 -1 0 0 1 }}
| 13.066
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[153/152|Ganassisma]], Ganassi's comma
| Nusoma
| 19u17oM
| 153/152
| {{Monzo| -3 2 0 0 0 0 1 -1 }}
| 11.352
| [[User:Flirora|+merlan #flirora]] (2021)
|-
| [[171/170|Malcolmisma]]
| Nosuguma
| 19o17ugM
| 171/170
| {{Monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.154
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[131072/130321|Undevicesimal diminished comma]], Hunt 19-cycle comma
| Saquadnuma
| s4(19u)M
| 131072 / 130321
| {{Monzo| 17 0 0 0 0 0 0 -4 }}
| 9.9479
|
|-
| [[Eye comma]]
| Nubisoluma
| 19u2(17o1u)M
| 2312/2299
| {{Monzo| 3 0 0 0 -2 0 2 -1 }}
| 9.7619
|
|-
| [[363/361|Godzillisma]]
| Binuloma
| 2(19u1o)M
| 363/361
| {{Monzo| 0 1 0 0 2 0 0 -2 }}
| 9.5649
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[190/189|Cotylisma]]
| Noruyoma
| 19oryM
| 190/189
| {{Monzo| 1 -3 1 -1 0 0 0 1 }}
| 9.1358
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[209/208|Yama comma]]
| Nothuloma
| 19o3u1oM
| 209/208
| {{Monzo| -4 0 0 0 1 -1 0 1 }}
| 8.3033
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[210/209|Spleen comma]]
| Nuluzoyoma
| 19u1uzyM
| 210/209
| {{Monzo| 1 1 1 1 -1 0 0 -1 }}
| 8.2637
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1083/1078|Bihendrixma]]
| Nonolururuma
| 19oo1urrM
| 1083/1078
| {{Monzo| -1 1 0 -2 -1 0 0 2 }}
| 8.0113
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[286/285|Chthonisma]]
| Nuthologuma
| 19u3o1ogM
| 286/285
| {{Monzo| 1 -1 -1 0 1 1 0 -1 }}
| 6.0639
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[324/323|Photisma]]
| Nusuma
| 19u17uM
| 324/323
| {{Monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.3516
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[343/342|Nutrisma]]
| Nutrizoma
| 19u3zM
| 343/342
| {{Monzo| -1 -2 0 3 0 0 0 -1 }}
| 5.0547
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Triraptor comma]]
| Trinuso-azoguma
| 3(19u17o)azgM
| 34391/34295
| {{Monzo|0 0 -1 1 0 0 3 -3}}
| 4.8394
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[361/360|Go comma]], dudon comma
| Nonoguma
| 19oogM
| 361/360
| {{Monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.8023
| [[User:Xenwolf|Xenwolf]] (2013) for ''go comma''
|-
| [[400/399|Devichroma]]
| Nuruyoyoma
| 19uryyM
| 400/399
| {{Monzo| 4 -1 2 -1 0 0 0 -1 }}
| 4.3335
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[456/455|Abnobisma]]
| Nothuruguma
| 19o3urgM
| 456/455
| {{Monzo| 3 1 -1 -1 0 -1 0 1 }}
| 3.8007
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[476/475|Hedwigma]]
| Nusozoguguma
| 19u17ozggM
| 476/475
| {{Monzo| 2 0 -2 1 0 0 1 -1 }}
| 3.6409
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[495/494|Eulalisma]]
| Nuthuloyoma
| 19u3u1oyM
| 495/494
| {{Monzo| -1 2 1 0 1 -1 0 -1 }}
| 3.5010
| [[User:Xenllium|Xenllium]] (2023)
|}

=== 23-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by

|-
| [[69/68|Large vicesimotertial 1/8-tone]]
| Twethosuma
| 23o17uM
| 69/68
| 2.3.17.23 {{monzo| -2 1 -1 1 }}
| 25.274
|[[User:Xenllium|Xenllium]] (2023)
|-
| [[70/69|Small vicesimotertial 1/8-tone]]
| Twethuzoyoma
| 23uzyM
| 70/69
| 2.3.5.7.23 {{monzo| 1 -1 1 1 -1 }}
| 24.910
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[92/91|Undinisma]]
| Twethothuruma
| 23o3urM
| 92/91
| 2.7.13.23 {{monzo| 2 -1 -1 1 }}
| 18.921
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[736/729|23-limit Tenney/Cage comma]]
| Satwethoma
| s23oM
| 736/729
| 2.3.23 {{monzo| 5 -6 1 }}
| 16.544
| [[Stephen Weigel]] (2023)
|-
| [[115/114|Yarmanisma]]
| Twethonuyoma
| 23o19uyM
| 115/114
| 2.3.5.19.23 {{monzo| -1 -1 1 -1 1 }}
| 15.120
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[161/160|Major kirnbergerisma]]
| Twethozoguma
| 23ozgM
| 161/160
| 2.5.7.23 {{monzo| -5 -1 1 1 }}
| 10.787
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[162/161|Minor kirnbergerisma]]
| Twethuruma
| 23urM
| 162/161
| 2.3.7.23 {{monzo| 1 4 -1 -1 }}
| 10.720
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[208/207|Vicetone comma]]
| Twethuthoma
| 23u3oM
| 208/207
| 2.3.13.23 {{monzo| 4 -2 1 -1 }}
| 8.3433
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[231/230|Major neutravicema]]
| Twethulozoguma
| 23u1ozgM
| 231/230
| {{monzo| -1 1 -1 1 1 0 0 0 -1 }}
| 7.5108
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Vicesimotertial schisma]]
| Lala-twethuma
| LL23uM
| 387420489 / 385875968
| 2.3.23 {{monzo| -24 18 -1 }}
| 6.9157
|
|-
| [[253/252|Middle neutravicema]]
| Twetholoruma
| 23o1orM
| 253/252
| 2.3.7.11.23 {{monzo| -2 -2 -1 1 1 }}
| 6.8564
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[276/275|Minor neutravicema]]
| Twetholuguguma
| 23o1uggM
| 276/275
| 2.3.5.11.23 {{monzo| 2 1 -2 -1 1 }}
| 6.2840
| [[User:Xenllium|Xenllium]] (2023)
|-
| 21-23-comma
| Trisa-septritwethuma
| 3s21(23u)M
| 281474976710656 / 280462473659039
| 2.23 {{monzo| 95 -21 }}
| 6.2387
|
|-
| [[300/299|Major naiadvicema]]
| Twethuthuyoyoma
| 23u3uyyM
| 300/299
| 2.3.5.13.23 {{monzo| 2 1 2 -1 -1 }}
| 5.7804
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[323/322|Major semivicema]]
| Twethunosoruma
| 23u19o17orM
| 323/322
| 2.7.17.19.23 {{monzo| -1 -1 1 1 -1 }}
| 5.3682
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[391/390|Minor naiadvicema]]
| Twethosothuguma
| 23o17o3ugM
| 391/390
| {{monzo| -1 -1 -1 0 0 -1 1 0 1 }}
| 4.4334
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[392/391|Minor semivicema]]
| Twethusuzozoma
| 23u17uzzM
| 392/391
| 2.7.17.23 {{monzo| 3 2 -1 -1 }}
| 4.4221
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[460/459|Scanisma, vicewolf comma]]
| Twethosuyoma
| 23o17uyM
| 460/459
| 2.3.5.17.23 {{monzo| 2 -3 1 -1 1 }}
| 3.7676
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[484/483|Pittsburghisma]]
| Twethuloloruma
| 23u1oorM
| 484/483
| 2.3.7.11.23 {{monzo| 2 -1 -1 2 -1 }}
| 3.5806
| [[User:Xenllium|Xenllium]] (2023)
|}

=== 29-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| Classical mediant of Didymus' and Archytas' commas
| Twenothuluyoma
| 29o3u1uyM
| 145/143
| 5.11.13.29 {{monzo| 1 -1 -1 1 }}
| 24.045
|
|-
| [[88/87|Farewell comma]]
| Twenuloma
| 29u1oM
| 88/87
| 2.3.11.29 {{monzo| 3 -1 1 -1 }}
| 19.786
|
|-
| [[116/115|Sironisma]]
| Twenotwethuguma
| 29o23ugM
| 116/115
| 2.5.23.29 {{monzo| 2 -1 -1 1 }}
| 14.989
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[117/116|Lomisma]]
| Twenuthoma
| 29u3oM
| 117/116
| 2.3.13.29 {{monzo| -2 2 1 -1 }}
| 14.860
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[145/144|29th-partial chroma]]
| Twenoyoma
| 29oyM
| 145/144
| 2.3.5.29 {{monzo| -4 -2 1 1 }}
| 11.981
| [[User:Flirora|Flirora]] (2023)
|-
| [[175/174|Major chthonovinema]]
| Twenuzoyoyoma
| 29uzyyM
| 175/174
| 2.3.5.7.29 {{monzo| -1 -1 2 1 -1 }}
| 9.9211
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[204/203|Kallistisma]]
| Twenusoruma
| 29u17orM
| 204/203
| 2.3.7.17.29 {{monzo| 2 1 -1 1 -1 }}
| 8.5073
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[232/231|Major paravinema]]
| Twenoluruma
| 29o1urM
| 232/231
| 2.3.7.11.29 {{monzo| 3 -1 -1 -1 1 }}
| 7.4783
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[Jackpot comma]]
| Laseptwenoma
| L7(29o)M
| 17249876309 / 17179869184
| 2.29 {{monzo| -34 7 }}
| 7.0404
| [[Eliora]] (2024)
|-
| [[261/260|Major vinetonema]]
| Twenothuguma
| 29o3ugM
| 261/260
| 2.3.5.13.29 {{monzo| -2 2 -1 -1 1 }}
| 6.6458
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[290/289|Brunisma]]
| Twenosusuyoma
| 29o17uuyM
| 290/289
| 2.5.17.29 {{monzo| 1 1 -2 1 }}
| 5.9801
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[320/319|Minor paravinema]]
| Twenuluyoma
| 29u1uyM
| 320/319
| 2.5.11.29 {{monzo| 6 1 -1 -1 }}
| 5.4186
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[378/377|Major semivinema]]
| Twenuthuzoma
| 29u3uzM
| 378/377
| 2.3.7.13.29 {{monzo| 1 3 1 -1 -1 }}
| 4.5861
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[406/405|Minor semivinema]]
| Twenozoguma
| 29ozgM
| 406/405
| 2.3.5.7.29 {{monzo| 1 -4 -1 1 1 }}
| 4.2694
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[494/493|Minor vinetonema]]
| Twenunosuthoma
| 29u19o17u3oM
| 494/493
| 2.13.17.19.29 {{monzo| 1 1 -1 1 -1 }}
| 3.5081
| [[User:Xenllium|Xenllium]] (2024)
|}

=== 31-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[63/62|Co-archytas comma]]
| Thiwuzoma
| 31uzM
| 63/62
| 2.3.7.31 {{monzo| -1 2 1 -1 }}
| 27.700
| [[Xenwolf]] (2020)
|-
| [[93/92|Tricema]]
| Thiwotwethuma
| 31o23uM
| 93/92
| 2.3.23.31 {{monzo| -2 1 -1 1 }}
| 18.716
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[125/124|Twizzler]]
| Thiwutriyoma
| 31u3yM
| 125/124
| 2.5.31 {{monzo| -2 3 -1 }}
| 13.906
| [[Xenwolf]] (2020)
|-
| [[155/154|Scyllisma]]
| Thiwoluruyoma
| 31o1uryM
| 155/154
| 2.5.7.11.31 {{monzo| -1 1 -1 -1 1 }}
| 11.205
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[156/155|Xanthippisma]]
| Thiwuthoguma
| 31u3ogM
| 156/155
| 2.3.5.13.31 {{monzo| 2 1 -1 1 -1 }}
| 11.133
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[187/186|Lambertisma]]
| Thiwusoloma
| 31u17o1oM
| 187/186
| 2.3.11.17.31 {{monzo| -1 -1 1 1 -1 }}
| 9.2828
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[217/216|Tricesimoprimal kleisma]]
| Thiwozoma
| 31ozM
| 217/216
| 2.3.7.31 {{monzo| -3 -3 1 1 }}
| 7.9965
|
|-
| [[Doctorsma]]
| Lathiwu-athuquadzoma
| L3(31u)3u4zM
| 388962/387283
| 2.3.7.13.31 {{Monzo|1 4 4 -1 -3}}
| 7.4892
| [[User:Stavats|Stavats]] (2026)
|-
| [[248/247|Lameisma]]
| Thiwonuthuma
| 31o19u3uM
| 248/247
| 2.13.19.31 {{monzo| 3 -1 -1 1 }}
| 6.9949
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[280/279|Tricetone comma]]
| Thiwuzoyoma
| 31uzyM
| 280/279
| 2.3.5.7.31 {{monzo| 3 -2 1 1 -1 }}
| 6.1940
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Junebug comma]]
| Thiwutwenotwethunusotholuzoyoma
| 31u29o23u19u17o3o1uzyM
| 448630/447051
| {{monzo| 1 -1 1 1 -1 1 1 -1 -1 1 -1 }}
| 6.1040
| [[Tristan Bay]] (2024)
|-
| [[341/340|Californisma]]
| Thiwosuloguma
| 31o17u1ogM
| 341/340
| 2.5.11.17.31 {{monzo| -2 -1 1 -1 1 }}
| 5.0844
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[342/341|Endymisma]]
| Thiwunoluma
| 31u19o1uM
| 342/341
| 2.3.11.19.31 {{monzo| 1 2 -1 1 -1 }}
| 5.0695
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[435/434|Chinthisma]]
| Thiwutwenoruyoma
| 31u29oryM
| 435/434
| 2.3.5.7.29.31 {{monzo| -1 1 1 -1 1 -1 }}
| 3.9844
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[465/464|Alektisma]]
| Thiwotwenuyoma
| 31o29uyM
| 465/464
| 2.3.5.29.31 {{monzo| -4 1 1 -1 1 }}
| 3.7271
| [[User:Xenllium|Xenllium]] (2025)
|}

=== Higher-limit commas ===
{| class="wikitable"
! Name(s)
! colspan="2" | [[Kite's color notation/Temperament names|Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[714984/704969|Lightyear comma]]
|
| 89u331o3M
| 714984/704969
| 2.3.31.89 {{monzo| 3 1 3 -3 }}
| 24.421
| [[Budjarn Lambeth]] (2023)
|-
| [[82/81|41-limit Johnston comma]]
| Fowoma
| 41oM
| 82/81
| 2.3.41 {{monzo| 1 -4 1 }}
| 21.242
| [[Stephen Weigel]] (2023)
|-
| [[2883/2848|Lilac comma]]
|
| 89u31ooM
| 2883/2848
| 2.3.31.89 {{monzo| -5 1 2 -1 }}
| 21.146
| [[Budjarn Lambeth]] (2023)
|-
| [[86/85|43-limit 10th-tone]], large quadracesimotertial 1/10-tone
| Fothosuguma
| 43o17ugM
| 86/85
| 2.5.17.43 {{monzo| 1 -1 -1 1 }}
| 20.249
| [[Mister Shaf]] (2025)
|-
| [[87/86|43-limit 10th-tone]], small quadracesimotertial 1/10-tone
| Fothutwenoma
| 43u29oM
| 87/86
| 2.3.29.43 {{monzo| -1 1 1 -1 }}
| 20.014
|
|-
| [[89/88|Tailwind comma]]
|
| 89o1uM
| 89/88
| 2.11.89 {{monzo| -3 -1 1 }}
| 19.562
| [[Budjarn Lambeth]] (2023)
|-
| [[389/385|Rebbe comma]]
|
| 389o1urgM
| 389/385
| 5.7.11.389 {{monzo| -1 -1 -1 1 }}
| 17.8794
| [[Mister Shaf]] (2025)
|-
| [[8277/8192|Lilly pilly comma]]
|
| 89o31oM
| 8277/8192
| 2.3.31.89 {{monzo| -13 1 1 1 }}
| 17.871
| [[Budjarn Lambeth]] (2023)
|-
| [[8000/7921|Incisor comma]]
|
| 89uu3yM
| 8000/7921
| 2.5.89 {{monzo| 6 3 -2 }}
| 17.181
| [[Budjarn Lambeth]] (2023)
|-
| [[129/128|43-limit Johnston comma]]
| Fothoma
| 43oM
| 129/128
| 2.3.43 {{monzo| -7 1 1 }}
| 13.473
| [[Stephen Weigel]] (2023)
|-
| [[979/972|Basement comma]]
|
| 89o1oM
| 979/972
| 2.3.11.89 {{monzo| -2 -5 1 1 }}
| 12.423
| [[Budjarn Lambeth]] (2023)
|-
| [[226/225|Reversed marvel comma]]
|
| 113oggM
| 226/225
| 2.3.5.113 {{monzo| 1 -2 -2 1 }}
| 7.6773
| [[User:Xenllium|Xenllium]] (2025)
|-
| [[Sidereal comma]]
|
| 73u61ogM
| 366/365
| 2.3.5.61.73 {{monzo| 1 1 -1 1 -1 }}
| 4.7366
| [[User:Frostburn|Frostburn]] (2024)
|-
| [[381/380|Five feet comma]]
|
| 127o19ugM
| 381/380
| 2.3.5.19.127 [-2 1 -1 -1 1⟩
| 4.5499
| [[Eliora]] (2022)
|-
| [[481/480|Semaphorisma]]
| Thisothoguma
| 37o3ogM
| 481/480
| 2.3.5.13.37 {{monzo| -5 -1 -1 1 1 }}
| 3.6030
| [[User:Kaiveran|Kaiveran]] (2023)
|}

== List of irrational commas ==
For intervals expressible as edosteps, see [[Interval size measure]]. We skip them here for brevity.

{| class="wikitable"
! Name(s)
! colspan="2" | [[Color name]]
! Ratio
! Monzo
! Cents
! Named by
|-
| [[Caffeinterval]]
|
|
| 2((7/12) - (1/sqrt(3)))
|
| 7.1797
| [[User:R-4981|R-4981]] (2023)
|}

== See also ==
* [[Comma and diesis]]

== References ==

[[Category:Small commas| ]] 
[[Category:Lists of commas]]
{{Todo|complete table|add color name}}