Xenharmonic Wiki - User contributions [en]

Talk:Ternary scale theorems

2026-06-07T21:38:49Z

Sintel: edit

== Citations ==
I'd be interested to know how much of these theorems are restatements of things that are already known in the literature on word combinatorics, versus what is original research. Currently this page does not have any citations, but I would be surprised if it is 100% original work.
– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 21:35, 7 June 2026 (UTC)
: EDIT: nevermind there are some in-text citations. Should use the citation footnote system! – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 21:38, 7 June 2026 (UTC)

Talk:Ternary scale theorems

2026-06-07T21:35:47Z

Sintel: Created page with "== Citations == I'd be interested to know how much of these theorems are restatements of things that are already known in the literature on word combinatorics, versus what is original research. Currently this page does not have any citations, but I would be surprised if it is 100% original work. ~~~~"

9/5

2026-06-07T14:15:39Z

Sintel: dubious

{{Infobox Interval
| Name = just minor seventh, classic(al) minor seventh, ptolemaic minor seventh
| Color name = g7, gu 7th
| Sound = jid_9_5_pluck_adu_dr220.mp3
}}
{{Wikipedia|Minor seventh}}

'''9/5''', the '''just''', '''classic(al)''', or '''ptolemaic minor seventh'''<ref>For reference, see [[5-limit]]. </ref> is often treated as a consonance in [[5-limit]] [[just intonation]], forming a part of such chords such as the 1-6/5-3/2-9/5 minor seventh chord, and the supermajor tetrad, 1-9/7-3/2-9/5 in the 7-limit.

Coincidentally, the ratio between a common "alternative" tuning frequency (A432) and the most common North American AC electrical frequency (60hz) is exactly 36/5, two octaves above 9/5. This is notably a more consonant interval than the 11/6 formed by the more common tuning frequency of A440, which may lead to a noticeable improvement in consonance when electrically powered instruments or amplifiers are interfered with by AC power.{{dubious}}
== Approximation ==
{{Interval edo approximation|9/5}}

== See also ==
* [[10/9]] – its [[octave complement]]
* [[5/3]] – its [[twelfth complement]]
* [[Ed9/5]]
* [[Gallery of just intervals]]

== Notes ==
<references/>

[[Category:Seventh]]
[[Category:Minor seventh]]
[[Category:Over-5 intervals]]

12edo

2026-06-07T00:01:20Z

Sintel: /* Scales */ typo'd

{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament, tempering out [[81/80]], equating four [[3/2|perfect fifths]] with the [[5/1|5th harmonic]].

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of [[just intonation]]. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|312/219]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. The [[duodene]] is an unequal 12-note scale in 5-limit just intonation which observes these commas, and can be considered a [[detempering|detemper]] of 12edo. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], and the schisma, [[32805/32768]], and in the 7-limit, the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented, as well as the [[16:19:24]] otonal minor triad.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite.

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations<ref group="note">Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations</ref>
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775) [[16/15]] (−11.731) [[25/24]] (+29.328)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443) [[17/16]] (−4.955), [[18/17]] (+1.045) [[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910) [[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802) [[17/15]] (−16.687), [[19/17]] (+7.442), [[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865) [[6/5]] (−15.641) [[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847) [[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820) [[5/4]] (+13.686) [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084) [[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955) [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283) [[36/25]] (-31.283) [[45/32]] (+9.776) [[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488) [[24/17]] (+3.000), [[17/12]] (−3.000) [[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955) [[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820) [[8/5]] (−13.686) [[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108) [[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865) [[5/3]] (+15.641) [[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847) [[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910) [[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802) [[30/17]] (+16.687), [[34/19]] (−7.442) [[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775) [[15/8]] (+11.731) [[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039) [[32/17]] (+4.955), [[17/9]] (−1.045) [[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
<references group="note" />

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1) Minor second (m2)
| D# Eb
|-
| 2
| 200
| '''Major second (M2)''' Diminished third (d3)
| '''E''' Fb
|-
| 3
| 300
| Augmented second (A2) '''Minor third (m3)'''
| E# '''F'''
|-
| 4
| 400
| Major third (M3) Diminished fourth (d4)
| F# Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4) Diminished fifth (d5)
| G# Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5) Minor sixth (m6)
| A# Bb
|-
| 9
| 900
| '''Major sixth (M6)''' Diminished seventh (d7)
| '''B''' Cb
|-
| 10
| 1000
| Augmented sixth (A6) '''Minor seventh (m7)'''
| B# '''C'''
|-
| 11
| 1100
| Major seventh (M7) Diminished octave (d8)
| C# Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (♯) and flats (♭) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]] (movable do)
! [[Uniform solfege]] (2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1) Ra (m2)
| Du (A1) Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2) Me (m3)
| Ru (A2) Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3) Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3) Fa (P4)
|-
| 6
| 600
| Fi (A4) Se (d5)
| Pa (A4) Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5) Le (m6)
| Su (A5) Fla (m6)
|-
| 9
| 900
| La
| La (M6) Tho (d7)
|-
| 10
| 1000
| Li (A6) Te (m7)
| Lu (A6) Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7) Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== 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
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this.
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| <abbr title="531441/524288">(12 digits)</abbr>
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| <abbr title="262144/253125">(12 digits)</abbr>
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| <abbr title="67108864/66430125">(16 digits)</abbr>
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| <abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| <abbr title="321489/320000">(12 digits)</abbr>
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| <abbr title="250047/250000">(12 digits)</abbr>
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
<references group="note" />

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
<nowiki>*</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

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 & 72), [[schismic]]/[[garibaldi]] (41 & 53), and [[diaschismic]] (46 & 58). Rank-3 temperaments to which 12et can be detempered include [[marvel]], [[starling]]/[[thrush]], [[aberschismic]], [[orthoschismic]], and [[cassaschismic]]. For more comprehensive lists, see:
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{See also| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic: [[5L 2s]] – 2221221 (generator = 7\12)
* Pentatonic: [[2L 3s]] – 22323 (generator = 7\12)

The diminished and augmented scales are also MOS scales.
* Diminished: [[4L 4s]] – 12121212 (generator = 1\12, period = 3\12)
* Augmented: [[3L 3s]] – 131313 (generator = 1\12, period = 4\12)

Other widely used scales include:
* Melodic minor – 2122221
* Harmonic minor – 2122131
* Harmonic major – 2212131
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] 
[[Category:Historical]]
[[Category:Meantone]]

12edo

2026-06-06T20:37:01Z

Sintel: /* Scales */ Wording, give more obvious examples, remove pdf link

{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament, tempering out [[81/80]], equating four [[3/2|perfect fifths]] with the [[5/1|5th harmonic]].

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of [[just intonation]]. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|312/219]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. The [[duodene]] is an unequal 12-note scale in 5-limit just intonation which observes these commas, and can be considered a [[detempering|detemper]] of 12edo. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], and the schisma, [[32805/32768]], and in the 7-limit, the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented, as well as the [[16:19:24]] otonal minor triad.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite.

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations<ref group="note">Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations</ref>
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775) [[16/15]] (−11.731) [[25/24]] (+29.328)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443) [[17/16]] (−4.955), [[18/17]] (+1.045) [[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910) [[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802) [[17/15]] (−16.687), [[19/17]] (+7.442), [[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865) [[6/5]] (−15.641) [[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847) [[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820) [[5/4]] (+13.686) [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084) [[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955) [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283) [[36/25]] (-31.283) [[45/32]] (+9.776) [[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488) [[24/17]] (+3.000), [[17/12]] (−3.000) [[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955) [[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820) [[8/5]] (−13.686) [[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108) [[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865) [[5/3]] (+15.641) [[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847) [[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910) [[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802) [[30/17]] (+16.687), [[34/19]] (−7.442) [[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775) [[15/8]] (+11.731) [[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039) [[32/17]] (+4.955), [[17/9]] (−1.045) [[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
<references group="note" />

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1) Minor second (m2)
| D# Eb
|-
| 2
| 200
| '''Major second (M2)''' Diminished third (d3)
| '''E''' Fb
|-
| 3
| 300
| Augmented second (A2) '''Minor third (m3)'''
| E# '''F'''
|-
| 4
| 400
| Major third (M3) Diminished fourth (d4)
| F# Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4) Diminished fifth (d5)
| G# Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5) Minor sixth (m6)
| A# Bb
|-
| 9
| 900
| '''Major sixth (M6)''' Diminished seventh (d7)
| '''B''' Cb
|-
| 10
| 1000
| Augmented sixth (A6) '''Minor seventh (m7)'''
| B# '''C'''
|-
| 11
| 1100
| Major seventh (M7) Diminished octave (d8)
| C# Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (♯) and flats (♭) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]] (movable do)
! [[Uniform solfege]] (2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1) Ra (m2)
| Du (A1) Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2) Me (m3)
| Ru (A2) Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3) Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3) Fa (P4)
|-
| 6
| 600
| Fi (A4) Se (d5)
| Pa (A4) Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5) Le (m6)
| Su (A5) Fla (m6)
|-
| 9
| 900
| La
| La (M6) Tho (d7)
|-
| 10
| 1000
| Li (A6) Te (m7)
| Lu (A6) Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7) Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== 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
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this.
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| <abbr title="531441/524288">(12 digits)</abbr>
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| <abbr title="262144/253125">(12 digits)</abbr>
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| <abbr title="67108864/66430125">(16 digits)</abbr>
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| <abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| <abbr title="321489/320000">(12 digits)</abbr>
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| <abbr title="250047/250000">(12 digits)</abbr>
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
<references group="note" />

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
<nowiki>*</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

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 & 72), [[schismic]]/[[garibaldi]] (41 & 53), and [[diaschismic]] (46 & 58). Rank-3 temperaments to which 12et can be detempered include [[marvel]], [[starling]]/[[thrush]], [[aberschismic]], [[orthoschismic]], and [[cassaschismic]]. For more comprehensive lists, see:
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{See also| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic: [[5L 2s]] – 2221221 (generator = 7\12)
* Pentatonic: [[2L 3s]] – 22323 (generator = 7\12)

The diminished and augmented scales are also MOS scales.
* Diminished: [[4L 4s]] – 12121212 (generator = 1\12, period = 3\12)
* Augmented: [[3L 3s]] – 13131313 (generator = 1\12, period = 4\12)

Other widely used scales include:
* Melodic minor – 2122221
* Harmonic minor – 2122131
* Harmonic major – 2212131
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] 
[[Category:Historical]]
[[Category:Meantone]]

Harmonic timbre

2026-06-05T21:40:54Z

Sintel: clarifications and corrections

{{Wikipedia|Harmonic spectrum}}

A [[timbre]] is '''harmonic''' when all [[overtone]]s are integer multiples of the lowest frequency.
All periodic waves are harmonic.
The [[fundamental]], which is (usually) the lowest overtone in the spectrum, determines the [[pitch]] of a musical tone.
When considered as intervals, the sequence of harmonic overtones is called the [[harmonic series]] and is the basis of [[just intonation]].

Examples of harmonic timbres include those of most instruments, the human voice, and most synthesized tones (like saw, square, and triangle waves).

A timbre can be said to be '''nearly harmonic''' if its overtones are approximately equal to integer multiples of the fundamental frequency. Examples include most plucked and hammered string instruments (like violin-family played ''pizzicato'', guitars, harpsichords, and pianos).

A timbre is '''inharmonic''' if its overtones deviate significantly from integer multiples of the fundamental. Examples include membranophones (drums) and idiophones (like xylophones, glockenspiels, and many of the instruments used in [[gamelan]]).

[[Category:Timbre]]

Talk:12edo

2026-06-05T16:29:11Z

Sintel: duodene

{{High priority}}

== Picardy thirds example ==
> Picardy thirds became less common, likely due to the stability of 16:19:24, 12edo's minor triad.

Sounds like a bit of a stretch. Other plausible explanations I can think of include the less pure major triad making the two triads more balanced, or plain aesthetic shifts that have little to do with tuning.

I'm not sure if we wanna go over this level of detail and be this assertive about 12edo's historical influence on music styles.

[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 15:33, 2 August 2025 (UTC)

: Yeah this is BS. Will remove it – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 14:38, 3 August 2025 (UTC)

== Detemperaments ==
Why compress useful detemperaments to a tiny paragraph? Why no rank 3? I doubt that the continuum is is more useful than a select bunch of great detemps. --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 11:09, 2 June 2026 (UTC)

: It's still a select bunch of great detemps, but without bloat. And with clear guidance for where to find more. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 17:21, 2 June 2026 (UTC)

:: Disagree, if anything, continua have a lot of chaff. Schismic/garibaldi/helmholtz, meantone, diaschismic, compton, definitely wheat. Misty, atomic, possibly wheat. The rest is arguably mid. Don't make the reader sort through pages of continua... give them the flour already milled! If they want to explore more, that's fine. As it stood before in my last edit, I believe that's a good list of A-tier 12edo specific detemperaments, which are also solid or extremely good temps either way. --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 18:33, 2 June 2026 (UTC)

::: Put my vote in for making the reader's life easier, although maybe organized further into sub-sections so that a reader that doesn't want all that can skip it without worrying they missed something else, while a reader who wants it can find it easily. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 20:39, 2 June 2026 (UTC)

:::: These are given in ''List of 12et rank two temperaments by complexity'' and ''List of 12et rank two temperaments by badness''. Both are well curated lists and are linked at the start of the rank-2 temperaments section. So plz don't make more duplicate contents. We've had too many of them already. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 21:19, 2 June 2026 (UTC) (updated [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 21:33, 2 June 2026 (UTC))

: Surprisingly [[duodene]] is not mentioned anywhere currently! I want to add it but not sure which section is most relevant. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 16:29, 5 June 2026 (UTC)

Talk:5120/5103

2026-06-01T12:30:30Z

Sintel:

== "Universal" name for 5120/5103 ==

I've noticed the conversation on the XA Discord about picking a name to replace "hemifamity comma". Suggestions I've seen include ''argent comma'' and ''pele comma''. I'm a bit biased towards ''aberschisma'' since I coined the name, but MidnightBlue pointed out that 6¢ is quite wide to be calling it a schisma, which I've also thought about. Maybe ''aberkleisma'' or even ''pentasept comma''?

: I'm not too invested into this comma, but will add my <strike>schisma</strike> 2c after seeing the Discord convo: ''pele comma'' and ''peleisma'' are fine with me, whereas I'm afraid ''argent comma'', while logical, may be confused with the [[argyria]] (which is more of an ''arg''ument for renaming the latter).

: On a side note, 5120/5103 does function like a kleisma for me, particularly because the ratio of the pental kleisma to it is the [[horwell comma]], which is among the staple commas in my 7-limit analysis of edos incl. 53. Because schismic x kleismic product words are among the best ways to make well-tempered 53-note scales, the pental kleisma is a chroma there, and when horwell tempered, it turns into 5120/5103 and is, among other roles, the scale-chroma between the 81/80 and 64/63 steps.

: Meanwhile, the ratio of 5120/5103 to the pental schisma is the [[garischisma]]. So for fans of the latter, which I'm not, it may act like a schisma instead, but that's less likely because the pental schisma flattens the fifth while the garischisma sharpens it, so if anything, the latter and 5120/5103 would be seen as 'negative schismas', which, btw, brings us to the concept of [[counterpyth]].

: Afaik, counterpyth has never been considered under this name without 5120/5103, whereas [[1216/1215]] works well together with other commas that stack slightly sharp fifths, such as the [[wilschisma]] and the [[symbiotic comma]], and the name ''Eratosthenes' comma'' is good, so I disagree with the assignment of the counterpyth family label to any temp with 1216/1215 in sintel's finder. I.e., to me, 5120/5103 is more related to counterpyth than 1216/1215 is. But I can't be sure of my judgment on this without FloraC's opinion. Either way, I don't mind ''counterpyth comma'' for 5120/5103, its 7-limit rank-3 then called counterpyth like its canonical extension to 2.3.5.7.19 already is.

: --[[User:VIxen|VIxen]] ([[User talk:VIxen|talk]]) 23:55, 1 June 2025 (UTC)

:: I'm all for ''argent comma''. The similarity with ''argyria'' isn't high enough to worry me. I'm against ''pele comma'' cuz that would set pele as canon which I don't think we should ever do. For the same reason I'd hesitate to call it ''counterpyth comma''. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 14:17, 2 June 2025 (UTC)

::: OK, then I settle on ''argent comma'' too. That matches my view of argent fifths as a distinct region that's roughly [65\111, 17\29] and sharp of the olympic / garischismic / symbiotic / wilschismic fifths region that's roughly [55\94, 65\111]. --[[User:VIxen|VIxen]] ([[User talk:VIxen|talk]]) 18:38, 2 June 2025 (UTC)

:::: What about 41edo and 46edo? Those are both notable tunings that temper out the comma and have fifths that fall outside of your *argent* range. -- [[User:Tristanbay|Tristanbay]] ([[User talk:Tristanbay|talk]]) 00:35, 12 June 2025 (UTC)

::::: On my part I include 41edo within the argent range; and there's a case to be made that 46edo and 53edo, while notable, aren't truly representative of the intonation of tempering out 5120/5103. "Argent" strictly speaking refers not to a particular tuning range, anyhow, but to a specific tuning that sets the logarithmic ratio of the perfect fifth to the perfect fourth to be sqrt(2):1, for which one can define bands of tolerance around, but which very closely corresponds to the most accurate tunings that temper out this comma. Perhaps "argentisma" -> argentic, argentismic would be clearer, so as not to imply an RTT interpretation for the term "argent temperament" which is already in use.

::::: Compare this to the intonation of counterpyth, which quite distinctly favors tunings of 3/2 far flatter than the optimum of tempering out 5120/5103 by itself: just 19/15 gives us roughly 1/16-comma hemifamity as opposed to just 15/14, 7/5, or 21/20 which provide 1/5, 1/6, and 1/7-comma tunings. For this reason, I oppose seeing counterpyth as a canonical extension to the 7-limit rank-3 {5120/5103} temperament. -- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 01:24, 12 June 2025 (UTC)

::::: I did think about making 24\41 the boundary instead, as rank-3 microtemps tend to have flatter fifths than that even if 152fg or 111 support them. My flat end of argent is surely not flatter than 24\41 and not sharper than 41\70. Between those are kwai fifths... that I may consider too damaged indeed on second thought, and so belonging to the "slightly exo" range codenamed argent. [[User:VIxen|VIxen]] ([[User talk:VIxen|talk]]) 20:23, 19 June 2025 (UTC)

:::: I prefer "Saruyo". It's the only name out of all these suggestions that directly indicates 5120/5103. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 09:19, 5 June 2025 (UTC)

: I have a dumb idea. Why not call it the *pell comma* after the Pell sequence of numbers, whose convergent ratio gives the approximate ratio between an octave and a perfect fourth for the optimal tuning of the temperament? [[User:Tristanbay|Tristanbay]] ([[User talk:Tristanbay|talk]]) 20:59, 18 September 2025 (UTC)

:: I think referencing the Pell sequence makes way more sense for a member of the family of commas going 50/49, 289/288, 1682/1681, 9801/9800, etc. The only relation of Pell numbers to 5120/5103 is the edo sequence, which seems rather secondary, as much as I'm a promoter of 239edo. --[[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 22:42, 18 September 2025 (UTC)

: I propose the name "interkleisma", since 5120/5103 is the difference between 64/63 and 81/80 (the main formal commas for primes 5 and 7), and is around a kleisma in size.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 03:02, 13 February 2026 (UTC)

:: It's a half kleisma in 270edo (and 311edo if you consider other kleismata such as 1029/1024) tho. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 09:20, 13 February 2026 (UTC)

::: Your essay on the 13-limit JI space considers 5120/5103~352/351~847/845, 325/324~385/384, 364/363~441/440, 540/539~729/728, and 351/350 as kleismas. Even if it is a half-kleisma in 270edo, the comma is close enough to the rough interval region, and also no single edo should decide the name.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 16:58, 13 February 2026 (UTC)

:::: Oh wow, my bad. I'll change them to hemikleismata. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 21:39, 13 February 2026 (UTC)

Should we do a poll here? The name was basically pre-maturely changed according to a poll on XA Discord. Besides, we need to decide what to do with the temp's name. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 12:51, 22 January 2026 (UTC)

: If enough people want to, then I guess. I like the current name of this comma, and I was thinking of the associated full 7-limit temperament being "argentic" and the 2.3.7/5 subgroup one being "argic". [[User:Tristanbay|Tristanbay]] ([[User talk:Tristanbay|talk]]) 06:12, 27 January 2026 (UTC)

: Why not [[64/63|S8]]/[[81/80|S9]]? Are there many properties of this comma that aren't explained by it being ((8/7)/(9/8)) / ((9/8)/(10/9)) = (64/63) / (81/80) and hence ([[10/7]])/([[9/8]])3? --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 12:27, 16 February 2026 (UTC)

:: Because "ess eight over ess nine" is too many syllables. [[User:Tristanbay|Tristanbay]] ([[User talk:Tristanbay|talk]]) 05:34, 17 February 2026 (UTC)

: I'm fine calling it saruyoma, y'all sort this out. --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 10:04, 6 May 2026 (UTC)

=== Petition to officialize ''aberschisma'', and change ''hemifamity'' to ''aberschismic'' ===
At this point, ''aberschisma'' and ''aberschismic'' seem like the most widely liked and used names in xenharmonic communities. Thereby I request ''aberschisma'' be set as the permanent, main name for 5120/5103, and ''hemifamity'' be officially changed to ''aberschismic''.

Please put your "yes" or "no" and signature below. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 05:56, 1 June 2026 (UTC)

# '''Yes'''. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 05:56, 1 June 2026 (UTC)
# Yes, sure, why not. --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 07:48, 1 June 2026 (UTC)

: I was under the impression that "argent" won out in the community? – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 12:30, 1 June 2026 (UTC)

Schisma

2026-06-01T01:16:35Z

Sintel: Undo revision 231381 by Godtone (talk)

{{Interwiki
| en = 32805/32768
| de = 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]]).

== Temperaments ==
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.

== 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.

== 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 is also very close in size—about 0.0013{{c}} off—from the difference between 3/2 and 7\12, which is about 1.9550009{{c}}. 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]]

Cangwu badness

2026-05-30T21:53:54Z

Sintel: fix headings and some math formatting

''Cangwu badness'' is a polynomial function [[Badness|badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham_Breed|Graham Breed]] thought it up.

== Definitions ==

In the following definitions, suppose that <math>C(T)</math> is the [[Tenney-Euclidean_temperament_measures#TE_Complexity|TE Complexity]], and <math>E_R(T)</math> is the [[Tenney-Euclidean_temperament_measures#TE_simple_badness|"relativized error"]] of temperament <math>T</math>. The "relativized error" for EDOs gives the error as a proportion of the step size, and yields a similar result for higher-rank temperaments; we may also call it the "TE Simple Badness." The Cangwu badness, then, can be most simply thought of as a weighted RMS of the two. The main feature of Cangwu badness is that there is a free '''weighting''' parameter determining how much we are weighting the error vs complexity into the calculation.

There are several different conventions for the Cangwu badness, usually corresponding to a choice of "units" for the free parameter. The most common is Graham's <math>E_k</math>, which can be thought of as describing the "intended target error in cents" in temperament searches, and also has sometimes written in terms of a simplified <math>\epsilon</math> parameter that makes the entire thing a weighted RMS. Gene has a separate derivation in terms of matrix determinants that is equivalent to the square of the preceeding definition. There are also some different conventions regarding scalar normalization of the rank, and whether the norm used is an <math>\ell_2</math> norm ("root-sum-squared") or an RMS ("root-mean-squared").

However, these definitions are all equivalent, up to some monotonic transformation that preserves the relative ratings of each temperament, given any choice of weighting and temperament rank. We will begin with Graham's <math>\epsilon</math> definition as it is simplest.

=== Simplified Version of Graham's Definition ===

Let <math>\epsilon</math> be a free parameter in <math>[0,1]</math>. The Cangwu badness is then simply defined via the weighted RMS

:<math>
\sqrt{(1-\epsilon^2)E_R(T)^2 + \epsilon^2 C(T)^2}
</math>

where the squaring is just due to convention and/or for numerical stability reasons.

This is equivalent to the definition used in Graham's regular temperament python library. It is also equivalent to the definitions in [http://x31eq.com/te.pdf "Tenney-Euclidean Formulas"], [http://x31eq.com/badness.pdf "Parametric Scalar Badness"], and [http://x31eq.com/primerr.pdf "Prime Based Error and Complexity Measures"], except for a multiplication by a scalar "rank-normalization" term that is constant for any choice of <math>\epsilon</math> or rank.

=== Simplified Version of Graham's Definition With "Intended Cents of Error" Parameter ===

Graham tends to use a change of variables so that the free parameter can be viewed as a number of cents representing the "intended target error" in temperament searches. To that extent, we define the change of variables

:<math>
\begin{aligned}
\epsilon^2 = \frac{E_k^2}{1+E_k^2} \\
E_k^2 = \frac{\epsilon^2}{1-\epsilon^2}
\end{aligned}
</math>

So that <math>E_k</math> can now be thought of as a real number in <math>[0,\infty)</math>. This leads to the formula

:<math>
\sqrt{\frac{E_R(T)^2 + E_k^2 C(T)^2}{1+E_k^2}}
</math>

This formula is also equivalent to the one used in Graham's python library in terms of the <math>E_k</math> parameter, and is again equivalent to the version in his articles except for the aforementioned scalar multiplication.

=== Graham's Original Definition ===

Graham's original definition, in terms of <math>E_k</math>, can be written as

:<math>
\sqrt{\frac{E_R(T)^2 + E_k^2 C(T)^2}{1+E_k^2}} \cdot \left(\sqrt{1+E_k^2}\right)^{\text{rank}(T)}
</math>

This can be shown to be identical to the metric given in Graham's three articles linked above, although he originally defined it in terms of a large matrix formula. It is also equivalent to the metrics previously given, except for the presence of the added term

:<math>
\left(\sqrt{1+E_k^2}\right)^{\text{rank}(T)}
</math>

which, for any choice of <math>E_k</math> or <math>\text{rank}(T)</math>, multiplies all temperaments only by a scalar.

=== Gene Smith's Matrix Determinant Definition ===

[[Gene Ward Smith]] wrote the below presentation of Cangwu badness in terms of the characteristic polynomial of a matrix. The definition is again equivalent to the above, except it is equivalent to the *square* of the above Cangwu badness, and again involving equivalence only up to a scalar multiplication.

Note that the notation below is not, in general, the same as the notation above; Gene's original notation has been left as written, where the free parameter is called <math>x</math>.

Gene's definition of the Cangwu badness is in terms

:<math>
C(x) = \det\left(\left[(1+x)\frac{v_i \cdot v_j}{n} - a_ia_j\right]\right)
</math>

where the vᵢ are r independent weighted vals of dimension n defining a rank r regular temperament, and the aᵢ are the average of the values of vᵢ; that is, the sum of the values of vᵢ divided by n.

From this definition, it follows that C(0) is proportional to the square of [[Tenney-Euclidean_temperament_measures#TE simple badness|simple badness]], aka relative error. The leading coefficient of the polynomial defining C(x) is proportional to the square of [[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]]. Hence as x grows larger, C(x) increasingly weights smaller complexity temperaments as better, and in the limit becomes a complexity measure, whereas at 0 C(0) is a badness measure which strongly favors more accurate temperaments, being in fact a measure of relative error.

== Cangwu Dominance ==
If Ca(x) is the Cangwu badness for rank r temperament 'a', and Cb(x) for 'b', we can say a ''dominates'' b if Cb(x) - Ca(x) is a positive function for x ≥ 0, which says that the badness of 'a' is always less than the badness of 'b' for every choice of the parameter x. If a temperament is not dominated by any temperament of the same rank (and on the same subgroup, if we are considering subgroups) we may say it is ''indomitable''. An alternative procedure is to divide the cangwu badness polynomial by the coefficient of the highest degree term, getting a monic polynomial with constant term proportional to the the square of TE absolute error. When one temperament dominates another using this polynomial, it is lower in both error and complexity.

=== Examples of Indomitable Temperaments ===

Examples of 5-limit indomitable temperaments are:

Father 16/15 |4 -1 -1>

Dicot 25/24 |-3 -1 2>

Meantone 81/80 |-4 4 -1>

Srutal/Diaschismic 2048/2025 |11 -4 -2>

Hanson/Kleismic 15625/15552 |-6 -5 6>

Helmholtz/Schismic 32805/32768 |-15 8 1>

Hemithirds |38 -2 -15>

Ennealimmal |1 -27 18>

Kwazy |-53 10 16>

Monzismic |54 -37 2>

Senior |-17 62 -35>

Pirate |-90 -15 49>

Atomic |161 -84 -12>

Some 7-limit examples are beep, meantone, miracle, pontiac and ennealimmal.

If two temperaments of the same rank are such that neither dominants the other, we may subtract one Cangwu badness polynomial from the other and find the positive root of the result. This gives a value of the parameter 'x' at which the two temperaments are rated equal in badness, which can be applied to rate other temperaments by badness. For example, if 5-limit father and helmholtz are made equally bad, then meantone, augmented, dicot, porcupine, srutal, diminished, magic, hanson and mavila, in that order, rate as better.
[[Category:badness]]

Talk:39edo

2026-05-30T19:47:57Z

Sintel:

== 39dfgijk ==

If you don't want the second row of odd harmonics (or prime harmonics if switched to that), you should also get rid of "39edo can be usefully mapped onto the val 39dfgijk", since the argument about higher harmonics being too inaccurate would make this val not so useful. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 03:04, 2 April 2026 (UTC)

: Octave compression certainly makes the higher harmonics more accurate, though one needs to be careful about intervals with many powers of 2 (and also 11, since it loses accuracy at that level of compression). A second row won't do too much harm, so I guess adding it back is fine. [[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 03:54, 2 April 2026 (UTC)

: Also, when you added the second table you added an extra line between the templates, which makes them more spaced apart than they should be. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 03:57, 2 April 2026 (UTC)

== 39 isn't a dual-7 edo ==
39d is clearly the best val up to the 11-limit and 39 patent should not be put in the interval table as a competing column (39df might be considered as the 13-limit mapping tho that's besides the point), for mostly the same reason 44d should not as I showed in Talk: 44edo.

To be clear, the question of a dual-prime edo concerns whether two mappings are nearly equally valid. If one mapping is considerably more accurate, it is hard for one to hear the other mapping as a valid approximation to the same set of intervals, since their presence in the same tuning system means the difference in quality is highlighted thru contrast. As such, for many edo articles we present a main mapping most useful for composition. This mapping is discussed at length in the theory section and put in the interval table. The distinction of a main mapping and various ancillary mappings is a consistent feature of edo articles on this wiki.

The ancillary mappings can also be used, and may be interesting for various reasons. I think they deserve to be discussed briefly in the theory section. However, we can't afford to put whatever we think is potentially or marginally useful in the interval table, cuz human readers have limited attention resource and wish to spend it on the best things. A less valid mapping in the interval table means divided attention and less efficiency of presenting information.

For example, in 145edo, there is this short sentence discussing the utility of a less accurate mapping: "The 145c val provides a tuning for magic which is nearly identical to the POTE tuning." But the main mapping is discussed in the rest of the article.

The reasons that 39d commends itself as the main mapping are mostly the same as that for 44. Specifically:
* The sharp 3, 5, 11 justifies the sharp 7. The interactions of 7 with 3, 5, 9, 11, and 15 all favor the sharp mapping. Iow 7 itself is the only inconsistently mapped interval in the 11-limit 15-odd-limit. While this is also true for 34edo, which is treated as dual-7, 39edo differs from 34edo in that the other primes and especially the 5 are very sharp, which brings us to …
* With the flat 7, the 7/5 will have 93% error and the 15/14 will have ''112%'' error, whereas with the sharp 7, the maximum error comes from 7 itself, only 51%.
* TE error for 39d: 2.43 cents; 39dee: 3.13 cents; 39: 3.79 cents. Note that 39dee has a lower error than 39, so if 39dee isn't reasonable to consider, neither is 39 logically.

The only difference here is that the flat-7 mapping is a patent val. On that account one might argue that the mapping is of some special importance. I think the value of patentness has been overstated in the community at large. What we mean by a patent val is really using the closest approximation for the basis elements, but basis elements can change. For example, many ppl consider 5/3 and/or 7/6 to be as important in composition as 5/4 and 7/4, and one can generate the 7-limit with 2, 3, 5/3, and 7/6. In this basis, the patent val for 39edo isn't the same as the one found for 2, 3, 5, and 7. In fact it's the sharp-7 mapping. That reveals the lack of unique significance of patent vals in practice (and in math, as every GPV is demonstrably patent in some way); as such the importance of a mapping solely from being a patent val in this specific case is baseless from a broader perspective.

—[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 21:51, 29 May 2026 (UTC)

: Overall, I agree that 39d feels more natural to use. However, the wiki is supposed to present info from a neutral point of view rather than pushing a perspective. Not everyone agrees that more accurate necessarily means "better", and that patent vals are completely arbitrary. People often think of the octave as the equivalence interval, so they want to keep it pure. The pure-octave patent val with prime harmonics as basis entries feels like the most natural mapping to use for many people, even if it is less accurate overall. The patent val isn't completely uninteresting, supporting structures like immunity and triforce. 39edo is a medium-sized edo, and someone who uses it very much may not be focused on accuracy.

: Overall, I think 39 and 39d should have about equal coverage, with structures in both presented. The page definitely should explain how 39d improves accuracy of many intervals, and it should be up to the reader to decide which perspective they agree with, and which mapping to use. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 17:08, 30 May 2026 (UTC)

:: That sounds very sensible — put my vote in for that as well. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 18:36, 30 May 2026 (UTC)

: I have no opinion on 39et in particular, but I will agree that "patentness" is a completely arbitrary quality and the procedure of rounding is merely heuristic to get something that usually works. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 19:47, 30 May 2026 (UTC)

Antares

2026-05-28T14:42:43Z

Sintel: -TempClean

'''Antares''' is the [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] of the 3.5.7 [[subgroup]] that [[tempering out|tempers out]] [[21875/19683]], the amount by which, assuming [[tritave]] [[equivalence]], 6 [[9/5]] fall short of [[7/5]] or equivalently the amount by which 6 [[5/3]] exceed [[15/7]]. It is basically the [[Arcturus]] equivalent of [[mavila]], and is even more inaccurate than mavila. It falls into [[exotemperament]] range with the "comma" it tempers out being a small major second (~182.8{{cent}}). 9/5 is extremely sharp and becomes a [[neutral]] seventh, and 7/5 is mapped to a [[supermajor]] third!

== Etymology ==
This temperament is named after the star {{w|Antares}}, following a series of [[nonoctave]] temperaments that are named after stars.
{{todo|add etymology|inline=1|text=Add name (person who coined the term) and year (when it was coined).}}

[[Category:Antares| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Non-octave temperaments]]

{{Todo|inline=1|expand|link}}

Miaplacidus

2026-05-28T14:42:16Z

Sintel: -TempClean

'''Miaplacidus''' is a [[tritave]]-repeating [[rank-2 temperament]] that tempers out the comma 5859375/5764801 in the 3.5.7 subgroup, the amount by which 4 [[15/7]]'s exceed [[9/7]]. It thus tunes the chord 7:9:15 accurately. Having a generator of ~[[15/7]], it possesses [[MOS scale]]s of the families 3L 4s<3/1>, 3L 7s<3/1>, 3L 10s<3/1>, and 13L 3s<3/1>. Good tunings for this temperament are [[13edt]], [[16edt]], and [[29edt]].
== Interval chain (POTE tuning) ==
{|class="wikitable"
|-
! Generators up
! Cents
! Generators down
! Cents
|-
| 0
| 0.00
| 0
| 0.00
|-
| 1
| 1316.10
| 1
| 585.86
|-
| 2
| 730.24
| 2
| 1171.72
|-
| 3
| 144.38
| 3
| 1757.58
|-
| 4
| 1460.48
| 4
| 441.48
|-
| 5
| 874.62
| 5
| 1027.34
|}

[[Category:Miaplacidus| ]] <-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Non-octave temperaments]]

{{Stub}}

Hyperpyth

2026-05-28T14:42:03Z

Sintel: -TempClean

'''Hyperpyth temperament''' is a pentave-based 5.9.13 subgroup temperament which tempers out 28561/28125 (quadtho-aquingu comma).

== Hyperpyth ==
Using the fifth harmonic ([[5/1]], pentave) as an interval of equivalence, instead of the more common octave or even [[tritave]], the first place to look for xenharmonies is isoharmonic chords, of which there are two. 1:2:3:4:5, and 5:9:13:17:21:25. The latter is more xenharmonic, though 16ed5 suits the former and can be strange. It turns out that the key to making a scale of this chord is to use as the generator ~13/5, which, in the fifth harmonic, results in a scale of the same shape as meantone and similar placement of consonances. Thus it is a [[Macrodiatonic and microdiatonic scales|macrodiatonic]] tuning. In this case, the most suitable scales are such as would have a sharp fifth, in octaves known as "superpythagorean", so I dub this "hyperpyth".

The quintessential comma of which is 28561/28125, wherein (13 the "perfect fifth")^4 = 9 (the "major third") and 5's are fungible. 13^3 (ie. a "major sixth") can also constitute the 17/5 interval, and 21/5 is liable to be an augmented sixth, (and a bonus, 19/5 can be found as a dominant seventh). This is eerily similar to the case in meantone (especially if you call the sixth a 13/8). [http://x31eq.com/cgi-bin/rt.cgi?ets=c22_c5&limit=5_9_13]

Good tunings for hyperpyth are:

* [[5ed5|5ED5]]
* [[10ed5|10ED5]]
* [[17ed5|17ED5]]
* [[22ed5|22ED5]]
* [[27ed5|27ED5]]
* [[29ed5|29ED5]]
* [[39ed5|39ED5]]

etc.

== Hyperreich? ==
{{main|Juggernaut}}

Looking at the primes, 7 and 11 (and 19) are "conspicuously absent" which begs comparison to the Meantone/Orgone dichotomy. The search being on, in the context of simple scales, 11/5 is close enough to the square root of 5, that one might as well just use it (1393 v the real 11/5 at 1365 cents); eventually as step sizes get closer to 60 cents or so, better approximations will abound. This would make a good period for a scale. The pure 7/5 then is around 582 cents, and among the simpler temperaments 557-cent (from 5ED5, 10ED5, 15ED5) and 596-cent (from [[14ed5|14ED5]], which is a slightly compressed [[6edo|6EDO]]) intervals are the closest approximations. That is, until [[19ed5|19ED5]] (14+5) which is a very slightly stretched [[13edt|13EDT]] (Bohlen-Pierce) scale, and [[24ed5|24ED5]] which is something completely different.

[[Category:Hyperpyth| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Non-octave temperaments]]

Diaschismic extensions

2026-05-28T14:41:53Z

Sintel: -TempClean

{{Breadcrumb|Diaschismic}}
In the [[5-limit]], '''diaschismic''' is a [[regular temperament]] (also known as ''srutal'', though they refer to different extensions in higher limits) defined by [[tempering out]] the comma [[2048/2025]] = [11 -4 -2⟩, the diaschisma. The octave is split into two periods, each representing [[~]][[45/32]]~[[64/45]]; and the [[generator]] can be considered to be a perfect fifth (~[[3/2]]), or a perfect fifth less a period, which is a diatonic semitone of ~[[16/15]]. Tempering out the diaschisma implies that two of these semitones are equated to [[9/8]], and as [[9/8]] = ([[18/17]])([[17/16]]), ~[[16/15]] can very naturally be equated to 17/16 and 18/17 as well, producing a 2.3.5.17 [[subgroup]] extension known as '''srutal archagall''', whose commas are [[136/135]] and [[256/255]]. There are multiple ways to extend diaschismic to primes [[7/1|7]], [[11/1|11]], and [[13/1|13]].

== 7-limit extensions ==
The two alternative names for this temperament are assigned to different strong extensions to the [[7-limit]]: srutal (34d&46) and diaschismic (46&58), though there are other mappings that are comparable in complexity and error: [[pajara]] (12&22) and keen (22&34).

=== Srutal ===
Srutal tempers out [[4375/4374]] in addition to the diaschisma, and therefore [[7/4]] is represented by 15 semitones less a half octave, or five [[6/5]]s less a half octave.

For technical data on 7-limit and higher-limit srutal: see [[Diaschismic family #Srutal]].

=== Diaschismic ===
Diaschismic sacrifices a slight amount of accuracy by tempering out [[126/125]], but slightly reduces complexity: [[8/7]] is represented by 8 semitones less a half-octave, or we can say 7/4 is equated to four [[5/4]]s less a half octave.

For technical data on 7-limit and higher-limit diaschismic: see [[Diaschismic family #Septimal diaschismic]].

Both of these can be extended straightforwardly to the [[11-limit|11-]], [[13-limit|13-]], and [[17-limit]] by adding 176/175, 352/351, and 221/220 to the comma list in this order. The extensions to prime [[11/1|11]] and [[13/1|13]] can be characterized by [[parapyth]], which makes sense as the fifth is tuned slightly sharp, and prime 17 is found via srutal archagall.

=== Pajara ===
[[Pajara]] combines diaschismic with [[archy]], tempering the fifth to about 709 cents. The interval of two stacked fifths is equated to 16/7, and the harmonic seventh [[7/4]] and the just major third [[5/4]] are separated by a perfect semioctave.

For technical data on 7-limit and higher-limit pajara, see [[Diaschismic family #Pajara]].

=== Keen ===
{{todo|inline=1|complete section}}

== Intervals ==
{{todo|cleanup|complete section|inline=1}}
=== Diaschismic (2.3.5.7.17) ===
<div><div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | First period
|-
! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate Ratios
|-
! 2.3.5.17 subgroup !! Intervals of 7
|-
| −8 || 370.6 || 100/81 || 21/17, 56/45
|-
| −7 || 474.2 || 125/96 || '''21/16''', 112/85
|-
| −6 || 577.9 || 25/18 || 7/5
|-
| −5 || 81.6 || 25/24 || 21/20
|-
| −4 || 185.3 || 10/9, 75/68 || 28/25
|-
| −3 || 289.0 || 20/17, 32/27 || 119/100
|-
| −2 || 392.6 || '''5/4''', 34/27, 64/51 || 63/50
|-
| −1 || 496.3 || 4/3, 45/34 || 168/125
|-
| 0 || 0.0 || '''1/1''' || 126/125
|-
| 1 || 103.7 || 16/15, '''17/16''', 18/17 ||
|-
| 2 || 207.4 || '''9/8''', 17/15 || 125/112
|-
| 3 || 311.0 || 6/5, 81/68 || 25/21
|-
| 4 || 414.7 || 32/25, 51/40, 81/64 || 80/63
|-
| 5 || 518.4 || 27/20, 34/25 || 75/56, 85/63
|-
| 6 || 22.1 || 51/50, 81/80 || 85/84
|-
| 7 || 125.8 || 27/25 || 15/14, 68/63
|-
| 8 || 229.5 || 144/125 || 8/7
|}
<nowiki />* In 7-limit [[POTE]] tuning
</div>

<div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Second period
|-
! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate Ratios
|-
! 2.3.5.17 subgroup !! Intervals of 7
|-
| −8 || 970.6 || 125/72 || '''7/4'''
|-
| −7 || 1074.2 || 50/27 || 28/15, 63/34
|-
| −6 || 1177.9 || 100/51, 160/81 || 168/85
|-
| −5 || 681.6 || 40/27, 25/17 || 112/75, 126/85
|-
| −4 || 785.3 || '''25/16''', 80/51, 128/81 || 63/40
|-
| −3 || 889.0 || 5/3, 136/81 || 42/25
|-
| −2 || 992.6 || 16/9, 30/17 || 224/125
|-
| −1 || 1096.3 || '''15/8''', 17/9, 32/17 ||
|-
| 0 || 600.0 || 17/12, 24/17, 45/32, 64/45 ||
|-
| 1 || 703.7 || '''3/2''', 68/45 || 125/84
|-
| 2 || 807.4 || 8/5, 27/17, 51/32 || 100/63
|-
| 3 || 911.0 || 17/10, '''27/16''' || 200/119
|-
| 4 || 1014.7 || 9/5, 136/75 || 25/14
|-
| 5 || 1118.4 || 48/25 || 40/21
|-
| 6 || 622.1 || 36/25 || 10/7
|-
| 7 || 725.8 || 192/125 || 32/21, 85/56
|-
| 8 || 829.5 || 81/50 || 34/21, 45/28
|}
<nowiki />* In 7-limit [[POTE]] tuning
</div>

=== Srutal (17-limit) ===
<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
<div style="line-height: 1.6;">'''Intervals of srutal {{nowrap|(34d & 46)}}'''</div>
<div class="mw-collapsible-content">
{| class="wikitable"
|-
! Generator
! −17
! −16
! −15
! −14
! −13
! −12
|-
! Cents*
| 17.73
| 122.57
| 227.40
| 332.24
| 437.08
| 541.92
|-
! Ratios
|
| 15/14
| 8/7
| 17/14
| 9/7
| 15/11
|-
! Generator
! −11
! −10
! −9
! −8
! −7
! −6
|-
! Cents*
| 46.76
| 151.60
| 256.44
| 361.28
| 466.12
| 570.96
|-
! Ratios
|
| 12/11
| 15/13
| 16/13
| 17/13
| 18/13
|-
! Generator
! −5
! −4
! −3
! −2
! −1
! 0
|-
! Cents*
| 75.80
| 180.64
| 285.48
| 390.32
| 495.16
| 600.00
|-
! Ratios
| 22/21
| 10/9
| 20/17, 13/11
| 5/4
| 4/3
| 24/17, 17/12
|-
! Generator
! 0
! 1
! 2
! 3
! 4
! 5
|-
! Cents*
| 0.00
| 104.84
| 209.68
| 314.52
| 419.36
| 524.20
|-
! Ratios
| 1/1
| 18/17, 17/16, 16/15
| 9/8, 17/15
| 6/5
| 14/11
|
|-
! Generator
! 6
! 7
! 8
! 9
! 10
! 11
|-
! Cents*
| 29.04
| 133.88
| 238.72
| 343.56
| 448.40
| 553.24
|-
! Ratios
|
| 14/13, 13/12
|
| 11/9
| 22/17, 13/10
| 11/8
|-
! Generator
! 12
! 13
! 14
! 15
! 16
! 17
|-
! Cents*
| 58.08
| 162.92
| 267.76
| 372.60
| 477.43
| 582.27
|-
! Ratios
|
| 11/10
| 7/6
| 21/17
| 21/16
| 7/5
|}
<nowiki />* In 17-limit POTE tuning
</div></div>

== Scales ==
* [[Srutal12]] – proper [[10L 2s]]
* [[Srutal22]] – improper [[12L 10s]]
* [[Diaschismic12]] – proper [[10L 2s]]
* [[Diaschismic22]] – improper [[12L 10s]]
* [[Diaschismic34]] – improper [[12L 22s]]

== See also ==
* [[Lumatone mapping for diaschismic]]

[[Category:Temperament extensions]]
[[Category:Srutal]]
[[Category:Diaschismic]]

3L 1s (3/2-equivalent)

2026-05-28T14:41:43Z

Sintel: -TempClean

{{Infobox MOS|Equalized=1|Collapsed=1|Pattern=LLLs}}
{{MOS intro}} The so-called "Super Ultra Hyper Mega Meta Lydian" is one mode of this mos.

The notation "<3/2>" means the period of the MOS is 3/2, disambiguating it from octave-repeating [[3L 1s]]. The name of the period interval is called the '''sesquitave''' (by analogy to the [[tritave]]). The generator range is 171.4 to 240{{c}}, placing it near the [[9/8|diatonic major second]], usually representing a major second of some type. The dark (chroma-negative) generator is, however, its fifth complement (480 to 514.3{{c}}).

In the fifth-repeating version of the diatonic scale, each tone has a 3/2 perfect fifth above it. The scale has two major chords and two minor chords.

'''Angel''' is a proposed name for this mos. [[Basic]] Angel is in [[7edf]], which is a very good fifth-based equal tuning similar to [[12edo]].

== Notation ==
There are 4 main ways to notate the angel scale. One method uses a simple sesquitave (fifth) repeating notation consisting of 4 naturals (eg. Do Re Mi Fa, Sol La Si Do). Given that {{dash|1, 5/4, 5/3}} is fifth-equivalent to a tone cluster of {{dash|1, 10/9, 5/4}}, it may be more convenient to notate angel scales as repeating at the double, triple or quadruple sesquitave (major ninth, thirteenth or seventeenth i. e. ~pentave), however it does make navigating the [[Generator|genchain]] harder. This way, 5/3 is its own pitch class, distinct from 10/9. Notating this way produces a major ninth which is the Aeolian mode of Napoli[6L 2s], a major thirteenth which is the Dorian mode of Bijou[9L 3s] or an ~pentave which is the Mixolydian mode of Hextone[12L 4s]. Since there are exactly 8 naturals in double sesquitave notation, 12 in triple sesquitave notation and 16 in quadruple sesquitave notation, letters A–H (FGABHCDEF) or dozenal or hex digits (0123456789XE0 or D1234567FGACD with flats written C molle or 0123456789ABCDEF0 or G123456789ABCDEFG with flats written F molle) may be used.

{| class="wikitable"
|+ style="font-size: 105%;" | Cents<ref name=":0">Fractions repeating more than 4 digits written as continued fractions</ref>
|-
! colspan="4" | Notation
! Supersoft
! Soft
! Semisoft
! Basic
! Semihard
! Hard
! Superhard
|-
! Diatonic
! Napoli
! Bijou
! Hextone
! ~15edf
! ~11edf
! ~18edf
! ~7edf
! ~17edf
! ~10edf
! ~13edf
|-
| Do#, Sol#
| F#
| 0#, D#
| 0#, G#
| 1\1546; 6.5
| 1\1163: 6.{{Overline|3}}
| 2\1877; 2, 2.6
| rowspan="2" | 1\7 100
| 3\17124; 7.25
| 2\10141; 5.{{Overline|6}}
| 3\13 163.{{Overline|63}}
|-
| Reb, Lab
| Gb
| 1b, 1c
| 1f
| 3\15138; 3.25
| 2\11126; 3.1{{Overline|6}}
| 3\18116; 7.75
| 2\1782; 1.3{{Overline|18}}
| 1\1070; 1.7
| 1\13 54.{{Overline|54}}
|-
| '''Re, La'''
| '''G'''
| '''1'''
| '''1'''
| '''4\15''''''184; 1.625'''
| '''3\11''''''189; 2.{{Overline|1}}'''
| '''5\18''''''193; 1, 1, 4.{{Overline|6}}'''
| '''2\7''' '''200'''
| '''5\17''''''206; 1, 8.{{Overline|6}}'''
| '''3\10''''''211; 1, 3.25'''
| '''4\13''' '''218.{{Overline|18}}'''
|-
| Re#, La#
| G#
| 1#
| 1#
| 5\15230; 1.3
| 4\11252; 1.58{{Overline|3}}
| 7\18270; 1.0{{Overline|3}}
| rowspan="2" | 3\7 300
| 8\17331; 29
| 5\10352; 1.0625
| 7\13 381.{{Overline|81}}
|-
| Mib, Sib
| Ab
| 2b, 2c
| 2f
| 7\15323; 13
| 5\11315; 1.2{{Overline|6}}
| 8\18309; 1, 2.1
| 7\17289; 1, 1.9
| 4\10282; 2.8{{Overline|3}}
| 5\13 272.{{Overline|72}}
|-
| Mi, Si
| A
| 2
| 2
| 8\15369; 4.{{Overline|3}}
| 6\11378; 1.0{{Overline|5}}
| 10\18387; 10.{{Overline|3}}
| 4\7 400
| 10\17413; 1, 3.8{{Overline|3}}
| 6\10423; 1.{{Overline|8}}
| 8\13 436.{{Overline|36}}
|-
| Mi#, Si#
| A#
| 2#
| 2#
| 9\15415; 2.6
| rowspan="2" | 7\11442; 9.5
| 12\18464; 1.0625
| 5\7 500
| 13\17537; 14.5
| 8\10564; 1.41{{Overline|6}}
| 11\13 600
|-
| Fab, Dob
| Bbb
| 3b, 3c
| 3f
| 10\15461; 1, 1.1{{Overline|6}}
| 11\18425; 1.24
| 4\7 400
| 9\17372; 2.41{{Overline|6}}
| 5\10352; 1.0625
| 6\13 327.{{Overline|27}}
|-
| '''Fa, Do'''
| '''Bb'''
| '''3'''
| '''3'''
| '''11\15''''''507; 1.{{Overline|4}}'''
| '''8\11''''''505; 3.8'''
| '''13\18''''''503; 4, 2.{{Overline|3}}'''
| '''5\7''' '''500'''
| '''12\17''''''496; 1.8125'''
| '''7\10''''''494; 8.5'''
| '''9\13''' '''490.{{Overline|90}}'''
|-
| Fa#, Do#
| B
| 3#
| 3#
| 12\15553; 1.{{Overline|18}}
| 9\11568; 2.375
| 15\18580; 1.55
| 6\7 600
| 15\17620; 1.45
| 9\10635; 3.4
| 12\13 654.{{Overline|54}}
|-
| Fax, Dox
| B#
| 3x
| 3x
| 13\15 600
| rowspan="2" | 10\11 631; 1.{{Overline|72}}
| 17\18 658; 15.5
| 7\7 700
| 18\17 744; 1.208{{Overline|3}}
| 11\10 776; 2.125
| 15\13 818.{{Overline|18}}
|-
| Dob, Solb
| Hb
| 4b, 4c
| 4f
| 14\15 646; 6.5
| 16\18 619; 2.{{Overline|81}}
| 6\7 600
| 14\17 579; 3.{{Overline|2}}
| 8\10564; 1.41{{Overline|6}}
| 10\13 545.{{Overline|45}}
|-
! Do, Sol
! H
! 4
! 4
! '''15\15''' '''692; 3.25'''
! '''11\11''' '''694; 1, 2.8'''
! '''18\18''' '''696; 1.291'''{{Overline|6}}
! '''7\7''' '''700'''
! '''17\17''' '''703; 2, 2.1'''{{Overline|6}}
! '''10\10''' '''705; 1.1'''{{Overline|3}}
! '''13\13''' '''709.'''{{Overline|09}}
|-
| Do#, Sol#
| Η#
| 4#
| 4#
| 16\15 738; 2.1{{Overline|6}}
| 12\11 757; 1, 8.5
| 20\18 774; 5, 6
| rowspan="2" | 8\8 800
| 20\17 827; 1, 1.41{{Overline|6}}
| 12\10 847; 17
| 16\13 872.{{Overline|72}}
|-
| Reb, Lab
| Cb
| 5b, 5c
| 5
| 18\15 830; 1.3
| 13\11 821; 19
| 21\18 812; 1, 9.{{Overline|3}}
| 19\17 786; 4.8{{Overline|3}}
| 11\10 776; 2.125
| 14\13 763.{{Overline|63}}
|-
| '''Re, La'''
| '''C'''
| '''5'''
| '''5'''
| '''19\15''' '''876; 1.08{{Overline|3}}'''
| '''14\11''' '''884; 4.75'''
| '''23\18''' '''890; 3.1'''
| '''9\5''' '''900'''
| '''22\17''' '''910; 2.9'''
| '''13\10''' '''917; 1.{{Overline|54}}'''
| '''17\13''' '''927.{{Overline|27}}'''
|-
| Re#, La#
| C#
| 5#
| 5#
| 20\15 923: 13
| 15\11 947; 2, 1.4
| 25\18 967; 1, 2.875
| rowspan="2" | 10\7 1000
| 25\17 1034; 2, 14
| 15\10 1058; 1, 4.{{Overline|6}}
| 20\13 1090.{{Overline|90}}
|-
| Mib, Sib
| Db
| 6b, 6c
| 6f
| 22\15 1015; 2.6
| 16\11 1010; 1.9
| 26\18 1006; 2, 4.{{Overline|6}}
| 24\17 993; 9.{{Overline|6}}
| 14\10 988; 4.25
| 18\13 981.{{Overline|81}}
|-
| Mi, Si
| D
| 6
| 6
| 23\15 1061; 1, 1.1{{Overline|6}}
| 17\11 1073; 1, 2.1{{Overline|6}}
| 28\18 1083; 1.{{Overline|148}}
| 11\7 1100
| 27\17 1117; 4, 7
| 16\10 1129; 2, 2.{{Overline|3}}
| 21\9 1145.{{Overline|45}}
|-
| Mi#, Si#
| D#
| 6#
| 6#
| 24\15 1107; 1.{{Overline|4}}
| rowspan="2" | 18\11 1136; 1.1875
| 30\18 1161; 3.{{Overline|4}}
| 12\7 1200
| 30\17 1241; 2.{{Overline|63}}
| 18\10 1270; 1.7
| 24\13 1309.{{Overline|09}}
|-
| Fab, Dob
| Ebb
| 7b, 7c
| 7f
| 25\15 1153; 1.{{Overline|18}}
| 29\18 1121; 1, 1, 2.6
| 11\7 1100
| 26\17 1075; 1.16
| 15\10 1058; 1, 4.{{Overline|6}}
| 19\13 1036.{{Overline|36}}
|-
| '''Fa, Do'''
| '''Eb'''
| '''7'''
| '''7'''
| '''26\15''' '''1200'''
| '''19\11''' '''1200'''
| '''31\18''' '''1200'''
| '''12\7''' '''1200'''
| '''29\17''' '''1200'''
| '''17\10''' '''1200'''
| '''22\13''' '''1200'''
|-
| Fa#, Do#
| E
| 7#
| 7#
| 27\15 1246; 6.5
| 20\11 1263; 6.{{Overline|3}}
| 33\18 1277; 2, 2.6
| 13\7 1300
| 32\17 1324; 7.25
| 19\10 1341; 5.{{Overline|6}}
| 25\13 1363.{{Overline|63}}
|-
| Fax, Dox
| E#
| 7x
| 7x
| 28\15 1292; 3.25
| rowspan="2" | 21\11 1326; 3.1{{Overline|6}}
| 35\18 1354; 1, 5.2
| 14\7 1400
| 35\17 1448; 3.625
| 21\10 1482; 2.8{{Overline|3}}
| 28\13 1527.{{Overline|27}}
|-
| Dob, Solb
| Fb
| 8b, Fc
| 8f
| 29\15 1338; 2.1{{Overline|6}}
| 34\18 1316; 7.75
| 13\7 1300
| 31\17 1282; 1.3{{Overline|18}}
| 18\10 1270; 1.7
| 23\13 1254.{{Overline|54}}
|-
! Do, Sol
! F
! 8, F
! 8
! 30\15 1384; 1.625
! 22\11 1389; 2.{{Overline|1}}
! 36\18 1393; 1, 1, 4.{{Overline|6}}
! 14\7 1400
! 34\17 1406; 1, 8.{{Overline|6}}
! 20\10 1411; 1, 3.25
! 26\13 1418.{{Overline|18}}
|-
| Do#, Sol#
| F#
| 8#, F#
| 8#
| 31\15 1430; 1.3
| 23\11 1452; 1.58{{Overline|3}}
| 38\18 1470; 1.0{{Overline|3}}
| rowspan="2" | 15\7 1500
| 37\17 1531; 29
| 22\10 1552; 1.0625
| 29\13 1581.{{Overline|81}}
|-
| Reb, Lab
| Gb
| 9b, Gc
| 9f
| 33\15 1523; 13
| 24\11 1515; 1.2{{Overline|6}}
| 39\18 1509; 1, 2.1
| 36\17 1489; 1, 1.9
| 21\10 1482; 2.8{{Overline|3}}
| 27\13 1472.{{Overline|72}}
|-
| '''Re, La'''
| '''G'''
| '''9, G'''
| 9
| '''34\15''' '''1569; 4.{{Overline|3}}'''
| '''25\11''' '''1578; 1.0{{Overline|5}}'''
| '''41\18''' '''1587; 10.{{Overline|3}}'''
| '''16\7''' '''1600'''
| '''39\17''' '''1613; 1, 3.8{{Overline|3}}'''
| '''23\10''' '''1623; 1.{{Overline|8}}'''
| '''30\13''' '''1636.{{Overline|36}}'''
|-
| Re#, La#
| G#
| 9#, G#
| 9#
| 35\15 1615; 2.6
| 26\11 1642; 9.5
| 43\18 1664; 1.0625
| rowspan="2" | 17\7 1700
| 42\17 1737; 14.5
| 25\10 1764; 1.41{{Overline|6}}
| 33\13 1800
|-
| Mib, Sib
| Ab
| Xb, Ac
| Af
| 37\15 1707; 1.{{Overline|4}}
| 27\11 1705; 3.8
| 44\18 1703; 4, 2.{{Overline|3}}
| 41\17 1696; 1.8125
| 24\10 1694; 8.5
| 31\13 1690.{{Overline|90}}
|-
| Mi, Si
| A
| X, A
| A
| 38\15 1753; 1.{{Overline|18}}
| 28\11 1768; 2.375
| 46\18 1780; 1.55
| 18\7 1800
| 44\17 1820; 1.45
| 26\10 1835; 3.4
| 34\13 1854.{{Overline|54}}
|-
| Mi#, Si#
| A#
| X#, A#
| A#
| 39\15 1800
| rowspan="2" | 29\11 1831; 1.{{Overline|72}}
| 48\18 1858; 15.5
| 19\7 1900
| 47\17 1944; 1.208{{Overline|3}}
| 28\10 1976; 2.125
| 37\13 2018.{{Overline|18}}
|-
| Fab, Dob
| Bbb
| Ebb, Ccc
| Bf
| 40\15 1846; 6.5
| 47\18 1819; 2.{{Overline|81}}
| 18\7 1800
| 43\17 1779; 3.{{Overline|2}}
| 25\10 1764; 1.41{{Overline|6}}
| 32\13 1745.{{Overline|45}}
|-
| '''Fa, Do'''
| '''Bb'''
| '''Eb, Cc'''
| '''B'''
| '''41\15''' '''1892; 3.25'''
| '''30\11''' '''1894; 1, 2.8'''
| '''49\18''' '''1896; 1.291{{Overline|6}}'''
| '''19\7''' '''1900'''
| '''46\17''' '''1903; 2.1{{Overline|6}}'''
| '''27\10''' '''1905; 1.1{{Overline|3}}'''
| '''35\13''' '''1909.{{Overline|09}}'''
|-
| Fa#, Do#
| B
| E, C
| B#
| 42\15 1938; 2.1{{Overline|6}}
| 31\11 1957; 1, 8.5
| 51\18 1974; 5.1{{Overline|6}}
| 20\7 2000
| 49\17 2027; 1, 1.41{{Overline|6}}
| 29\10 2047; 17
| 38\13 2072.{{Overline|72}}
|-
| Fax, Dox
| B#
| Ex, Cx
| Bx
| 43\15 1984; 1.625
| rowspan="2" | 32\11 2021; 19
| 53\18 2051; 1, 1, 1, 1.4
| 21\7 2100
| 52\17 2151; 2.625
| 31\10 2188; 4.25
| 41\13 2236.{{Overline|36}}
|-
| Dob, Solb
| Hb
| 0b, Dc
| Cf
| 44\15 2030; 1.3
| 52\18 2012; 1, 9,{{Overline|3}}
| 20\7 2000
| 48\17 1986; 4.8{{Overline|3}}
| 28\10 1976; 2.125
| 36\13 1963.{{Overline|63}}
|-
! Do, Sol
! H
! 0, D
! C
! 45\15 2076; 1.08'''{{Overline|3}}'''
! 33\11 2084; 4.75
! 54\18 2090; 3.1
! 21\7 2100
! 51\17 2110; 2.9
! 30\10 2117; 1.{{Overline|54}}
! 39\13 2127.{{Overline|27}}
|-
| Do#, Sol#
| Η#
| 0#, D#
| C#
| 46\152123; 13
| 34\112147; 2, 1.4
| 56\182167; 1, 2.875
| rowspan="2" | 22\72200
| 54\172234; 2, 14
| 32\102258; 1, 4.{{Overline|6}}
| 42\132090.{{Overline|90}}
|-
| Reb, Lab
| Cb
| 1b, 1c
| Df
| 48\152215; 2.6
| 35\112210; 1.9
| 57\182206; 2, 4.{{Overline|6}}
| 53\172193; 9.{{Overline|6}}
| 31\10 2188; 4.25
| 40\132181.{{Overline|81}}
|-
| '''Re, La'''
| '''C'''
| '''1'''
| '''D'''
| '''49\15''''''2261; 1, 1.1{{Overline|6}}'''
| '''36\11''''''2273; 1, 2.1{{Overline|6}}'''
| '''59\18''''''2283; 1.{{Overline|148}}'''
| '''23\7''''''2300'''
| '''56\17''''''2317; 4, 7'''
| '''33\10''''''2329; 2, 2.{{Overline|3}}'''
| '''43\13''''''2245.{{Overline|45}}'''
|-
| Re#, La#
| C#
| 1#
| D#
| 50\152307; 1.{{Overline|4}}
| 37\112336; 1.1875
| 61\182361; 3.{{Overline|4}}
| rowspan="2" | 24\72400
| 59\172441; 2.{{Overline|63}}
| 35\102470; 1.7
| 46\132509.{{Overline|09}}
|-
| Mib, Sib
| Db
| 2b, 2c
| Ef
| 52\152400
| 38\112400
| 62\182400
| 58\172400
| 34\102400
| 44\132400
|-
| Mi, Si
| D
| 2
| E
| 53\152446; 6.5
| 39\112463; 6.{{Overline|3}}
| 64\182477; 2, 2.6
| 25\72500
| 61\172524; 7.25
| 36\102541; 5.{{Overline|6}}
| 47\132563.{{Overline|63}}
|-
| Mi#, Si#
| D#
| 2#
| E#
| 54\152492; 3.25
| rowspan="2" | 40\112526; 3.1
| 66\182554; 1, 5.2
| 26\72600
| 64\172648; 2.625
| 38\102682; 2.8{{Overline|3}}
| 50\132727.{{Overline|27}}
|-
| Fab, Dob
| Ebb
| 3b, 3c
| Fff
| 55\152538; 2.1{{Overline|6}}
| 65\182516; 7.75
| 25\72500
| 60\172482; 1.3{{Overline|18}}
| 35\102470; 1.7
| 45\132454.{{Overline|54}}
|-
| '''Fa, Do'''
| '''Eb'''
| '''3'''
| '''Ff'''
| '''56\15''''''2584; 1.625'''
| '''41\11''''''2589; 2.{{Overline|1}}'''
| '''67\18''''''2593; 1, 1, 4.{{Overline|6}}'''
| '''26\7''''''2600'''
| '''63\17''''''2606; 1, 8.{{Overline|6}}'''
| '''37\10''''''2611; 1, 3.25'''
| '''48\13''''''2618.{{Overline|18}}'''
|-
| Fa#, Do#
| E
| 3#
| F
| 57\152630; 1.3
| 42\112652; 1.58{{Overline|3}}
| 69\182670; 1.0{{Overline|3}}
| 27\72700
| 66\172731; 29
| 39\102752; 1.0625
| 51\132781.{{Overline|81}}
|-
| Fax, Dox
| E#
| 3x
| F#
| 58\152676; 1.08{{Overline|3}}
| rowspan="2" | 43\112715; 1.2{{Overline|6}}
| 71\182748; 2.58{{Overline|3}}
| 28\72800
| 69\172855; 4.8
| 41\102894; 8.5
| 54\132945.{{Overline|45}}
|-
| Dob, Solb
| Fb
| 4b, 4c
| 0f, Gf
| 59\152723; 13
| 70\182709; 1, 2.1
| 27\72700
| 65\172689; 1, 1.9
| 38\102682; 2.8{{Overline|3}}
| 49\132672.{{Overline|72}}
|-
! Do, Sol
! F
! 4
! 0, G
! 60\152769; 4.'''{{Overline|3}}'''
! 44\112778; 1.0{{Overline|5}}
! 72\182787; 3.1
! 28\72800
! 68\172813; 1, 3.8{{Overline|3}}
! 40\102823; 1.{{Overline|8}}
! 52\132836.{{Overline|36}}
|}

== Modes ==
The mode names are based on the species of fifth:
{{MOS modes
| Mode Names=
Lydian $
Minor $
Major $
Phrygian $
|}

== Temperaments ==
The most basic rank-2 temperament interpretation of angel is '''Napoli'''. The name "Napoli" comes from the “Neapolitan” sixth triad spelled <code>root-(p-2g)-(2p-3g)</code> ({{nowrap|p {{=}} 3/2|g {{=}} the whole tone}}) which serves as its minor triad approximating 5:6:8 in pental interpretations or 18:21:28 in septimal ones. Basic ~7edf fits both interpretations.

=== Napoli-Meantone ===
[[Subgroup]]: 3/2.6/5.8/5

[[Comma]] list: [[81/80]]

[[POL2]] generator: ~9/8 = 192.6406

[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]

[[Optimal ET sequence]]: ~(7edf, 11edf, 18edf)

=== Napoli-Archy ===
[[Subgroup]]: 3/2.7/6.14/9

[[Comma]] list: [[64/63]]

[[POL2]] generator: ~8/7 = 218.6371

[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]

[[Optimal ET sequence]]: ~(7edf, 10edf, 13edf, 16edf)

=== Scale tree ===
The spectrum looks like this:
{{MOS tuning spectrum
| 3/2 = Napoli-Meantone starts here
| 2/1 = Napoli-Meantone ends, Napoli-Pythagorean begins
| 5/2 = Napoli-Neogothic heartland is from here...
| 8/3 = ...to here
| 3/1 = Napoli-Pythagorean ends, Napoli-Archy begins
| 5/1 = Napoli-Archy ends
}}

Ennealimmal

2026-05-28T13:52:42Z

Sintel: -TempClean

{{Infobox regtemp
| Title = Ennealimmal
| Subgroups = 2.3.5.7
| Comma basis = [[2401/2400]], [[4375/4374]]
| Edo join 1 = 27 | Edo join 2 = 45
| Mapping = 9; 2 3 2
| Generators = 5/3 | Generators tuning = 884.322 | Optimization method = CWE
| MOS scales = [[18L 9s]], [[27L 18s]], [[27L 45s]]
| Pergen = (P8/9, P5/2)
| Odd limit 1 = 9 | Mistuning 1 = 0.204 | Complexity 1 = 45
| Odd limit 2 = 7-limit 81 | Mistuning 2 = 0.408 | Complexity 2 = 99
}}
'''Ennealimmal''' is a [[regular temperament|temperament]] with a period of {{frac|1|9}} octave and tempers out [[2401/2400]] and [[4375/4374]]. Edos that support ennealimmal include {{EDOs| 27, 45, 72, 99, 171, 270, 441, and 612 }}.

See [[Septiennealimmal clan #Ennealimmal]] for technical data.

Ennealimmal scales are built from a ''period'' (which is exactly {{frac|1|9}} of an octave), and a ''generator'' (which is approximately 49 cents and represents several small intervals including 36/35). Depending on the size of the generator and the period in steps, the above listed edos make sense:

{| class="wikitable"
|- style="white-space: nowrap;"
! Period (steps) !! Generator (steps) !! Generator (cents) (pure octave) !! Edo
|-
| 3 || 1 || 44.444 || 27
|-
| 11 || 4 || 48.485 || 99
|-
| 30 || 11 || 48.889 || 270
|-
| 19 || 7 || 49.123 || 171
|-
| 8 || 3 || 50.000 || 72
|-
| 5 || 2 || 53.333 || 45
|}

Ennealimmal extends less well to the [[11-limit]]. Extensions include enneabiotic (99e & 270), ennealympic (171 & 270), ennealimnic (72 & 99e), and ennealiminal (72 & 171e).

See [[Ennealimmal extensions]] for a discussion on 11-limit extensions.

== Interval chain ==
In the following table, odd harmonics 1–9 are labeled in '''bold'''.

{| class="wikitable center-1 right-2 right-4 right-6 right-8"
! rowspan="2" | Period
! colspan="2" | Generator 0
! colspan="2" | Generator 1
! colspan="2" | Generator 2
! colspan="2" | Generator 3
|-
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
|-
| 0
| 0.000
| '''1/1'''
|
|
|
|
|
|
|-
| 1
| 133.333
| 27/25
| 84.322
| 21/20
| 35.310
| 49/48, 50/49
| 1186.298
| 125/63
|-
| 2
| 266.667
| 7/6
| 217.655
| 245/216
| 168.643
| 54/49
| 119.631
| 15/14
|-
| 3
| 400.000
| 63/50
| 350.988
| 49/40, 60/49
| 301.976
| 25/21
| 252.965
| 81/70, 125/108
|-
| 4
| 533.333
| 49/36
| 484.322
| 250/189
| 435.310
| 9/7
| 386.298
| '''5/4'''
|-
| 5
| 666.667
| 72/49
| 617.655
| 10/7
| 568.643
| 25/18
| 519.631
| 27/20
|-
| 6
| 800.000
| 100/63
| 750.988
| 54/35
| 701.976
| '''3/2'''
| 652.965
| 35/24
|-
| 7
| 933.333
| 12/7
| 884.322
| 5/3
| 835.310
| 81/50
| 786.298
| 63/40
|-
| 8
| 1066.667
| 50/27
| 1017.655
| 9/5
| 968.643
| '''7/4'''
| 919.631
| 245/144
|-
| 9
| 1200.000
| 2/1
| 1150.988
| 35/18
| 1101.976
| 189/100
| 1052.965
| 147/80
|}
<nowiki>*</nowiki> In 7-limit CWE tuning, octave reduced

== Scales ==
* [[Ennealimmal27]] – proper [[18L 9s]]. Ninth-octave analog of haplotonic scale
* [[Ennealimmal45]] – improper [[27L 18s]]. Ninth-octave analog of mega-haplotonic scale
** [[Ennealimmal45trans]] – symmetric 5-limit transversal version
* [[Ennealimmal72]] – proper [[27L 45s]]. Ninth-octave analog of albitonic scale
* [[Ennealimmal99]] – proper [[72L 27s]]. Ninth-octave analog of chromatic scale
* [[Ennealimmal171]] – [[99L 72s]] scale. The boundary of propriety is [[270edo]].

== Music ==
; [[Gene Ward Smith]]
* [https://archive.org/details/fingers_201403 ''The 45000 fingers of Dr. S''] (2003) – Ennealimmal[54] in TOP tuning

[[Category:Ennealimmal| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Microtemperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Landscape microtemperaments]]

Arcturus

2026-05-28T13:52:30Z

Sintel: -TempClean

'''Arcturus''' is the [[non-octave]] [[rank]]-2 [[regular temperament]] of the 3.5.7 [[subgroup]] that [[tempering out|tempers out]] the arcturus comma, [[15625/15309]]. Having an ~[[5/3]] as a generator, this temperament is the application of the [[Pythagorean]] principle of tuning a stack of the next higher prime number and then factoring out powers of the equivalence to [[tritave]] composition. However, a heptatonic {{mos scalesig|2L 5s<3/1>|link=1}} [[MOS]] will not suffice to produce an understandable rendition of it because a very close ~5/3 generates a L:s ratio between 4:1 and 5:1, which is beginning to get too lopsided to still be a complete presentation of a temperament.

{{tdlink|No-twos subgroup temperaments #Arcturus}}

== Etymology ==
This temperament is named after the star {{w|Arcturus}}, following a series of non-octave temperaments that are named after stars.
{{todo|add etymology|inline=1|text=Add name (person who coined the term) and year (when it was coined).}}

== Chords ==
Arcturus contains the triad 5:7:9 (used in [[Bohlen–Pierce]] harmony) and the triad 27:35:45 which divides 5/3 into two nearly-equal parts.

== Tuning spectrum ==
Below is a list of MOS families which present it completely (however smearily) using a generator of 845.3 to 951.0{{c}}:

{| class="wikitable" style="text-align: center;"
|-
! colspan="7" | Generator
! Cents Hekts
! L
! s
! 2g
! Notes
|-
| 6\13
|
|
|
|
|
|
| 877.825 600
| 146.304 100
| 0
| 1755.651 1200
| {{nowrap|L {{=}} 1|s {{=}} 0}}
|-
|
|
|
|
|
|
| 43\93
| 879.399 601.075
| 143.158 97.8495
| 20.451 13.9785
| 1758.797 1202.151
| {{nowrap|L {{=}} 7|s {{=}} 1}}
|-
|
|
|
|
|
| 37\80
|
| 879.654 601.25
| 142.647 97.5
| 23.774 16.25
| 1759.38 1202.5
| {{nowrap|L {{=}} 6|s {{=}} 1}}
|-
|
|
|
|
|
|
| 68\147
| 879.816 601.3605
| 142.323 97.279
| 25.877 17.687
| 1759.632 1202.721
|
|-
|
|
|
|
| 31\67
|
|
| 880.009 601.4925
| 141.937 97.015
| 28.387 19.403
| 1760.081 1202.985
| {{nowrap|L {{=}} 5|s {{=}} 1}}
|-
|
|
|
|
|
|
| 87\188
| 880.16 601.596
| 141.634 96.8085
| 30.35 20.745
| 1760.32 1203.191
|
|-
|
|
|
|
|
| 56\121
|
| 880.243 601.653
| 141.468 96.694
| 31.437 21.488
| 1760.487 1203.306
|
|-
|
|
|
|
|
|
| 81\175
| 880.3335 601.714
| 141.288 96.571
| 32.605 22.286
| 1760.667 1203.429
|
|-
|
|
|
| 25\54
|
|
|
| 880.535 601.852
| 140.886 96.296
| 35.221 24.074
| 1761.069 1203.704
| {{nowrap|L {{=}} 4|s {{=}} 1}}
|-
|
|
|
|
|
|
| 94\203
| 880.708 601.97
| 140.5385 96.059
| 37.477 25.616
| 1761.4165 1203.971
|
|-
|
|
|
|
|
| 69\149
|
| 880.711 602.013
| 140.413 95.973
| 38.294 26.1745
| 1761.542 1204.027
|
|-
|
|
|
|
|
|
| 113\244
| 880.823 602.049
| 140.308 95.902
| 38.9745 26.639
| 1761.647 1204.098
|
|-
|
|
|
|
| 44\95
|
|
| 880.9055 602.105
| 140.144 95.7895
| 40.041 27.368
| 1761.811 1204.2105
| {{nowrap|L {{=}} 7|s {{=}} 2}}
|-
|
|
|
|
|
|
| 107\231
| 880.992 602.1645
| 139.971 95.671
| 41.168 28.1385
| 1761.984 1204.329.
|
|-
|
|
|
|
|
| 63\136
|
| 881.053 602.206
| 139.85 95.588
| 41.955 28.6765
| 1762.105 1204.412
|
|-
|
|
|
|
|
|
| 82\177
| 881.132 602.26
| 139.692 95.48
| 42.982 22.034
| 1762.263 1204.52
|
|-
|
|
| 19\41
|
|
|
|
| 881.394 602.439
| 139.167 95.122
| 46.389 31.707
| 1762.788 1204.878
| {{nowrap|L {{=}} 3|s {{=}} 1}}
|-
|
|
|
|
|
|
| 89\192
| 881.635 602.604
| 138.684 94.792
| 49.53 33.854
| 1763.271 1205.208
|
|-
|
|
|
|
|
| 70\151
|
| 881.701 602.649
| 138.553 94.702
| 50.383 25.828
| 1763.402 1205.298
|
|-
|
|
|
|
|
|
| 121\261
| 881.794 602.682
| 138.4565 89.655
| 51.01 34.866
| 1763.4985 1205.362
|
|-
|
|
|
|
| 51\110
|
|
| 881.8155 602.727
| 138.324 94.5455
| 51.8715 35.4545
| 1763.631 1205.4545
|
|-
|
|
|
|
|
|
| 134\289
| 881.875 602.768
| 138.204 94.464
| 52.649 35.986
| 1763.751 1205.536
|
|-
|
|
|
|
|
| 83\179
|
| 881.912 602.793
| 138.131 94.413
| 53.172 36.313
| 1763.824 1205.586
|
|-
|
|
|
|
|
|
| 115\248
| 881.955 602.823
| 138.045 94.355
| 53.684 36.6935
| 1763.91 1205.645
|
|-
|
|
|
| 32\69
|
|
|
| 882.066 602.899
| 137.823 94.203
| 55.129 37.681
| 1764.132 1205.797
| {{nowrap|L {{=}} 5|s {{=}} 2}}
|-
|
|
|
|
|
|
| 109\235
| 882.183 602.979
| 137.588 94.043
| 56.654 38.723
| 1764.367 1205.957
|
|-
|
|
|
|
|
| 77\166
|
| 882.232 603.012
| 137.491 93.976
| 57.288 39.157
| 1764.464 1206.024
|
|-
|
|
|
|
|
|
| 122\263
| 882.276 603.042
| 137.404 93.916
| 57.854 39.544
| 1764.551 1206.084
|
|-
|
|
|
|
| 45\97
|
|
| 882.35 603.093
| 137.2545 93.814
| 58.823 40.206
| 1764.7005 1203.185
| {{nowrap|L {{=}} 7|s {{=}} 3}}
|-
|
|
|
|
|
|
| 103\222
| 882.439 603.153
| 137.078 93.694
| 59.972 40.991
| 1764.877 1206.306
|
|-
|
|
|
|
|
| 58\125
|
| 882.507 603.2
| 136.941 93.6
| 60.863 41.6
| 1765.014 1206.4
|
|-
|
|
|
|
|
|
| 71\153
| 882.607 603.268
| 136.742 93.464
| 62.155 42.484
| 1765.213 1206.536
|
|-
|
| 13\28
|
|
|
|
|
| 883.0505 603.571
| 135.854 92.857
| 67.93 46.429
| 1766.101 1207.143
| {{nowrap|L {{=}} 2|s {{=}} 1}}
|-
|
|
|
|
|
|
| 72\155
| 883.489 603.871
| 134.9775 92.258
| 73.624 50.323
| 1766.9775 1207.742
|
|-
|
|
|
|
|
| 59\127
|
| 883.585 603.937
| 134.784 92.126
| 74.88 51.181
| 1767.171 1207.574
|
|-
|
|
|
|
|
|
| 105\226
| 883.652 603.982
| 134.652 92.035
| 75.742 51.77
| 1767.303 1207.964
|
|-
|
|
|
|
| 46\99
|
|
| 883.737 604.04
| 134.482 91.919
| 76.847 52.525
| 1767.473 1208.081
| {{nowrap|L {{=}} 7|s {{=}} 4}}
|-
|
|
|
|
|
|
| 125\269
| 883.808 604.089
| 134.339 91.822
| 77.775 53.16
| 1767.616 1208.178
|
|-
|
|
|
|
|
| 79\170
|
| 883.85 604.118
| 134.256 91.765
| 78.316 53.529
| 1767.699 1208.235
|
|-
|
|
|
|
|
|
| 112\241
| 883.896 604.149
| 134.163 91.701
| 78.919 53.942
| 1767.792 1208.299
|
|-
|
|
|
| 33\71
|
|
|
| 884.007 604.225
| 133.94 91.549
| 80.364 54.93
| 1768.0145 1208.451
| {{nowrap|L {{=}} 5|s {{=}} 3}}
|-
|
|
|
|
|
|
| 119\256
| 884.112 604.297
| 133.731 91.406
| 81.725 55.859
| 1768.224 1208.594
|
|-
|
|
|
|
|
| 86\185
|
| 884.152 604.324
| 133.651 91.351
| 82.247 56.216
| 1768.304 1208.649
|
|-
|
|
|
|
|
|
| 139\299
| 884.186 604.348
| 133.582 91.304
| 82.694 56.522
| 1768.373 1208.696
| Golden Arcturus is near here
|-
|
|
|
|
| 53\114
|
|
| 884.24 604.386
| 133.4705 91.228
| 83.419 57.0175
| 1768.4845 1208.772
|
|-
|
|
|
|
|
|
| 126\271
| 884.303 604.428
| 133.347 91.144
| 84.219 57.565
| 1768.608 1208.856
|
|-
|
|
|
|
|
| 73\157
|
| 884.3485 604.459
| 133.258 91.083
| 84.8005 57.962
| 1768.697 1208.917
| style="text-align:center;" | 5/3-Pythagorean is near here
|-
|
|
|
|
|
|
| 93\200
| 884.409 604.5
| 133.137 91
| 85.588 58.5
| 1768.818 1209
|
|-
|
|
| 20\43
|
|
|
|
| 884.63 604.651
| 132.6945 90.698
| 88.463 60.465
| 1769.2605 1209.302
| {{nowrap|L {{=}} 3|s {{=}} 2}}
|-
|
|
|
|
|
|
| 87\187
| 884.867 604.813
| 132.2215 90.374
| 91.538 62.567
| 1769.7335 1209.626
|
|-
|
|
|
|
|
| 67\144
|
| 884.937 604.861
| 132.08 90.278
| 92.456 63.194
| 1769.875 1209.722
|
|-
|
|
|
|
|
|
| 114\245
| 884.991 604.898
| 131.972 90.204
| 93.157 52.6735
| 1769.983 1209.896
|
|-
|
|
|
|
| 47\101
|
|
| 885.068 604.9505
| 131.819 90.099
| 94.156 64.356
| 1770.136 1209.901
| {{nowrap|L {{=}} 7|s {{=}} 5}}
|-
|
|
|
|
|
|
| 121\260
| 885.141 605
| 131.674 90
| 95.098 65
| 1770.281 1210
|
|-
|
|
|
|
|
| 74\159
|
| 885.187 605.031
| 131.582 89.937
| 95.696 65.409
| 1770.373 1210.063
|
|-
|
|
|
|
|
|
| 101\217
| 885.242 605.069
| 131.4715 89.862
| 96.4125 65.899
| 1770.4835 1210.138
|
|-
|
|
|
| 27\58
|
|
|
| 885.393 605.172
| 131.169 89.655
| 98.377 67.241
| 1770.786 1210.345
| {{nowrap|L {{=}} 4|s {{=}} 3}}
|-
|
|
|
|
|
|
| 88\189
| 885.566 605.291
| 130.822 89.418
| 100.6325 68.783
| 1771.133 1210.582
|
|-
|
|
|
|
|
| 61\131
|
| 885.643 605.3435
| 130.669 89.313
| 101.631 69.466
| 1771.286 1210.687
|
|-
|
|
|
|
|
|
| 95\204
| 885.714 605.392
| 130.526 89.216
| 102.556 70.098
| 1771.429 1210.784
|
|-
|
|
|
|
| 34\73
|
|
| 885.842 605.4795
| 130.271 89.041
| 104.217 71.233
| 1771.684 1210.959
| {{nowrap|L {{=}} 5|s {{=}} 4}}
|-
|
|
|
|
|
|
| 75\161
| 886.004 605.59
| 129.947 88.82
| 106.3205 72.671
| 1772.008 1211.18
|
|-
|
|
|
|
|
| 41\88
|
| 886.138 605.682
| 129.679 88.636
| 108.065 73.864
| 1772.276 1211.364
| {{nowrap|L {{=}} 6|s {{=}} 5}}
|-
|
|
|
|
|
|
| 48\103
| 886.348 605.825
| 129.259 88.3495
| 110.7935 75.728
| 1772.696 1211.6505
| {{nowrap|L {{=}} 7|s {{=}} 6}}
|-
| 7\15
|
|
|
|
|
|
| 887.579 606.667
| colspan="2" | 126.797 86.667
| 1775.158 1213.333
| {{nowrap|L {{=}} 1|s {{=}} 1}}
|}

== Scales ==
* {{mos scalesig|9L 2s<3/1>|link=1}} (mini chromatic, aka sub-Arcturus)
* {{mos scalesig|11L 2s<3/1>|link=1}} (anti-chromatic, aka anti-Arcturus)
* {{mos scalesig|15L 2s<3/1>|link=1}} (mini enharmonic, aka super-Arcturus 15L 2s)
* {{mos scalesig|17L 2s<3/1>|link=1}} (enharmonic, aka super-Arcturus 17L 2s)
* {{mos scalesig|2L 17s<3/1>|link=1}} (anti-enharmonic, aka trans-Arcturus 2L 7s)

[[Category:Arcturus| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Non-octave temperaments]]

Template talk:Under review by Wikiproject TempClean

2026-05-27T14:00:42Z

Sintel:

== Wording ==

The wording off this template is kind of odd. "Jurisdiction", really?
– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 15:58, 16 April 2025 (UTC)

: Can we remove this? I don't see what purpose it serves. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 14:00, 27 May 2026 (UTC)

Michael Harrison

2026-05-26T23:30:01Z

Sintel: tense

{{Wikipedia|Michael Harrison (musician)}}
'''Michael Harrison''' (1958-2026) was an American contemporary classical music composer and pianist living in New York City.

Harrison created dedicated [[tuning]] systems for many of his works. He pioneered a structural approach to composition in which the proportions of harmonic relationships organically determine other musical elements such as pitch, duration, and dynamics. He also invented the "harmonic piano", a grand piano that plays 24 notes per [[octave]], documented in the Grove Dictionary of Musical Instruments.

== External links ==
* [https://www.michaelharrison.com/ Official website]

{{DEFAULTSORT:Harrison, Michael}}
[[Category:People]]
[[Category:Composers]]
[[Category:Musicians]]

Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments

2026-05-26T22:23:14Z

Sintel: /* Duality, nullspace, commas, bases, canonicalization */ Fix the row operations

{{breadcrumb}}
This is article 4 of 9 in [[Dave Keenan]] & [[Douglas Blumeyer]]'s guide to RTT, or "[[D&D's guide]]" for short.

{{Subpage|Mappings|prev|text=In an earlier article of this series we gave an introduction to mappings}}, the most important objects in RTT. As part of that effort, we couldn't help but touch upon some important underlying properties of these objects — in particular how one-row mappings could merge together to create multi-row mappings, and how this affects which commas are made to vanish by the mapping. These concepts run pretty deep, and in this article we'll be diving deeper into them. By the end of this article, you will understand the relationship between mapping matrices, as well as between mapping matrices and their commas, and how to find them from each other.

== Projective tuning space ==
In this section, we will be going to go into (potentially excruciating) detail about how to read the projective tuning space diagram featured prominently in Paul Erlich's Middle Path paper. Basically, JI, ETs, and the higher-rank temperaments in between can all be plotted in space, which can be visualized on a diagram, to help you navigate their interrelationships. That's what projective tuning space is.

For me personally (Douglas here), attaining total understanding of this visualization was critical before a lot of the linear algebra stuff started to mean much to me. But other people might not work that way, and the extent of detail we go into in this section is not necessary to become competent with RTT (in fact, to our delight, one of the points we make in this section was news to Paul himself). So if you're already confident about reading the PTS diagram, you may try skipping ahead to the second half of this article.

=== Intro to PTS ===
[[File:Pts-2-3-5-e2-twtop-tlin.jpg|center|thumb|800px|'''Figure 1a.''' 5-limit projective tuning space]]

This is 5-limit [[projective tuning space]], or PTS for short ''(see Figure 1a)''. This diagram was created by RTT pioneer [[Paul Erlich]]. It compresses a huge amount of valuable information into a small space. If at first it looks overwhelming or insane, do not despair. It may not be instantly easy to understand, but once you learn the tricks for navigating it from these materials, you will find it is very powerful. Perhaps you will even find patterns in it which others haven't found yet.

We suggest you open this diagram in another window and keep it open as you proceed through these next few sections, as we will be referring to it frequently.

[[File:JI scale 2.png|thumb|right|150px|'''Figure 1b.''' Just an example JI scale]]

If you've worked with 5-limit JI before, you're probably aware that it is three-dimensional. You've probably reasoned about it as a 3D lattice, where one axis is for the factors of prime 2, one axis is for the factors of prime 3, and one axis is for the factors of prime 5. This way, you can use vectors, such as {{vector|-4 4 -1}} or {{vector|1 -2 1}}, just like coordinates.

PTS can be thought of as a projection of 5-limit JI map space, which similarly has one axis each for 2, 3, and 5. But PTS is not a JI pitch lattice. In fact, in a sense, it is the opposite! This is because the coordinates in map space aren't prime-counts, but generator-counts-per-prime, as found in maps such as {{map|12 19 28}}. That particular map is seen here as the biggish, slightly tilted numeral 12 just to the left of the center point.

[[File:PTS with axes.png|300px|left|thumb|'''Figure 1c.''' PTS, with axes]]

And the two 17-ETs we looked at can be found here too. {{map|17 27 40}}, 17c, is the slightly smaller numeral 17 found on the line labeled "meantone" which the 12 is also on, thus representing the fact we mentioned earlier that they both vanish the meantone comma {{sfrac|81|80}}. The other 17, {{map|17 27 39}}, is found on the other side of the center point, aligned horizontally with the first 17. So you could say that map space plots ETs, showing how they are related to each other.

Of course, PTS looks nothing like this JI lattice ''(see Figure 1b)''. This diagram has a ton more information, and as such, Paul needed to get creative about how to structure it. It's a little tricky, but we'll get there. For starters, the axes are not actually shown on the PTS diagram; if they were, they would look like this ''(see Figure 1c)''.

The 2-axis points toward the bottom right, the 3-axis toward the top right, and the 5-axis toward the left. These are the positive halves of each of these axes; we don't need to worry about the negative halves of any of them, because every term of every ET map is positive.

And so it makes sense that {{map|17 27 40}} and {{map|17 27 39}} are aligned horizontally, because the only difference between their maps is in the 5-term, and the 5-axis is horizontal.

=== Scaled axes ===
You might guess that to arrive at that tilted numeral 12, you would start at the origin in the center, move 12 steps toward the bottom right (along the 2-axis), 19 steps toward the top right (not along, but parallel to the 3-axis), and then 28 steps toward the left (parallel to the 5-axis). And if you guessed this, you'd probably also figure that you could perform these moves in any order, because you'd arrive at the same ending position regardless ''(see Figure 1d)''.

[[File:PTS with finding 12-ET.png|400px|right|thumb|'''Figure 1d.''' arriving at 12-ET by moving in any of the 6 possible axis orders (Note: this is a visualization of an early guess at how things work. They're different and more complicated than this. Keep reading!)]]

If you did guess this, you are on the right track, but the full truth is a bit more complicated than that.

The first difference to understand is that each axis's steps have been scaled proportionally according to their prime ''(see Figure 1e)''. We will see in a moment that the scaling factor, to be precise, is the inverse of the logarithm of the prime. To illustrate this, let's choose an example ET and compare its position with the positions of three other closely-related ETs:

# The one which is one step away from it on the 5-axis,
# The one which is one step away from it on the 3-axis, and
# The one which is one step away from it on the 2-axis.

[[File:Shape_of_scale_of_movements_on_axes.png|thumb|left|200px|'''Figure 1e.''' the basic shape the scaled axes make between neighbor maps (maps with only 1 difference between their terms)]]

Our example ET will be 40. We'll start out at the map {{map|40 63 93}}. This map is a default of sorts for 40-ET, because it's the map where all three terms are as close as possible to JI when prime 2 is exact (we'll be calling it a [[simple map]] here; it has more commonly been called a "[[patent val]]", but we are critical of that terminology.<ref group="note">See our thoughts on that here: https://en.xen.wiki/w/Talk:Patent_val</ref>).

From here, let's move by a single step on the 5-axis by adding 1 to the 5-term of our map, from 93 to 94, therefore moving to the map {{map|40 63 94}}. This map is found directly to the left. This makes sense because the orientation of the 5-axis is horizontal, and the positive direction points out from the origin toward the left, so increases to the 5-term move us in that direction.

Back from our starting point, let's move by a single step again, but this time on the 3-axis, by adding 1 to the 3-term of our map, from 63 to 64, therefore moving to the map {{map|40 64 93}}. This map is found up and to the right. Again, this direction makes sense, because it's the direction the 3-axis points.

Finally, let's move by a single step on the 2-axis, from 40 to 41, moving to the map {{map|41 63 93}}, which unsurprisingly is in the direction the 2-axis points. This move actually takes us off the chart, way down here.

[[File:40-ET distances example.png|400px|right|thumb|'''Figure 1f.''' Distances between 40-ET's neighbors in PTS]]

Now let's observe the difference in distances ''(see Figure 1f)''. Notice how the distance between the maps separated by a change in 5-term is the smallest, the maps separated by a change in 3-term have the medium-sized distance, and maps separated by a change in the 2-term have the largest distance. This tells us that steps along the 3-axis are larger than steps along the 5-axis, and steps along the 2-axis are larger still. The relationship between these sizes is that the 3-axis step has been divided by the binary logarithm of 3, written <math>\log_2{3}</math>, which is approximately 1.585, while the 5-axis step has been divided by the binary logarithm of 5, written <math>\log_2{5}</math>, and which is approximately 2.322. The 2-axis step can also be thought of as having been divided by the binary logarithm of its prime, but because <math>\log_2{2}</math> is exactly 1, and dividing by 1 does nothing, the scaling has no effect on the 2-axis.

The reason Paul chose this particular scaling scheme<ref group="note">Really, he didn't ''choose an axis scaling scheme'', though, as much as this scaling scheme arises as one of many possible ways to understanding the end result. We felt was an easier introduction to the image than Paul's original conceptualization for it would have been. Paul's projective intentions are described a bit later on in this article.</ref> is that it causes those ETs which are closer to JI to appear closer to the center of the diagram (and this is a useful property to organize ETs by). How does this work? Well, let's look into it.

Remember that near-just ETs have maps whose terms are in close proportion to <math>\small\log(2\!:\!3\!:\!5)</math>. ET maps use only integers, so they can only approximate this ideal, but a theoretical pure JI map would be {{map|<math>\small\log_2{\!2}\;</math> <math>\small\log_2{\!3}\;</math> <math>\small\log_2{\!5}</math>}}. If we scaled this theoretical JI map by this scaling scheme, then, we'd get <math>\small 1\!:\!1\!:\!1</math>, because we're just dividing things by themselves:

<math>\dfrac{\log_2{2}}{\log_2{2}} \!:\! \dfrac{\log_2{3}}{\log_2{3}} \!:\! \dfrac{\log_2{5}}{\log_2{5}} = 1\!:\!1\!:\!1</math>

This tells us that we should find this theoretical JI map at the point arrived at by moving exactly the same amount along the 2-axis, 3-axis, and 5-axis. Well, if we tried that, these three movements would cancel each other out: we'd draw an equilateral triangle and end up exactly where we started, at the origin, or in other words, at pure JI. Any other ET approximating but not exactly <math>\small\log(2\!:\!3\!:\!5)</math> will be scaled to proportions not exactly <math>\small 1\!:\!1\!:\!1</math>, but approximately so, like maybe <math>\small 1.000\!:\!0.999\!:\!1.002</math>, and so you'll move in something close to an equilateral triangle, but not exactly, and land in some interesting spot that's not quite in the center. In other words, we scale the axes this way so that we can compare the maps not in absolute terms, but in terms of what direction and by how much they deviate from JI ''(see Figure 1g)''.

[[File:Scaling.png|600px|right|thumb|'''Figure 1g.''' a visualization of how scaling axes illuminates deviations from JI]]

For example, let's scale our 12-ET example:

* <math>\frac{12}{\log_2{2}} = 12</math>
* <math>\frac{19}{\log_2{3}} \approx 11.988</math>
* <math>\frac{28}{\log_2{5}} \approx 12.059</math>

Clearly, 12:11.988:12.059 is quite close to 1:1:1. This checks out with our knowledge that it is close to JI, at least in the 5-limit.

But if instead we picked some random alternate mapping of 12-ET, like {{map|12 23 25}}, looking at those integer terms directly, it may not be obvious how close to JI this map is. However, upon scaling them:

* <math>\frac{12}{\log_2{2}} = 12</math>
* <math>\frac{23}{\log_2{3}} \approx 14.511</math>
* <math>\frac{25}{\log_2{5}} \approx 10.767</math>

It becomes clear how far this map is from JI.

So what really matters here are the little differences between these numbers. Everything else cancels out. That 12-ET's scaled 3-term, at ≈11.988, is ever-so-slightly less than 12, indicates that prime 3 is mapped ever-so-slightly narrow. And that its 5-term, at ≈12.059, is slightly more than 12, indicates that prime 5 is mapped slightly wide in 12. This checks out with the placement of 12 on the diagram: ever-so-slightly below and to the left of the horizontal midline, due to the narrowness of the 3, and slightly further still to the left, due to the wideness of the 5.

We can imagine that if we hadn't scaled the steps, as in our initial naive guess, we'd have ended up nowhere near the center of the diagram. How could we have, if the steps are all the same size, but we're moving 28 of them to the left, but only 12 and 19 of them to the bottom left and top right? We'd clearly end up way, way further to the left, and also above the horizontal midline. And this is where pretty much any near-just ET would get plotted, because 3 being bigger than 2 would dominate its behavior, and 5 being larger still than 3 would dominate its behavior.

=== Perspective ===
The truth about distances between related ETs on the PTS diagram is actually slightly even more complicated than that, though; as we mentioned, the scaled axes are only the first difference from our initial guess. In addition to the effect of the scaling of the axes, there is another effect, which is like a perspective effect. Basically, as ETs get more complex, you can think of them as getting farther and farther away; to suggest this, they are printed smaller and smaller on the page, and the distances between them appear smaller and smaller too.

Remember that 5-limit JI is 3D, but we're viewing it on a 2D page. It's not the case that its axes are flat on the page. They're not literally occupying the same plane, 120° apart from each other. That's just not how axes typically work, and it's not how they work here either! The 5-axis is perpendicular to the 2-axis and 3-axis just like typical Cartesian space. Again, we're looking only at the positive coordinates, which is to say that this is only the +++ [[Wikipedia:Octant_(solid_geometry)|octant]] of space, which comes to a point at the origin (0,0,0) like the corner of a cube. So you should think of this diagram as showing that cubic octant sticking its corner straight out of the page at us, like a triangular pyramid. So we're like a tiny little bug, situated right at the tip of that corner, pointing straight down the octant's interior diagonal, or in other words the line equidistant from three axes, the line which we understand represents theoretically pure JI. So we see that in the center of the page, represented as a red hexagram, and then toward the edges of the page is our peripheral vision. ''(See Figure 1h.)''

[[File:Understanding projection.png|600px|thumb|left|'''Figure 1h.''' Visualization of the projection process. (In real life, the cube is infinite in size. We made it smaller here to help make the shape clearer.)]]

PTS doesn't show the entire tuning cube. You can see evidence of this in the fact that some numerals have been cut off on its edges. We've cropped things around the central region of information, which is where the ETs best approximating JI are found (note how close 53-ET is to the center!). Paul added some concentric hexagons to the center of his diagram, which you could think of as concentric around that interior diagonal, or in other words, are defined by gradually increasing thresholds of deviations from JI for any one prime at a time.

No maps past [[99edo|99-ET]] are drawn on this diagram. ETs with that many steps are considered too complex (read: big numbers, impractical) to bother cluttering the diagram with. Better to leave the more useful information easier to read.

Okay, but what about the perspective effect? Right. So every step further away on any axis, then, appears a bit smaller than the previous step, because it's just a bit further away from us. And how much smaller? Well, the perspective effect is such that, as seen on this diagram, the distances between n-ETs are twice the size of the distances between 2n-ETs.

Moreover, there's a special relationship between the positions of n-ETs and 2n-ETs, and indeed between n-ETs and 3n-ETs, 4n-ETs, etc. To understand why, it's instructive to plot it out ''(see Figure 1i)''.

[[File:Hiding vals.png|500px|thumb|right|'''Figure 1i.''' Redundant maps hiding behind their simpler counterparts. The eye is the origin; the same as in Figure 1h. Projective tuning space is the plane resting at the bottom that we are projecting onto. The portion we see in the Middle Path version is only a tiny part right in the middle. The dotted lines just above where the PTS plane is drawn are there to indicate the elision of an infinitude of space; potentially you could go way up to insanely large ETs and they would all be between the origin-eye and this projective plane.]]

For simplicity, we're looking at the octant cube here from the angle straight on to the 2-axis, so changes to the 2-terms don't matter here. At the top is the origin; that's the point at the center of PTS. Close-by, we can see the map {{map|3 5 7}}, and two closely related maps {{map|3 4 7}} and {{map|3 5 8}}. Colored lines have been drawn from the origin through these points to the black line in the top-right, which represents the page; this is portraying how if our eye is at that origin, where on the page these points would appear to be.

In between where the colored lines touch the maps themselves and the page, we see a cluster of more maps, each of which starts with 6. In other words, these maps are about twice as far away from us as the others. Let's consider {{map|6 10 14}} first. Notice that each of its terms is exactly 2x the corresponding term in {{map|3 5 7}}. In effect, {{map|6 10 14}} is redundant with {{map|3 5 7}}. If you imagine doing a mapping calculation or two, you can easily convince yourself that you'll get the same answer as if you'd just done it with {{map|3 5 7}} instead and then simply divided by 2 one time at the end. It behaves in the exact same way as {{map|3 5 7}} in terms of the relationships between the intervals it maps, the only difference being that it needlessly includes twice as many steps to do so, never using every other one. So we don't really care about {{map|6 10 14}}. Which is great, because it's hidden exactly behind {{map|3 5 7}} from where we're looking.

The same is true of the map pair {{map|3 4 7}} and {{map|6 8 14}}, as well as of {{map|3 5 8}} and {{map|6 10 16}}. Any map whose terms have a common factor other than 1 is going to be redundant in this sense, and therefore hidden. You can imagine that even further past {{map|3 5 7}} you'll find {{map|9 15 21}}, {{map|12 20 28}}, and so on, and these we could call "enfactored" maps.<ref group="note">Elsewhere you may see these called "contorted", but as you can read on the page [[defactoring]], this is not technically correct, but has historically been frequently confused.</ref><ref group="note">On some versions of PTS which Paul prepared, these enfactored ETs are actually printed on the page.</ref> More on those later. What's important to realize here is that Paul found a way to collapse 3 dimensions worth of information down to 2 dimensions without losing anything important. Each of these lines connecting redundant ETs have been [[Wikipedia:Projection_(mathematics)|projected]] onto the page as a single point. That's why the diagram is called "projective" tuning space.

Now, to find a 6-ET with anything new to bring to the table, we'll need to find one whose terms don't share a common factor. That's not hard. We'll just take one of the ones halfway between the ones we just looked at. How about {{map|6 11 14}}, which is halfway between {{map|6 10 14}} and {{map|6 12 14}}. Notice that the purple line that runs through it lands halfway between the red and blue lines on the page. Similarly, {{map|6 10 15}} is halfway between {{map|6 10 14}} and {{map|6 10 16}}, and its yellow line appears halfway between the red and green lines on the page. What this is demonstrating is that halfway between any pair of n-ETs on the diagram, whether this pair is separated along the 3-axis or 5-axis, you will find a 2n-ET. We can't really demonstrate this with 3-ET and 6-ET on the diagram, because those ETs are too inaccurate; they've been cropped off. But if we return to our 40-ET example, that will work just fine.

[[File:Plot of 5 10 20 40 80.png|800px|thumb|left|'''Figure 1j.'''Plot of 40-ETs with 80-ETs halfway between each pair, including the enfactored 40-ETs hiding behind 20-ET and 10-ET]]

I've circled every 40-ET visible in the chart ''(see Figure 1j)''. And you can see that halfway between each one, there's an [[80edo|80-ET]] too. Well, sometimes it's not actually printed on the diagram<ref group="note">The reason is that Paul's diagram, in addition to cutting off beyond 99-ET, also filters out maps that aren't uniform maps.</ref>, but it's still there. You will also notice that if we also land right about on top of [[20edo|20-ET]] and [[10edo|10-ET]]. That's no coincidence! Hiding behind that 20-ET is a redundant 40-ET whose terms are all 2x the 20-ET's terms, and hiding behind the 10-ET is a redundant 40-ET whose terms are all 4x the 40-ET's terms (and also a redundant 20-ET and a [[30edo|30-ET]], and [[50edo|50-ET]], [[60edo|60-ET]], etc. etc. etc.)

Also, check out the spot halfway between our two 17-ETs: there's the 34-ET we briefly mused about earlier, which would solve 17's problem of approximating prime 5 by subdividing each of its steps in half. We can confirm now that this 34-ET does a superb job at approximating prime 5, because it is almost vertically aligned with the JI red hexagram.

Just as there are 2n-ETs halfway between n-ETs, there are 3n-ETs a third of the way between n-ETs. Look at these two [[29edo|29-ET]]s here. The [[58edo|58-ET]] is here halfway between them, and two [[87edo|87-ET]]s are here each a third of the way between.

=== Map space vs. tuning space ===
So far, we've been describing PTS as a projection of map space, which is to say that we've been thinking of maps as the coordinates. We should be aware that tuning space is a slightly different structure. In tuning space, coordinates are not maps, but tunings, specified in cents, octaves, or some other unit of pitch. So a coordinate might be {{map|6 10 14}} in map space, but {{map|1200 2000 2800}} in tuning space.

Both tuning space and map space project to the identical result as seen in Paul's diagram, which is how we've been able to get away without distinguishing them thus far.

Why did we do this to you? Well, we decided map space was conceptually easier to introduce than tuning space. Paul himself prefers to think of this diagram as a projection of tuning space, however, so we don't want to leave this material before clarifying the difference. Also, there are different helpful insights you can get from thinking of PTS as tuning space. Let's consider those now.

The first key difference to notice is that we can standardize coordinates in tuning space, so that the first term of every coordinate is the same, namely, one octave, or 1200 cents. For example, note that while in map space, {{map|3 5 7}} is located physically in front of {{map|6 10 14}}, in tuning space, these two points collapse to literally the same point, {{map|1200 2000 2800}}. This can be helpful in a similar way to how the scaled axes of PTS help us visually compare maps' proximity to the central JI spoke: they are now expressed closer to in terms of their deviation from JI, so we can more immediately compare maps to each other, as well as individually directly to the pure JI primes, as long as we memorize the cents values of those (they're 1200, 1901.955, and 2786.314). For example, in map space, it may not be immediately obvious that {{map|6 9 14}} is halfway between {{map|3 5 7}} and {{map|3 4 7}}, but in tuning space it is immediately obvious that {{map|1200 1800 2800}} is halfway between {{map|1200 2000 2800}} and {{map|1200 1600 2800}}.

So if we take a look at a cross-section of projection again, but in terms of tuning space now ''(see Figure 1k)'', we can see how every point is about the same distance from us.

[[File:Tuning space version.png|400px|thumb|right|'''Figure 1k.''' Demonstration of projection in terms of ''tuning'' space (compare with Figure 1i, which shows projection in terms of ''map'' space). As you can see here, all the points are in about the same region of space, since tuning space tends toward JI.]]

The other major difference is that tuning space is continuous, where map space is discrete. In other words, to find a map between {{map|6 10 14}} and {{map|6 9 14}}, you're subdividing it by 2 or 3 and picking a point in between, that sort of thing. But between {{map|1200 2000 2800}} and {{map|1200 1800 2800}} you've got an infinitude of choices smoothly transitioning between each other; you've basically got knobs you can turn on the proportions of the tuning of 2, 3, and 5. Everything from from {{map|1200 1999.999 2800}} to {{map|1200 1901.955 2800}} to {{map|1200 1817.643 2800}} is along the way.

[[File:Tuning projection.png|400px|thumb|left|'''Figure 1l.''' Demonstration of tuning projection. As long as the tunings change in a fixed proportion, the tuning will project to the same point on PTS.]]

But perhaps even more interesting than this continuous tuning space that appears in PTS between points is the continuous tuning space that does not appear in PTS because it exists within each point, that is, exactly out from and deeper into the page at each point. In tuning space, as we've just established, there are no maps in front of or behind each other that get collapsed to a single point. But there are still many things that get collapsed to a single point like this, but in tuning space they are different tunings ''(see Figure 1l)''. For example, {{map|1200 1900 2800}} is the way we'd write 12-ET in tuning space. But there are other tunings represented by this same point in PTS, such as {{map|1200.12 1900.19 2800.28}} (note that in order to remain at the same point, we've maintained the exact proportions of all the prime tunings). That tuning might not be of particular interest. We just used it as a simple example to illustrate the point. A more useful example would be {{map|1198.440 1897.531 2796.361}}, which by some algorithm is the optimal tuning for 12-ET (minimizes error across primes or intervals); it may not be as obvious from looking at that one, but if you check the proportions of those terms with each other, you will find they are still exactly 12:19:28.

The key point here is that, as we mentioned before, the problems of tuning and tempering are largely separate. PTS projects all tunings of the same temperament to the same point. This way, issues of tuning are completely hidden and ignored on PTS, so we can focus instead on tempering.

=== Regions ===
We've shown that ETs with the same number that are horizontally aligned differ in their mapping of 5, and ETs with the same number that are aligned on the 3-axis running bottom left to top right differ in their mapping of 3. These basic relationships can be extrapolated to be understood in a general sense. ETs found in the center-left map 5 relatively big and 2 and 3 relatively small. ETs found in the top-right map 3 relatively big and 2 and 5 relatively small. ETs found in the bottom-right map 2 relatively big and 3 and 5 relatively small. And for each of these three statements, the region on the opposite side maps things in the opposite way.

So: we now know which point is {{map|12 19 28}}, and we know a couple of 17's, 40's and a 41. But can we answer in the general case? Given an arbitrary map, like {{map|7 11 16}}, can we find it on the diagram? Well, you may look to the first term, 7, which tells you it's [[7edo|7-ET]]. There's only one big 7 on this diagram, so it's probably that. (You're right). But that one's easy. The 7 is huge.

What if we gave you {{map|43 68 100}}. Where's [[43edo|43-ET]]? I'll bet you're still complaining: the map expresses the tempering of 2, 3, and 5 in terms of their shared generator, but doesn't tell us directly which primes are sharp, and which primes are flat, so how could we know in which region to look for this ET?

The answer to that is, unfortunately: that's just how it is. It can be a bit of a hunt sometimes. But the chances are, in the real world, if you're looking for a map or thinking about it, then you probably already have at least some other information about it to help you find it, whether it's memorized in your head, or you're reading it off the results page for an automatic temperament search tool.

Probably you have the information about the primes' tempering; maybe you get lucky and a 43 jumps out at you but it's not the one you're looking for, but you can use what you know about the perspectival scaling and axis directions and log-of-prime scaling to find other 43's relative to it.

Or maybe you know some comma that is made to vanish by the map {{map|43 68 100}}, so you can find it along the line for that comma's temperament.

=== Temperament lines ===
So we understand the shape of projective tuning space. And we understand what points are in this space. But what about the magenta lines, now?

So far, we've only mentioned one of these lines: the one labeled "meantone", noting that the fact that 12 and 17c appear on it means that either of them makes the meantone comma vanish. In other words, this line represents the meantone temperament.

For another example, the line on the right side of the diagram running almost vertically which has the other 17-ET we looked at, as well as 10-ET and 7-ET, is labeled "dicot", and so this line represents the dicot temperament, and unsurprisingly all of these ET's make the dicot comma vanish.

Simply put, lines on PTS are temperaments.

But hold up now: points are ETs, which are temperaments, too, right? Well, yes, that's still true. But while points are equal temperaments, or '''[[List_of_rank_one_temperaments_by_step_size|rank-1 temperaments]]''', the lines represent '''[https://en.xen.wiki/index.php?title=Rank_two_temperament rank-2 temperaments]'''. It may be helpful to differentiate the names in your mind in terms of their geometric dimensionality. Recall that projective tuning space has compressed all our information by one dimension; every point on our diagram is actually a line radiating out from our eye. So a rank-1 temperament is really a line, which is one-dimensional; rank-1, 1D. And the rank-2 temperaments, which are seen as lines in our diagram, are truly planes coming up out of the page, and planes are of course two-dimensional; rank-2, 2D. If you wanted to, you could even say 5-limit JI was a rank-3 temperament, because that's this entire space, which is 3-dimensional; rank-3, 3D.

"[[Rank]]" has a slightly different meaning than dimension, but that's not important yet. For now, it's enough to know that each temperament line on this 5-limit PTS diagram is defined by making a comma with the same name vanish. For now, we're still focusing on visually how to navigate PTS. So the natural thing to wonder next, then, is what's up with the slopes of all these temperament lines?

[[File:Diagrams to understand PTS for RTT.png|thumb|left|400px|'''Figure 1m.''' How the vanishing comma affects slope on PTS. A temperament defined by a comma with a 0 for a prime will be perpendicular to that prime's axis, because the tuning of that prime does not affect whether or not the comma vanishes. Therefore the prime corresponding to the 0 in the comma is represented by x, which can be anything, while the proportion between the other two primes must remain fixed.]]

Let's begin with a simple example: the perfectly horizontal line that runs through just about the middle of the page, through the numeral 12, labeled "[[compton]]". What's happening along this line? Well, as we know, moving to the left means tuning 5 sharper, and moving to the right means tuning 5 flatter. But what about 2 and 3? Well, they are changing as well: 2 is sharp in the bottom right, and 3 is sharp in the top right, so when we move exactly rightward, 2 and 3 are both getting sharper (though not as directly as 5 is getting flatter). But the critical thing to observe here is that 2 and 3 are sharpening at the exact same rate. Therefore the approximations of primes 2 and 3 are in a constant ratio with each other along horizontal lines like this. Said another way, if you look at the 2 and 3 terms for any ET's map on this line, the ratio between its term for 2 and 3 will be identical.

Let's grab some samples to confirm. We already know that 12-ET here looks like {{map|12 19}} (I'm dropping the 5 term for now). The 24-ET here looks like {{map|24 38}}, which is simply 2×{{map|12 19}}. The 36-ET here looks like {{map|36 57}} = 3×{{map|12 19}}. And so on. So that's why we only see multiples of 12 along this line: because 12 and 19 are co-prime, so the only other maps which could have them in the same ratio would be multiples of them.

Let's look at the other perfectly horizontal line on this diagram. It's found about a quarter of the way down the diagram, and runs through the 10-ET and 20-ET we looked at earlier. This one's called "[[blackwood]]". Here, we can see that all of its ETs are multiples of 5. In fact, [[5edo|5-ET]] itself is on this line, though we can only see a sliver of its giant numeral off the left edge of the diagram. Again, all of its maps have locked ratios between their mappings of prime 2 and prime 3: {{map|5 8}}, {{map|10 16}}, {{map|15 24}}, {{map|20 32}}, {{map|40 64}}, {{map|80 128}}, etc. You get the idea.

So what do these two temperaments have in common such that their lines are parallel? Well, they're defined by commas, so why don't we compare their commas. The compton comma is {{vector|-19 12 0}}, and the blackwood comma is {{vector|8 -5 0}}.<ref group="note">Yes, these are the same as the [[Pythagorean comma]] and [[Pythagorean diatonic semitone]], respectively.</ref> What sticks out about these two commas is that they both have a 5-term of 0. This means that when we ask the question "how many steps does this comma map to in a given ET", the ET's mapping of 5 is irrelevant. Whether we check it in {{map|40 63 93}} or {{map|40 63 94}}, the result is going to be the same. So if {{map|40 63 93}} makes the blackwood comma vanish, then so does {{map|40 63 94}}. And if {{map|24 38 56}} makes the compton comma vanish, then so does {{map|24 38 55}}. And so on.

Similar temperaments can be found which include only 2 of the 3 primes at once. Take "[[augmented (temperament)|augmented]]", for instance, running from bottom-left to top-right. This temperament is aligned with the 3-axis. This tells us several equivalent things: that relative changes to the mapping of 3 are irrelevant for augmented temperament, that the augmented comma has no 3's in its prime factorization, and the ratios of the mappings of 2 and 5 are the same for any ET along this line. Indeed we find that the augmented comma is {{vector|7 0 -3}}, or [[128/125]], which has no 3's. And if we sample a few maps along this line, we find {{map|12 19 28}}, {{map|9 14 21}}, {{map|15 24 35}}, {{map|21 33 48}}, {{map|27 43 63}}, etc., for which there is no pattern to the 3-term, but the 2- and 5-terms for each are in a 3:7 ratio.

There are even temperaments whose comma includes only 3's and 5's, such as "[[bug]]" temperament, which makes [[27/25]] vanish, or {{vector|0 3 -2}}. If you look on this PTS diagram, however, you won't find bug. Paul chose not to draw it. There are infinite temperaments possible here, so he had to set a threshold somewhere on which temperaments to show, and bug just didn't make the cut in terms of how much it distorts harmony from JI. If he had drawn it, it would have been way out on the left edge of the diagram, completely outside the concentric hexagons. It would run parallel to the 2-axis, or from top-left to bottom-right, and it would connect the 5-ET (the huge numeral which is cut off the left edge of the diagram so that we can only see a sliver of it) to the [[9edo|9-ET]] in the bottom left, running through the 19-ET and [[14edo|14-ET]] in-between. Indeed, these ET maps — {{map|9 14 21}}, {{map|5 8 12}}, {{map|19 30 45}}, and {{map|14 22 33}} — lock the ratio between their 3-terms and 5-terms, in this case to 2:3.

Those are the three simplest slopes to consider, i.e. the ones which are exactly parallel to the axes ''(see Figure 1m)''. But all the other temperament lines follow a similar principle. Their slopes are a manifestation of the prime factors in their defining comma. If having zero 5's means you are perfectly horizontal, then having only one 5 means your slope will be close to horizontal, such as meantone {{vector|-4 4 -1}} or [[Helmholtz (temperament)|helmholtz]] {{vector|-15 8 1}}. Similarly, magic {{vector|-10 -1 5}} and [[würschmidt]] {{vector|17 1 -8}}, having only one 3 apiece, are close to parallel with the 3-axis, while porcupine {{vector|1 -5 3}} and [[ripple]] {{vector|-1 8 -5}}, having only one 2 apiece, are close to parallel with the 2-axis.

Think of it like this: for meantone, a change to the mapping of 5 doesn't make near as much of a difference to the outcome as does a change to the mapping of 2 or 3, therefore, changes along the 5-axis don't have near as much of an effect on that line, so it ends up roughly parallel to it.

=== Scale trees ===
Patterns, patterns, everywhere. PTS is chock full of them. One pattern we haven't discussed yet is the pattern made by the ETs that fall along each temperament line.

Let's consider meantone as our first example. Notice that between 12 and 7, the next-biggest numeral we find is 19, and 12+7=19. Notice in turn that between 12 and 19 the next-biggest numeral is 31, and 12+19=31, and also that between 19 and 7 the next-biggest numeral is 26, and 19+7=26. You can continue finding deeper ETs indefinitely following this pattern: 43 between 12 and 31, 50 between 31 and 19, 45 between 19 and 26, 33 between 26 and 7. In fact, if we step back a bit, remembering that the huge numeral just off the left edge is a 5, we can see that 12 is there in the first place because 5+7=12.

This effect is happening on every other temperament line. Look at dicot. 10+7=17. 10+17=27. 17+7=24. Etc.<ref group="note">There's an extension of this pattern. Pick any ET. Maybe start with a prominent one like 7, or 12. Notice that you can find lines radiating out from it of aligned ETs. These would all be rank-2 temperaments, though they're not all drawn. You'll see that if you pick any size of numeral and follow consecutive numerals of continuously changing size, that the values decrease by the ET number you're radiating out from. That's because each step you can think of subtracting that ET number over and over, because moving inward you'd be doing the opposite: repeatedly adding that ET number, per the rules of the scale tree.</ref>

To fully understand why this is happening, we need a crash course in [[mediants]], and the [[Stern-Brocot_ancestors_and_rank_2_temperaments|The Stern-Brocot tree]].

The mediant of two fractions <math>\frac{n_1}{d_1}</math> and <math>\frac{n_2}{d_2}</math> is <math>\frac{n_1 + n_2}{d_1 + d_2}</math>. It's sometimes called the freshman's sum because it's an easy mistake to make when first learning how to add fractions. And while this operation is certainly not equivalent to adding two fractions, it does turn out to have other important mathematical properties. The one we're leveraging here is that the mediant of two numbers is always greater than one and less than the other. For example, the mediant of {{sfrac|3|5}} and {{sfrac|2|3}} is {{sfrac|5|8}}, and it's easy to see in decimal form that 0.625 is between 0.6 and 0.666.

The Stern-Brocot tree is a helpful visualization of all these mediant relations. Flanking the part of the tree we care about — which comes up in the closely-related theory of [[MOS_scale|MOS scales]], where it is often referred to as the "scale tree" — are the extreme fractions {{sfrac|0|1}} and {{sfrac|1|1}}. Taking the mediant of these two gives our first node: {{sfrac|1|2}}. Each new node on the tree drops an infinitely descending line of copies of itself on each new tier. Then, each node branches to either side, connecting itself to a new node which is the mediant of its two adjacent values. So {{sfrac|0|1}} and {{sfrac|1|2}} become {{sfrac|1|3}}, and {{sfrac|1|2}} and {{sfrac|1|1}} become {{sfrac|2|3}}. In the next tier, {{sfrac|0|1}} and {{sfrac|1|3}} become {{sfrac|1|4}}, {{sfrac|1|3}} and {{sfrac|1|2}} become {{sfrac|2|5}}, {{sfrac|1|2}} and {{sfrac|2|3}} become {{sfrac|3|5}}, and {{sfrac|2|3}} and {{sfrac|1|1}} become {{sfrac|3|4}}.<ref group="note">Each tier of the Stern-Brocot tree is the next [[Wikipedia:Farey_sequence|Farey sequence]].</ref> The tree continues forever.

So what does this have to do with the patterns along the temperament lines in PTS? Well, each temperament line is kind of like its own section of the scale tree. The key insight here is that in terms of meantone temperament, there's more to 7-ET than simply the number 7. The 7 is just a fraction's denominator. The numerator in this case is 3. So imagine a {{sfrac|3|7}} floating on top of the 7-ET there. And there's more to 5-ET than simply the number 5, in that case, the fraction is the {{sfrac|2|5}}. So the mediant of {{sfrac|2|5}} and {{sfrac|3|7}} is {{sfrac|5|12}}. And if you compare the decimal values of these numbers, we have 0.4, 0.429, and 0.417. Success: {{sfrac|5|12}} is between {{sfrac|2|5}} and {{sfrac|3|7}} on the meantone line. You may verify yourself that the mediant of {{sfrac|5|12}} and {{sfrac|3|7}}, {{sfrac|8|19}}, is between them in size, as well as {{sfrac|7|17}} being between {{sfrac|2|5}} and {{sfrac|5|12}} in size.

In fact, if you followed this value along the meantone line all the way from {{sfrac|2|5}} to {{sfrac|3|7}}, it would vary continuously from 0.4 to 0.429; the ET points are the spots where the value happens to be rational.

Okay, so it's easy to see how all this follows from here. But where the heck did we get {{sfrac|2|5}} and {{sfrac|3|7}} in the first place? We seemed to pull them out of thin air. And what the heck is this value?

Different generator sizes also gives us a new way to think about the scale tree patterns. Remember how earlier we pointed out that {{map|12 19 28}} was simply {{map|5 8 12}} + {{map|7 11 16}}? Well, if {{rket|{{map|5 8 12}} {{map|7 11 16}}}} is a way of expressing meantone in terms of its two generators, you can imagine that 12-ET is the point where those two generators converge on being the same exact size.<ref group="note">For real numbers <math>p,q</math> we can make the two generators respectively <math>\frac{p}{5p+7q}</math> and <math>\frac{q}{5p+7q}</math> of an octave, e.g. <math>(p,q)=(1,0)</math> for 5-ET, <math>(0,1)</math> for 7-ET, <math>(1,1)</math> for 12-ET, and many other possibilities.</ref> If they become the same size, then they aren't truly two separate generators, or at least there's no effect in thinking of them as separate. And so for convenience you can simply combine their mappings into one. You could imagine gradually increasing the size of one generator and decreasing the size of the other until they were both 100¢. As long as you maintain the correct proportion, you'll stay along the meantone line.

=== Generators ===
[[File:Generator sizes in PTS.png|800px|thumb|'''Figure 1n.''' Generator sizes of rank-2 temperaments in PTS. Don't worry too much about the valid ranges yet; we'll discuss that part later. And we didn't break down what's happening along the blackwood, compton, augmented, dimipent, and some other lines which are labeled on the original PTS diagram. In some cases, it's just because we got lazy and didn't want to deal with fitting more numbers on this thing. But in the case of all those that we just listed, it's because although they are rank-2, those temperaments all have non-octave periods, and so it doesn't make enough sense to compare their generators here. You'll learn about periods in the next section.]]

The answer to both of those questions is: it's the generator (in this case, the meantone generator).

A generator is an interval which generates a temperament. Again, if you're already familiar with MOS scales, this is the same concept. If not, all this means is that if you repeatedly move by this interval, you will visit the pitches you can include in your tuning.

We looked at generators in the earlier article. We saw how the generator for 12-ET was about 1.059, because repeated movement is like repeated multiplication (1.059 × 1.059 × 1.059 ...) and <math>1.059^{12} \approx 2</math>, <math>1.059^{19} \approx 3</math>, and <math>1.059^{28} \approx 5</math>. This meantone generator is the same basic idea, but there's a couple of important differences we need to cover.

First of all, and this difference is superficial, it's in a different format. We were expressing 12-ET's generator 1.059 as a frequency multiplier; it's like 2, 3, or 5, and this could be measured in Hz, say, by multiplying by 440 if A4 was our 1/1 (1.059 away from A is 466Hz, which is #A). But the meantone generators we're looking at now in forms like {{sfrac|2|5}}, {{sfrac|3|7}}, or {{sfrac|5|12}}, are expressed as fractional octaves, i.e. they're in terms of pitch, something that could be measured in cents if we multiplied by 1200 (2/5 × 1200 ¢ = 480 ¢). And we have that special way of writing fractional octaves, and that's with a backslash instead of a slash, like this: 2\5, 3\7, 5\12.

Cents and hertz values can readily be converted between one form and the other, so it's the second difference which is more important. It's their size. If we do convert 12-ET's generator to cents so we can compare it with meantone's generator at 12-ET, we can see that 12-ET's generator is 100 ¢ (<math>\log_2{1.059}</math> × 1200 ¢ = 100 ¢) while meantone's generator at 12-ET is 500 ¢ (5/12 × 1200 ¢ = 500 ¢). When we look at 12-ET in terms of itself, rather than in terms of any particular rank-2 temperament, its generator is 1\12; that's the simplest, smallest generator which if we iterate it 12 times will touch every pitch in 12-ET. But when we look at 12-ET not as the end goal, but rather as a foundation upon which we could work with a given temperament, things change; we don't necessarily need to include every pitch in 12-ET to realize a temperament it supports. Instead, we just need to make sure the temperament's generator is a multiple of the ET's generator, as we have with 500 ¢ = 100 ¢ × 5.

The fact that the meantone temperament line passes through 12-ET, and also the augmented temperament line passes through 12-ET, doesn't mean that you need the entirety of 12-ET to play either one. It means something more like this: if you had an instrument locked into 12-ET, you could use it to play some kind of meantone and some kind of augmented, but 12-ET is not necessarily the most interesting manifestation of either meantone or augmented. It's merely the case that it technically supports either one. The most interesting manifestations of meantone or augmented may lay between ETs, and/or boast far more than 12 notes.

We mentioned that the generator value changes continuously as we move along a temperament line. So just to either side of 12-ET along the meantone line, the tuning of 2, 3, and 5 supports a generator size which in turn supports meantone, but it wouldn't support augmented. And just to either side of 12-ET along the augmented line, the tuning of 2, 3, and 5 supports a generator which still supports augmented, but not meantone. 12-ET, we could say, is a convergence point between the meantone generator and the augmented generator. But it is not a convergence point because the two generators become identical in 12-ET, but rather because they can both be achieved in terms of 12-ET's generator. In other words, 5\12 ≠ 4\12, but they are both multiples of 1\12.

Here's a diagram that shows how the generator size changes gradually across each line in PTS. It may seem weird how the same generator size appears in multiple different places across the space. But keep in mind that pretty much any generator is possible pretty much anywhere here. This is simply the generator size pattern you get when you lock the octave to exactly 1200 cents, to establish a common basis for comparison. This is what enables us to produce maps of temperaments such as the one found at [[Map_of_linear_temperaments|this Xen wiki page]], or this chart here ''(see Figure 1n)''.

=== Periods and generators ===
Let's bring up MOS theory again. We mentioned earlier that you might have been familiar with the scale tree if you'd worked with MOS scales before, and if so, the connection was scale cardinalities, or in other words, how many notes are in the resultant scales when you continuously iterate the generator until you reach points where there are only two scale step sizes. At these points scales tend to sound relatively good, and this is in fact the definition of a MOS scale. There's a mathematical explanation for how to know, given a ratio between the size of your generator and period, the cardinalities of scales possible; we won't re-explain it here. The point is that the scale tree can show you that pattern visually. And so if each temperament line in PTS is its own segment of the scale tree, then we can use it in a similar way.

For example, if we pick a point along the meantone line between 12 and 19, the cardinalities will be 5, 7, 12, 19, 31, 50, etc. If we chose exactly the point at 12 then the cardinality pattern would terminate there, or in other words, eventually we'll hit a scale with 12 notes and instead of two different step sizes there would only be one, i.e. you've got an ET, and there's no place else to go from there. The system has circled back around to its starting point, so it's a closed system. Further generator iterations will only retread notes you've already touched. The same would be true if you chose exactly the point at 19, except ''that'''s where you'd hit an ET instead, at 19 notes.

Between ETs, in the stretches of rank-2 temperament lines where the generator is not a rational fraction of the octave, theoretically those temperaments could have infinite pitches; you could continuously iterate the generator and you'd never exactly circle back to the point where you started. If bigger numbers were shown on PTS, you could continue to use those numbers to guide your cardinalities forever.

The structure when you stop iterating the meantone generator with five notes is called meantone[5]. If you were to use the entirety of 12-ET as meantone then that'd be meantone[12]. But you can also realize meantone[12] in 19-ET; in the former you have only one step size, but in the latter you have two. You can't realize meantone[19] in 12-ET, but you could also realize it in 31-ET.

=== Temperament merging ===
We've seen how 12-ET is found at the intersection of the meantone and augmented temperament lines, and therefore supports both at the same time. In fact, no other ET can boast this feat. Therefore, we can even go so far as to describe 12-ET as the "intersection" of meantone and augmented. Using the pipe operator "|" to mean "[[comma-merging|comma-merge]]", then, we could call 12-ET "meantone|augmented", read "meantone comma-merge augmented", or "meantone or augmented" for short. In other words, we express a rank-1 temperament in terms of two rank-2 temperaments.

For another rank-1 example, we could call 7-ET "meantone|dicot", because it is the comma-merge of meantone and dicot temperaments.

We can conclude that there's no "blackwood|compton" temperament, because those two lines are parallel. In other words, it's impossible to make the blackwood comma and compton comma vanish simultaneously. How could it ever be the case that 12 fifths take you back where you started yet also 5 fifths take you back where you started?<ref group="note">As you can confirm using the matrix tools you'll learn soon, technically speaking you ''can'' make them both vanish at the same time... but it'll only be by using 0-EDO, i.e. a system with only a single pitch. For more information see [[trivial temperaments]].</ref>

Similarly, we can express rank-2 temperaments in terms of rank-1 temperaments. Have you ever heard the expression "two points make a line"? Well, if we choose two ETs from PTS, then there is one and only one line that runs through both of them. So, by choosing those ETs, we can be understood to be describing the rank-2 temperament along that line, or in other words, the one and only temperament whose comma both of those ETs make vanish. This is just what we saw in the earlier article where e.g. 7&22 made porcupine.

For another example, we could choose 5-ET and 7-ET. Looking at either 7-ET or 5-ET, we can see that many temperament lines pass through them individually. Even more pass through them which Paul chose (via a complexity threshold) not to show. But there's only one line which runs through both 5-ET and 7-ET, and that's the meantone line. So of all the commas that 5-ET makes vanish, and all the commas that 7-ET makes vanish, there's only a single one which they have in common, and that's the meantone comma. Therefore we could give meantone temperament another name, and that's "5&7"; in this case we use the ampersand operator, not the pipe. We can call this operator "[[map-merging|map-merge]]", so we can read that "5 map-merge 7", or "5 and 7" for short.

When specifying a rank-1 temperament in terms of two rank-2 temperaments, an obvious constraint is that the two rank-2 temperaments cannot be parallel. When specifying a rank-2 temperament in terms of two rank-1 temperaments, it seems like things should be more open-ended. Indeed, however, there is a special additional constraint on either method, and they're related to each other. Let's look at rank-2 as the map-merge of rank-1 first.

5&7 is valid for meantone. So is 7&12. 12&19 and 19&7 are both fine too, and so are 5&17 and 17&12. Yes, these are all literally the same thing (though you may connote a meantone generator size on the meantone line somewhere between these two ETs). So how could we mess this one up, then? Well, here are our first counterexamples: 5&19, 7&17, and 17&19. And what problem do all these share in common? The problem is that between 5 and 19 on the meantone line we find 12, and 12 is a smaller number than 19 (or, if you prefer, on PTS, it is printed as a larger numeral). It's the same problem with 17&19, and with 7&17 the problem is that 12 is smaller than 17. It's tricky, but you have to make sure that between the two ETs you map-merge there's not a smaller ET (which you should be map-merging instead). The reason why is out of scope to explain here, but we'll get to it eventually.

And the related constraint for rank-1 from two rank-2 is that you can't choose two temperaments whose names are printed smaller on the page than another temperament between them. More on that later.

== Duality, nullspace, commas, bases, canonicalization ==
From the PTS diagram, we can visually pick out rank-1 temperaments as the comma-mergings of rank-2 temperaments as well as rank-2 temperaments as the map-mergings of rank-1 temperaments. But we can also understand these results through maps and vectors. And we're going to need to learn how, because PTS can only take us so far. 5-limit PTS is good for humans because we live in a physically 3-dimensional world (and spend a lot of time sitting in front of 2D pages on paper and on computer screens), but as soon as you want to start working in 7-limit harmony, which is 4D, visual analogies will begin to fail us, and if we're not equipped with the necessary mathematical abstractions, we'll no longer be able to effectively navigate.

Don't worry: we're not going 4D just yet. We've still got plenty we can cover using only the 5-limit. But we're going to set aside PTS for now. It's matrix time. By the end of this section, you'll understand how to represent a temperament in matrix form, how to interpret them, notate them, and use them, as well as how to apply important transformations between different kinds of these matrices. So you can imagine what you're doing another way, if not visually.

=== Mapping-row-bases and comma bases ===
19-ET's map is {{map|19 30 44}}. We also now know that we could call it "meantone|magic", because we find it at the intersection of the meantone and magic temperament lines. But how would we mathematically, non-visually make this connection?

The first critical step is to recall that a temperament can be defined by its vanishing commas, which can be expressed as vectors. So, we can represent meantone using the meantone comma, {{vector|-4 4 -1}}, and magic using the magic comma {{vector|-10 -1 5}}.

The comma-merge of two vectors can be represented as a matrix. Technically, vectors are vertical lists of numbers, or columns, so when we put meantone and magic together, we get a matrix that looks like this:

<math>
\left[ \begin{array} {c|c}
{-4} & {-10} \\
4 & {-1} \\
{-1} & 5
\end{array} \right]
</math>

We call such a matrix a '''[[comma basis]]'''. The plural of "basis" is "bases", but pronounced like BAY-seez (/ˈbeɪsiz/).

Now how in the world could that matrix represent the same temperament as {{map|19 30 44}}? Well, they're two different ways of describing it. {{map|19 30 44}}, as we know, tells us how many generator steps it takes to reach each prime approximation. This new matrix, it turns out, is an equivalent way of stating the same information; it is a minimal representation of the ''nullspace'' of that mapping, or in other words, of all the commas it makes vanish.
This was a bit tricky for me (Douglas) to get my head around, so let me hammer this point home: when you say "the nullspace", you're referring to ''the entire infinite set of all commas that a mapping makes vanish'', ''not only'' the two commas you see in any given basis for it. Think of the comma basis as one of many valid sets of instructions to find every possible comma, the "comma space", by adding or subtracting (integer multiples of) these two commas from each other.<ref group="note">To be clear, because what you are adding and subtracting in interval vectors are exponents (as you know), the commas are actually being multiplied by each other; e.g. {{nowrap|{{vector|-4 4 -1}} + {{vector|10 1 -5}} {{=}} {{vector|6 5 -6}}}}, which is the same thing as <math>\frac{81}{80} × \frac{3072}{3125} = \frac{15552}{15625}</math></ref> The math term for adding and subtracting vectors like this, which you will certainly see plenty of as you explore RTT, is "linear combination". It should be visually clear from the PTS diagram that this 19-ET comma basis couldn't be listing every single comma 19-ET makes vanish, because we can see there are at least four temperament lines that pass through it (there are actually infinity of them!). But so it turns out that picking two commas is perfectly enough; every other comma that 19-ET makes vanish could be expressed in terms of these two!

It's the analogous effect to what we've seen with mapping-rows and how there are many forms to any given temperament's mapping. It's more relative than it is absolute. As long as you add and subtract whole multiples of commas from each other, if the comma vanished before, it'll still vanish now. In more technical terms, any other interval arrived at through linear combinations of the commas in a basis would also be a valid column in the basis; any of these interval vectors, by definition, is mapped to zero steps by the mapping. So any combination of them will also map to zero steps, and thus be a comma that is made to vanish by the temperament.

Try one. How about the kleisma, {{vector|6 5 -6}}. Well that one's too easy! Clearly if you go down by one magic comma to {{vector|10 1 -5}} and then up by one meantone comma you get one kleisma. What you're doing when you're adding and subtracting multiples of commas from each other like this are technically called [[Wikipedia:Elementary_matrix|elementary column operations]]. Feel free to work through any other examples yourself.

A good way to explain why we don't need three of these commas is that if you had three of them, you could use any two of them to create the third, and then subtract the result from the third, turning that comma into a zero vector, or a vector with only zeroes, which is pretty useless, so we could just discard it.

[[File:Different nestings.png|400px|thumb|left|'''Figure 2a.''' How to write matrices in terms of either columns/vectors/commas or rows/row-vectors/maps. (Apologies for the unnecessary 7-limit here, i.e. these matrices are 2×4 and 4×2. Everything we've been doing so far has been 5-limit, i.e. 2×3 and 3×2. But we haven't got around to correcting this asset.)]]

We can [[extended bra-ket notation|extend our angle bracket notation]] to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 2a)''. For example, we could have written our comma basis like this: [{{vector|-4 4 -1}} {{vector|-10 -1 5}}]. Starting from the outside, the square brackets tell us to think in terms of a list. It's just that this list isn't a list of ''numbers'', like the ones we've gotten used to by now, but rather a list of ''vectors''. So this list includes two such vectors. Alternatively, we could have written this same matrix like {{ket|[-4 -10] [4 -1] [-1 5]}}, but that would obscure the fact that it is the combination of two familiar commas (but that notation ''would'' be useful for expressing a matrix built out of multiple maps, as we will soon see).

Sometimes a comma basis may have only a single comma. That's okay. It's just like how ET mappings have only a single row. A single vector can become a matrix. To disambiguate this situation, if necessary, just as we did with single-row mappings, you could put the vector inside row brackets, like this: [{{vector|-4 4 -1}}]. Similarly, a single row vector (map) can become a matrix, by nesting inside column brackets, like this: {{rket|{{map|19 30 44}}}}<ref group="note">The reasons for the different styles of bracket are explained here: [[Extended_bra-ket_notation#Variant_including_curly_and_square_brackets|Extended bra-ket notation: Variant including curly and square brackets]].</ref>.

Why isn't a mapping a "basis", you ask? Well, it can be thought of as a basis too. It depends on the context. When you use the word "mapping" for it, you're treating it like a function, or a machine: it takes in intervals, and spits out new forms of intervals. That's how we've been using it here. But in other places, you may be thinking of this matrix as a basis for the infinite space of possible maps that could be combined to produce a matrix which works the same way as a given mapping, i.e. it makes the same commas vanish. In these contexts, it might make more sense to call such a mapping matrix a "mapping-row-basis".

And now you wonder why it's not just "map basis". Well, that's answerable too. It's because "map" is the analogous term to an "interval", but we're looking for the analogous term to a "comma". A comma is an interval which vanishes. So we need a word that means a map that makes a comma vanish, and that term is "mapping-row".

So, yes, that's right: maps are similar to commas insofar as — once you have more than one of them in your matrix — the possibilities for individual members immediately go infinite. Technically speaking, though, while a comma basis is a basis of the nullspace of the mapping, a mapping-row-basis is a ''row''-basis of the ''row''-space of the mapping.

=== Duality ===
The fact that we can define a temperament by either a mapping or a comma basis is referred to as ''duality''. We say the comma basis is the ''dual'' of the mapping, and the mapping is the dual of the comma basis.

We could imagine drawing a diagram for a temperament with a line of duality down the center, with a mapping on the left, and a comma basis on the right. Either side ultimately gives the same information, but sometimes you want to come at it in terms of the maps, and sometimes in terms of the commas.

One last note on the bracket notation before we proceed: you will regularly see matrices across the wiki that use only square brackets on the outside, whether it's a mapping or a comma basis e.g. [{{map|5 8 12}} {{map|7 11 16}}] or [{{vector|-4 4 -1}} {{vector|-10 -1 5}}]. That's fine because it's unambiguous; if you have a list of rows, it's fairly obvious you've arranged them vertically, and if you've got a list of columns, it's fairly obvious you've arranged them horizontally. For the mapping, we prefer the style of using a left angle bracket for the rows and a right curly bracket on the outside — for slightly more effort, it raises slightly less questions — but using only square brackets on the outside should not be said to be wrong. And as for comma bases, they are perhaps best thought of as a list of vectors, rather than a matrix, so we try to capture that in the notation, by using only square brackets on the outside, though it is often helpful also to think of them all smooshed together as a matrix.

Our preferred notation is explained further in [[Extended_bra-ket_notation#Variant_including_curly_and_square_brackets|Extended bra-ket notation: Variant including curly and square brackets]].

=== Nullspace ===
We learned above, that we can merge the maps for 5-ET and 7-ET to obtain this meantone mapping:

<math>
\left[ \begin{matrix}
5 & 8 & 12 \\
7 & 11 & 16
\end{matrix} \right]
</math>

We're going to show you how to start with this mapping and find a corresponding comma basis. This is sometimes called "finding the nullspace", even though what it's really finding is a ''basis'' for this nullspace. Another word you may see used for the nullspace, that we prefer to avoid, is the ''kernel''. A more specific name for the "nullspace" in our RTT application is the "comma space".

Working this out by hand goes like this (it is a standard linear algebra operation, so if you're comfortable with it already, you can skip this and other similar parts of these materials).

First, augment our mapping with an "identity matrix".

<math>
\left[ \begin{matrix}
5 & 8 & 12 \\
7 & 11 & 16 \\
\hline
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{matrix} \right]
</math>

What's an identity matrix? Here are some examples:

<math>
\left[ \begin{matrix}
1 \\
\end{matrix} \right]
</math>

<math>
\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right]
</math>

<math>
\left[ \begin{matrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{matrix} \right]
</math>

etc. No matter its size, it's always a square matrix consisting of all <math>0</math>'s except for its main diagonal which has <math>1</math>'s. It's called an identity matrix because when you multiply something by it, you still have the identical thing. It doesn't change.<ref group="note">"Identity" has special meaning in some math situations. For some simple examples, 0 can be called the "additive identity", because with respect to the addition operation, it's the thing which if you add it to something else, you don't change it, or in other words, the output is identical to the input. 5 + 0 = 5, 7.98<math>\pi</math> + 0 = 7.98<math>\pi</math>, and in general <math>x</math> + 0 = <math>x</math>. Similarly we have the multiplicative identity, which is 1, because anything times 1 is itself. Hopefully you see the similarity here. It's helpful to have "identities" like this for any given mathematical operation, and matrix multiplication is no exception. So "identity matrices" are the identities for matrix multiplication. For matrix multiplication, there's more than one identity, because it depends on the row or column count of the matrix you're multiplying with. You can test it for yourself. Any matrix multiplied by an appropriately-sized identity matrix will give you the same matrix back, unchanged. Other examples of algebraic identities are the additive identity, <math>0</math>, and the multiplicative identity, <math>1</math>. The reason why <math>0</math> is the additive identity is because <math>n + 0 = n</math>, and <math>1</math> is the multiplicative identity because <math>n × 1 = n</math>. An identity ''matrix'' is the multiplicative identity for matrices in the same way: <math>AI = A</math>. There's a different identity matrix for each size of square matrix: <math>(1, 1)</math>-shaped, <math>(2, 2)</math>-shaped, <math>(3, 3)</math>-shaped, etc. but they all follow the same pattern: their entries are all <math>0</math>'s except for <math>1</math>'s along the main diagonal. Here's the <math>(2, 2)</math>-shaped <math>I</math> for comparison:
 
<math>
I =
\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right]
</math>
 
</ref>

Now, add and subtract integer multiples of columns from each other until you can get one of the columns to be all zeroes above the line, and what's left below the line will be our comma:

There are many different ways of attacking this, but the the general strategy is just to try and make the numbers as small as possible.
Let's start with subtracting the second column from the last.

<math>
\left[ \begin{matrix}
5 & 8 & 4 \\
7 & 11 & 5 \\
\hline
1 & 0 & 0 \\
0 & 1 & -1 \\
0 & 0 & 1
\end{matrix} \right]</math>

Then we can subtract the first column from the second.

<math>\left[ \begin{matrix}
5 & 3 & 4 \\
7 & 4 & 5 \\
\hline
1 & -1 & 0 \\
0 & 1 & -1 \\
0 & 0 & 1
\end{matrix} \right]</math>

Columns two and three now differ by one, so subtract the second from the last again.

<math>\left[ \begin{matrix}
5 & 3 & 1 \\
7 & 4 & 1 \\
\hline
1 & -1 & 1 \\
0 & 1 & -2 \\
0 & 0 & 1
\end{matrix} \right]</math>

We're getting closer! Now subtract the second column from the first ''twice'', which also leaves us with ones.

<math>\left[ \begin{matrix}
-1 & 3 & 1 \\
-1 & 4 & 1 \\
\hline
3 & -1 & 1 \\
-2 & 1 & -2 \\
0 & 0 & 1
\end{matrix} \right]</math>

Now finally, we can add the last column to the first.

<math>\left[ \begin{matrix}
0 & 3 & 1 \\
0 & 4 & 1 \\
\hline
4 & -1 & 1 \\
-4 & 1 & -2 \\
1 & 0 & 1
\end{matrix} \right]</math>

And we've done it! A column with all zeros above the line.
We can now grab the part of that column that's below the line:

<math>
\left[ \begin{matrix}
\color{green}4 \\
\color{green}{-4} \\
\color{green}1
\end{matrix} \right]
</math>

And ta-da! You've found a comma basis given a mapping, and it is [{{vector|4 -4 1}}] which represents {{sfrac|80|81}}. In other words, for this temperament, you have converted a row-basis for its mapping row-space into a basis for its nullspace. Feel free to try this with any of the other combinations of two ET maps mentioned above. You could try to show that [{{vector|4 -4 1}}] is a basis for the nullspace of any other combination of ETs we found that could specify meantone, such as 7&12, or 12&19.

Throughout this section of this article we will be referring to examples implemented in [https://www.wolfram.com/language/ Wolfram Language] (formerly Mathematica), a popular and capable programming language for working with math. We encourage you to try them out, to get a feel for things in another way, and get yourself started exploring temperaments yourself! If you're interested, you can run them right on the web without downloading or setting anything up on your computer: just go to https://www.wolframcloud.com, sign up for free, create a new computational notebook, paste in the contents from [https://github.com/cmloegcmluin/RTT/blob/main/main.m this file], and Shift+Enter to run it, which will load up all the functions. Then open a new tab to use them; you'll be computing in no time. (And of course you're encouraged to look over the implementations of the functions if that may help you.) FYI, any notebook you create has a lifespan of 60 days before Wolfram Cloud will recycle it, so you'll have to copy and paste them to new notebooks or wherever if you don't want to lose your work.<ref group="note">It's not as simple as select-all, copy, paste, because of how computational notebooks can (and should) be broken down into many cells. However there is a handy way to copy all cells, including all of each of their output: just click in the top right to select the first cell (it should highlight along the right edge in blue), then shift-click the same area but for the bottom cell, copy, and paste. Voilà!</ref>
If, on the other hand, you're not interested in code examples, that's no big deal. They're not necessary to follow along.
Let's try it out in Wolfram Language:

<nowiki>In: nullSpaceBasis["[⟨5 8 12] ⟨7 11 16]}"]
Out: "[4 -4 1⟩"</nowiki>

When we looked at a comma-merge corresponding to 19-ET above, there was nothing special about the pairing of meantone and magic. We could have chosen meantone|hanson, or magic|negri, etc. A matrix formed out of the comma-merge of any two of these particular commas will capture the same exact nullspace of the mapping {{rket|{{map|19 30 44}}}}.

We already have the tools to check that each of these commas' vectors is made to vanish individually by the mapping-row {{map|19 30 44}}; all we have to do is make sure that the comma is mapped to zero steps by it. But that doesn't indicate a special relationship between 19-ET and any of these commas ''individually''; each of these commas are made to vanish by many different ETs, not just 19-ET. 19-ET does have a special relationship to a nullspace, but it's not to a nullspace which can be expressed in basis form as a ''single'' comma; rather, it is to a nullspace which can be expressed in basis form as the comma-merge of ''two'' commas (at least in the 5-limit; more on this later). In this way, comma bases which represent the comma-merge of two commas are greater than the sum of their individual parts.

We can confirm the relationship between an ET and its nullspace by converting back and forth between them. In the next section we'll look at how to go from any one of these comma bases to the mapping {{rket|{{map|19 30 44}}}}, thus demonstrating the various bases' equivalence with respect to it.

=== ...And nullspace again ===
Interestingly, the same operation that takes us from a mapping to its dual comma basis also takes us from a comma basis back to its dual mapping. They're both done with the nullspace operation!

We'll demonstrate working this one out by hand too. The only difference between doing the nullspace for a wide matrix like a mapping and a tall matrix like a comma basis is that for a tall matrix everything will look sideways from how it looked for the wide matrix. We'll do everything the same way, just rotated by 90°.

Here's our starting point, a meantone|magic comma basis:

<math>
\left[ \begin{array} {c|c}
{-4} & {-10} \\
4 & {-1} \\
{-1} & 5 \\
\end{array} \right]
</math>

Now, augment it with an "identity matrix". (We've been separating the commas with a vertical bar, but to better distinguish the commas from the identity matrix here, we'll only use the vertical bar between them and the identity matrix.)

<math>
\left[ \begin{array} {cc|ccc}
{-4} & {-10} & 1 & 0 & 0 \\
4 & {-1} & 0 & 1 & 0 \\
{-1} & 5 & 0 & 0 & 1 \\
\end{array} \right]
</math>

Now, add and subtract multiples of rows from each other until you can get one of the rows to be all zeroes to the left the line. We show this step in a different way to how we showed it above, in case one way of showing it works for some people and the other way works for others:

<math>
\left[ \begin{array} {cc|ccc}
{-4} + \left(\style{background-color:#F2B2B4;padding:5px}{2} × \style{background-color:#98CC70;padding:5px}{-1}\right) &
{-10} + \left(\style{background-color:#F2B2B4;padding:5px}{2} × \style{background-color:#98CC70;padding:5px}{5}\right) &
1 + \left(\style{background-color:#F2B2B4;padding:5px}{2} × \style{background-color:#98CC70;padding:5px}{0}\right) &
0 + \left(\style{background-color:#F2B2B4;padding:5px}{2} × \style{background-color:#98CC70;padding:5px}{0}\right) &
0 + \left(\style{background-color:#F2B2B4;padding:5px}{2} × \style{background-color:#98CC70;padding:5px}{1}\right) \\
4 & {-1} & 0 & 1 & 0 \\
\style{background-color:#98CC70;padding:5px}{-1} & \style{background-color:#98CC70;padding:5px}{5} & \style{background-color:#98CC70;padding:5px}{0} & \style{background-color:#98CC70;padding:5px}{0} & \style{background-color:#98CC70;padding:5px}{1} \\
\end{array} \right] =
</math>

<math>
\left[ \begin{array} {cc|ccc}
{-6} & 0 & 1 & 0 & 2 \\
4 & {-1} & 0 & 1 & 0 \\
\style{background-color:#98CC70;padding:5px}{-1} & \style{background-color:#98CC70;padding:5px}{5} & \style{background-color:#98CC70;padding:5px}{0} & \style{background-color:#98CC70;padding:5px}{0} & \style{background-color:#98CC70;padding:5px}{1} \\
\end{array} \right]
</math>

What we've done here is add two times the 3rd row to the 1st row. Hence the <math>\style{background-color:#F2B2B4;padding:5px}{2}</math> we see for each column. The reason we did this is because our goal is get one of the rows in the left half to be all zeros. So we're already halfway there! We've got a 0 in the top-right cell of the left half.

This first one was easy. But as one approaches a target like the one we're approaching, things often get trickier the closer one gets. In this case, we still have to somehow change that -6 to the left of our new 0 into a 0, too. But there's no row that we can add or subtract from this row that can change the -6 into a 0 without also messing up the 0 to its left that we already accomplished. So we can't go straight to the finish line from here. First, we have to create a row that we can use.

Basically we need to create a row which has the power to change the -6 without also changing the 0 too. So that means we need a row which also has a 0 in the same column as that 0 we want to keep. We can create this row either by modifying the 2nd row or the 3rd row. In the following step, we've chosen to modify the 3rd row.

So if we need to change the 5 in the 3rd row to a 0, there's only one way to do that: add 5 times the 2nd row to the 3rd row:

<math>
\left[ \begin{array} {cc|ccc}
{-6} & 0 & 1 & 0 & 2 \\
4 & {-1} & 0 & 1 & 0 \\
19 & 0 & 0 & 5 & 1 \\
\end{array} \right]
</math>

So now we've created the row we need in order to destroy that -6 without messing up the 0. But wait... maybe we're not quite there yet. The problem is that 19 and doesn't divide evenly into 6. So if we want to use a 19 to wipe out a 6, we'll actually need to multiply the row with the 6 by 19, and the row with the 19 by 6, so that their values match and can be canceled out. So that's why we do this:

<math>
\left[ \begin{array} {cc|ccc}
{-114} & 0 & 19 & 0 & 38 \\
4 & {-1} & 0 & 1 & 0 \\
19 & 0 & 0 & 5 & 1 \\
\end{array} \right]
</math>

And then add the 3rd row to the 1st row 6 times:

<math>
\left[ \begin{array} {cc|ccc}
\color{lime}0 & \color{lime}0 & \color{green}19 & \color{green}30 & \color{green}44 \\
4 & {-1} & 0 & 1 & 0 \\
19 & 0 & 0 & 5 & 1 \\
\end{array} \right]
</math>

And we've done it! A row with all zeros to the left of the line. So now we're ready to grab the rest of the row from the right side of the line:

<math>
\left[ \begin{matrix}
\color{green}19 & \color{green}30 & \color{green}44 \\
\end{matrix} \right]
</math>

Great. We've found a mapping given a comma basis, and it is {{rket|{{map|19 30 44}}}}. In other words, for this temperament, we have converted a basis for its nullspace to a row-basis for its mapping row-space. Feel free to try this with any other combination of two commas made to vanish by this mapping-row.

Now just to convince ourselves of the nullspace-both-ways relationship between the mapping and the comma basis, let's do the nullspace function to take us from this mapping {{rket|{{map|19 30 44}}}} back to its comma basis. Start at the augmentation step:

<math>
\left[ \begin{matrix}
19 & 30 & 44 \\
\hline
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{matrix} \right]
</math>

This time we need to get two of the columns to have (all) zeroes above the line.

<math>
\left[ \begin{matrix}
19 & 30 & 836 \\
\hline
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 19
\end{matrix} \right]

→

\left[ \begin{matrix}
19 & 30 & \color{lime}0 \\
\hline
1 & 0 & \color{green}{-44} \\
0 & 1 & \color{green}0 \\
0 & 0 & \color{green}19
\end{matrix} \right]

→

\left[ \begin{matrix}
19 & 570 & \color{lime}0 \\
\hline
1 & 0 & \color{green}{-44} \\
0 & 19 & \color{green}0 \\
0 & 0 & \color{green}19
\end{matrix} \right]

→

\left[ \begin{matrix}
19 & \color{lime}0 & \color{lime}0 \\
\hline
1 & \color{green}{-30} & \color{green}{-44} \\
0 & \color{green}19 & \color{green}0 \\
0 & \color{green}0 & \color{green}19
\end{matrix} \right]
</math>

Now grab the parts of the columns from below the line.

<math>
\left[ \begin{array} {c|c}
\color{green}{-30} & \color{green}{-44} \\
\color{green}19 & \color{green}0 \\
\color{green}0 & \color{green}19
\end{array} \right]
</math>

So that's not any of the commas we've looked at so far (it's the [[19-comma|19-edo-comma]] and the [[acute limma]]). But it is clear to see that either of them would be made to vanish by 19-ET (no need to map by hand — just look at these commas side-by-side with the mapping-row {{rket|{{map|19 30 44}}}} and it should be apparent). We're done!

And let's try that one in Wolfram Language, too:

<nowiki>In: nullSpaceBasis["⟨19 30 44]"]
Out: "[[-44 0 19⟩ [-30 19 0⟩]"</nowiki>

So we've gotten right back where we've started.

The RTT library for Wolfram Language includes a function called <code>dual[]</code>. This will give you a comma basis for a mapping, or a mapping for a comma basis, depending on which you put it:

<nowiki>In: dual["⟨19 30 44]"]
Out: "[[-44 0 19⟩ [-30 19 0⟩]"
In: dual["[[-4 4 -1⟩ [-10 -1 5⟩]"]
Out: "⟨19 30 44]"</nowiki>

It's great to have Wolfram and other such tools to compute these things for us, once we understand them. But we think it's a very good idea to work through these operations by hand at least a couple times, to demystify them and give you a feel for them.

=== JI as a temperament ===
Two points make a line. By the same logic, three points make a plane. Does this carry any weight in RTT? Yes it does.

Our hypothesis might be: this represents the entirety of 5-limit JI. If two rank-1 temperaments — each of which can be described as making 2 commas vanish — when map-merged result in a rank-2 temperament — which is defined as making 1 comma vanish — then when we map-merge three rank-1 temperaments, we should expect to get a rank-3 temperament, which makes 0 commas vanish. The rank-1 temperaments appear as 0D points in PTS but are understood to be a 1D line coming straight at us; the rank-2 temperaments appear as 1D lines in PTS but are understood to be 2D planes coming straight at us; the rank-3 temperament appear as the 2D plane of the entire PTS diagram but is understood to be the entire 3D space.

Let's check our hypothesis using the PTS navigation techniques and matrix math we've learned.

Let's say we pick three ETs from PTS: 12, 15, and 22. The same constraint applies here that we can't choose ETs for which there is a smaller number between them on the line that connects them. Each pair of these pass that test. Done.

Their combined matrix is:

<math>
\left[ \begin{matrix}
12 & 19 & 28 \\
15 & 24 & 35 \\
22 & 35 & 51
\end{matrix} \right]
</math>

We explain later about [[#Canonical_form|canonical form]], but for now you'll have to take our word for it that the canonical form of the above mapping is this:

<math>
\left[ \begin{matrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{matrix} \right]
</math>

Hey, that looks like an identity matrix! Well, in this case the best interpretation can be found by checking its mapping of {{sfrac|2|1}}, {{sfrac|3|1}}, and {{sfrac|5|1}}, or in other words {{vector|1}}, {{vector|0 1}}, and {{vector|0 0 1}}. Each prime is generated by a different generator, independently. And if you think about the implications of that, you'll realize that this is simply another way of expressing the idea of 5-limit JI! Because the three generators are entirely independent, we are capable of exactly generating literally any 5-limit interval. Which is another way of confirming our hypothesis that no commas vanish.

=== Tempered lattice ===
Let's make sure we establish what exactly the tempered lattice is. This is something like the JI lattice we looked at very early on, except instead of one axis per prime, we have one axis per generator. As we saw just a moment ago, these two situations are not all that different; the JI lattice could be viewed as a tempered lattice, where each prime is a generator.

In this rank-2 example of 5-limit meantone, we have 2 generators, so the lattice is 2D, and can therefore be viewed on a simple square grid on the page. Up and down correspond to movements by one generator, and left and right correspond to movements by the other generator.

The next step is to understand our primes in terms of this temperament's generators. Meantone's mapping is {{rket|{{map|1 0 -4}} {{map|0 1 4}}}}. This maps prime 2 to one of the first generator and zero of the second generator. This can be seen plainly by slicing the first column from the matrix; we could even write it as the vector {{vector|1 0}}. Similarly, this mapping maps prime 3 to zero of the first generator and one of the second generator, or in vector form {{vector|0 1}}. Finally, this mapping maps prime 5 to negative four of the first generator and four of the second generator, or {{vector|-4 4}}.

So we could label the nodes with a list of approximations. For example, the node at {{vector|-4 4}} would be ~5. We could label ~9/8 on {{vector|-3 2}} just the same as we could label {{vector|-3 2}} 9/8 in JI, however, here, we can also label that node ~10/9, because {{vector|1 -2 1}} → 1×{{vector|1 0}} + -2×{{vector|0 1}} + 1×{{vector|-4 4}} = {{vector|1 0}} + {{vector|0 -2}} + {{vector|-4 4}} = {{vector|-3 2}}. Cool, huh? Because conflating 9/8 and 10/9 is a quintessential example of the effect of making the meantone comma vanish ''(see Figure 2b)''.

[[File:Mapping to tempered vector.png|400px|thumb|right|'''Figure 2b.''' Converting from a JI interval vector to a tempered interval vector, with one less rank, conflating intervals related by the vanished comma.]]

Sometimes it may be more helpful to imagine slicing your mapping matrix the other way, by columns (vectors) corresponding to the different primes, rather than rows (maps) corresponding to generators. Meaning we can look at {{rket|{{map|1 0 -4}} {{map|0 1 4}}}} as a matrix of three vectors, [{{vector|1 0}} {{vector|0 1}} {{vector|-4 4}}] which tells us that 2/1 is {{vector|1 0}}, 3/1 is {{vector|0 1}}, and 5/1 is {{vector|-4 4}}.

And so we can see that tempering has reduced the dimensionality of our lattice by 1. Or in other words, the dimensionality of our lattice was always the rank; it's just that in JI, the rank was equal to the dimensionality. And what's happened by reducing this rank is that we eliminated one of the primes in a sense, by making it so we can only express things in terms of it via combinations of the other remaining primes.

=== Rank and nullity ===
Let's review what we've seen so far. 5-limit JI is 3-dimensional. When we have a rank-3 temperament of 5-limit JI, 0 commas vanish. When we have a rank-2 temperament of 5-limit JI, 1 comma vanishes. When we have a rank-1 temperament of 5-limit JI, 2 commas vanish.<ref group="note">Probably, a rank-0 temperament of 5-limit JI would make 3 commas vanish. All we can think a rank-0 temperament could be is a single pitch, or in other words, every interval vanishes (becomes a unison).</ref>

There's a straightforward formula here: <math>d - n = r</math>, where <math>d</math> is dimensionality, <math>n</math> is nullity, and <math>r</math> is rank.<ref group="note">If you wanted a trio of words that all end in "-ity" as a mnemonic device, you could use "rowity" or "rangity" for <math>r</math>, where "row" refers to rows of mappings, and "range" refers to the domain/range distinction of functions such as mappings (and along those lines, you'd also get "domainity" for <math>d</math> if you like).</ref> We've seen every one of those words so far except '''nullity'''. [[Nullity]] simply means the minimum number of vanishing commas whose multiples can be added and subtracted to span the entire space of vanishing commas, or in other words, the count of commas in a basis for the nullspace ''(see Figure 2c)''.

So far, everything we've done has been in terms of 5-limit, which has dimensionality of 3. Before we generalize our knowledge upwards, into the 7-limit, let's take a look at how things one step downwards, in the simpler direction, in the 3-limit, which is only 2-dimensional.

We don't have a ton of options here! The PTS diagram for 3-limit JI could be a simple line. This axis would define the relative tuning of primes 2 and 3, which are the only harmonic building blocks available. Along this line we'll find some points, which familiarly are ETs. For example, we find 12-ET. Its map here is {{map|12 19}}; no need to mention the 5-term because we have no vectors that will use it here. At this ET, being a rank-1 temperament, <math>r</math> = 1. So if <math>d</math> = 2, then solve for <math>n</math> and we find that it only makes a single comma vanish (unlike the rank-1 temperaments in 5-limit JI, which made two commas vanish). We can use our familiar nullspace function to find what this comma is:

<math>
\left[ \begin{matrix}
12 & 19 \\
\hline
1 & 0 \\
0 & 1
\end{matrix} \right]

→

\left[ \begin{matrix}
12 & 228 \\
\hline
1 & 0 \\
0 & 12
\end{matrix} \right]

→

\left[ \begin{matrix}
12 & 0 \\
\hline
1 & {-19} \\
0 & 12
\end{matrix} \right]
</math>

Let's try it out in Wolfram Language:

<nowiki>In: nullSpaceBasis["⟨12 19]"]
Out: "[-19 12⟩"</nowiki>

Unsurprisingly, the comma is {{vector|-19 12}}, the compton comma. Basically, any 3-limit comma that would vanish is going to be obvious from the ET's map. Another option would be the blackwood comma, {{vector|-8 5}} which vanishes in 5-ET, {{map|5 8}}. Exciting stuff! Okay, not really. But good to ground yourself with.

But now you shouldn't be afraid even of 11-limit or beyond. The 11-limit is 5D. So if you make 2 commas vanish there, you'll have a rank-3 temperament.

[[File:Mapping and comma basis dnr.png|400px|thumb|right|'''Figure 2c.''' The relationship between dimensionality d, rank r, and nullity n]]

Here we've mentioned the term "rank". We warned you that it wasn't actually the same thing as dimensionality, even though we could use dimensionality in the PTS to help differentiate rank-2 from rank-1 temperaments. Now it's time to learn the true meaning of rank: it's how many generators a temperament has. So, it ''is'' the dimensionality of the ''tempered'' lattice; but it's still important to stay clear about the fact that it's different from the dimensionality of the original system from which you are tempering.

=== Beyond the 5-limit ===
So far we've only been dealing with RTT in terms of prime limits, which is by far the most common and simplest way to use it. But nothing is stopping you from using other types of spaces. What is a space? Well, I'll explain in terms of what we already know: prime limits. Prime limits are basically the simplest type of space. A prime limit is shorthand for the space whose basis consists of all the primes up to that prime which is your limit; for example, the 7-limit is the same thing as the domain with the basis "2.3.5.7". So a [[basis]] is just a set of JI intervals, and they are notated by separating the selected intervals with dots.

Sometimes you may want to use a [[Domain basis#Nonstandard domains|nonstandard domain]], i.e. anything other than a prime limit. For example, you could create a 3D tuning space out of primes 2, 3, and 7 instead, skipping prime 5. You would call it "the 2.3.7 domain".<ref group="note">Some people refer to these as subspaces. But even standard domains are subspaces of all of the primes, so who really cares that it's a subspace of another space or not?</ref>

You could even choose a domain with combinations of primes, such as the 2.5/3.7 space. Note that since the dots are what separate intervals, that should be parsed as 2.(5/3).7. Here, we still care about approximating primes 2, 3, 5, and 7, however there's something special about 3 and 5: we don't specifically care about approximating 3 or 5 individually, but only about approximating their combination. Note that this is different yet from the 2.15.7 space, where the combinations of 3 and 5 we care about approximating are when they're on the same side of the fraction bar.

As you can see from the 2.15.7 example, you don't even have to use primes. Simple and common examples of this situation are the 2.9.5 or the 2.3.25 spaces, where you're targeting multiples of the same prime, rather than combinations of different primes.

You can even use irrationals, like the 2.<math>\phi</math>.5.7 space! But now you won't be tempering JI, but that's fine, if that's what you want. The sky is the limit. Whatever you choose, though, this core structural rule <math>d - n = r</math> holds strong ''(see Figure 2d)''.

The order you list the pitches you're approximating with your temperament is not standardized; generally you increase them in size from left to right, though as you can see from the 2.9.5 and 2.15.7 examples above it can often be less surprising to list the numbers in prime limit order instead. Whatever order you choose, the important thing is that you stay consistent about it, because that's the only way any of your vectors and maps are going to match up correctly!

[[File:Temperaments by rnd.png|400px|thumb|left|'''Figure 2d.''' Some temperaments by dimensionality, rank, and nullity]]

Alright, here's where things start to get pretty fun. 7-limit JI is 4D. We can no longer refer to our 5-limit PTS diagram for help. Maps and vectors here will have four terms; the new fourth term being for prime 7. So the map for 12-ET here is {{map|12 19 28 34}}.

Because we're starting in 4D here, if we make one comma vanish, we still have a rank-3 temperament, with 3 independent generators. Make two commas vanish, and we have a rank-2 temperament, with 2 generators (remember, one of them is the period, which is usually the octave). And we'd need to make 3 commas vanish here to pinpoint a single ET.

The particular case I'd like to focus our attention on here is the rank-2 case. This is the first situation we've been able to achieve which boasts both an infinitude of matrices made from comma vectors which can represent the temperament by a comma basis, as well as an infinitude of matrices made from ET maps which can represent a temperament by a mapping-row-basis. These are not contradictory. Let's look at an example: septimal meantone.

Septimal meantone may be thought of as the temperament which makes the meantone comma and the starling comma (126/125) vanish, or "meantone|starling". But it may also be thought of as "meantone|marvel", where the marvel comma is 225/224. We don't even necessarily need the meantone comma at all: it can even be "starling|marvel"! This speaks to the fact that any temperament with a nullity greater than 1 has an infinitude of equivalent comma bases. It's up to you which one to use.

On the other side of duality, septimal meantone's mapping-row-basis has two rows, corresponding to its two generators. We don't have PTS for 7-limit JI handy, but because septimal meantone includes, or extends plain meantone, we can still refer to 5-limit PTS, and pick ETs from the meantone line there. The difference is that this time we need to include their 7-term. So the map-merge of {{map|12 19 28 34}} and {{map|19 30 44 53}} would work. But so would {{map|19 30 44 53}} and {{map|31 49 72 87}}. We have an infinitude of options on this side of duality too, but here it's not because our nullity is greater than 1, but because our rank is greater than 1.

=== Canonical form ===
Recently we reduced

<math>
\left[ \begin{matrix}
5 & 8 & 12 \\
7 & 11 & 16 \\
\end{matrix} \right]
</math>

to

<math>
\left[ \begin{matrix}
1 & 1 & 0 \\
0 & 1 & 4
\end{matrix} \right]
</math>

In this form, as we observed, the period is an octave and the generator is a fifth, which is a popular and convenient way to think about meantone. But there are other good forms this mapping could be put into.

For example, you might want the form that Graham Breed's temperament finder puts them in, where all values in a mapping-row may be negative, but this is in the service of the generator being positive, and less than half the size of the period. For example, for meantone, we'd want the fourth instead of the fifth, and we can see that

<math>
\left[ \begin{matrix}
1 & 2 & 4 \\
0 & {-1} & {-4}
\end{matrix} \right]
</math>

maps the fourth (4/3, {{vector|2 -1 0 }}) to {{vector|0 1}}. That form is called [[mingen]] form.

But there are still more forms! One very important form is called [[defactored Hermite form]] (DHF), or we may call it here '''canonical form'''.

It's often the case that a temperament's rank is greater than 1, and therefore we have an infinitude of equivalent ways of expressing the mapping.<ref group="note">The same is true of the comma basis when the nullity is greater than 1, but we'll deal with that in a later article.</ref> This can be problematic, if we want to efficiently communicate about and catalog temperaments. It's good to have a standardized form in these cases. The approach RTT takes here is to get these matrices into canonical form. In plain words, this just means: we have a function which takes in a matrix and spits out a matrix of the same shape, and no matter which matrix we input from a set of matrices which we consider all to be equivalent to each other, it will spit out the same result. This output is thereby "canonical", and it can therefore uniquely identify a temperament.

To be clear, canonical form isn't necessary to avoid ambiguity: you will never find a mapping that could represent more than one temperament.

For example, the canonical form of meantone is:

<math>
\left[ \begin{matrix}
1 & 0 & {-4} \\
0 & 1 & 4
\end{matrix} \right]
</math>

So if you take the canonical form of {{rket|{{map|5 8 12}} {{map|7 11 16}}}}, that's what you get. It's also what you get if you take the canonical form of {{rket|{{map|12 19 28}} {{map|19 30 44}}}}, or any equivalent other mapping. That's the power of canonicalization.

Let's try it out in Wolfram Language:

<nowiki>In: canonicalForm["[⟨5 8 12] ⟨7 11 16]}"]
Out: "[⟨1 0 -4] ⟨0 1 4]}"</nowiki>

Canonical form can be done by hand, but it's a bit involved, because it requires first [[Defactoring_algorithms#Column_Hermite_defactoring|defactoring]] and then putting into [[Hermite normal form|Hermite Normal Form]]. We've demonstrated how to do these processes at the links provided.

Canonicalization used to be achieved in RTT through the use of the "wedgie", an object that involves more advanced math. So while you may see "wedgies" around on the wiki and elsewhere, don't worry — you don't need to worry about them in order to do RTT. If you want to learn more anyway, we've gathered up everything we figured out about those here: [[Intro to exterior algebra for RTT]].

We can also use canonical form for comma bases. One comma basis for 12-ET is:

<nowiki>In: dual[{{{12,19,28}},"mapping"}]
Out: {{{-19,12,0},{-15,8,1}},"comma basis"}</nowiki>

So that's the Pythagorean comma and the schisma. We might think 12-ET is defined by the meantone comma and augmented comma, though. Well…

<nowiki>In: canonicalForm[{{{-4, 4,-1},{7,0,-3}},"comma basis"}]
Out: {{{-19,12,0},{-15,8,1}},"comma basis"}</nowiki>

So those are the same thing.

There's one difference when we find the canonical form of a comma basis: we enclose the DHF operation in an antitranspose sandwich, which is to say we perform an antitranspose operation before and after the DHF operation.

What's an antitranspose, you ask? Well, while an ordinary transpose operation flips a matrix about its main diagonal — which is the diagonal that begins in the upper left corner — an antitranspose operation flips it about its antidiagonal, which is perpendicular to the main diagonal.

For more information on why we do that, see: [[Normal lists#Antitransposed Defactored Hermite form]]. Basically, it's because we want our canonical comma basis to have zeros in its bottom-left corner just like a mapping, and an antitranspose sandwich coaxes the HNF into giving us just that.

=== Summary table ===
{| class="wikitable" style="text-align: center;"
|+ style="font-size: 105%;" | RTT terminology
|- style="white-space: nowrap;"
! Terminology category
! Building block →
! Temperament ID
! Temperament ID dual
! ← Building block
|-
! RTT application
| Map (often an ET), mapping-row
| Mapping, mapping-row-basis
| Comma basis
| Interval, comma
|-
! RTT structure
| (Generator-count-per-prime) map
| List of maps
| List of vectors
| (Prime-count) vector
|-
! Linear algebra structure
| Row vector, matrix row
| Matrix, list of row vectors
| Matrix, list of vectors
| (Column) vector, matrix column, vector
|-
! Extended bra-ket notation representation
| Bra
| Ket of bras
| Bra of kets
| Ket
|-
! RTT jargon
| Val
| List of vals
| List of monzos
| Monzo
|}

== See also ==
Congrats! You've made it to the end of the basic section of our article series. You have plenty of information now to go and make some music using RTT. If you just want to keep learning all the things, though (or just want to procrastinate, like us) then please check out the intermediate section of our series here:
* 5. {{subpage|Units analysis|prev}}: To look at temperament and tuning in a new way, think about the units of the values in frequently used matrices
* 6. {{subpage|Tuning computation|prev}}: For methods and derivations; learn how to compute tunings, and why these methods work
* 7. {{subpage|All-interval tuning schemes|prev}}: The variety of tuning scheme that is most commonly named and written about on the Xenharmonic wiki

You may also be interested in checking out Chris Kline's web app for exploring PTS, available here: https://www.projectivetuningspace.com

== Footnotes ==
<references group="note" />

[[Category:Dave Keenan & Douglas Blumeyer's guide to RTT]]

7L 8s

2026-05-24T18:03:20Z

Sintel: i dont care

{{Infobox MOS}}
{{MOS intro}}

It is notable for supporting [[Porcupine]], of the [[Porcupine_family|porcupine family]].

== Scale properties ==
{{TAMNAMS use}}

=== Intervals ===
{{MOS intervals}}

=== Generator chain ===
{{MOS genchain}}

=== Modes ===
{{MOS mode degrees}}

== Scale tree ==
{{todo|inline=1|complete table|text=There was previously octachord info in the old scale tree, in the form of the step pattern LsLsLsL. Please add it to the new scale tree.}}
{{MOS tuning spectrum
| Depth = 6
| 3/2 = Optimal rank range ({{nowrap|L/s {{=}} 3/2}}) porcupine
| 13/8 = Golden porcupine {{nowrap|L/s {{=}} φ}}
}}

{{stub}}

[[Category:Porcupine]]
[[Category:Abstract MOS patterns]]

Numerary nexus

2026-05-22T22:04:07Z

Sintel: Give a basic definition and some examples

In the theory of [[Harry Partch]], the '''numerary nexus''' is a number that serves as the shared numerical identity in a set of ratios that determines them as a tonality.<ref>Harry Partch, ''Genesis of a Music'', (1974)</ref>
It is the common factor that appears in a set of ratios, appearing in either the numerator or denominator. In an otonality, the nexus appears in the denominator of all the ratios; in a utonality, it appears in the numerator.

The numerary nexus is central to the [[tonality diamond]], where every row and column is organized by a shared nexus.

== Examples ==

In the [[otonal]] set 7/7 (= [[1/1]]), [[7/6]], [[7/5]], [[7/4]], the number 7 appears in the numerator of all intervals, so it is the ''nexus'' linking them.

The [[11-limit]] [[tonality diamond]] contains six identities: (1, 3, 5, 7, 9, 11). The [[utonal]] set based on 5 as the numerary nexus is:
: 1/5, 3/5, 5/5, 7/5, 9/5, 11/5
Which, which [[octave-reduced]] and sorted by their size, gives:
: [[1/1]], [[11/10]], [[6/5]], [[7/5]], [[8/5]], [[9/5]]

== See also ==
* [[Tonality diamond]]
* [[Otonality and utonality]]

== External links ==
* [http://www.tonalsoft.com/enc/n/nexus.aspx Numerary nexus] on [[Tonalsoft Encyclopedia]]

== References ==
<references />

[[Category:Terms]]
[[Category:Otonality and utonality]]
[[Category:Harry Partch]]

User talk:Stalefleas

2026-05-22T12:26:05Z

Sintel: fix redirect

User talk:Stalefleas

2026-05-22T12:22:50Z

Sintel: Sintel moved page User talk:Stalefleas to User:Stalefleas

#REDIRECT [[User:Stalefleas]]

User:Stalefleas

2026-05-22T12:22:50Z

Sintel: Sintel moved page User talk:Stalefleas to User:Stalefleas

[this page is still in progress]

todo: add a link to kite's notation

== 22edo chord notation ==
The following ideas have been formed out as part of the first Tuning of the Year project (for year 2025), hereafter referred to as TOTY. The focus of the original TOTY was 22edo, which was voted on by the Xenharmonic Alliance discord.

Our objective was to flesh out 22edo music theory, arrive at a deeper understanding, and provide resources to help beginners learn how to think about and play in 22edo. The most rigorous work in this tuning had already been accomplished. Ups and downs notation sufficiently identifies the notes and chords, and the most important chords have already been identified and labelled.

Most of our work was in assessing our overall impressions of the tuning, discovering what language was available to discuss harmonies in this tuning, and to check existing nomenclature against our intuitive grasp of what things sound like. Ultimately we arrived at a nomenclature that explains 22edo in terms of how it sounds, as opposed to how it conforms to the logic of the pythagorean circle of fifths. Note names have been unaltered, but some chord and interval names have been altered and potentially refined.

When TOTY began, there was one existing system of chord nomenclature that covers 22edo, Kite Giedraitis' ups and down notation. Most of us were unfamiliar with the naming convention of this system. In developing our own systems of nomenclature, Kite reached out to us with information on his existing system.

Including ups and downs notation, we have at least four distinct methods of identifying chords in 22edo. These methods are all comprehensive, meaning one can reasonably identify any chord just as well as one can identify chords in 12edo using standard chord labels. In sum, we have ups and downs notation, classic notation, double-qualifier notation, and temperament notation.

= Intervals =
Using a "sounds-like" system of interval naming, we can identify 22edo as having the following intervals:
{| class="wikitable"
|+
!edosteps
!symbol
!spoken name
!size (in cents)
|-
|1\22
|s2
|subminor second
|54.5
|-
|2\22
|m2
|minor second
|109.1
|-
|3\22
|n2
|neutral second
|163.6
|-
|4\22
|M2
|major second
|218.2
|-
|5\22
|s3
|subminor third
|272.7
|-
|6\22
|m3
|minor third
|327.3
|-
|7\22
|M3
|major third
|381.8
|-
|8\22
|S3
|supermajor third
|436.4
|-
|9\22
|p4
|perfect fourth
|490.9
|-
|10\22
|M4
|major fourth
|545.5
|-
|11\22
|S4/s5
|supermajor fourth
subminor fifth
|600
|-
|12\22
|m5
|minor fifth
|654.5
|-
|13\22
|P5
|perfect fifth
|709.1
|-
|14\22
|s6
|subminor sixth
|763.6
|-
|15\22
|m6
|minor sixth
|818.2
|-
|16\22
|M6
|major sixth
|872.7
|-
|17\22
|S6
|supermajor sixth
|927.3
|-
|18\22
|m7
|minor seventh
|981.8
|-
|19\22
|n7
|neutral seventh
|1036.4
|-
|20\22
|M7
|major seventh
|1090.9
|-
|21\22
|S7
|supermajor seventh
|1145.5
|-
|22\22
|p8
|octave
|1200
|}
Note that this is contrary to the established convention of referring to the 6\22 interval as the "upminor" and the 7\22 interval as the "downmajor". Generally speaking, we felt that these intervals simply sound like minor and major intervals, respectively. Also, their lesser and greater counterparts, 5\22 and 8\22, which are labelled on the 22edo page as "minor" and "major," sound more like subminor and supermajor intervals. The nearest LCJI approximations to these two intervals, 7/6 (266.8c) and 9/7 (435c) are typically defined as subminor and supermajor.

One can see that there is some inconsistency in qualities across the intervals. Seconds and sevenths are defined as minor, neutral, major, and supermajor, while thirds and sixths are defined as subminor, minor, major, and supermajor. This simply seems more accurate to the sound. Note that symmetry is still preserved across the tritone.

While obviously, there is some subjectivity here, and the possibility of an even more refined perspective. But this seems good enough to talk about 22edo in a way that is intuitive to what we are hearing.

= Triads =
In order to communicate the chords of 22edo, it is necessary to identify the names of the tertian triads. Adding sevenths to our triads gives us many practical chords that will be used by most composers and musicians. Adding our standard modifications to these, adding the sevenths, and alterations, gives us a complete system with which to name any chord. Here a triad is defined explicitly as a tertian triad, being a chord made out of two stacked thirds. Only the subminor, minor, major, and supermajor thirds (5\22, 6\22, 7\22, 8\22) are considered.

There are (at least) four comprehensive systems for naming chords in 22edo: ups and downs notation, classic notation, double-qualifier notation, and temperament notation. Ups and downs notation adheres to pythagorean logic in naming all chords and intervals, and is oldest and most established of the four. TOTY was not involved in its inception. The other three were created in tandem, and are quite closely related.

For consistency and convenience, all chords are defined on the root of C.

=== Classic Notation ===
The classic notation system is intended to be intuitive and clear. Chord labels are very similar to 12edo chord symbols. With the exception of ups and downs, which are standard accidentals in many microtonal systems, no new chord symbols are introduced.

Many tertian triads in 22edo have altered fifths. In classic notation, the alteration is spelled out explicitly. The chord C - E - vG would be spelled as a CS(v5).

The tertian triads in classic notation would be spelled thusly:
{| class="wikitable"
|+
!Note names
!Edosteps
!Interval sizes
!Chord label
!Spoken name
|-
|C Eb Gb
|0 5 10
|s3 s3
|Csb5
|C subdiminished
|-
|C Eb ^Gb
|0 5 11
|s3 m3
|Cs^b5
|C subminor up-flat five
|-
|C ^Eb ^Gb
|0 6 11
|m3 s3
|Cdim
|C diminished
|-
|C Eb vG
|0 5 12
|s3 M3
|Csv5
|C subminor down five
|-
|C ^Eb vG
|0 6 12
|m3 m3
|Cmv5
|C minor down five
|-
|C vE vG
|0 7 12
|M3 s3
|Cv5
|C down-five
|-
|C Eb G
|0 5 13
|s3 S3
|Cs
|C subminor
|-
|C vEb G
|0 6 13
|m3 M3
|Cm
|C minor
|-
|C vE G
|0 7 13
|M3 m3
|C
|C (major)
|-
|C E G
|0 8 13
|S3 s3
|CS
|C supermajor
|-
|C vEb ^G
|0 6 14
|m3 S3
|Cm^5
|C minor up five
|-
|C vE ^G
|0 7 14
|M3 M3
|Caug
|C augmented
|-
|C E ^G
|0 8 14
|S3 m3
|CS^5
|C supermajor up five
|-
|C vE vG#
|0 7 15
|M3 S3
|Cv#5
|C down-sharp five
|-
|C E vG#
|0 8 15
|S3 M3
|CSv#5
|C supermajor down-sharp five
|-
|C E G#
|0 8 16
|S3 S3
|CS#5
|C supermajor sharp five
|}
In this system, chords are quite clear. For maximum clarity, one could also opt to call the diminished chord "C minor up-flat five" and call the augmented chord "C major up five".

While the chord labels can be completely unambiguous written in this way, many people do not write ups and downs before the note they modify. In this case, it would be necessary to include the alterations of the fifth either in parenthesis or otherwise demarcated. However, it's quite clean if ups and downs are written before the note they modify. For instance, written correctly, ^D - F# ^A would be written as:

^D5

or written without respect to this convention:

D^(5)

since if this chord were written as D^5, it would be unclear if the root was being modified or the fifth.

=== Double-Qualifier Notation ===
In order to avoid using many alterations, we can actually describe every triad based on the quality of its third and the quality of its fifth. Using the interval system above, and the necessary triads, we have seven qualities of fifth: diminished, subminor, minor, perfect, major, supermajor, and augmented. These intervals are 10\22, 11\22, 12\22, 13\22, 14\22, 15\22, and 16\22.

While double-qualifier notation is not immediately obvious, it is quite easy to understand, and is basically interchangeable with classic notation. The advantage to using double-qualifier notation it does not require writing alterations for every type of fifth. In actuality, this is somewhat consistent with 12-edo nomenclature, which usually doesn't explicitly notate altered fifths in chord symbols (except in modern styles like jazz). For instance, the diminished fifth is communicated by the symbol for diminished triad, and the augmented fifth is communicated by the symbol for the augmented triad.

{| class="wikitable"
!Note names
!Edosteps
!Interval sizes
!DQ Label
!DQ spoken name
!Classic label
|-
|C Eb Gb
|0 5 10
|s3 s3
|Csd
|C subminor diminished
|Csb5
|-
|C Eb ^Gb
|0 5 11
|s3 m3
|Css
|C double subminor
|Cs^b5
|-
|C ^Eb ^Gb
|0 6 11
|m3 s3
|Cms
|C minor subminor
|Cdim
|-
|C Eb vG
|0 5 12
|s3 M3
|Csm
|C subminor minor
|Csv5
|-
|C ^Eb vG
|0 6 12
|m3 m3
|Cmm
|C doubleminor
|Cmv5
|-
|C vE vG
|0 7 12
|M3 s3
|CMm
|C major minor
|Cv5
|-
|C Eb G
|0 5 13
|s3 S3
|Cs
|C subminor
|Cs
|-
|C vEb G
|0 6 13
|m3 M3
|Cm
|C minor
|Cm
|-
|C vE G
|0 7 13
|M3 m3
|C
|C (major)
|C
|-
|C E G
|0 8 13
|S3 s3
|CS
|C supermajor
|CS
|-
|C vEb ^G
|0 6 14
|m3 S3
|CmM
|C minor major
|Cm^5
|-
|C vE ^G
|0 7 14
|M3 M3
|CMM
|C double major
|Caug
|-
|C E ^G
|0 8 14
|S3 m3
|CSM
|C supermajor major
|CS^5
|-
|C vE vG#
|0 7 15
|M3 S3
|CMS
|C major supermajor
|Cv#5
|-
|C E vG#
|0 8 15
|S3 M3
|CSS
|C double supermajor
|CSv#5
|-
|C E G#
|0 8 16
|S3 S3
|CSA
|C supermajor augmented
|CS#5
|}
If writing triads, including the 5 is optional. One could opt to include the 5, especially to maintain clarity when writing seventh chords or extended chords (explored in more detail below). For instance, Css could be written as Css5 or Cs(s5).

An advantage to this system is that ups and downs can be used to define alterations beyond fifths and sevenths. A disadvantage is that one must learn the various qualities of the fifth as defined by this system, and the chord names sound very similar. It is also contrary to the usual method of chord qualification, where two qualifiers typically define the quality of the triad and the seventh. For instance, C minor-major is shorthand in 12edo for a minor triad with a major seventh. In this double qualifier system, the same chord should be explicitly called a C minor major-seventh, as opposed to C minor major-fifth.

=== Temperament Notation ===
The least conventional of these notation systems is the temperament notation. In this system, unique symbols are prescribed for every triad. The basic triads are given the same symbols as classic notation. But altered fifths are given specific names. There are also plusses and minuses added as qualifiers for various triads.

While temperament notation has the steepest learning curve, it has the advantage of highlighting relationships between chords that might otherwise be less obvious. It also is, at least in my opinion, actually quite intuitive, at least for 22edo.
{| class="wikitable"
!Note names
!Edosteps
!Interval sizes
!Temp label
!Temp spoken name
!Classic label
|-
|C Eb Gb
|0 5 10
|s3 s3
|Cw
|C orwell / subdiminished
|Csb5
|-
|C Eb ^Gb
|0 5 11
|s3 m3
|Cd-
|C utonal diminished
|Cs^b5
|-
|C ^Eb ^Gb
|0 6 11
|m3 s3
|Cd+
|C otonal diminished
|Cdim
|-
|C Eb vG
|0 5 12
|s3 M3
|Cw-
|C orwell minor
|Csv5
|-
|C ^Eb vG
|0 6 12
|m3 m3
|Ck
|C keemic
|Cmv5
|-
|C vE vG
|0 7 12
|M3 s3
|Cw+
|C orwell major
|Cv5
|-
|C Eb G
|0 5 13
|s3 S3
|Cs
|C subminor
|Cs
|-
|C vEb G
|0 6 13
|m3 M3
|Cm
|C minor
|Cm
|-
|C vE G
|0 7 13
|M3 m3
|C
|C (major)
|C
|-
|C E G
|0 8 13
|S3 s3
|CS
|C supermajor
|CS
|-
|C vEb ^G
|0 6 14
|m3 S3
|CZ-
|C sensaminor
|Cm^5
|-
|C vE ^G
|0 7 14
|M3 M3
|CJ
|C magic
|Caug
|-
|C E ^G
|0 8 14
|S3 m3
|CZ+
|C sensamajor
|CS^5
|-
|C vE vG#
|0 7 15
|M3 S3
|CJ-
|C magic minor
|Cv#5
|-
|C E vG#
|0 8 15
|S3 M3
|CJ+
|C magic major
|CSv#5
|-
|C E G#
|0 8 16
|S3 S3
|CZ
|C sensamagic
|CS#5
|}
The three orwell triads: subdiminished, orwell minor, and orwell major, are the only existing triads in the orwell [5] MOS scale in 22edo. The keemic triad is a reduced form of the keemic seventh chord. The magic MOS also includes the magic triads. CJ inverts to CJ- which inverts to CJ+. Similarly, CZ inverts to CZ+ and to CZ-.

This naming convention highlights the variety of diminished and augmented chords in 22edo. Instead of calling any specific triad the augmented triad, we have opted to consider the magic triads and the sensamagic triad their own distinct identities, though they are all functionally augmented triads in the right context.

An advantage to using temperament notation is that it is fairly clean and elegant. It is actually consistent with 12edo chord labels and logic, since 12edo uses unique symbols for every tertian triad. The disadvantage is that the names are not immediately obvious, especially to beginners who are unaware of temperaments.

Some of the chord symbols are also more or less arbitrary. It seemed necessary to use symbols that would not be confused for existing accidentals or existing note names. I chose letters that sound like they belong to the word they represent. K for keemic is quite obvious, but J for magic is a little less obvious--really it is just because the G in "magic" sounds like a J. As for Z, this seems perhaps the most arbitrary symbol, but I felt like the sensamagic triad is quite a big chord, and Z is quite a big letter, and Z looks like a backwards S.

=== Chord Families ===
In examining the 16 tertian triads of 22edo, we can see identify four categories: diminished, basic, augmented, and hybrid.

==== Diminished ====
The orwell subdiminished triad, the utonal and otonal diminished triads, and the keemic triad can all function as a diminished triad in the proper context. The orwell triad is the natural diminished triad of the super-pythagorean diatonic scale (the 5L2s mos-diatonic). The otonal diminished triad is the natural diminished triad of nicetone, as well as the upper triad of the harmonic seventh chord. The utonal diminished is the natural diminished ii chord of nicetone. Keemic does not occur in either of these diatonic scales of 22edo, but being a stack of two minor thirds, can sound quite like a diminished chord.

==== Basic ====
Triads that form a perfect fifth can be considered basic. They are in all probability the most commonly used and explored chords. These include the subminor, minor, major, and supermajor triads. Of these, the supermajor triad in 22edo is probably the most difficult to use.

==== Augmented ====
The magic triads, including magic major and magic minor, can all function as augmented triads in the proper context. The sensamagic triad can also function as an augmented triad. Augmented triads are not naturally occurring in any diatonic scale, but the sensamagic triad would be the logical augmented triad of a super-pythagorean diatonic, and the magic minor would be the natural extension of the nicetone diatonic.

==== Hybrid ====
Four chords of less obvious function remain. These might sound like wolf triads, or less conventional versions of basic triads. Orwell major and orwell minor sound somewhat like major and minor chords, as do sensamajor and sensaminor. These chords are not quite diminished or augmented, and not quite basic. Thus, they can be conceptualized as hybrid triads.

=== Triads Overview ===
The following table displays the tertian triads, their respective labels, and the families to which they belong.
{| class="wikitable"
!Note names
!Edosteps
!Interval sizes
!Classic
!DQ
!Kite
!Temp
!Chord family
|-
|C Eb Gb
|0 5 10
|s3 s3
|Csb5
|Csd
|Cd
|Cw
|diminished
|-
|C Eb ^Gb
|0 5 11
|s3 m3
|Cs^b5
|Css
|Cd(^5)
|Cd-
|diminished
|-
|C ^Eb ^Gb
|0 6 11
|m3 s3
|Cdim
|Cms
|C^d(^5)
|Cd+
|diminished
|-
|C Eb vG
|0 5 12
|s3 M3
|Csv5
|Csm
|Cm(v5)
|Cw-
|hybrid
|-
|C ^Eb vG
|0 6 12
|m3 m3
|Cmv5
|Cmm
|C^m(v5)
|Ck
|diminished
|-
|C vE vG
|0 7 12
|M3 s3
|Cv5
|CMm
|Cv(v5)
|Cw+
|hybrid
|-
|C Eb G
|0 5 13
|s3 S3
|Cs
|Cs
|Cm
|Cs
|basic
|-
|C vEb G
|0 6 13
|m3 M3
|Cm
|Cm
|C^m
|Cm
|basic
|-
|C vE G
|0 7 13
|M3 m3
|C
|C
|Cv
|C
|basic
|-
|C E G
|0 8 13
|S3 s3
|CS
|CS
|C
|CS
|basic
|-
|C vEb ^G
|0 6 14
|m3 S3
|Cm^5
|CmM
|C^m(^5)
|CZ-
|hybrid
|-
|C vE ^G
|0 7 14
|M3 M3
|Caug
|CMM
|Cv(^5)
|CJ
|augmented
|-
|C E ^G
|0 8 14
|S3 m3
|CS^5
|CSM
|C(^5)
|CZ+
|hybrid
|-
|C vE vG#
|0 7 15
|M3 S3
|Cv#5
|CMS
|Cv(v#5)
|CJ-
|augmented
|-
|C E vG#
|0 8 15
|S3 M3
|CSv#5
|CSS
|C(v#5)
|CJ+
|augmented
|-
|C E G#
|0 8 16
|S3 S3
|CS#5
|CSA
|Ca
|CZ
|augmented
|}

= Seventh Chords =
Notating seventh chords is fairly straightforward. One simply identifies the type of triad in their preferred system, and appends the quality of the seventh.

If we follow the convention of 12edo, neither major triads nor minor sevenths require clarification. So, an unqualified triad is assumed to be major, and an unqualified seventh is assumed to be minor.

If we define the sevenths as being either minor, neutral, major, and supermajor, some of our common seventh chords might include:
{| class="wikitable"
|+
!Notes
!Chord label
!Spoken name
|-
|C vE G vB
|CM7
|C major seventh
|-
|C ^Eb G ^Bb
|Cmn7
|C minor neutral seventh
|-
|C vE G Bb
|C7
|C dominant / C seven
|-
|C E G B
|CSS7
|C supermajor supermajor seventh
|-
|C Eb G Bb
|Css7
|C subminor subminor seventh
|}
Some less common chords would be expressed differently in different notation systems:
{| class="wikitable"
|+
!Notes
!Classic
!Classic spoken
!DQ
!DQ spoken
!Temp
!Temp spoken
|-
|C vE vG B
|Cv5(S7)
|C down five supermajor seventh
|CMm5(S7)
|C major minor fifth super seventh
|Cw+S7
|C orwell major super seventh
|-
|C E ^G ^Bb
|C^5(n7)
|C up five neutral seventh
|CMM5(n7)
|C major fifth neutral seventh
|CJn7
|C magic neutral seventh
|-
|C Eb ^G Bb
|Cs^5(m7)
|C subminor up five minor seventh
|CsM5(m7)
|C subminor major fifth minor seventh
|CZ-7
|C sensaminor seven
|}

=== Disambiguation ===
In 22edo, the chord that would typically be expressed as the basic "minor seventh" chord actually has a neutral seventh, and the chord that would be expressed as the basic dominant chord has the subminor seventh, here notated simply as the minor seventh. One would expect that many will simply call the minor neutral seventh chord a minor seventh chord for short, and even call the chord C ^Eb G Bb the minor subminor seventh chord. This is fine for conversation, conceptualization, and disambiguation. However, here we are defining the lesser seventh as the minor seventh and the larger minor seventh as the neutral seventh, in part because this preserves tritone symmetry with the seconds, and in part because this seems like an adequate description of the sound of these chords.

It is common to call a chord with a supermajor third and a supermajor seventh simply a supermajor seventh chord, and even to notate it as CS7. However, this is not a clearly defined chord. Here the S could be modifying the triad or the seventh.

So we might have either

C - vE - G - B

or

C - E - G - Bb

By explicitly defining the quality of the triad and the seventh as CSS7, we make it clear that we want the chord

C - E - G - B

What if we did want the chord C - vE - G - B? CS7 would still not suffice. In this chord, it is necessary to separate the seventh from the triad. So here we could write C(S7) or even C.S7 to clarify the seventh is in fact supermajor, and not the triad. And if we wanted the other chord C - E - G - Bb, we could write CS(7) or CS.7 or even CSm7

Disambiguation is not necessary for all chords, but defaulting to using it could lead to greater consistency in notation.

= Extended chords =
Following the convention of 12edo notation, we have qualifiers for both triads and sevenths. In 22edo these have been described above. To add additional extensions, we can also follow the example provided by standard notation. All extensions will be assumed to be mos-diatonic (super-pythagorean) unless otherwise qualified. So, seconds are supermajor by default, fourths are perfect, and sixths are supermajor.

Thus the chord C11 could include the following notes:

C - vE - G - Bb - D - F

However, in 22edo, it is quite likely that we might want the ^11, as it approximates the eleventh harmonic and want the chord that in total approximates the 4:5:6:7:9:11 chord:

C - vE - G - Bb - D - ^F

This could be notated as C9(^11)

Various extended diminished chords can be specified by appending a sixth. If we alternate minor and subminor thirds, we have

C - ^Eb - ^Gb - vBb

since vBb is enharmonically A, we can consider this chord Cd+6, Cdim6. Or, if one prefers, they might opt for the unwieldy Cm^b5(vb7)

We can also add extensions to triads using "add" or by using the comma from Kite notation.

C - vE - G - D

could be written as Cadd9 or C,9

= Non-tertian chords =
Of course, any chord that exists in 12edo can still be expressed in 22edo. We have many non-tertian triads including suspended chords, chords with no thirds, chords with no fifths, etc. We can adopt the same conventions from standard chord notation and apply them to 22edo.

C - F - G

would still simply be a Csus4 chord. But of course in 22edo we also have options like

C - ^F - G

which we might write as Csus^4.

While this is not standard practice, one could borrow the convention of writing powerchords with a "5" to notate chords without thirds. So for instance

C - G - vB

might be major or minor depending on the context, but devoid of such a context (or in a situation where specificity is desired) this could be written as a C5M7. Or, one could follow the existing convention of writing this as CM7(no3).

Some chords are difficult to write, for instance, quartal chords like

C - F - Bb

which is often notated as C7sus4, and could be done here. Or, one could opt to write this as Fsus4/C.

Highly complex chords can be notated using polychord notation. For instance

C - Db - E - F - Gb - G - B

could be written as DbS(7)/CS.

Of course, there are instances where notating a chord might be difficult, if there are large clusters of notes, for instance. Non-standard labels might need to be referred to or invented. Or, once could simply use standard sheet music notation, which can express any number of notes with absolute clarity.

= Additional resources =
Video of the tertian triads of 22edo:

https://www.youtube.com/watch?v=fMeRlKlreV8

= Contributors =

Official members of TOTY 2025, as identified by their discord handles at the time:<blockquote>Nick Vuci (project lead)

stalefleas (project lead)

Sevish

pailiaq

maily

Wrenharmonic

Domin

Ebooone

bartonius

Colonizor48

ground

x e n o [ i ] n d e x

dogwithabome

YoVariable

54edo Pajara

CompactStar

Samikata

MidnightBlue

j87j]oi:bu

recentlymaterialized

MR. LEAKY 666

Romeolz

don page commata

The Universe of Music

Alexei</blockquote>

Comma

2026-05-20T10:13:24Z

Sintel: impressive to make three errors in a single sentence

{{interwiki
| de = Komma
| en = Comma
| es =
| ja =
| ro = Come raționale
}}
{{Wikipedia|Comma (music)}}
In just intonation, a '''comma''' is a small [[interval]] that occurs between two intervals which are close in [[pitch]].

Commas are often considered [[dissonant]] due to their small but noticeable [[interval size measure|size]] which induces an audible [[beat]]. In addition, certain chord progressions are [[comma pump]]s, which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as features to be desired or as problems to be solved. Examples of approaches that try to solve these problems include [[adaptive just intonation]], [[temperament]]s, and [[fudging]].

In [[regular temperament theory]], a comma is something to be ''[[tempered out]]'', equating two mathematically distinct intervals. For example, the [[syntonic comma]] (81/80), which occurs between [[10/9]] and [[9/8]] as well as between [[81/64]] and [[5/4]], is tempered out by [[meantone]] temperament and thus conflates these two pairs of intervals.
[[File:EufaJICommasFinalV2.svg|thumb|A visualization of intervals commonly used as commas, built from stacked intervals, to scale.]]
Commas are usually written as [[frequency ratio]]s, but they can also be written as products of primes, sometimes called [[monzo]]s or '''unison vectors'''. The [[color name]] refers to both the comma and the temperament created when it is tempered out, except for 3-limit commas, which create [[edo]]s, commonly called n-commas, such as the [[29-comma]], or the [[41-comma]].

== As an interval region ==
{{Main| Comma and diesis }}

''Comma'' can be used to refer to any interval bigger than 0 [[cents]], but smaller than around 30 cents.

== Lists of commas by size ==
Commas can theoretically have any size, but in practice most are much smaller than a [[12edo]] semitone (100{{cent}}). The following categories, while arbitrary, are used on the Xenharmonic Wiki to classify commas by size.
* [[Unnoticeable comma]]: under 3.5 cents in size; below the rough boundary of melodic [[just-noticeable difference]] and thus imperceptible.
* [[Small comma]]: between 3.5 and 30 cents.
* [[Medium comma]]: between 30 and 100 cents.
* [[Large comma]]: over 100 cents in size; much wider than what is typically considered to be a comma.

== See also ==
* [[PIFE comma]]
* [[TIFE comma]]
* [[Superparticular ratio]]
* [[Wolf interval]]

== External links ==
* [http://www.tonalsoft.com/enc/c/comma.aspx Comma] on [[Tonalsoft Encyclopedia]]

[[Category:Comma| ]]

Kleisma (interval region)

2026-05-20T10:10:11Z

Sintel: weird claim

{{Infobox interval region
| Name = Kleisma
| Cents lower = 6
| Cents upper = 9
| JI intervals = 15625/15552, 243/242, 225/224, 1029/1024, 16875/16807, 65536/65219
| Lower region = [[Unnoticeable comma]]
| Superregions = [[Small comma]] [[Comma (interval region)|Comma]] [[Comma and diesis]]
| Higher region = [[Small comma]]
}}
A '''kleisma''' is an interval of about 8.1 [[cent]]s, roughly the size of the interval [[15625/15552]], which is called the kleisma in [[just intonation]]. In [[Sagittal notation]], a kleisma is specifically defined as between half of the Pythagorean 200-fifths kleisma {{monzo| 317 -200 }} and half of the [[Pythagorean comma]] {{monzo| -19 12 }}, about 4.5{{c}} to 11.7{{c}}.

The kleisma is significant as it is a limit of intonational fidelity when playing on some physical instruments. That is, on free-pitch instruments, there is a level of precision to which one can be expected to play a note or interval "correctly": that level of precision is the kleisma.{{cn}} Another significance is that a lot of commas are about 3–4 kleismas in size.

Kleismas belong to the larger interval region of [[Comma (interval region)|commas]], which are part of the [[Comma and diesis|"comma and diesis"]] category.

== See also ==
* [[Kleisma]] (disambiguation page)

{{Navbox intervals}}

Interval region

2026-05-20T00:42:14Z

Sintel: /* Schulter system */ We don't need the table twice

There are infinite possible intervals (both tempered and just), even within a single [[2/1|octave]]. It can be helpful to group these intervals into a finite number of '''interval regions''' or '''interval categories'''.

== Concrete regions vs abstract categories ==
An ''interval region'' usually implies it is concrete, defined by concrete boundaries of interval sizes. The boundaries are usually fuzzy to allow some vagueness, in line with how we perceive them. Which region an interval falls into solely depends on the interval's size.

An ''interval category'' is usually meant to be abstract. It uses some [[mapping]] to determine which category an interval falls into, short-circuiting the question of where exactly to place the boundaries. It also takes account of an interval's [[prime factorization|prime components]], allowing us to find a composite interval's category through [[interval arithmetic]].

The [[5L 2s|diatonic]] interval category system commonly used to categorize JI intervals consists of a [[interval quality|quality]] and a diatonic scale degree.

== Extended-diatonic interval names ==
{{Main|Extended-diatonic interval names}}
Many interval naming systems extend the diatonic interval names by adding new [[interval qualities]] to the usual set. While some systems preserve the fifth-based structure entirely, other systems define regions based on the proximity to the intervals associated with the diatonic intervals, which are then divided into finer subregions.

== Latitude ==
When describing interval regions in terms of size relative to a (possibly tempered) fifth, it leads to the system of [[Latitude|latitude and medial intervals]].

== Schulter system ==
[[Margo Schulter]] describes her system for categorizing intervals in ''[http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt Regions of the Interval Spectrum]'', which begins:

: In naming categories of intervals, or regions of the spectrum in which they are found, there may be many valid and desirable schemes reflecting the diversity of viewpoints and styles to be found in world musics. What I describe here is merely one possible solution, and one influenced by my own musicmaking experience and philosophy which seeks an equitable and inclusive balance between intervals at or near simple integer ratios, and those having a more complex or active nature.

Schulter proposes the following categories and gives a tentative range of cents values for intervals that fall within those categories. In ''Regions'', she points out, "A main caution is that the borders are inevitably 'fuzzy,' so that one region shades into another and suggested values in cents are more illustrative than definitive."

{| class="wikitable"
|-
! | Interval Category
! | Approx. Cents Ranges
! colspan="2" |sub-category
|-
| | [[Unison|Pure Unison]] (1:1)
| | 0
|
|
|-
| | [[Comma]]s
| | 0-30
|
|
|-
| | [[Diesis|Dieses]]
| | 30-60
|
|
|-
| rowspan="3"| [[Minor second|Minor Second]]s
| rowspan="3"| 60-125
|small
|60-80
|-
|middle
|80-100
|-
|large
|100-125
|-
| rowspan="3"|[[Neutral second|Neutral Second]]s
| rowspan="3"| 125-170
|small
|125-135
|-
|middle
|135-160
|-
|large
|160-170
|-
| |[[Equable heptatonic|Equable Heptatonic]]
| | 160-182
|
|
|-
| rowspan="3"|[[Major second|Major Second]]s
| rowspan="3"| 180-240
|small
|180-200
|-
|middle
|200-220
|-
|large
|220-240
|-
| |[[Interseptimal interval|Interseptimal]] (Maj2-min3)
| | 240-260
|
|
|-
| rowspan="3"|[[Minor third|Minor Third]]s
| rowspan="3"| 260-330
|small
|260-280
|-
|middle
|280-300
|-
|large
|300-330
|-
| rowspan="3"| [[Neutral third|Neutral Third]]s
| rowspan="3"| 330-372
|small
|330-342
|-
|middle
|342-360
|-
|large
|360-372
|-
| rowspan="3"|[[Major third|Major Third]]s
| rowspan="3"| 372-440
|small
|372-400
|-
|middle
|400-423
|-
|large
|423-440
|-
| |[[Interseptimal interval|Interseptimal]] (Maj3-4)
| | 440-468
|
|
|-
| rowspan="3"|[[Perfect fourth|Perfect Fourth]]s
| rowspan="3"| 468-528
|small
|468-491
|-
|middle
|491-505
|-
|large
|505-528
|-
| |[[Superfourth]]s
| | 528-560
|
|
|-
| rowspan="3"|[[Tritone|Tritonic Region]]
| rowspan="3"| 560-640
|small
|560-577
|-
|middle
|577-623
|-
|large
|623-640
|-
| |[[Subfifth]]s
| | 640-672
|
|
|-
| rowspan="3"|[[Perfect fifth|Perfect Fifth]]s
| rowspan="3"| 672-732
|small
|672-695
|-
|middle
|695-709
|-
|large
|709-732
|-
| | [[Interseptimal interval|Interseptimal]] (5-min6)
| | 732-760
|
|
|-
| rowspan="3"|[[Minor sixth|Minor Sixth]]s
| rowspan="3"| 760-828
|small
|760-777
|-
|middle
|777-800
|-
|large
|800-828
|-
| rowspan="3"|[[Neutral sixth|Neutral Sixth]]s
| rowspan="3"| 828-870
|small
|828-840
|-
|middle
|840-858
|-
|large
|858-870
|-
| rowspan="3"|[[Major sixth|Major Sixth]]s
| rowspan="3"| 870-940
|small
|870-900
|-
|middle
|900-920
|-
|large
|920-940
|-
| | [[Interseptimal interval|Interseptimal]] (Maj6-min7)
| | 940-960
|
|
|-
| rowspan="3"|[[Minor seventh|Minor Seventh]]s
| rowspan="3"| 960-1025
|small
|960-987
|-
|middle
|987-1000
|-
|large
|1000-1025
|-
| | [[Equable heptatonic|Equable Heptatonic]]
| | 1018-1040
|
|
|-
| rowspan="3"|[[Neutral seventh|Neutral Seventh]]s
| rowspan="3"| 1030-1075
|small
|1030-1043
|-
|middle
|1043-1065
|-
|large
|1065-1075
|-
| rowspan="3"| [[Major seventh|Major Seventh]]s
| rowspan="3"| 1075-1140
|small
|1075-1100
|-
|middle
|1100-1120
|-
|large
|1120-1140
|-
| | Octave less diesis
| | 1140-1170
|
|
|-
| | Octave less comma
| | 1170-1200
|
|
|-
| | [[Octave|Pure Octave]] (2:1)
| | 1200
|
|
|}

== See also ==
; Interval region naming schemes
* [[Mike Sheiman's Alternative Interval Categorizations]]
* [[SKULO interval names]]
* [[User:VectorGraphics/Walker brightness notation|Walker brightness notation]]
* [[5L 2s/Interval categories]]

; Other related concepts
* [[Supermajor and subminor]]
* [[Interval size measure]]
* [[Table of MOSes]]
* [[Gallery of just intervals]]
* [[Universal solfege]] - [[solfege]] based on the Schulter system

[[Category:Interval regions| ]]

[[Category:Classification]]
[[Category:Distance measure]]

625/384

2026-05-19T14:11:18Z

Sintel: typo

{{Infobox Interval
| Name = (smaller) pental neutral sixth, tetraptolemaic double-augmented fifth
| Color name = y45, quadyo 5th
}}
'''625/384''', the '''(smaller) pental neutral sixth''' or '''tetraptolemaic double-augmented fifth''' is a [[5-limit]] [[interval]] of about 843.3 [[cent]]s. It is flat of the Pythagorean double-augmented fifth by four [[81/80|syntonic comma]]s. Equivalently, it is equal to an [[octave reduction|octave-reduced]] stack of four [[5/4|classical major thirds]] minus a [[3/2|fifth]], or equal to a [[5/3|classical major sixth]] minus a [[128/125|diesis]]. In the 11-limit it is 6912/6875 flat of [[18/11]], and [[5632/5625]] flat of [[44/27]]. In the 13-limit it is [[625/624]] sharp of [[13/8]].

== See also ==
* [[768/625]] – its [[octave complement]]

[[Category:Sixth]]
[[Category:Fifth]]
[[Category:Neutral sixth]]
[[Category:Augmented fifth]]

Subgroup temperaments

2026-05-17T18:24:57Z

Sintel: Be serious. Undo revision 225412 by Tristanbay (talk)

{{Technical data page}}
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.

For temperaments that omit various prime harmonics, see:
* [[No-thirteens subgroup temperaments]]
* [[No-elevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
* [[No-fives subgroup temperaments]]
* [[No-threes subgroup temperaments]]
* [[No-twos subgroup temperaments]] (additionally, [[Catalog of 3.5.7 subgroup rank two temperaments]]).

Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].

= Composite subgroup temperaments =
== 2.9.5.7 subgroup ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].

=== Commatose ===
Commatose is a [[Dual-fifth temperaments|dual-fifth temperament]] which uses the Pythagorean comma as a generator. It was developed by [[Eliora]] to highlight the near-perfect expression of 9/8 by [[1789edo]], while at the same time the fact that it completely misses 3/2. It is described as the 460 & 1329 temperament. In the 13-limit extension 24 generators are equal to [[~]][[13/9]].

[[Subgroup]]: 2.9.5.7

[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}

{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765

{{Optimal ET sequence|legend=1| 460, 869, 1329 }}

[[Badness]]: 0.611

==== 2.9.5.7.11 ====
Subgroup: 2.9.5.7.11

Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}

Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}

Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.165

==== 2.9.5.7.11.13 ====
Subgroup: 2.9.5.7.11.13

Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156

Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}

Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.0564

=== Daemotertiaschis ===
{{See also|Schismatic family#Tertiaschis}}
Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a [[7L 4s|daemotonic 7L 4s]] scale of reasonable hardness, which is notoriously difficult to approximate with simple JI or RTT methods.

Subgroup: 2.9.5.7.33.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}

Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982

[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}

=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}

Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.

[[Subgroup]]: 2.9.5.7

[[Comma list]]: 225/224, 3125/3087

{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170

{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}

Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]

==== 2.9.5.7.13 ====
{{See also|Chromatic pairs #Baldy}}

Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 & 35 temperament.

[[Subgroup]]: 2.9.5.7.13

[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]

{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}

{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}

: [[gencom]]: [2 9/8; 225/224 325/324 640/637]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090

{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents

Related temperament: [[Schismatic family #Garibaldi|Cassandra]]

==== Baldanders ====
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.

Subgroup: 2.9.5.7.11

Comma list: 100/99, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 | 0 1 -4 -7 -9 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743

{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}

Related temperament: [[Schismatic family #Garibaldi|Andromeda]]

===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414

{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}

== 2.9.5.11 subgroup ==
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}

[[Subgroup]]: 2.9.5.11.13

[[Comma list]]: 45/44, 65/64, 81/80

{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}

{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}

: [[gencom]]: [2 9/8; 45/44 65/64 81/80]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151

{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}

[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents

Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)

== 2.9.7 subgroup ==
=== Mabon ===
Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.

Subgroup: 2.9.7

Comma basis: 44957696/43046721

Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]

Optimal tuning (CTE): ~729/448 = 870.792

{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...

==== 2.9.7.11 subgroup ====
Subgroup: 2.9.7.11

Comma basis: 896/891, 1331/1296

Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]

Optimal tuning (CTE): ~16/11 = 870.966

{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}

== 2.9.7.11 subgroup ==
=== Apparatus ===
[[Subgroup]]: 2.9.7.11

[[Comma list]]: 41503/41472, 322102/321489

{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}

: mapping generators: ~2, ~77/72

{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}

: [[gencom]]: [2 77/72; 41503/41472 322102/321489]

[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685

{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}

[[Badness]]: 0.00263

=== Joan ===
{{See also| Chromatic pairs #Joan }}

Joan is related to [[casablanca]] as well as to [[orwell]].

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 99/98, 9317/9216

{{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }}

{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }}

: [[gencom]]: [2 11/8; 99/98 9317/9216]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents

{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}

[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents

=== Machine ===
Machine is every other step of [[supra]], most interesting for its scale patterns.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 64/63, 99/98

{{Mapping|legend=2| 1 0 6 13 | 0 1 -1 -3 }}

: sval mapping generators: ~2, ~9

{{Mapping|legend=3| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}

: [[gencom]]: [2 8/7; 64/63 99/98]

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843

{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}

[[Badness]]: 0.00233

=== Penta a.k.a. mechanism ===
Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 896/891, 26411/26244

{{Mapping|legend=2| 1 0 -1 6 | 0 5 6 -4 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}

: [[gencom]]: [2 9/7; 896/891 26411/26244]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents

[[Badness]]: 0.00439

Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]

== 2.9.11 subgroup ==
=== Demon ===
Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[1331/1296]]

{{Mapping|legend=2|1 1 2|0 3 2}}

[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060

{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}

=== Genius ===

Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[131769/131072]]

{{Mapping|legend=2|1 1 4|0 4 -1}}

[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863

{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]

== 2.9.15.7 subgroup ==
=== Stacks (a.k.a. 2magic) ===
Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].

[[Subgroup]]: 2.9.15.7

[[Comma list]]: 225/224, 245/243

{{Mapping|legend=2| 1 0 2 -1 | 0 5 3 6 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}

: [[gencom]]: [2 9/7; 225/224 245/243]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

==== 2.9.15.7.11 ====
Subgroup: 2.9.15.7.11

Comma list: 100/99, 225/224, 245/243

Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}

: gencom: [2 9/7; 100/99 225/224 245/243]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393

Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}

RMS error: 1.226 cents

==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}

: gencom: [2 9/7; 100/99 105/104 144/143 196/195]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023

Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}

RMS error: 1.540 cents

== 2.9.21 subgroup ==
=== A-team ===
A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].

[[Subgroup]]: 2.9.21

[[Comma list]]: 1029/1024

{{Mapping|legend=2| 1 2 4 | 0 3 1 }}

: sval mapping generators: ~2, ~21/16

{{Mapping|legend=3| 1 1 0 3 | 0 3/2 0 -1/2 }}

: [[gencom]]: [2 21/16; 1029/1024]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375

{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents

==== 2.9.5.21 ====
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''

Subgroup: 2.9.5.21

[[Comma]] list: 81/80, 1029/1024

Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}

Mapping generators: ~2, ~21/16

Optimal ([[Lp tuning|POL2]]) generator: 464.3865

{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}

===== 2.9.5.21.11 =====
Subgroup: 2.9.5.21.11

Comma list: 81/80, 99/98, 385/384

Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}

Gencom mapping: {{mapping| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}

: gencom: [2 21/16; 81/80 99/98 385/384]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956

{{Optimal ET sequence|legend=1| 5, 13, 31 }}

==== B-team ====
B-team (23 & 41) is every other step of [[rodan]].

Subgroup: 2.9.15.21.33

Comma list: 245/243, 385/384, 441/440

Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918

{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}

== 4.3.5 subgroup ==
=== Tetrahanson ===
{{Main| Tetrahanson }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 15625/15552

{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}

: Mapping generators: ~4, ~5/3

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941

[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}

=== Tetrameantone ===
{{Main| Tetrameantone }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 81/80

{{Mapping|legend=2| 1 1 2 | 0 -1 -4 }}

: Mapping generators: ~4, ~4/3

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761

[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}

=== Tetramagic ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 3125/3072

{{Mapping|legend=2| 1 0 1 | 0 5 1 }}

: Mapping generators: ~4, ~5/4

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059

[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}

=== Blacktetra ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 256/243

{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}

: Mapping generators: ~4, ~16/15

[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062

[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}

== 4.6.5 subgroup ==
=== Meanquad ===
{{Main| Meanquad }}

[[Subgroup]]: 4.6.5

[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}

{{Mapping|legend=2| 1 0 -4| 0 1 4 }}

: mapping generators: ~4, ~6

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214

[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69

<nowiki />* Wart for 4

==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7

[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}

{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622

[[Support]]ing [[ET]]s: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]

<nowiki />* Wart for 4

=== Fourwar ===
The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.

Fourwar is named after the closely related [[hemiwar]] temperament.

{{Todo|inline=1|cleanup}}

<pre>
Reduced Mapping
4 6 5
[ ⟨ 1 0 1 ]
⟨ 0 16 2 ] ⟩

TE Generator Tunings (cents)
⟨2399.3973, 193.8643]

TE Step Tunings (cents)
⟨25.21211, 47.81337]

TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]

TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]

Complexity 1.369085
Adjusted Error 0.692892 cents
TE Error 0.268047 cents/octave

Unison Vector
[8, 1, -8⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>

==== 4.6.5.7 ====
<pre>
Reduced Mapping
4 6 5 7
[ ⟨ 1 0 1 1 ]
⟨ 0 16 2 5 ] ⟩

TE Generator Tunings (cents)
⟨2399.4195, 193.8654]

TE Step Tunings (cents)
⟨25.23883, 47.79592]

TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]

TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]

Complexity 1.192044
Adjusted Error 0.653313 cents
TE Error 0.232715 cents/octave

Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[3, 0, -5, 2⟩ (3136:3125)
[5, 1, -3, -2⟩ (6144:6125)
[8, 1, -8, 0⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>

==== 4.6.5.7.11 ====
<pre>
Reduced Mapping
4 6 5 7 11
[ ⟨ 1 0 1 1 1 ]
⟨ 0 16 2 5 9 ] ⟩

TE Generator Tunings (cents)
⟨2400.1097, 193.9498]

TE Step Tunings (cents)
⟨24.18752, 48.52491]

TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]

TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]

Complexity 1.068792
Adjusted Error 2.926965 cents
TE Error 0.846083 cents/octave

Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[2, 0, -2, -1, 1⟩ (176:175)
[-3, -1, 1, 1, 1⟩ (385:384)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[1, 0, 1, -4, 2⟩ (2420:2401)

Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>

==== 4.6.5.7.11.13 ====

<pre>
Reduced Mapping
4 6 5 7 11 13
[ ⟨ 1 0 1 1 1 0 ]
⟨ 0 16 2 5 9 23 ] ⟩

TE Generator Tunings (cents)
⟨2401.2305, 193.5378]

TE Step Tunings (cents)
⟨42.79107, 35.98524]

TE Tuning Map (cents)
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]

TE Mistunings (cents)
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]

Complexity 1.219191
Adjusted Error 6.699599 cents
TE Error 1.810487 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1⟩ (66:65)
[-1, -1, -1, 0, 2, 0⟩ (121:120)
[1, 2, 0, 0, -1, -1⟩ (144:143)
[2, 0, -2, -1, 1, 0⟩ (176:175)
[-2, 1, 1, 1, 0, -1⟩ (105:104)
[-3, -1, 1, 1, 1, 0⟩ (385:384)
[-3, 0, 0, 1, 2, -1⟩ (847:832)
[1, 3, -1, 0, 0, -2⟩ (864:845)
[-1, 0, 3, -3, 1, 0⟩ (1375:1372)

Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>

==== 4.6.5.7.11.13.17 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17
[ ⟨ 1 0 1 1 1 0 1 ]
⟨ 0 16 2 5 9 23 13 ] ⟩

TE Generator Tunings (cents)
⟨2400.4701, 193.4599]

TE Step Tunings (cents)
⟨43.39350, 35.55764]

TE Tuning Map (cents)
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]

TE Mistunings (cents)
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]

Complexity 1.129881
Adjusted Error 8.082725 cents
TE Error 1.977443 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)

Subsets
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>

==== 4.6.5.7.11.13.17.19 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19
[ ⟨ 1 0 1 1 1 0 1 1 ]
⟨ 0 16 2 5 9 23 13 14 ] ⟩

TE Generator Tunings (cents)
⟨2399.9219, 193.3952]

TE Step Tunings (cents)
⟨44.14256, 35.03670]

TE Tuning Map (cents)
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]

TE Mistunings (cents)
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]

Complexity 1.058472
Adjusted Error 8.712222 cents
TE Error 2.050935 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)

Subsets
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>

==== 4.6.5.7.11.13.17.19.23 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19 23
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩

TE Generator Tunings (cents)
⟨2399.3286, 193.5316]

TE Step Tunings (cents)
⟨37.31613, 39.63311]

TE Tuning Map (cents)
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]

TE Mistunings (cents)
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]

Complexity 1.115920
Adjusted Error 9.502017 cents
TE Error 2.100561 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)

Subsets
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>

== 4.9.25 subgroup ==
=== Meansquared ===
[[Subgroup]]: 4.9.25

[[Comma list]]: [[6561/6400]]

{{Mapping|legend=2| 1 3 4 | 0 1 4 }}

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429

[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]

== 4.9.49 subgroup ==
=== Archsquared ===
[[Subgroup]]: 4.9.49

[[Comma list]]: 4096/3969

{{Mapping|legend=2| 1 3 0 | 0 1 -2 }}

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190

[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49

== 8.9.7 subgroup ==
=== Sixscared ===
Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."

[[Subgroup]]: 8.9.7

[[Comma list]]: 64/63

{{Mapping|legend=2| 1 0 2 | 0 1 -1 }}

: sval mapping generators: ~8, ~9

: [[gencom]]: [8 9/8; 64/63]

[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898

[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}

[[Badness]]: 0.0215 × 10-3

= Fractional subgroup temperaments =
== 2.5/3.… subgroups ==
=== Magicaltet ===
{{See also| Chromatic pairs #Magicaltet }}

Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.

[[Subgroup]]: 2.5/3.7.11

[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})

{{Mapping|legend=2| 1 0 5 2 | 0 1 -3 2 }}
: mapping generators: ~2, ~5/3

{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
: [[gencom]]: [2 6/5; 100/99 385/384]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351

{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
: <nowiki/>* wart for 5/3

[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents

=== Starlingtet ===
{{See also | Chromatic pairs #Starlingtet }}

Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.

[[Subgroup]]: 2.5/3.7/3

[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})

{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}

: mapping generators: ~2, ~5/3

{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
: [[gencom]]: [2 6/5; 126/125]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846

{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents

==== Greeley ====
{{See also| Chromatic pairs #Greeley }}

Greeley is related to [[opossum]] as well as to [[nusecond]].

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})

{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}

{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
: [[gencom]]: [2 11/10; 121/120 126/125]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776

{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
: <nowiki/>* wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents

==== Skateboard ====
{{See also| Chromatic pairs #Skateboard }}

Skateboard is related to [[thrasher]].

[[Subgroup]]: 2.5/3.7/3.11.13/9

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})

{{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }}

{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158

{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}

[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents

=== Gariberttet ===
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].

==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
{{See also | Chromatic pairs #Gariberttet }}

Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].

[[Subgroup]]: 2.5/3.7/3.13/11

[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})

{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}

{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
: [[gencom]]: [2 13/11; 275/273 847/845]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679

{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
: <nowiki/>* wart for 13/11

[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents

==== Indium ====
{{See also | Chromatic pairs #Indium }}

Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})

{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}

{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010

{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents

==== Ammon ====
{{See also| Chromatic pairs #Ammon }}

Ammon can be described as the {{nowrap| 8 & 29 }} temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[tridec]], and is related to [[ammonite]]. It is generated by a semidiminished fourth, hence the old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name.

[[Subgroup]]: 2.5/3.7/3.11/3.13/3

[[Comma list]]: [[121/120]] ({{monzo| -3 -1 0 2 }}), [[169/168]] ({{monzo| -3 0 -1 0 2 }}), [[275/273]] ({{monzo| 0 2 -1 1 -1 }})

{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}

{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242

{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents

=== Sentry ===
{{See also | Chromatic pairs #Sentry }}

Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].

[[Subgroup]]: 2.5/3.9/7

[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})

{{Mapping|legend=2| 1 0 0 | 0 2 1 }}

{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902

{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166†, 197*†, 215†, 264*† }}
: <nowiki/>* wart for 5/3
: † wart for 9/7

[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents

=== Marveltwintri ===
{{See also| Chromatic pairs #Marveltwintri }}

Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]].

[[Subgroup]]: 2.5/3.13/9

[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})

{{Mapping|legend=2| 1 0 2 | 0 1 -2 }}

{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
: [[gencom]]: [2 6/5; 325/324]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861

{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents

== 2.….7/3.… subgroups ==
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}

Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.

[[Subgroup]]: 2.5.7/3.11/3

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})

{{Mapping|legend=2| 1 0 2 -2 | 0 3 -1 5 }}
: mapping generators: ~2, ~12/7

{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
: [[gencom]]: [2 7/6; 176/175 540/539]

[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~12/7 = 929.545
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~12/7 = 929.907

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3

[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents

==== Laz ====
{{See also | Chromatic pairs #Laz }}

Laz is related to [[avalokita]] as well as to [[winston]].

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}

{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}

{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700

{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents

=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}

Kryptonite is related to [[krypton]].

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }})

{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}
: mapping generators: ~2, ~13/12

{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428

{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents

=== Kiribati ===
{{See also| Chromatic pairs #Kiribati }}

Kiribati is related to [[nakika]] as well as to [[octacot]].

[[Subgroup]]: 2.9/5.7/3.11/9

[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})

{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20

{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
: [[gencom]]: [2 21/20; 100/99 245/242]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892

{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents

=== Mothwelltri ===
{{See also| Chromatic pairs #Mothwelltri }}

Mothwelltri, the {{nowrap| 1 & 4 }} temperament in the 2.7/3.11 subgroup, is related to [[orwell]]. The tonic and the first two generator steps make a [[mothwellsmic triad]], hence the name.

[[Subgroup]]: 2.7/3.11

[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})

{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
: mapping generators: ~2, ~7/3

{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [2 7/6; 99/98]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~7/6 = 273.695
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~7/6 = 273.174

{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents

== 2.….9/7.… subgroups ==
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}

Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.

[[Subgroup]]: 2.5.9/7

[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})

{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~5

{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638

{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 9/7

[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents

==== Sulis ====
Sulis is related to [[minerva]] and [[würschmidt]].

[[Subgroup]]: 2.5.9/7.11/9

[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})

{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558

{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

== 2.….7/5.… subgroups ==
=== Hydrothermal ===
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.

[[Subgroup]]: 2.3.7/5

[[Comma list]]: [[50/49]]

{{Mapping|legend=2| 2 3 1 | 0 1 0 }}

[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962

[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}

=== Edson ===
Edson is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].

==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{See also| Chromatic pairs #Edson }}

Edson is related to [[pele]] and [[andromeda]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}

{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
: mapping generators: ~2, ~3

{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414

{{Optimal ET sequence|legend=1| 12, 17, 29 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents

==== Haumea ====
{{See also| Chromatic pairs #Haumea }}

Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]

{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}

{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491

{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents

=== Historical ===
{{distinguish|Historical temperaments}}
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.

Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: 364/363, 441/440, 1001/1000

{{Mapping|legend=2| 1 2 0 1 2 | 0 -6 7 2 -9 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016

{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents

=== Terrain ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Terrain }}

Terrain, the 6 & 21 temperament in the 2.7/5.9/5 subgroup, is related to [[domain (temperament)|domain]]. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.

[[Subgroup]]: 2.7/5.9/5

[[Comma list]]: [[250047/250000]]

{{Mapping|legend=2| 3 1 3 | 0 1 -1 }}

{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461

{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents

=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}

Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].

[[Subgroup]]: 2.7/5.11/5.13/5

[[Comma list]]: [[847/845]], [[1001/1000]]

{{Mapping|legend=2| 1 2 0 1 | 0 -4 3 1 }}

{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}

[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents

==== Naiadec ====
[[Subgroup]]: 2.7/5.11/5.13/5.17/5

[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]

{{Mapping|legend=2| 1 2 0 1 1 | 0 -4 3 1 2 }}

{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }}
: [[gencom]]: [2 13/10; 170/169 221/220 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95t, 124t }}
: t wart for 17/5

[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents

== 2.….11/5.… subgroups ==
=== Petrtri ===
{{See also| Chromatic pairs #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}

Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup.

[[Subgroup]]: 2.11/5.13/5

[[Comma list]]: [[2200/2197]]

{{Mapping|legend=2| 1 0 1| 0 3 1 }}

{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
: [[gencom]]: [2 13/10; 2200/2197]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012

{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}

[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents

==== Bridgetown ====
{{See also| Chromatic pairs #Bridgetown }}

Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].

[[Subgroup]]: 2.3.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]]

{{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }}

{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
: [[gencom]]: [2 15/13; 352/351 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents

=== Hypnosis ===
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]

[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 169/168, 540/539, 729/728

{{Mapping|legend=2| 1 0 -3 8 0 | 0 3 11 -13 7 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518

{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents

=== Trisect ===
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].

Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[portent]] or [[landscape]] results in the [[weak extension]] known as [[tritikleismic]].

[[Subgroup]]: 2.3.7.11/5

[[Comma list]]: 1029/1024, 4000/3993

{{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742

{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

==== 2.3.7.11/5.13 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 1029/1024, 1575/1573, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918

{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

==== 2.3.7.11/5.13.17 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13.17

[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820

{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== Trisector =====
[[Subgroup]]: 2.3.7.11/5.13.17.19

[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894

{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== 2.3.7.11/5.13.17.19.23 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23

[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038

{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29

[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102

{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}

[[Tp tuning #T2 tuning|RMS error]]: ???

== 2.….11/7.… subgroups ==
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}

Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.

[[Subgroup]]: 2.3.11/7.13/7

[[Comma list]]: 352/351, 364/363

{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}

{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
: [[gencom]]: [2 3/2; 352/351 364/363]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856

{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*† }}
: <nowiki />* wart for 11/7
: † wart for 13/7

[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents

== 2.….13/5.… subgroups ==
=== Barbados ===
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

[[Subgroup]]: 2.3.13/5

[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}

[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}

[[Badness]]: 0.002335

; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]

==== Tobago ====
{{See also| Chromatic pairs #Tobago }}

Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].

[[Subgroup]]: 2.3.11.13/5

[[Comma list]]: [[243/242]], [[676/675]]

{{Mapping|legend=2| 2 0 -1 -2 | 0 2 5 3 }}

{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
: [[gencom]]: [55/39 15/13; 243/242 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312

{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents

==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17

[[Comma list]]: 221/220, 243/242, 289/288

{{Mapping|legend=2| 2 0 -1 -2 5 | 0 2 5 3 2 }}

[[Optimal tuning]]s:
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)

{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki />* wart for 13/5

=== Oceanfront ===
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]

[[Subgroup]]: 2.3.7.13/5

[[Comma list]]: 64/63, 91/90

{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910

{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents

Scales: [[Oceanfront scales]]

== 2.….49/5.… subgroups ==
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]

[[Subgroup]]: 2.3.49/5

[[Comma list]]: 2401/2400

{{Mapping|legend=2| 1 1 3 | 0 2 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966

{{Optimal ET sequence|legend=1|7, 10, 17}}

[[Tp tuning #T2 tuning|RMS error]]: ?

== 2.….17/5.… subgroups ==
=== Fiventeen ===
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.

[[Subgroup]]: 2.3.17/5

[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})

{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}

{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}

== 2.….19/7.… subgroups ==
=== Surprise ===
This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version.

[[Subgroup]]: 2.3.19/7

[[Comma list]]: [[57/56]] ({{monzo| -3 1 1 }})

{{Mapping|legend=2| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}

{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}

<nowiki/>* wart for 19/7

[[Badness]] (Sintel): 0.082

== 3/2.5/2.… subgroups ==
{{Main|Half-prime subgroup}}

=== Hemihemi ===
[[Subgroup]]: 3/2.5/2.7/2

[[Comma list]]: [[10976/10935]]

{{Mapping|legend=2| 1 2 3 | 0 3 1 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~[[3/2]] = 1\[[1edf]], ~[[28/27]] = 60.909

[[Support]]ing [[ET]]s: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]

=== Halftone ===
{{Main| Halftone }}

[[Subgroup]]: 3/2.5/2.7/2

[[Comma list]]: 9604/9375

{{Mapping|legend=2| 1 3 4 | 0 -4 -5 }}
: sval mapping generators: ~3/2, ~15/14

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 128.783

Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
: <nowiki />* wart for 3/2

==== 3/2.5/2.7/2.11/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2

[[Comma list]]: 1232/1215, 27783/27500

{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }}
: sval mapping generators: ~3/2, ~15/14

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.186

[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
: <nowiki />* wart for 3/2

==== 3/2.5/2.7/2.11/2.13/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2.13/2

[[Comma list]]: 275/273, 1232/1215, 1323/1300

{{Mapping|legend=2| 1 3 4 4 5 | 0 -4 -5 1 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.381

[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
: <nowiki />* wart for 3/2

=== Semiwolf ===
[[Subgroup]]: 3/2.5/2.7/4

[[Comma list]]: 245/243

{{Mapping|legend=2| 1 1 2 | 0 2 -1 }}

: sval mapping generators: ~3/2, ~9/7

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 262.1728

[[Optimal ET sequence]]: [[3edf]], [[5edf]], [[8edf]]

==== Semilupine ====
[[Subgroup]]: 3/2.5/2.7/4.11/4

[[Comma list]]: 100/99, 245/243

{{Mapping|legend=2| 1 1 2 0 | 0 2 -1 4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 264.3771

[[Optimal ET sequence]]: [[8edf]], [[13edf]]

==== Hemilycan ====
[[Subgroup]]: 3/2.5/2.7/4.11/4

[[Comma list]]: 245/243, 441/440

{{Mapping|legend=2| 1 1 2 5 | 0 2 -1 -4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 261.5939

[[Optimal ET sequence]]: [[8edf]], [[11edf]]

== 3/2.5/4.… subgroups ==
=== Poseidon ===
'''This temperament will be subjected to renaming due to a conflict.'''

[[Subgroup]]: 3/2.5/4.11/8

[[Comma list]]: 121/120

{{Mapping|legend=2| 1 1 1 | 0 2 -1 }}]

: [[gencom]]: [3/2 12/11; 121/120]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2, ~12/11 = 158.29

{{Optimal ET sequence|legend=1|9, 5, 13, 22, 14, 31, 17, 6[+5/4], 23, 40, 35, 21[-5/4], 19[+5/4], 49}}

== Other 3/2-equave subgroups ==
=== Auk ===
[[Subgroup]]: 3/2.7.13

[[Comma list]]: 87808/85293

{{Mapping|legend=2| 1 0 -8 | 0 1 3 }}

: sval mapping generators: ~3/2, ~7

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~28/9 = 1950.859

Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
: <nowiki />* wart for 3/2

=== Doubleton ===
[[Subgroup]]: 3/2.7.13

[[Comma list]]: 1352/1323

{{Mapping|legend=2| 2 0 3 | 0 1 1 }}

: sval mapping generators: ~26/21, ~7

[[Optimal tuning]] (subgroup [[CTE]]): ~26/21 = 1\2edf, ~28/9 = 1971.772

Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
: <nowiki />* wart for 3/2

== 5/2-equave subgroups ==
=== Hyperion ===
[[Subgroup]]: 5/2.7.11

[[Comma list]]: {{monzo| 11 1 -5 }}

{{Mapping|legend=2| 1 4 3 | 0 -5 -1 }}

: [[gencom]]: [5/2 125/88; 341796875/329832448]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/2 = 1586.3137, ~125/88 = 593.6668

Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
: <nowiki />* wart for 5/2

= Related temperament collections =
* [[Dual-fifth temperaments]]
* [[Equalizer subgroup]] temperaments
* [[Substitute harmonic]] temperaments

[[Category:Subgroup temperaments| ]] 
[[Category:Temperament collections]]
{{Todo| review | cleanup }}

User talk:Francium/647edo

2026-05-14T17:37:46Z

Sintel:

== Deletion ==

I vote '''keep''', it has at least one song written in it!
– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 00:18, 13 May 2026 (UTC)

Or maybe move to Francium's userspace. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 00:24, 13 May 2026 (UTC)

: I don't believe that simply having music written in it constitutes a reason to avoid deletion, particularly in cases such as this where it's extremely unclear how exactly the song uses the structures associated with the edo/tuning. 647 might as well just be an arbitrary number. --[[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 04:18, 14 May 2026 (UTC)

:: Fair, I guess my point is that there's a thousand more pages that don't even meet this very meager standard. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 17:37, 14 May 2026 (UTC)

Talk:Ed7/3

2026-05-14T13:39:39Z

Sintel:

Anybody know what this is about? I propose we delete it, or else someone should really break it down and explain what this is. [[User:Keenan Pepper|Keenan Pepper]] ([[User talk:Keenan Pepper|talk]]) 23:35, 20 September 2018 (UTC)

== Request for deletion ==

Since the comment above was never adressed, I will just say we rewrite this current page (I'm willing to take a shot), and delete these:
* [[8edX]]
* [[9edX]]
* [[15edX]]
* [[16edX]]
* [[17edX]]
* [[19edX]]

As they are all nonsensical tables with no practical use. If someone wants to replace them with a simple templated table that generates the intervals (as on the EDO pages) that'd be fine too.

EDIT: signed -- [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 21:16, 19 January 2022 (UTC)

: I do not like all these mentioned tables in any way, and for me personally their value is not evident, at all am against trying to prove hypotheses by handmade tables. But since users have spent time on these tables, I am against simply deleting the pages in question. I would rather move them to the main editor's username space, where they will have time to elaborate them so that they are ready for the article namespace from the other users' point of view as well. I am doing the necessary work for this right now... --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 13:43, 26 February 2022 (UTC)

== Deletion, again ==
Almost all ed7/3 pages are completely devoid of information, and this page itself is filled with what is clearly nonsense.
Since MMTM has been banned, and apparently nobody cares about any of this, I think we should revisit this.

– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 00:36, 13 May 2026 (UTC)

: What about ed5/3? —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 09:27, 13 May 2026 (UTC)

:: Probably same story. Honestly I think we should be somewhat careful in applying [[XW:NG]] retroactively, but the part about:
:::Entries consisting solely of automatically generated content (e.g. interval tables, infoboxes) without context or explanation can be deleted without relocation.
:: Can be enforced without much issues, since we don't lose anything.
:: – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]])

::: I'm in support of deleting most of the ed5/3 and ed7/3 pages based on that guideline. I think this page itself should be considered under a different standard, and we'll have a clearer idea if we clean it up from MMTMisms (like "Middletown Valley" etc.).
::: Regards. -- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 20:43, 13 May 2026 (UTC)

Using this thread for a wider discussion about the proliferation of various edX pages.

Another complaint by users is the pollution of search results for common edos.
For example, the search string "31ed" brings up:
: 31ed4
: 31ed5/2
: 31ed5/4
: 31ed6
: 31ed7/3
: 31ed7/4
: 31ed8
: 31edf
: 31edo
When presumably most users are looking for 31edo. Now, some of these might actually be useful, but surely not all of them.

– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 13:39, 14 May 2026 (UTC)

User:Sintel/Expected Dirichlet coefficient for temperaments

2026-05-14T01:51:02Z

Sintel:

In the context of [[regular temperament theory]], a natural question is how well a given temperament approximates [[just intonation]] relative to its [[complexity]].
The '''Dirichlet coefficient''' gives a quantitative way to measure this. This is the same as the "[[TE logflat badness|badness]]" used on the wiki, though the derivation here is given for the regular Euclidean norm for clarity.

Given a target vector <math>y \in \mathbb{R}^n</math>, such as the [[JIP | vector of log-primes]] in some ''p''-limit, and a rank-''k'' temperament ''X'', the Dirichlet coefficient is defined as:

:<math>
d(y, X) \cdot H(X)^{\frac{n}{n-k}}
</math>

where <math>d(y, X)</math> is the projective distance between the target vector and the temperament, and <math>H(X)</math> is the height (or [[complexity]]) of the temperament. Both of these quantities can be computed straightforwardly using the temperament's [[Plücker coordinates]].

This coefficient generalizes {{w|Dirichlet's approximation theorem}}. A fundamental result in Diophantine approximation by W. M. Schmidt<ref name="Schmidt1967"/> states that for any valid target vector ''y'', there exists a constant <math>C_{n,k}</math> such that there are infinitely many rational subspaces ''X'' which satisfy:
:<math>
d(y, X) \cdot H(X)^{\frac{n}{n-k}} \le C_{n,k}
</math>

The exponent is ''critical'' or ''sharp'' for this problem: if we replace the exponent by <math>\tfrac{n}{n-k} + \epsilon</math> for some <math>\epsilon > 0</math>, then we find only finitely many solutions.

By analyzing the distribution of rational points on the {{w|Grassmannian}}, we can derive the expected value for this coefficient, giving us a baseline to determine whether a temperament is a "good" or "bad" approximation relative to its complexity.
Importantly, this measure does not take into account any kind of psychoacoustics, so it is not in any way "calibrated" to human tolerance to tuning error.
Instead, it is a purely mathematical metric of coincidence. The upside is that this is robust over an arbitrary range of complexities and does not rely on any free parameters or empirical weights.

== Motivating example: equal temperaments in the 5-limit ==

To understand the Dirichlet coefficient, let's look at rank-1 temperaments (equal temperaments) in the 5-limit (''n''=3).
Our target vector is the standard [[JIP|just intonation vector]] of log-primes:
:<math>
y =[\log_2(2), \log_2(3), \log_2(5)] \approx [1, 1.585, 2.322]
</math>

An equal temperament is defined by a line through the origin with a rational slope. For example, [[12edo|12-equal temperament]] corresponds to the line passing through the integer vector <math>X_{12} =[12, 19, 28]</math>.
This approximation is good in the sense that the ratios of its coordinates closely match the target vector:
:<math>
\frac{19}{12} \approx \log_2(3), \quad \frac{28}{12} \approx \log_2(5)
</math>

Because we are in 3D, the wedge product used to define projective distance reduces to the standard cross product.
The projective distance is the sine of the angle <math>\theta</math> between the temperament line and the JI vector:
:<math>
d(X_{12}, y) = \sin(\theta) = \frac{\|X_{12} \times y\|}{\|X_{12}\| \|y\|} \approx 0.00276
</math>
Since the angle is extremely small (which is always the case for any reasonable temperament) we can take <math>\sin(\theta) \approx \theta</math>.

The height is simply the Euclidean norm of the integer vector:
:<math>
H(X_{12}) = \sqrt{12^2 + 19^2 + 28^2} \approx 35.902
</math>

For an equal temperament (''k''=1) in the 5-limit (''n''=3), the critical exponent is 3/2. This is equivalent to the classical Dirichlet theorem for simultaneous approximation of two irrational numbers.

Plugging this into our formula gives the Dirichlet coefficient for 12-ET:
:<math>
C_{12} = d(X_{12}, y) \cdot H(X_{12})^{3/2} \approx 0.00276 \times 35.902^{1.5} \approx 0.595
</math>

We can compare this to the coefficients of some other 5-limit equal temperaments. Lower values indicate that the temperament is exceptionally accurate for its size, while higher values indicate poor approximations.

{| class="wikitable"
|-
! Temperament !! Dirichlet coefficient
|-
| 53-ET || 0.467
|-
| 12-ET || 0.595
|-
| 34-ET || 0.716
|-
| 20-ET || 3.855
|-
| 33-ET || 4.621
|-
| 52-ET || 6.125
|}
Note: the point here is not to argue over which of these is "better", just that this measure generally agrees on which equal temperaments contain good approximations to 5-limit JI relative to their size.

== Generalizing to higher rank ==

For higher-rank temperaments, we rely on Schmidt's general formula. A rank-''k'' temperament in an ''n''-prime limit is viewed as a rational ''k''-dimensional subspace <math>X \in \mathrm{Gr}(k, n)</math>. By slight abuse of notation, we will identify ''X'' directly with its Plücker coordinates.

As defined in the article on [[Plücker coordinates]], the height is simply the Euclidean norm of the (reduced) Plücker coordinates, and the projective distance is given by:
:<math>
d(X, y) = \frac{\|X \wedge y\|}{\|X\| \|y\|}
</math>
This has the same interpretation in terms of the sine of the minimal angle between the subspace and the target.

According to Schmidt's theorem on metric Diophantine approximation<ref name="Schmidt1967"/>, the critical exponent balancing distance and height for approximating a target vector ''y'' by a ''k''-dimensional rational subspace ''X'' is exactly <math>\tfrac{n}{n-k}</math>, so we immediately obtain:
:<math>
d(y, X) \cdot H(X)^{\frac{n}{n-k}}
</math>

While we typically assume ''y'' to be the log-primes for some ''p''-limit, this property holds for any target vector in <math>\mathbb{R}^n</math>, and so it cleanly generalizes to any subgroup.

== Deriving the heuristic constant ==

For the case ''n''=2, {{w|Hurwitz's theorem (number theory)|Hurwitz's theorem}} states that the best possible constant is <math>\tfrac{1}{\sqrt{5}} \approx 0.447</math>.
Not much is known about the exact constant needed to obtain a tight bound in the general case.

=== Counting temperaments ===

To determine the expected bound, we must first know how many temperaments exist up to a certain complexity.
Another classical result by Schmidt<ref name="Schmidt1968"/> gives the asymptotic distribution of primitive (i.e., [[torsion]]-free) sublattices, which directly correspond to temperaments.

The number of rank-''k'' temperaments with a complexity bounded by <math>H(X) \le H_{\max}</math> grows as:
:<math>
\#\left\{ X: H(X) \le H_{\max} \right\} \sim c_{n, k} H_{\max}^n
</math>

The constant ''cn,k'' is given by the formula:
:<math>
c_{n, k} = \frac{1}{n} \binom{n}{k} \prod_{i=1}^{k} \frac{V(n-i+1)}{V(i)} \cdot \frac{\zeta(i)}{\zeta(n-i+1)}
</math>
where <math>V(m) = \frac{\pi^{m/2}}{\Gamma(m/2+1)}</math> is the volume of the ''m''-dimensional Euclidean unit ball, and <math>\zeta</math> is our good old friend, the {{w|Riemann zeta function}} (with the convention that the pole at <math>\zeta(1)</math> is treated as 1).

=== The distribution of random temperaments ===
We can find the expected minimum distance for a given maximum height by treating the temperaments as being randomly distributed on the Grassmannian manifold.
This is asymptotically true by the equidistribution of rational points.<ref name="Schmidt1998"/>

By rotational invariance, projecting a fixed target vector ''y'' onto a random ''k''-plane is statistically identical to projecting a random unit vector onto a fixed plane.

The squared projective distance is the complement of the squared length of the projection (<math>\cos^2 \theta</math>).
From standard probability theory, the squared length of a projection of a random unit vector in <math>\mathbb{R}^n</math> onto a plane follows a {{w|Beta distribution}}.
Therefore, the squared distance is distributed as:
:<math>
\sin^2(\theta) \sim \mathrm{Beta}\left(\frac{n-k}{2}, \frac{k}{2}\right)
</math>

For some search distance ''r'', the volume (with respect to the normalized {{w|Haar measure}} of the Grassmannian) is given by the cumulative Beta distribution:
:<math>
\text{Vol}(d(X,y) \le r) = \frac{1}{B\!\left(\frac{n-k}{2}, \frac{k}{2}\right)} \int_0^{r^2} s^{\frac{n-k}{2}-1}(1-s)^{\frac{k}{2}-1}\, ds
</math>

When ''r'' is small, <math>(1-s)^{k/2 - 1} \to 1</math>, so the leading term is:
:<math>
\begin{aligned}
\mathrm{Vol}(d(X,y) \le r)
&\approx \frac{1}{B\!\left(\frac{n-k}{2}, \frac{k}{2}\right)} \cdot \frac{r^{n-k}}{\frac{n-k}{2}} \\[10pt]
&= \frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}\, r^{n-k}
\end{aligned}
</math>

Since there are <math>c_{n, k} H_{\max}^n</math> planes available, the expected number that fall in this radius is:
:<math>
\mathbb{E}[\#\{X : H(X) \leq H_{\max},\ d(y,X) \leq r\}] \approx
c_{n,k}
\cdot H_{\max}^n
\cdot \frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}
\cdot r^{n-k}
</math>

Setting this equal to 1 and taking the ''(n-k)''-th root, we find:
:<math>
r \cdot H_{\max}^{\frac{n}{n-k}} \approx \left( c_{n,k}
\frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}
\right) ^ {\frac{-1}{n-k}}
</math>
which recovers the same critical exponent, but now with an explicit constant ''Ch'' for our Dirichlet bound.

A temperament with a coefficient much better than this is exceptional: the heuristic says you would need to search through exponentially more planes to find it by chance.

The following table gives the values of ''Ch'' for some small dimensions:
{| class="wikitable"
|-
! n !! ''k'' = 1 !! ''k'' = 2 !! ''k'' = 3 !! ''k'' = 4
|-
| 2 || 1.645
|-
| 3 || 1.071 || 0.574
|-
| 4 || 1.011 || 0.400 || 0.345
|-
| 5 || 1.012 || 0.435 || 0.235 || 0.263
|}

== References ==
<references>
<ref name="Schmidt1967">Wolfgang M. Schmidt. ''On Heights of Algebraic Subspaces and Diophantine Approximations''. Annals of Mathematics, Vol. 85, No. 3 (1967), pp. 430-472, theorem 15 [https://doi.org/10.2307/1970352 doi:10.2307/1970352]</ref>
<ref name="Schmidt1968">Wolfgang M. Schmidt. ''Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height''. Duke Mathematical Journal Vol. 35 No. 2, pp. 327-339 (1968), theorem 1 [https://doi.org/10.1215/S0012-7094-68-03532-1 doi:10.1215/S0012-7094-68-03532-1]</ref>
<ref name="Schmidt1998">Wolfgang M. Schmidt. ''The distribution of sub-lattices of Zm''. Monatshefte für Mathematik Vol. 125 No. 1, pp 37–81 (1998) [https://doi.org/10.1007/BF01489457 doi:10.1007/BF01489457]</ref>
</references>

Talk:Ed7/3

2026-05-13T13:17:23Z

Sintel: /* Deletion, again */

Talk:Ed7/3

2026-05-13T00:36:16Z

Sintel: Deletion, again

Ed7/3

2026-05-13T00:34:51Z

Sintel: move the warning to the top

{{todo|inline=1|cleanup|explain edonoi|text=Most people do not think 7/3 sounds like an equivalence, so there must be some other reason why people are dividing it — some property ''other than'' equivalence that makes people want to divide it. Please add to this page an explanation of what that reason is... The page also needs a general overall cleanup.}}

The '''equal division of 7/3''' ('''ed7/3''') is a [[tuning]] obtained by dividing the [[7/3|septimal minor tenth (7/3)]] in a certain number of [[equal]] steps.

== Applications ==
Division of 7/3 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, ed7/3 scales have a perceptually important [[Pseudo-octave|false octave]], with various degrees of accuracy.

The structural utility of 7/3 (or another tenth) is apparent by being the absolute widest range most generally used in popular songs{{citation needed}} (and even the range of a {{w|Dastg%C4%81h-e_M%C4%81hur|dastgah}}{{citation needed}}).

== Chords and harmonies ==
{{main|Pseudo-traditional harmonic functions of enneatonic scale degrees}}
[[:Category:9-tone scales|Enneatonic scale]]s, especially those equivalent at 7/3, can sensibly take [[tetrad]]s as the fundamental complete sonorities of a pseudo-traditional functional harmony due to their seventh degree being as structurally important as it is. Many, though not all, of these scales have a perceptually important [[Pseudo-octave|pseudo (false) octave]], with various degrees of accuracy.

Incidentally, one way to treat 7/3 as an equivalence is the use of the 3:4:5:(7) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in [[meantone]]. Whereas in meantone it takes four [[3/2]] to get to [[5/1]], here it takes two [[28/15]] to get to [[7/2]] (tempering out the comma [[225/224]]). So, doing this yields 15-, 19-, and 34-note [[mos]] 2/1 apart. While the notes are rather farther apart, the scheme is uncannily similar to meantone. [[Joseph Ruhf]] named this scheme "macrobichromatic".

== Middletown ==
{{idiosyncratic terms}}
7/3 provides a fairly trivial point to split the difference between the [[octave]] and the [[tritave]], which is why Ruhf has named the region of intervals between 6 and 7 degrees of [[5edo]] the "[[Middletown valley]]".

The proper [[Middletown family|Middletown temperament family]] is based on an [[enneatonic]] scale [[generator|generated]] by a third or a fifth optionally with a [[period]] of a [[Wolf interval|wolf]] fourth at most 560 [[cents]] wide) and, as is the twelfth (tritave), an alternative interval where {{w|Inversion (music) #Counterpoint|invertible counterpoint}} has classically occurred.

The branches of the Middletown family are named thus:
* 3&6: Tritetrachordal
* 4&5: Montrose (between 5\4edo and 4\3edo in particular, MOS generated by [pseudo] octaves belong to this branch)
* 2&7: Terra Rubra

The family of interlaced [[octatonic scale]]-based temperaments in the "Middletown valley" is called Vesuvius (i.e. the volcano east of Naples).

The Middlebury temperament falls in the "Middletown valley", but its enneatonic scales are "[[generator-remainder]]".

The temperaments neighboring Middletown proper are named thus:
* 5&6: Rosablanca
* 4&7: Saptimpun (10 1/2)
* 5&7: 8bittone (Old Middetown)

The [[pyrite]] tuning of [[edX]]s will turn out to divide a barely mistuned [[5/2]] of almost exactly 45\[[34edo]].

== Individual pages for ed7/3's ==
{| class="wikitable center-all"
|+ style=white-space:nowrap | 0…99
| [[0ed7/3|0]]
| [[1ed7/3|1]]
| [[2ed7/3|2]]
| [[3ed7/3|3]]
| [[4ed7/3|4]]
| [[5ed7/3|5]]
| [[6ed7/3|6]]
| [[7ed7/3|7]]
| [[8ed7/3|8]]
| [[9ed7/3|9]]
|-
| [[10ed7/3|10]]
| [[11ed7/3|11]]
| [[12ed7/3|12]]
| [[13ed7/3|13]]
| [[14ed7/3|14]]
| [[15ed7/3|15]]
| [[16ed7/3|16]]
| [[17ed7/3|17]]
| [[18ed7/3|18]]
| [[19ed7/3|19]]
|-
| [[20ed7/3|20]]
| [[21ed7/3|21]]
| [[22ed7/3|22]]
| [[23ed7/3|23]]
| [[24ed7/3|24]]
| [[25ed7/3|25]]
| [[26ed7/3|26]]
| [[27ed7/3|27]]
| [[28ed7/3|28]]
| [[29ed7/3|29]]
|-
| [[30ed7/3|30]]
| [[31ed7/3|31]]
| [[32ed7/3|32]]
| [[33ed7/3|33]]
| [[34ed7/3|34]]
| [[35ed7/3|35]]
| [[36ed7/3|36]]
| [[37ed7/3|37]]
| [[38ed7/3|38]]
| [[39ed7/3|39]]
|-
| [[40ed7/3|40]]
| [[41ed7/3|41]]
| [[42ed7/3|42]]
| [[43ed7/3|43]]
| [[44ed7/3|44]]
| [[45ed7/3|45]]
| [[46ed7/3|46]]
| [[47ed7/3|47]]
| [[48ed7/3|48]]
| [[49ed7/3|49]]
|-
| [[50ed7/3|50]]
| [[51ed7/3|51]]
| [[52ed7/3|52]]
| [[53ed7/3|53]]
| [[54ed7/3|54]]
| [[55ed7/3|55]]
| [[56ed7/3|56]]
| [[57ed7/3|57]]
| [[58ed7/3|58]]
| [[59ed7/3|59]]
|-
| [[60ed7/3|60]]
| [[61ed7/3|61]]
| [[62ed7/3|62]]
| [[63ed7/3|63]]
| [[64ed7/3|64]]
| [[65ed7/3|65]]
| [[66ed7/3|66]]
| [[67ed7/3|67]]
| [[68ed7/3|68]]
| [[69ed7/3|69]]
|-
| [[70ed7/3|70]]
| [[71ed7/3|71]]
| [[72ed7/3|72]]
| [[73ed7/3|73]]
| [[74ed7/3|74]]
| [[75ed7/3|75]]
| [[76ed7/3|76]]
| [[77ed7/3|77]]
| [[78ed7/3|78]]
| [[79ed7/3|79]]
|-
| [[80ed7/3|80]]
| [[81ed7/3|81]]
| [[82ed7/3|82]]
| [[83ed7/3|83]]
| [[84ed7/3|84]]
| [[85ed7/3|85]]
| [[86ed7/3|86]]
| [[87ed7/3|87]]
| [[88ed7/3|88]]
| [[89ed7/3|89]]
|-
| [[90ed7/3|90]]
| [[91ed7/3|91]]
| [[92ed7/3|92]]
| [[93ed7/3|93]]
| [[94ed7/3|94]]
| [[95ed7/3|95]]
| [[96ed7/3|96]]
| [[97ed7/3|97]]
| [[98ed7/3|98]]
| [[99ed7/3|99]]
|}
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | 100…199
| [[100ed7/3|100]]
| [[101ed7/3|101]]
| [[102ed7/3|102]]
| [[103ed7/3|103]]
| [[104ed7/3|104]]
| [[105ed7/3|105]]
| [[106ed7/3|106]]
| [[107ed7/3|107]]
| [[108ed7/3|108]]
| [[109ed7/3|109]]
|-
| [[110ed7/3|110]]
| [[111ed7/3|111]]
| [[112ed7/3|112]]
| [[113ed7/3|113]]
| [[114ed7/3|114]]
| [[115ed7/3|115]]
| [[116ed7/3|116]]
| [[117ed7/3|117]]
| [[118ed7/3|118]]
| [[119ed7/3|119]]
|-
| [[120ed7/3|120]]
| [[121ed7/3|121]]
| [[122ed7/3|122]]
| [[123ed7/3|123]]
| [[124ed7/3|124]]
| [[125ed7/3|125]]
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| [[128ed7/3|128]]
| [[129ed7/3|129]]
|-
| [[130ed7/3|130]]
| [[131ed7/3|131]]
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| [[133ed7/3|133]]
| [[134ed7/3|134]]
| [[135ed7/3|135]]
| [[136ed7/3|136]]
| [[137ed7/3|137]]
| [[138ed7/3|138]]
| [[139ed7/3|139]]
|-
| [[140ed7/3|140]]
| [[141ed7/3|141]]
| [[142ed7/3|142]]
| [[143ed7/3|143]]
| [[144ed7/3|144]]
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| [[146ed7/3|146]]
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| [[148ed7/3|148]]
| [[149ed7/3|149]]
|-
| [[150ed7/3|150]]
| [[151ed7/3|151]]
| [[152ed7/3|152]]
| [[153ed7/3|153]]
| [[154ed7/3|154]]
| [[155ed7/3|155]]
| [[156ed7/3|156]]
| [[157ed7/3|157]]
| [[158ed7/3|158]]
| [[159ed7/3|159]]
|-
| [[160ed7/3|160]]
| [[161ed7/3|161]]
| [[162ed7/3|162]]
| [[163ed7/3|163]]
| [[164ed7/3|164]]
| [[165ed7/3|165]]
| [[166ed7/3|166]]
| [[167ed7/3|167]]
| [[168ed7/3|168]]
| [[169ed7/3|169]]
|-
| [[170ed7/3|170]]
| [[171ed7/3|171]]
| [[172ed7/3|172]]
| [[173ed7/3|173]]
| [[174ed7/3|174]]
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| [[178ed7/3|178]]
| [[179ed7/3|179]]
|-
| [[180ed7/3|180]]
| [[181ed7/3|181]]
| [[182ed7/3|182]]
| [[183ed7/3|183]]
| [[184ed7/3|184]]
| [[185ed7/3|185]]
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| [[189ed7/3|189]]
|-
| [[190ed7/3|190]]
| [[191ed7/3|191]]
| [[192ed7/3|192]]
| [[193ed7/3|193]]
| [[194ed7/3|194]]
| [[195ed7/3|195]]
| [[196ed7/3|196]]
| [[197ed7/3|197]]
| [[198ed7/3|198]]
| [[199ed7/3|199]]
|}

[[Category:Ed7/3's| ]]

[[Category:Lists of scales]]

User talk:Francium/647edo

2026-05-13T00:18:13Z

Sintel: Created page with "== Deletion == I vote '''keep''', it has at least one song written in it! ~~~~"

== Deletion ==

I vote '''keep''', it has at least one song written in it!
– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 00:18, 13 May 2026 (UTC)

User:Sintel/Expected Dirichlet coefficient for temperaments

2026-05-11T11:48:56Z

Sintel: /* References */ fix link

In the context of [[regular temperament theory]], a natural question is how well a given temperament approximates [[just intonation]] relative to its [[complexity]].
The '''Dirichlet coefficient''' gives a quantitative way to measure this. This is the same as the "[[TE logflat badness|badness]]" used on the wiki, though the derivation here is given for the regular Euclidean norm for clarity.

Given a target vector <math>y \in \mathbb{R}^n</math>, such as the [[JIP | vector of log-primes]] in some ''p''-limit, and a rank-''k'' temperament ''X'', the Dirichlet coefficient is defined as:

:<math>
d(y, X) \cdot H(X)^{\frac{n}{n-k}}
</math>

where <math>d(y, X)</math> is the projective distance between the target vector and the temperament, and <math>H(X)</math> is the height (or [[complexity]]) of the temperament. Both of these quantities can be computed straightforwardly using the temperament's [[Plücker coordinates]].

This coefficient generalizes {{w|Dirichlet's approximation theorem}}. A fundamental result in Diophantine approximation by W. M. Schmidt<ref name="Schmidt1967"/> states that for any valid target vector ''y'', there exists a constant <math>C_{n,k}</math> such that there are infinitely many rational subspaces ''X'' which satisfy:
:<math>
d(y, X) \cdot H(X)^{\frac{n}{n-k}} \le C_{n,k}
</math>

The exponent is ''critical'' or ''sharp'' for this problem: if we replace the exponent by <math>\tfrac{n}{n-k} + \epsilon</math> for some <math>\epsilon > 0</math>, then we find only finitely many solutions.

By analyzing the distribution of rational points on the {{w|Grassmannian}}, we can derive the expected value for this coefficient, giving us a baseline to determine whether a temperament is a "good" or "bad" approximation relative to its complexity.
Importantly, this measure does not take into account any kind of psychoacoustics, so it is not in any way "calibrated" to human tolerance to tuning error.
Instead, it is a purely mathematical metric of coincidence. The upside is that this is robust over an arbitrary range of complexities and does not rely on any free parameters or empirical weights.

== Motivating example: equal temperaments in the 5-limit ==

To understand the Dirichlet coefficient, let's look at rank-1 temperaments (equal temperaments) in the 5-limit (''n''=3).
Our target vector is the standard [[JIP|just intonation vector]] of log-primes:
:<math>
y =[\log_2(2), \log_2(3), \log_2(5)] \approx [1, 1.585, 2.322]
</math>

An equal temperament is defined by a line through the origin with a rational slope. For example, [[12edo|12-equal temperament]] corresponds to the line passing through the integer vector <math>X_{12} =[12, 19, 28]</math>.
This approximation is good in the sense that the ratios of its coordinates closely match the target vector:
:<math>
\frac{19}{12} \approx \log_2(3), \quad \frac{28}{12} \approx \log_2(5)
</math>

Because we are in 3D, the wedge product used to define projective distance reduces to the standard cross product.
The projective distance is the sine of the angle <math>\theta</math> between the temperament line and the JI vector:
:<math>
d(X_{12}, y) = \sin(\theta) = \frac{\|X_{12} \times y\|}{\|X_{12}\| \|y\|} \approx 0.00276
</math>
Since the angle is extremely small (which is always the case for any reasonable temperament) we can take <math>\sin(\theta) \approx \theta</math>.

The height is simply the Euclidean norm of the integer vector:
:<math>
H(X_{12}) = \sqrt{12^2 + 19^2 + 28^2} \approx 35.902
</math>

For an equal temperament (''k''=1) in the 5-limit (''n''=3), the critical exponent is 3/2. This is equivalent to the classical Dirichlet theorem for simultaneous approximation of two irrational numbers.

Plugging this into our formula gives the Dirichlet coefficient for 12-ET:
:<math>
C_{12} = d(X_{12}, y) \cdot H(X_{12})^{3/2} \approx 0.00276 \times 35.902^{1.5} \approx 0.595
</math>

We can compare this to the coefficients of some other 5-limit equal temperaments. Lower values indicate that the temperament is exceptionally accurate for its size, while higher values indicate poor approximations.

{| class="wikitable"
|-
! Temperament !! Dirichlet coefficient
|-
| 53-ET || 0.467
|-
| 12-ET || 0.595
|-
| 34-ET || 0.716
|-
| 20-ET || 3.855
|-
| 33-ET || 4.621
|-
| 52-ET || 6.125
|}
Note: the point here is not to argue over which of these is "better", just that this measure generally agrees on which equal temperaments contain good approximations to 5-limit JI relative to their size.

== Generalizing to higher rank ==

For higher-rank temperaments, we rely on Schmidt's general formula. A rank-''k'' temperament in an ''n''-prime limit is viewed as a rational ''k''-dimensional subspace <math>X \in \mathrm{Gr}(k, n)</math>. By slight abuse of notation, we will identify ''X'' directly with its Plücker coordinates.

As defined in the article on [[Plücker coordinates]], the height is simply the Euclidean norm of the (reduced) Plücker coordinates, and the projective distance is given by:
:<math>
d(X, y) = \frac{\|X \wedge y\|}{\|X\| \|y\|}
</math>
This has the same interpretation in terms of the sine of the minimal angle between the subspace and the target.

According to Schmidt's theorem on metric Diophantine approximation<ref name="Schmidt1967"/>, the critical exponent balancing distance and height for approximating a target vector ''y'' by a ''k''-dimensional rational subspace ''X'' is exactly <math>\tfrac{n}{n-k}</math>, so we immediately obtain:
:<math>
d(y, X) \cdot H(X)^{\frac{n}{n-k}}
</math>

While we typically assume ''y'' to be the log-primes for some ''p''-limit, this property holds for any target vector in <math>\mathbb{R}^n</math>, and so it cleanly generalizes to any subgroup.

== Deriving the heuristic constant ==

For the case ''n''=2, {{w|Hurwitz's theorem (number theory)|Hurwitz's theorem}} states that the best possible constant is <math>\tfrac{1}{\sqrt{5}} \approx 0.447</math>.
Not much is known about the exact constant needed to obtain a tight bound in the general case.

=== Counting temperaments ===

To determine the expected bound, we must first know how many temperaments exist up to a certain complexity.
Another classical result by Schmidt<ref name="Schmidt1968"/> gives the asymptotic distribution of primitive (i.e., [[torsion]]-free) sublattices, which directly correspond to temperaments.

The number of rank-''k'' temperaments with a complexity bounded by <math>H(X) \le H_{\max}</math> grows as:
:<math>
\#\left\{ X: H(X) \le H_{\max} \right\} \sim c_{n, k} H_{\max}^n
</math>

The constant ''cn,k'' is given by the formula:
:<math>
c_{n, k} = \frac{1}{n} \binom{n}{k} \prod_{i=1}^{k} \frac{V(n-i+1)}{V(i)} \cdot \frac{\zeta(i)}{\zeta(n-i+1)}
</math>
where <math>V(m) = \frac{\pi^{m/2}}{\Gamma(m/2+1)}</math> is the volume of the ''m''-dimensional Euclidean unit ball, and <math>\zeta</math> is our good old friend, the {{w|Riemann zeta function}} (with the convention that the pole at <math>\zeta(1)</math> is treated as 1).

=== The distribution of random temperaments ===
We can find the expected minimum distance for a given maximum height by treating the temperaments as being randomly distributed on the Grassmannian manifold.
This is asymptotically true by the equidistribution of rational points.<ref name="Schmidt1998"/>

By rotational invariance, projecting a fixed target vector ''y'' onto a random ''k''-plane is statistically identical to projecting a random unit vector onto a fixed plane.

The squared projective distance is the complement of the squared length of the projection (<math>\cos^2 \theta</math>).
From standard probability theory, the squared length of a projection of a random unit vector in <math>\mathbb{R}^n</math> onto a plane follows a {{w|Beta distribution}}.
Therefore, the squared distance is distributed as:
:<math>
\sin^2(\theta) \sim \mathrm{Beta}\left(\frac{n-k}{2}, \frac{k}{2}\right)
</math>

For some search distance ''r'', the volume (with respect to the normalized {{w|Haar measure}} of the Grassmannian) is given by the cumulative Beta distribution:
:<math>
\text{Vol}(d(X,y) \le r) = \frac{1}{B\!\left(\frac{n-k}{2}, \frac{k}{2}\right)} \int_0^{r^2} s^{\frac{n-k}{2}-1}(1-s)^{\frac{k}{2}-1}\, ds
</math>

When ''r'' is small, <math>(1-s)^{k/2 - 1} \to 1</math>, so the leading term is:
:<math>
\begin{aligned}
\mathrm{Vol}(d(X,y) < r)
&\approx \frac{1}{B\!\left(\frac{n-k}{2}, \frac{k}{2}\right)} \cdot \frac{r^{n-k}}{\frac{n-k}{2}} \\[10pt]
&= \frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}\, r^{n-k}
\end{aligned}
</math>

Since there are <math>c_{n, k} H_{\max}^n</math> planes available, the expected number that fall in this radius is:
:<math>
\mathbb{E}[\#\{X : H(X) \leq H_{\max},\ d(y,X) \leq r\}] \approx
c_{n,k}
\cdot H_{\max}^n
\cdot \frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}
\cdot r^{n-k}
</math>

Setting this equal to 1 and taking the ''(n-k)''-th root, we find:
:<math>
r \cdot H_{\max}^{\frac{n}{n-k}} \approx \left( c_{n,k}
\frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}
\right) ^ {\frac{-1}{n-k}}
</math>
which recovers the same critical exponent, but now with an explicit constant ''Ch'' for our Dirichlet bound.

A temperament with a coefficient much better than this is exceptional: the heuristic says you would need to search through exponentially more planes to find it by chance.

The following table gives the values of ''Ch'' for some small dimensions:
{| class="wikitable"
|-
! n !! ''k'' = 1 !! ''k'' = 2 !! ''k'' = 3 !! ''k'' = 4
|-
| 2 || 1.645
|-
| 3 || 1.071 || 0.574
|-
| 4 || 1.011 || 0.400 || 0.345
|-
| 5 || 1.012 || 0.435 || 0.235 || 0.263
|}

== References ==
<references>
<ref name="Schmidt1967">Wolfgang M. Schmidt. ''On Heights of Algebraic Subspaces and Diophantine Approximations''. Annals of Mathematics, Vol. 85, No. 3 (1967), pp. 430-472, theorem 15 [https://doi.org/10.2307/1970352 doi:10.2307/1970352]</ref>
<ref name="Schmidt1968">Wolfgang M. Schmidt. ''Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height''. Duke Mathematical Journal Vol. 35 No. 2, pp. 327-339 (1968), theorem 1 [https://doi.org/10.1215/S0012-7094-68-03532-1 doi:10.1215/S0012-7094-68-03532-1]</ref>
<ref name="Schmidt1998">Wolfgang M. Schmidt. ''The distribution of sub-lattices of Zm''. Monatshefte für Mathematik Vol. 125 No. 1, pp 37–81 (1998) [https://doi.org/10.1007/BF01489457 doi:10.1007/BF01489457]</ref>
</references>

User:Sintel/Expected Dirichlet coefficient for temperaments

2026-05-11T01:45:21Z

Sintel: link badness article

In the context of [[regular temperament theory]], a natural question is how well a given temperament approximates [[just intonation]] relative to its [[complexity]].
The '''Dirichlet coefficient''' gives a quantitative way to measure this. This is the same as the "[[TE logflat badness|badness]]" used on the wiki, though the derivation here is given for the regular Euclidean norm for clarity.

Given a target vector <math>y \in \mathbb{R}^n</math>, such as the [[JIP | vector of log-primes]] in some ''p''-limit, and a rank-''k'' temperament ''X'', the Dirichlet coefficient is defined as:

:<math>
d(y, X) \cdot H(X)^{\frac{n}{n-k}}
</math>

where <math>d(y, X)</math> is the projective distance between the target vector and the temperament, and <math>H(X)</math> is the height (or [[complexity]]) of the temperament. Both of these quantities can be computed straightforwardly using the temperament's [[Plücker coordinates]].

This coefficient generalizes {{w|Dirichlet's approximation theorem}}. A fundamental result in Diophantine approximation by W. M. Schmidt<ref name="Schmidt1967"/> states that for any valid target vector ''y'', there exists a constant <math>C_{n,k}</math> such that there are infinitely many rational subspaces ''X'' which satisfy:
:<math>
d(y, X) \cdot H(X)^{\frac{n}{n-k}} \le C_{n,k}
</math>

The exponent is ''critical'' or ''sharp'' for this problem: if we replace the exponent by <math>\tfrac{n}{n-k} + \epsilon</math> for some <math>\epsilon > 0</math>, then we find only finitely many solutions.

By analyzing the distribution of rational points on the {{w|Grassmannian}}, we can derive the expected value for this coefficient, giving us a baseline to determine whether a temperament is a "good" or "bad" approximation relative to its complexity.
Importantly, this measure does not take into account any kind of psychoacoustics, so it is not in any way "calibrated" to human tolerance to tuning error.
Instead, it is a purely mathematical metric of coincidence. The upside is that this is robust over an arbitrary range of complexities and does not rely on any free parameters or empirical weights.

== Motivating example: equal temperaments in the 5-limit ==

To understand the Dirichlet coefficient, let's look at rank-1 temperaments (equal temperaments) in the 5-limit (''n''=3).
Our target vector is the standard [[JIP|just intonation vector]] of log-primes:
:<math>
y =[\log_2(2), \log_2(3), \log_2(5)] \approx [1, 1.585, 2.322]
</math>

An equal temperament is defined by a line through the origin with a rational slope. For example, [[12edo|12-equal temperament]] corresponds to the line passing through the integer vector <math>X_{12} =[12, 19, 28]</math>.
This approximation is good in the sense that the ratios of its coordinates closely match the target vector:
:<math>
\frac{19}{12} \approx \log_2(3), \quad \frac{28}{12} \approx \log_2(5)
</math>

Because we are in 3D, the wedge product used to define projective distance reduces to the standard cross product.
The projective distance is the sine of the angle <math>\theta</math> between the temperament line and the JI vector:
:<math>
d(X_{12}, y) = \sin(\theta) = \frac{\|X_{12} \times y\|}{\|X_{12}\| \|y\|} \approx 0.00276
</math>
Since the angle is extremely small (which is always the case for any reasonable temperament) we can take <math>\sin(\theta) \approx \theta</math>.

The height is simply the Euclidean norm of the integer vector:
:<math>
H(X_{12}) = \sqrt{12^2 + 19^2 + 28^2} \approx 35.902
</math>

For an equal temperament (''k''=1) in the 5-limit (''n''=3), the critical exponent is 3/2. This is equivalent to the classical Dirichlet theorem for simultaneous approximation of two irrational numbers.

Plugging this into our formula gives the Dirichlet coefficient for 12-ET:
:<math>
C_{12} = d(X_{12}, y) \cdot H(X_{12})^{3/2} \approx 0.00276 \times 35.902^{1.5} \approx 0.595
</math>

We can compare this to the coefficients of some other 5-limit equal temperaments. Lower values indicate that the temperament is exceptionally accurate for its size, while higher values indicate poor approximations.

{| class="wikitable"
|-
! Temperament !! Dirichlet coefficient
|-
| 53-ET || 0.467
|-
| 12-ET || 0.595
|-
| 34-ET || 0.716
|-
| 20-ET || 3.855
|-
| 33-ET || 4.621
|-
| 52-ET || 6.125
|}
Note: the point here is not to argue over which of these is "better", just that this measure generally agrees on which equal temperaments contain good approximations to 5-limit JI relative to their size.

== Generalizing to higher rank ==

For higher-rank temperaments, we rely on Schmidt's general formula. A rank-''k'' temperament in an ''n''-prime limit is viewed as a rational ''k''-dimensional subspace <math>X \in \mathrm{Gr}(k, n)</math>. By slight abuse of notation, we will identify ''X'' directly with its Plücker coordinates.

As defined in the article on [[Plücker coordinates]], the height is simply the Euclidean norm of the (reduced) Plücker coordinates, and the projective distance is given by:
:<math>
d(X, y) = \frac{\|X \wedge y\|}{\|X\| \|y\|}
</math>
This has the same interpretation in terms of the sine of the minimal angle between the subspace and the target.

According to Schmidt's theorem on metric Diophantine approximation<ref name="Schmidt1967"/>, the critical exponent balancing distance and height for approximating a target vector ''y'' by a ''k''-dimensional rational subspace ''X'' is exactly <math>\tfrac{n}{n-k}</math>, so we immediately obtain:
:<math>
d(y, X) \cdot H(X)^{\frac{n}{n-k}}
</math>

While we typically assume ''y'' to be the log-primes for some ''p''-limit, this property holds for any target vector in <math>\mathbb{R}^n</math>, and so it cleanly generalizes to any subgroup.

== Deriving the heuristic constant ==

For the case ''n''=2, {{w|Hurwitz's theorem (number theory)|Hurwitz's theorem}} states that the best possible constant is <math>\tfrac{1}{\sqrt{5}} \approx 0.447</math>.
Not much is known about the exact constant needed to obtain a tight bound in the general case.

=== Counting temperaments ===

To determine the expected bound, we must first know how many temperaments exist up to a certain complexity.
Another classical result by Schmidt<ref name="Schmidt1968"/> gives the asymptotic distribution of primitive (i.e., [[torsion]]-free) sublattices, which directly correspond to temperaments.

The number of rank-''k'' temperaments with a complexity bounded by <math>H(X) \le H_{\max}</math> grows as:
:<math>
\#\left\{ X: H(X) \le H_{\max} \right\} \sim c_{n, k} H_{\max}^n
</math>

The constant ''cn,k'' is given by the formula:
:<math>
c_{n, k} = \frac{1}{n} \binom{n}{k} \prod_{i=1}^{k} \frac{V(n-i+1)}{V(i)} \cdot \frac{\zeta(i)}{\zeta(n-i+1)}
</math>
where <math>V(m) = \frac{\pi^{m/2}}{\Gamma(m/2+1)}</math> is the volume of the ''m''-dimensional Euclidean unit ball, and <math>\zeta</math> is our good old friend, the {{w|Riemann zeta function}} (with the convention that the pole at <math>\zeta(1)</math> is treated as 1).

=== The distribution of random temperaments ===
We can find the expected minimum distance for a given maximum height by treating the temperaments as being randomly distributed on the Grassmannian manifold.
This is asymptotically true by the equidistribution of rational points.<ref name="Schmidt1998"/>

By rotational invariance, projecting a fixed target vector ''y'' onto a random ''k''-plane is statistically identical to projecting a random unit vector onto a fixed plane.

The squared projective distance is the complement of the squared length of the projection (<math>\cos^2 \theta</math>).
From standard probability theory, the squared length of a projection of a random unit vector in <math>\mathbb{R}^n</math> onto a plane follows a {{w|Beta distribution}}.
Therefore, the squared distance is distributed as:
:<math>
\sin^2(\theta) \sim \mathrm{Beta}\left(\frac{n-k}{2}, \frac{k}{2}\right)
</math>

For some search distance ''r'', the volume (with respect to the normalized {{w|Haar measure}} of the Grassmannian) is given by the cumulative Beta distribution:
:<math>
\text{Vol}(d(X,y) \le r) = \frac{1}{B\!\left(\frac{n-k}{2}, \frac{k}{2}\right)} \int_0^{r^2} s^{\frac{n-k}{2}-1}(1-s)^{\frac{k}{2}-1}\, ds
</math>

When ''r'' is small, <math>(1-s)^{k/2 - 1} \to 1</math>, so the leading term is:
:<math>
\begin{aligned}
\mathrm{Vol}(d(X,y) < r)
&\approx \frac{1}{B\!\left(\frac{n-k}{2}, \frac{k}{2}\right)} \cdot \frac{r^{n-k}}{\frac{n-k}{2}} \\[10pt]
&= \frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}\, r^{n-k}
\end{aligned}
</math>

Since there are <math>c_{n, k} H_{\max}^n</math> planes available, the expected number that fall in this radius is:
:<math>
\mathbb{E}[\#\{X : H(X) \leq H_{\max},\ d(y,X) \leq r\}] \approx
c_{n,k}
\cdot H_{\max}^n
\cdot \frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}
\cdot r^{n-k}
</math>

Setting this equal to 1 and taking the ''(n-k)''-th root, we find:
:<math>
r \cdot H_{\max}^{\frac{n}{n-k}} \approx \left( c_{n,k}
\frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}
\right) ^ {\frac{-1}{n-k}}
</math>
which recovers the same critical exponent, but now with an explicit constant ''Ch'' for our Dirichlet bound.

A temperament with a coefficient much better than this is exceptional: the heuristic says you would need to search through exponentially more planes to find it by chance.

The following table gives the values of ''Ch'' for some small dimensions:
{| class="wikitable"
|-
! n !! ''k'' = 1 !! ''k'' = 2 !! ''k'' = 3 !! ''k'' = 4
|-
| 2 || 1.645
|-
| 3 || 1.071 || 0.574
|-
| 4 || 1.011 || 0.400 || 0.345
|-
| 5 || 1.012 || 0.435 || 0.235 || 0.263
|}

== References ==
<references>
<ref name="Schmidt1967">Wolfgang M. Schmidt. ''On Heights of Algebraic Subspaces and Diophantine Approximations''. Annals of Mathematics, Vol. 85, No. 3 (1967), pp. 430-472, theorem 15 [https://doi.org/10.2307/1970352 doi:10.2307/1970352]</ref>
<ref name="Schmidt1968">Wolfgang M. Schmidt. ''Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height''. Duke Mathematical Journal Vol. 35 No. 2, pp. 327-339 (1968), theorem 1 [https://doi.org/10.2307/1970352 doi:10.2307/1970352]</ref>
<ref name="Schmidt1998">Wolfgang M. Schmidt. ''The distribution of sub-lattices of Zm''. Monatshefte für Mathematik Vol. 125 No. 1, pp 37–81 (1998) [https://doi.org/10.1007/BF01489457 doi:10.1007/BF01489457]</ref>
</references>

User:Sintel/Expected Dirichlet coefficient for temperaments

2026-05-11T01:22:46Z

Sintel: publish

In the context of [[regular temperament theory]], a natural question is how well a given temperament approximates [[just intonation]] relative to its [[complexity]].
The '''Dirichlet coefficient''' gives a quantitative way to measure this.

Given a target vector <math>y \in \mathbb{R}^n</math>, such as the [[JIP | vector of log-primes]] in some ''p''-limit, and a rank-''k'' temperament ''X'', the Dirichlet coefficient is defined as:

:<math>
d(y, X) \cdot H(X)^{\frac{n}{n-k}}
</math>

where <math>d(y, X)</math> is the projective distance between the target vector and the temperament, and <math>H(X)</math> is the height (or [[complexity]]) of the temperament. Both of these quantities can be computed straightforwardly using the temperament's [[Plücker coordinates]].

This coefficient generalizes {{w|Dirichlet's approximation theorem}}. A fundamental result in Diophantine approximation by W. M. Schmidt<ref name="Schmidt1967"/> states that for any valid target vector ''y'', there exists a constant <math>C_{n,k}</math> such that there are infinitely many rational subspaces ''X'' which satisfy:
:<math>
d(y, X) \cdot H(X)^{\frac{n}{n-k}} \le C_{n,k}
</math>

The exponent is ''critical'' or ''sharp'' for this problem: if we replace the exponent by <math>\tfrac{n}{n-k} + \epsilon</math> for some <math>\epsilon > 0</math>, then we find only finitely many solutions.

By analyzing the distribution of rational points on the {{w|Grassmannian}}, we can derive the expected value for this coefficient, giving us a baseline to determine whether a temperament is a "good" or "bad" approximation relative to its complexity.
Importantly, this measure does not take into account any kind of psychoacoustics, so it is not in any way "calibrated" to human tolerance to tuning error.
Instead, it is a purely mathematical metric of coincidence. The upside is that this is robust over an arbitrary range of complexities and does not rely on any free parameters or empirical weights.

== Motivating example: equal temperaments in the 5-limit ==

To understand the Dirichlet coefficient, let's look at rank-1 temperaments (equal temperaments) in the 5-limit (''n''=3).
Our target vector is the standard [[JIP|just intonation vector]] of log-primes:
:<math>
y =[\log_2(2), \log_2(3), \log_2(5)] \approx [1, 1.585, 2.322]
</math>

An equal temperament is defined by a line through the origin with a rational slope. For example, [[12edo|12-equal temperament]] corresponds to the line passing through the integer vector <math>X_{12} =[12, 19, 28]</math>.
This approximation is good in the sense that the ratios of its coordinates closely match the target vector:
:<math>
\frac{19}{12} \approx \log_2(3), \quad \frac{28}{12} \approx \log_2(5)
</math>

Because we are in 3D, the wedge product used to define projective distance reduces to the standard cross product.
The projective distance is the sine of the angle <math>\theta</math> between the temperament line and the JI vector:
:<math>
d(X_{12}, y) = \sin(\theta) = \frac{\|X_{12} \times y\|}{\|X_{12}\| \|y\|} \approx 0.00276
</math>
Since the angle is extremely small (which is always the case for any reasonable temperament) we can take <math>\sin(\theta) \approx \theta</math>.

The height is simply the Euclidean norm of the integer vector:
:<math>
H(X_{12}) = \sqrt{12^2 + 19^2 + 28^2} \approx 35.902
</math>

For an equal temperament (''k''=1) in the 5-limit (''n''=3), the critical exponent is 3/2. This is equivalent to the classical Dirichlet theorem for simultaneous approximation of two irrational numbers.

Plugging this into our formula gives the Dirichlet coefficient for 12-ET:
:<math>
C_{12} = d(X_{12}, y) \cdot H(X_{12})^{3/2} \approx 0.00276 \times 35.902^{1.5} \approx 0.595
</math>

We can compare this to the coefficients of some other 5-limit equal temperaments. Lower values indicate that the temperament is exceptionally accurate for its size, while higher values indicate poor approximations.

{| class="wikitable"
|-
! Temperament !! Dirichlet coefficient
|-
| 53-ET || 0.467
|-
| 12-ET || 0.595
|-
| 34-ET || 0.716
|-
| 20-ET || 3.855
|-
| 33-ET || 4.621
|-
| 52-ET || 6.125
|}
Note: the point here is not to argue over which of these is "better", just that this measure generally agrees on which equal temperaments contain good approximations to 5-limit JI relative to their size.

== Generalizing to higher rank ==

For higher-rank temperaments, we rely on Schmidt's general formula. A rank-''k'' temperament in an ''n''-prime limit is viewed as a rational ''k''-dimensional subspace <math>X \in \mathrm{Gr}(k, n)</math>. By slight abuse of notation, we will identify ''X'' directly with its Plücker coordinates.

As defined in the article on [[Plücker coordinates]], the height is simply the Euclidean norm of the (reduced) Plücker coordinates, and the projective distance is given by:
:<math>
d(X, y) = \frac{\|X \wedge y\|}{\|X\| \|y\|}
</math>
This has the same interpretation in terms of the sine of the minimal angle between the subspace and the target.

According to Schmidt's theorem on metric Diophantine approximation<ref name="Schmidt1967"/>, the critical exponent balancing distance and height for approximating a target vector ''y'' by a ''k''-dimensional rational subspace ''X'' is exactly <math>\tfrac{n}{n-k}</math>, so we immediately obtain:
:<math>
d(y, X) \cdot H(X)^{\frac{n}{n-k}}
</math>

While we typically assume ''y'' to be the log-primes for some ''p''-limit, this property holds for any target vector in <math>\mathbb{R}^n</math>, and so it cleanly generalizes to any subgroup.

== Deriving the heuristic constant ==

For the case ''n''=2, {{w|Hurwitz's theorem (number theory)|Hurwitz's theorem}} states that the best possible constant is <math>\tfrac{1}{\sqrt{5}} \approx 0.447</math>.
Not much is known about the exact constant needed to obtain a tight bound in the general case.

=== Counting temperaments ===

To determine the expected bound, we must first know how many temperaments exist up to a certain complexity.
Another classical result by Schmidt<ref name="Schmidt1968"/> gives the asymptotic distribution of primitive (i.e., [[torsion]]-free) sublattices, which directly correspond to temperaments.

The number of rank-''k'' temperaments with a complexity bounded by <math>H(X) \le H_{\max}</math> grows as:
:<math>
\#\left\{ X: H(X) \le H_{\max} \right\} \sim c_{n, k} H_{\max}^n
</math>

The constant ''cn,k'' is given by the formula:
:<math>
c_{n, k} = \frac{1}{n} \binom{n}{k} \prod_{i=1}^{k} \frac{V(n-i+1)}{V(i)} \cdot \frac{\zeta(i)}{\zeta(n-i+1)}
</math>
where <math>V(m) = \frac{\pi^{m/2}}{\Gamma(m/2+1)}</math> is the volume of the ''m''-dimensional Euclidean unit ball, and <math>\zeta</math> is our good old friend, the {{w|Riemann zeta function}} (with the convention that the pole at <math>\zeta(1)</math> is treated as 1).

=== The distribution of random temperaments ===
We can find the expected minimum distance for a given maximum height by treating the temperaments as being randomly distributed on the Grassmannian manifold.
This is asymptotically true by the equidistribution of rational points.<ref name="Schmidt1998"/>

By rotational invariance, projecting a fixed target vector ''y'' onto a random ''k''-plane is statistically identical to projecting a random unit vector onto a fixed plane.

The squared projective distance is the complement of the squared length of the projection (<math>\cos^2 \theta</math>).
From standard probability theory, the squared length of a projection of a random unit vector in <math>\mathbb{R}^n</math> onto a plane follows a {{w|Beta distribution}}.
Therefore, the squared distance is distributed as:
:<math>
\sin^2(\theta) \sim \mathrm{Beta}\left(\frac{n-k}{2}, \frac{k}{2}\right)
</math>

For some search distance ''r'', the volume (with respect to the normalized {{w|Haar measure}} of the Grassmannian) is given by the cumulative Beta distribution:
:<math>
\text{Vol}(d(X,y) \le r) = \frac{1}{B\!\left(\frac{n-k}{2}, \frac{k}{2}\right)} \int_0^{r^2} s^{\frac{n-k}{2}-1}(1-s)^{\frac{k}{2}-1}\, ds
</math>

When ''r'' is small, <math>(1-s)^{k/2 - 1} \to 1</math>, so the leading term is:
:<math>
\begin{aligned}
\mathrm{Vol}(d(X,y) < r)
&\approx \frac{1}{B\!\left(\frac{n-k}{2}, \frac{k}{2}\right)} \cdot \frac{r^{n-k}}{\frac{n-k}{2}} \\[10pt]
&= \frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}\, r^{n-k}
\end{aligned}
</math>

Since there are <math>c_{n, k} H_{\max}^n</math> planes available, the expected number that fall in this radius is:
:<math>
\mathbb{E}[\#\{X : H(X) \leq H_{\max},\ d(y,X) \leq r\}] \approx
c_{n,k}
\cdot H_{\max}^n
\cdot \frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}
\cdot r^{n-k}
</math>

Setting this equal to 1 and taking the ''(n-k)''-th root, we find:
:<math>
r \cdot H_{\max}^{\frac{n}{n-k}} \approx \left( c_{n,k}
\frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{k}{2}\right)\,\Gamma\!\left(\frac{n-k+2}{2}\right)}
\right) ^ {\frac{-1}{n-k}}
</math>
which recovers the same critical exponent, but now with an explicit constant ''Ch'' for our Dirichlet bound.

A temperament with a coefficient much better than this is exceptional: the heuristic says you would need to search through exponentially more planes to find it by chance.

The following table gives the values of ''Ch'' for some small dimensions:
{| class="wikitable"
|-
! n !! ''k'' = 1 !! ''k'' = 2 !! ''k'' = 3 !! ''k'' = 4
|-
| 2 || 1.645
|-
| 3 || 1.071 || 0.574
|-
| 4 || 1.011 || 0.400 || 0.345
|-
| 5 || 1.012 || 0.435 || 0.235 || 0.263
|}

== References ==
<references>
<ref name="Schmidt1967">Wolfgang M. Schmidt. ''On Heights of Algebraic Subspaces and Diophantine Approximations''. Annals of Mathematics, Vol. 85, No. 3 (1967), pp. 430-472, theorem 15 [https://doi.org/10.2307/1970352 doi:10.2307/1970352]</ref>
<ref name="Schmidt1968">Wolfgang M. Schmidt. ''Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height''. Duke Mathematical Journal Vol. 35 No. 2, pp. 327-339 (1968), theorem 1 [https://doi.org/10.2307/1970352 doi:10.2307/1970352]</ref>
<ref name="Schmidt1998">Wolfgang M. Schmidt. ''The distribution of sub-lattices of Zm''. Monatshefte für Mathematik Vol. 125 No. 1, pp 37–81 (1998) [https://doi.org/10.1007/BF01489457 doi:10.1007/BF01489457]</ref>
</references>

Meantone family

2026-05-09T11:18:23Z

Sintel: Undo revision 229909 by Xenllium (talk)

{{interwiki
| de = Mitteltönig
| en = Meantone family
| es =
| ja =
}}
{{Technical data page}}
The '''meantone family''' is the family of [[rank-2 temperament]]s that [[tempering out|temper out]] the syntonic comma, [[81/80]], and thus can all be seen as [[extension]]s of [[meantone]].

== Meantone ==
{{Main| Meantone }}

Meantone is characterized by an [[2/1|octave]] [[period]], a [[3/2|fifth]] [[generator]], and the relationship that four fifths go to make up a [[5/1|5th harmonic]].

[[Subgroup]]: 2.3.5

[[Comma list]]: 81/80

{{Mapping|legend=1| 1 0 -4 | 0 1 4 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.3906{{c}}, ~3/2 = 697.0455{{c}}
: [[error map]]: {{val| +1.391 -3.519 +1.868 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6512{{c}}
: error map: {{val| 0.000 -5.304 +0.291 }}

[[Minimax tuning]]:
* [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

[[Tuning ranges]]:
* 5-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 720.000] (4\7 to 3\5)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)

{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }}

[[Badness]] (Sintel): 0.173

=== Overview to extensions ===
The second comma of the normal comma list defines which [[7-limit]] family member we are looking at.
* Flattertone adds {{monzo| -24 17 0 -1 }}, finding the [[~]][[7/4]] at the double-augmented sixth, for a tuning between 33edo and 26edo.
* Flattone adds {{monzo| -17 9 0 1 }}, finding the ~7/4 at the diminished seventh, for a tuning between 26edo and 19edo.
* Septimal meantone adds [[Harrison's comma|{{monzo| -13 10 0 -1 }}]], finding the ~7/4 at the augmented sixth, for a tuning between 19edo and 12edo.
* Dominant adds [[64/63|{{monzo| 6 -2 0 -1 }}]], finding the ~7/4 at the minor seventh, for a tuning between 12edo and 5edo.
* Sharptone adds [[28/27|{{monzo| 2 -3 0 1 }}]], finding the ~7/4 at the major sixth, for an [[exotemperament]] never exactly well-tuned, and where 5edo is the only [[diamond monotone]] tuning, with a terrible 5-limit part.
Those all have a fifth as generator.
* Injera adds {{monzo| -7 8 0 -2 }} with a half-octave period.
* Mohajira adds {{monzo| -23 11 0 2 }} and splits the fifth in two.
* Godzilla adds [[49/48|{{monzo| -4 -1 0 2 }}]] with an ~[[8/7]] generator, two of which give the [[4/3|fourth]].
* Mothra adds [[1029/1024|{{monzo| -10 1 0 3 }}]] with an ~8/7 generator, three of which give the fifth.
* Liese adds {{monzo| -9 11 0 -3 }} with a ~[[10/7]] generator, three of which give the [[3/1|twelfth]].
* Squares adds {{monzo| -3 9 0 -4 }} with a ~[[9/7]] generator, four of which give the [[8/3|eleventh]].
* Jerome adds {{monzo| 3 7 0 -5 }} and slices the fifth in five.

==== Strong extensions ====
For any meantone generator tuning between 7\12 and 11\19, the augmented sixth is sharper than the diminished seventh and flatter than the minor seventh, befitting an approximation to interval class of 7. This coincides with interpreting the tritone (~9/8)3 as [[7/5]], leading to septimal meantone, a very elegant extension to the 7-limit.

For any tuning flatter than 11\19, the augmented sixth and diminished seventh swap their orders, so the diminished seventh becomes a better approximation to the interval class of 7, resulting in flattone. Likewise, for any tuning sharper than 7\12, the minor seventh is the proper approximation instead, resulting in dominant.

Another way to extend meantone to higher limits involves decomposing the meantone comma into products of smaller commas, or expressing some other comma of interest in terms of the ratio between the meantone comma and another comma. However, this often results in [[weak extension]]s. Another opportunity given by the meantone fifth being flat is that the most obvious ways of dividing it into ''n'' parts leave the part closer to just than usual, because we can allow – and indeed want – more flatwards tempering on the fifth, so may be recommended for this reason.

==== Splitting the meantone fifth into two (243/242) ====
By tempering out [[243/242]] we equate the distance from 9/8 to 10/9 (= [[81/80|S9]]) with the distance between 11/10 to 12/11 (= [[121/120|S11]]), leading to [[mohaha]] which is in some sense thus a trivial tuning of [[rastmic]] (as 81/80 and 121/120 vanish), but an important one, as it leads to the 11/9 being a more in-tune "hemififth" than in non-meantone [[rastmic]] temperaments (which require sharper fifths in good tunings), and it has a natural extension to the full [[11-limit]] by finding [[7/4]] as the semi-diminished seventh, leading to [[mohajira]], which inflates [[64/63]] to equate it with a small quarter-tone, which is characteristic. Mohajira can also be thought of as equating a slightly sharpened [[25/16|(5/4)2]] with [[11/7]], which is also natural as meantone tempering usually has [[5/4]] slightly sharp. There is also the consideration that tempering out [[121/120]] leads to similarly high damage in the 11-limit as tempering [[81/80]] in the 5-limit, because both erase key distinctions of their respective JI subgroups.

==== Splitting the meantone fifth into three (1029/1024) ====
By tempering out [[1029/1024]] we equate the distance from 7/6 to 8/7 (= [[49/48|S7]]) with the distance from 8/7 to 9/8 (= [[64/63|S8]]), so that ([[8/7]])3 is equated with [[3/2]], because of being able to be rewritten as (9/8)(8/7)(7/6) – this observation can be generalized to define the family of [[ultraparticular]] commas. This is an unusually natural extension, with a surprising coincidence: ([[36/35]])/([[64/63]]) = [[81/80]], or using the shorthand notation, S6/S8 = S9. As S6/S8 is already tempered out, it is natural to want [[49/48]] (S7), which is bigger than S8 and smaller than S6 to be equated with both, to avoid inconsistent mappings. This has the surprising consequence of meaning that splitting the meantone fifth into three 8/7's is equivalent to splitting 8/5 into three 7/6's by tempering (8/5)/(7/6)3 = [[1728/1715]] (S6/S7), the orwellisma.

This strategy leads to the 7-limit version of [[mothra]], which is also sometimes called '''cynder'''. Though undecimal mothra is the simplest extension by a small margin, when measured in terms of generators required to reach 11, there is another extension that is perhaps more obvious, by noticing that because we have S6~S7~S8 with S9 tempered out, we can try S8~S10 by tempering out [[176/175]] (S8/S10), which is (11/7)/(5/4)2, taking advantage of 10/9 being tempered sharp in meantone so that we can distinguish 11/10 from it, thus finding 16/11 at 100/99 above the meantone diminished fifth, ([[6/5]])2 = [[36/25]] = ([[3/2]])/([[25/24]]).

==== 31edo as splitting the fifth into two, three and nine ====
[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering out 225/224, which interestingly, though a rank-2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering out 225/224).

Temperaments discussed elsewhere include
* ''[[Plutus]]'' (+15/14) → [[Very low accuracy temperaments #Plutus|Very low accuracy temperaments]]
* [[Godzilla]] (+49/48) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
* [[Mothra]] (+1029/1024) → [[Gamelismic clan #Mothra|Gamelismic clan]]
* ''[[Mohaha]]'' (+121/120) → [[Rastmic clan #Mohaha|Rastmic clan]]

The rest are considered below.

== Septimal meantone ==
<div style="float:right">[[:de:septimal-mitteltönig|Deutsch]]</div>
{{Main| Meantone #Septimal meantone}}
{{Wikipedia| Septimal meantone temperament }}

In septimal meantone, ten fifths get to the interval class for 7, so that [[7/4]] is an augmented sixth (C–A♯), [[7/6]] is an augmented second (C–D♯), [[7/5]] is an augmented fourth (C–F♯), and [[21/16]] is an augmented third (C–E♯). This mapping is rationalized by the fact that 81/80 factors as ([[126/125]])⋅([[225/224]]), and septimal meantone tempers out both of these commas as well as their difference, [[3136/3125]]. In fact it can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125, 225/224, and 3136/3125.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 126/125

{{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2358{{c}}, ~3/2 = 697.2122{{c}}
: [[error map]]: {{val| +1.236 -3.507 +2.535 -0.412 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6562{{c}}
: error map: {{val| 0.000 -5.299 +0.311 -2.264 }}

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

[[Algebraic generator]]: Cybozem, the real root of 15''x''3 - 10''x''2 - 18, 503.4257 cents. The recurrence converges quickly.

{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}

[[Badness]] (Sintel): 0.347

=== Undecimal meantone (huygens) ===
{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}}
{{See also| Huygens vs meanpop }}

Undecimal meantone<ref name="meantone & meanpop 2003">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6048.html#6052 Yahoo! Tuning Group | ''good 11-limit meantones'']</ref> a.k.a. huygens<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10437.html Yahoo! Tuning Group | ''The meantone family'']</ref><ref name="meantone & meanpop 2004">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10864.html#10870 Yahoo! Tuning Group | ''names and definitions: meantone'']</ref> maps the [[11/8]] to the double-augmented third (C–E𝄪). See [[chords of huygens]] for a list of dyadic chords in this temperament.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 126/125

Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}

Optimal tunings:
* WE: ~2 = 1200.7636{{c}}, ~3/2 = 697.4122{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0315{{c}}

Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/16 -1/8 0 0 1/16 }}, {{monzo| 9/4 -1/2 0 0 1/4 }}, {{monzo| 21/8 -5/4 0 0 5/8 }}, {{monzo| 25/8 -9/4 0 0 9/8 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

Algebraic generator: Traverse, the positive real root of ''x''4 + 2''x'' - 13, or 696.9529 cents.

{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }}

Badness (Sintel): 0.563

; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 ''Twinkle canon – 74 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]

==== Grosstone ====
Grosstone, named for tempering out the [[grossma]], is the main extension of interest that extends undecimal meantone to the 13-limit. It maps 13/8 to the double-diminished seventh (C–B♭♭♭). Note also that 11/10 is a double-augmented unison; 12/11~13/12 is a double-diminished third; and 14/13 is a triple-augmented seventh octave reduced. Grosstone is flexible with its tunings; among the good tunings are [[31edo]], [[43edo]], and [[74edo]].

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 144/143

Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}

Optimal tunings:
* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}

Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }}
: eigenmonzo basis (unchanged-interval basis): 2.13/7

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}

Badness (Sintel): 1.07

===== 17-limit =====
This extension maps 17/16 to the minor second (C–D♭), and 19/16 to the minor third (C–E♭), suitable for a system generated by a mildly tempered fifth.

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 120/119, 126/125, 144/143

Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}

Optimal tunings:
* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}

{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}

Badness (Sintel): 1.06

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143

Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}

Optimal tunings:
* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}

{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}

Badness (Sintel): 1.07

==== Fokkertone ====
Fokkertone maps the [[13/8]] to the double-augmented fifth (C–G𝄪). Note that the minor third conflates 13/11 with 6/5, and that 11/10~13/12 is a double-augmented unison; 12/11 is a double-diminished third; and 14/13 is a minor second. 31edo can be recommended as a tuning since it is the only 13-odd-limit diamond monotone tuning.

This extension used to be known as ''tridecimal meantone'', but was decanonicalized in 2025.

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 99/98, 105/104

Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}

Optimal tunings:
* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}

Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9

{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}

Badness (Sintel): 0.746

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 66/65, 81/80, 99/98, 105/104, 120/119

Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}

Optimal tunings:
* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}}

{{Optimal ET sequence|legend=0| 12f, 31 }}

Badness (Sintel): 1.02

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119

Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}

Optimal tunings:
* WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}}

{{Optimal ET sequence|legend=0| 12f, 31 }}

Badness (Sintel): 1.10

==== Meridetone ====
Meridetone maps the 13/8 to the quadruple-augmented fourth (C–F𝄪𝄪). 43edo can be recommended as a tuning since it is the only 13-odd-limit diamond monotone tuning.

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 99/98, 126/125

Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}

Optimal tunings:
* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}}

Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }}
: unchanged-interval (eigenmonzo) basis: 2.13/9

{{Optimal ET sequence|legend=0| 12f, 31f, 43 }}

Badness (Sintel): 1.09

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 81/80, 99/98, 120/119, 126/125

Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}

Optimal tunings:
* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}}

{{Optimal ET sequence|legend=0| 12f, 43 }}

Badness (Sintel): 1.22

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125

Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}

Optimal tunings:
* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}

{{Optimal ET sequence|legend=0| 12f, 43 }}

Badness (Sintel): 1.25

==== Hemimeantone ====
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 169/168

Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}

: mapping generators: ~2, ~26/15

Optimal tunings:
* WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5065{{c}}

{{Optimal ET sequence|legend=0| 19e, 43, 62 }}

Badness (Sintel): 1.30

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 126/125, 169/168, 221/220

Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}

Optimal tunings:
* WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}}

{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}

Badness (Sintel): 1.19

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220

Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}

Optimal tunings:
* WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}}

{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}

Badness (Sintel): 1.15

==== Semimeantone ====
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 847/845

Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}

: mapping generators: ~55/39, ~3

Optimal tunings:
* WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}}

{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }}

Badness (Sintel): 1.68

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 126/125, 221/220, 289/288

Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}

Optimal tunings:
* WE: ~17/12 = 600.5426{{c}}, ~3/2 = 697.5571{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9858{{c}}

{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}

Badness (Sintel): 1.60

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220

Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}

Optimal tunings:
* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}}

{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}

Badness (Sintel): 1.47

=== Meanpop ===
{{See also| Huygens vs meanpop }}

Meanpop<ref name="meantone & meanpop 2003"/><ref name="meantone & meanpop 2004"/> maps the 11/8 to the double-diminished fifth (C–G𝄫), and tridecimal meanpop maps the 13/8 to the double-augmented fifth (C–G𝄪), tempering out 144/143 like in grosstone. Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 385/384

Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}

: mapping generator: ~2, ~3

Optimal tunings:
* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}}

Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }}
: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| -3 0 5/2 0 0 }}, {{monzo| 11 0 -13/4 0 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''3 + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}

Badness (Sintel): 0.712

; Music
* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 ''Twinkle canon – 50 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]

==== Tridecimal meanpop ====
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 126/125, 144/143

Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}

Optimal tunings:
* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}

Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }}
: unchanged-interval (eigenmonzo) basis: 2.13/11

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }}

Badness (Sintel): 0.863

===== Meanpoppic =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 126/125, 144/143, 273/272

Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}

Optimal tunings:
* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}}

{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}

Badness (Sintel): 1.02

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272

Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}

Optimal tunings:
* WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}}

{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}

Badness (Sintel): 1.08

===== Meanpoid =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 120/119, 126/125, 144/143

Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}

Optimal tunings:
* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}}

{{Optimal ET sequence|legend=0| 19, 31 }}

Badness (Sintel): 1.17

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125

Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}

Optimal tunings:
* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}}

{{Optimal ET sequence|legend=0| 19, 31 }}

Badness (Sintel): 1.25

==== Semimeanpop ====
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 126/125, 385/384, 847/845

Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }}

: mapping generators: ~55/39, ~3

Optimal tunings:
* WE: ~55/39 = 600.6704{{c}}, ~3/2 = 697.2151{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.4341{{c}}

{{Optimal ET sequence|legend=0| 12e, 50, 62, 112b }}

Badness (Sintel): 1.78

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 126/125, 221/220, 273/272, 289/288

Mapping: {{mapping| 2 0 -8 -26 48 39 5 | 0 1 4 10 -13 -10 1 }}

Optimal tunings:
* WE: ~17/12 = 600.7232{{c}}, ~3/2 = 697.2820{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4411{{c}}

{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bg }}

Badness (Sintel): 1.45

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 126/125, 153/152, 209/208, 221/220, 273/272

Mapping: {{mapping| 2 0 -8 -26 48 39 5 -1 | 0 1 4 10 -13 -10 1 3 }}

Optimal tunings:
* WE: ~17/12 = 600.7527{{c}}, ~3/2 = 697.3244{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4525{{c}}

{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bgh }}

Badness (Sintel): 1.28

=== Meanenneadecal ===
Meanenneadecal maps the 11/8 to the augmented fourth (C–F♯), and tridecimal meanenneadecal maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a major second; 12/11~14/13, minor second; and 13/12, double-augmented unison.

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55, 81/80

Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}

Optimal tunings:
* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377]

{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}

Badness (Sintel): 0.708

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 78/77, 81/80

Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}

Optimal tunings:
* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}

{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}

Badness (Sintel): 0.875

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 56/55, 78/77, 81/80, 120/119

Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}

Optimal tunings:
* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}

{{Optimal ET sequence|legend=0| 12f, 19, 31e }}

Badness (Sintel): 1.17

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119

Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}

Optimal tunings:
* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}

{{Optimal ET sequence|legend=0| 12f, 19, 31e }}

Badness (Sintel): 1.23

==== Vincenzo ====
Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 65/64, 81/80

Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}

Optimal tunings:
* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}

{{Optimal ET sequence|legend=0| 7d, 12, 19 }}

Badness (Sintel): 1.02

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 52/51, 56/55, 65/64, 81/80

Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}

Optimal tunings:
* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}

{{Optimal ET sequence|legend=0| 12, 19 }}

Badness (Sintel): 1.30

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80

Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}

Optimal tunings:
* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}

{{Optimal ET sequence|legend=0| 12, 19 }}

Badness (Sintel): 1.36

=== Bimeantone ===
11/8 is mapped to half octave minus the [[128/125|meantone diesis]].

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 245/242

Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}

: mapping generators: ~63/44, ~3

Optimal tunings:
* WE: ~63/44 = 600.7492{{c}}, ~3/2 = 696.8853{{c}}
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 696.1908{{c}}

{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}

Badness (Sintel): 1.26

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 126/125, 245/242

Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}

Optimal tunings:
* WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}}

{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}

Badness (Sintel): 1.19

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 126/125, 189/187, 221/220

Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}

Optimal tunings:
* WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}}

{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}

Badness (Sintel): 1.15

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220

Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}

Optimal tunings:
* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}

{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}

Badness (Sintel): 1.08

=== Trimean ===
{{See also| No-sevens subgroup temperaments #Superpine }}

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 1344/1331

Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}

: mapping generators: ~2, ~11/10

Optimal tunings:
* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}

{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}

Badness (Sintel): 1.68

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 126/125, 144/143, 364/363

Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}

Optimal tunings:
* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}

{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}

Badness (Sintel): 1.46

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 126/125, 144/143, 189/187, 221/220

Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}

Optimal tunings:
* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}

{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}

Badness (Sintel): 1.28

=== Migration ===
See [[Rastmic clan #Migration|Rastmic clan]].

== Flattone ==
{{Main| Flattone }}

In flattone, 9 fourths get to the interval class for 7, so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a double-diminished fifth (C–G𝄫). In general, septimal subminor intervals are diminished and septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. The fifth in flattone is typically flatter than that of [[19edo]]. Good tunings for flattone include [[45edo]], [[64edo]], and [[71edo]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 525/512

{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.6308{{c}}, ~3/2 = 695.8782{{c}}
: [[error map]]: {{val| +3.631 -2.446 -2.801 -2.684 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.7334{{c}}
: error map: {{val| 0.000 -8.222 -11.380 -12.426 }}

[[Minimax tuning]]:
* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}
: [[projection map]]: [{{monzo| 1 0 0 0 }}, {{monzo| 21/13 0 1/13 -1/13 }}, {{monzo| 32/13 0 4/13 -4/13 }}, {{monzo| 32/13 0 -9/13 9/13 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
: [[projection map]]: [{{monzo| 1 0 0 0 }}, {{monzo| 17/11 2/11 0 -1/11 }}, {{monzo| 24/11 8/11 0 -4/11 }}, {{monzo| 34/11 -18/11 0 9/11 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]

[[Algebraic generator]]: Squarto, the positive root of 8''x''2 - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.

{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}

[[Badness]] (Sintel): 0.976

=== 11-limit ===
This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]].

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 385/384

Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}

Optimal tuning:
* WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}}

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]

{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}

Badness (Sintel): 1.12

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 81/80

Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}

Optimal tunings:
* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]

{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}

Badness (Sintel): 0.920

=== Ptolemy ===
See [[Rastmic clan #Ptolemy|Rastmic clan]].

== Dominant ==
{{Main| Dominant (temperament) }}
{{See also| Archytas clan }}

The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh (C–Bb). The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with the Pythagorean tuning of pure [[3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]].

Because dominant entails a near-pure perfect fifth, a small number of generators will not land on an interval close to prime 11. The canonical 11-limit extension identifies 11/8 with the diminished fifth. Domination tempers out 77/75 and identifies 11/8 with the augmented third. Domineering identifies 11/8 with the augmented fourth, which is a very inaccurate mapping; it is however, notable for having the lowest badness among the extensions. Arnold tempers out 33/32 and identifies 11/8 with the perfect fourth. None of them are nearly as good as the weak extension [[neutrominant]], splitting the fifth as well as the chromatic semitone in two like in all [[rastmic clan|rastmic]] temperaments.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 36/35, 64/63

{{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1195.3384{{c}}, ~3/2 = 698.8478{{c}}
: [[error map]]: {{val| -4.662 -7.769 +9.077 +14.832 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.1125{{c}}
: error map: {{val| 0.000 -0.842 +18.136 +28.949 }}

[[Tuning ranges]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 715.587]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]

{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}

[[Badness]] (Sintel): 0.524

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 64/63

Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]

Optimal tunings:
* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}}

{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}

Badness (Sintel): 0.799

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 56/55, 64/63, 66/65

Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}

Optimal tunings:
* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}}

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]

{{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }}

Badness (Sintel): 0.996

==== Dominion ====
Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 56/55, 64/63

Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}

Optimal tunings:
* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}

{{Optimal ET sequence|legend=0| 5, 12, 17c }}

Badness (Sintel): 1.13

=== Domination ===
Subgroup: 2.3.5.7.11

Comma list: 36/35, 64/63, 77/75

Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}

Optimal tunings:
* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}

{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}

Badness (Sintel): 1.21

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 64/63, 66/65

Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}

Optimal tunings:
* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}

{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}

Badness (Sintel): 1.13

=== Domineering ===
Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 64/63

Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}

Optimal tunings:
* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}}

{{Optimal ET sequence|legend=0| 5e, 7, 12 }}

Badness (Sintel): 0.727

=== Arnold ===
Subgroup: 2.3.5.7.11

Comma list: 22/21, 33/32, 36/35

Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}

Optimal tunings:
* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}}

{{Optimal ET sequence|legend=0| 5, 7, 12e }}

Badness (Sintel): 0.864

=== Neutrominant ===
See [[Rastmic clan #Neutrominant|Rastmic clan]].

== Flattertone ==
In flattertone, 17 fifths get to the interval class for 7, so that [[7/4]] is a double-augmented sixth (C–Ax). The fifth in flattertone is typically at least as flat as [[26edo]]. Here, 26edo and [[33edo|33cd-edo]] are the two primary flattertone tunings. [[1/2-comma meantone]] is also encompassed within flattertone's range. Any flatter than this, the meantone mapping for 5/4 is too inaccurate (it becomes more of a [[16/13]] or [[27/22]]), and [[deeptone]] temperament's mapping is more logical.

Flattertone was named by [[Flora Canou]] in 2024.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 1875/1792

{{Mapping|legend=1| 1 0 -4 -24 | 0 1 4 17 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1204.4511{{c}}, ~3/2 = 694.3258{{c}}
: [[error map]]: {{val| +4.451 -3.178 -9.011 +3.554 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 692.0479{{c}}
: error map: {{val| 0.000 -9.907 -18.122 -4.012 }}

{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}

[[Badness]] (Sintel): 2.43

==== 11-limit ====
Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 1375/1344

Mapping: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }}

Optimal tunings:
* WE: ~2 = 1203.4653{{c}}, ~3/2 = 693.8144{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 692.0422{{c}}

{{Optimal ET sequence|legend=0| 7d, 19d, 26 }}

Badness (Sintel): 1.53

; Music
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024)

== Sharptone ==
Sharptone is a low-accuracy temperament tempering out [[21/20]] and [[28/27]]. In sharptone, 7/4 is a major sixth, 7/6 a whole tone, and 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.

However, while 12edo ends up near-optimal, the only valid [[diamond monotone]] tuning for sharptone is [[5edo]]. Anything flat of it has ~12/7 and ~7/4 in the wrong order (and so should be dominant) and anything sharp of it has ~5/4 and ~4/3 in the wrong order (and so should not be meantone).

The 11-limit extension was named by Gene Ward Smith in 2004<ref name="meantone & meanpop 2004"/>.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 21/20, 28/27

{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1204.2961{{c}}, ~3/2 = 702.6463{{c}}
: [[error map]]: {{val| +4.296 +4.987 +24.271 -56.591 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4928{{c}}
: error map: {{val| 0.000 -0.462 +19.657 -64.347 }}

{{Optimal ET sequence|legend=1| 5, 7d, 12d }}

[[Badness]] (Sintel): 0.629

=== Meanertone ===
Subgroup: 2.3.5.7.11

Comma list: 21/20, 28/27, 33/32

Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }}

Optimal tunings:
* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}}

{{Optimal ET sequence|legend=0| 5, 7d, 12de }}

Badness (Sintel): 0.832

== Mildtone ==
Mildtone tempers out [[16128/15625]] and finds the interval class of 7 at 22 generators up, as a triple-augmented fifth (C–G#x). [[55edo]] and [[67edo]] are among the possible tunings.

Mildtone was named by [[User: Lucius Chiaraviglio|Lucius Chiaraviglio]] in 2024.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 16128/15625

{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7304{{c}}, ~3/2 = 698.3953{{c}}
: [[error map]]: {{val| -0.270 -3.829 +7.267 -1.434 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.5397{{c}}
: error map: {{val| 0.000 -3.415 +7.845 -0.952 }}

{{Optimal ET sequence|legend=1| 12, 43d, 55, 67 }}

[[Badness]] (Sintel): 2.67

=== 11-limit ===

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 81/80, 176/175, 7058/6875

{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.816{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.455{{c}}

{{Optimal ET sequence|legend=1| 12, 43de, 55, 67 }}

[[Badness]] (Sintel): 2.15

=== 13-limit ===

[[Subgroup]]: 2.3.5.7.11.13

[[Comma list]]: 81/80, 176/175, 196/195, 832/825

{{Mapping|legend=1| 1 0 -4 -32 -44 | 0 1 4 22 30}}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.788{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.471{{c}}

{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}

[[Badness]] (Sintel): 2.04

=== 17-limit ===

[[Subgroup]]: 2.3.5.7.11.13.17

[[Comma list]]: 81/80, 176/175, 189/197, 196/195, 832/825

{{Mapping|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}}

{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}

[[Badness]] (Sintel): 1.98

=== 19-limit ===

[[Subgroup]]: 2.3.5.7.11.13.19

[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825

{{Mapping|legend=1| 1 0 -4 -32 -44 12 9| 0 1 4 22 30 -5 -3}}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.371{{c}}, ~3/2 = 698.164{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.519{{c}}

{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}

[[Badness]] (Sintel): 1.95

{{Todo|unify precision|review}}

== Supermean ==
Supermean tempers out 672/625 and finds the interval class of 7 at 15 generators up, as a double-augmented fifth (C–Gx). As such, it extends [[leapfrog]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 672/625

{{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1195.4372{{c}}, ~3/2 = 702.2086{{c}}
: [[error map]]: {{val| -4.563 -4.309 +22.521 -8.319 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5375{{c}}
: error map: {{val| 0.000 +2.583 +31.836 -0.763 }}

{{Optimal ET sequence|legend=1| 5d, 12d, 17c }}

[[Badness]] (Sintel): 3.40

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 132/125

Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }}

Optimal tunings:
* WE: ~2 = 1195.7270{{c}}, ~3/2 = 702.5848{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7471{{c}}

{{Optimal ET sequence|legend=0| 5de, 12de, 17c }}

Badness (Sintel): 2.09

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 56/55, 66/65, 81/80

Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}

Optimal tunings:
* WE: ~2 = 1196.3958{{c}}, ~3/2 = 702.9766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7940{{c}}

{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }}

Badness (Sintel): 1.67

== Mohajira ==
{{Main| Mohajira }}

Mohajira can be viewed as derived from [[mohaha]] which maps the interval half a [[chromatic semitone|chroma]] flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the [[porwell comma]]. It can be described as {{nowrap| 24 & 31 }}; its ploidacot is dicot. [[31edo]] makes for an excellent mohajira tuning, with generator 9\31. Note that while 24 + 31 = [[55edo]] doesn't apear in the optimal ET sequence, it is a [[patent val]] tuning and recommendable if you prefer a light meantone tempering.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 6144/6125

{{Mapping|legend=1| 1 1 0 6 | 0 2 8 -11 }}

: mapping generators: ~2, ~128/105

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8160{{c}}, ~128/105 = 348.6518{{c}}
: [[error map]]: {{val| +0.816 -3.835 +2.901 +0.900 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 348.4194{{c}}
: error map: {{val| 0.000 -5.116 +1.041 -1.439 }}

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{monzo| 0 0 1/8 }}
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 6 0 -11/8 0 }}
: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.5

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978]
* 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]

[[Algebraic generator]]: Mohabis, real root of 3''x''3 - 3''x''2 - 1, 348.6067 cents. Corresponding recurrence converges quickly.

{{Optimal ET sequence|legend=1| 7, 24, 31 }}

[[Badness]] (Sintel): 1.41

Scales: [[mohaha7]], [[mohaha10]]

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120, 176/175

Mapping: {{mapping| 1 1 0 6 2 | 0 2 8 -11 5 }}

Optimal tunings:
* WE: ~2 = 1201.1562{{c}}, ~11/9 = 348.8124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.4910{{c}}

Minimax tuning:
* 11-odd-limit: ~11/9 = {{monzo| 0 0 1/8 }}
: projection map: [{{Monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 6 0 -11/8 0 0 }}, {{monzo| 2 0 5/8 0 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:
* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]

{{Optimal ET sequence|legend=0| 7, 24, 31 }}

Badness (Sintel): 0.862

Scales: [[mohaha7]], [[mohaha10]]

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 105/104, 121/120

Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 8 -11 5 -1 }}

Optimal tunings:
* WE: ~2 = 1200.4256{{c}}, ~11/9 = 348.6819{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5622{{c}}

{{Optimal ET sequence|legend=0| 7, 24, 31 }}

Badness (Sintel): 0.966

Scales: [[mohaha7]], [[mohaha10]]

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 66/65, 81/80, 105/104, 121/120, 154/153

Mapping: {{mapping| 1 1 0 6 2 4 7 | 0 2 8 -11 5 -1 -10 }}

Optimal tunings:
* WE: ~2 = 1200.0382{{c}}, ~11/9 = 348.7471{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.7360{{c}}

{{Optimal ET sequence|legend=0| 7, 24, 31 }}

Badness (Sintel): 1.05

Scales: [[mohaha7]], [[mohaha10]]

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152

Mapping: {{mapping| 1 1 0 6 2 4 7 6 | 0 2 8 -11 5 -1 -10 -6 }}

Optimal tunings:
* WE: ~2 = 1199.7469{{c}}, ~11/9 = 348.7367{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.8117{{c}}

{{Optimal ET sequence|legend=0| 7, 24, 31, 55 }}

Badness (Sintel): 1.05

Scales: [[mohaha7]], [[mohaha10]]

== Mohamaq ==
Mohamaq is a lower-accuracy alternative to mohajira that favors tunings sharp of 24edo. It may be described as {{nowrap| 17c & 24 }}; its ploidacot is dicot, the same as mohajira.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 392/375

{{Mapping|legend=1| 1 1 0 -1 | 0 2 8 13 }}

: mapping generators: ~2, ~25/21

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.0661{{c}}, ~25/21 = 350.3127{{c}}
: [[error map]]: {{val| -0.934 -2.264 +16.188 -13.827 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 350.4856{{c}}
: error map: {{val| 0.000 -0.984 +17.571 -12.513 }}

{{Optimal ET sequence|legend=1| 7d, 17c, 24 }}

[[Badness]] (Sintel): 1.97

Scales: [[mohaha7]], [[mohaha10]]

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 243/242

Mapping: {{mapping| 1 1 0 -1 2 | 0 2 8 13 5 }}

Optimal tunings:
* WE: ~2 = 1199.1924{{c}}, ~11/9 = 350.3286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4821{{c}}

{{Optimal ET sequence|legend=0| 7d, 17c, 24 }}

Badness (Sintel): 1.20

Scales: [[mohaha7]], [[mohaha10]]

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 77/75, 243/242

Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 8 13 5 -1 }}

Optimal tunings:
* WE: ~2 = 1198.5986{{c}}, ~11/9 = 350.3353{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.6459{{c}}

{{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }}

Badness (Sintel): 1.19

Scales: [[mohaha7]], [[mohaha10]]

== Liese ==
[[:de:Liese|Deutsch]]

Liese splits the [[3/1|perfect twelfth]] into three generators of ~[[10/7]], using the comma [[1029/1000]]. It also tempers out [[686/675]], the senga. It may be described as {{nowrap| 17c & 19 }}; its ploidacot is alpha-tricot. It is a very natural 13-limit tuning, given the generator is so near 13/9. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with mos scales: 7, 9, 11, 13, 15, 17, 19, 36, 55.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 686/675

{{Mapping|legend=1| 1 0 -4 -3 | 0 3 12 11 }}

: mapping generators: ~2, ~10/7

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.5548{{c}}, ~10/7 = 633.2251{{c}}
: [[error map]]: {{val| +1.555 -2.280 +6.168 -8.015 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 632.5640{{c}}
: error map: {{val| 0.000 -4.263 +4.454 -10.622 }}

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/7 = {{monzo| 1/3 0 1/12 }}
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 2/3 0 11/12 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

[[Algebraic generator]]: Radix, the real root of ''x''5 - 2''x''4 + 2''x''3 - 2''x''2 + 2''x'' - 2, also a root of ''x''6 - ''x''5 - 2. The recurrence converges.

{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}

[[Badness]] (Sintel): 1.18

=== Liesel ===
Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 540/539

Mapping: {{mapping| 1 0 -4 -3 4 | 0 3 12 11 -1 }}

Optimal tunings:
* WE: ~2 = 1198.8507{{c}}, ~10/7 = 632.4668{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 632.9963{{c}}

{{Optimal ET sequence|legend=0| 17c, 19, 36 }}

Badness (Sintel): 1.35

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 81/80, 91/90

Mapping: {{mapping| 1 0 -4 -3 4 0 | 0 3 12 11 -1 7 }}

Optimal tunings:
* WE: ~2 = 1199.4968{{c}}, ~10/7 = 632.7766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.0082{{c}}

{{Optimal ET sequence|legend=0| 17c, 19, 36 }}

Badness (Sintel): 1.13

=== Elisa ===
Subgroup: 2.3.5.7.11

Comma list: 77/75, 81/80, 99/98

Mapping: {{mapping| 1 0 -4 -3 -5 | 0 3 12 11 16 }}

Optimal tunings:
* WE: ~2 = 1201.0489{{c}}, ~10/7 = 633.6147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1644{{c}}

{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}

Badness (Sintel): 1.37

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 77/75, 81/80, 99/98

Mapping: {{mapping| 1 0 -4 -3 -5 0 | 0 3 12 11 16 7 }}

Optimal tunings:
* WE: ~2 = 1201.4815{{c}}, ~10/7 = 633.7720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1281{{c}}

{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}

Badness (Sintel): 1.11

=== Lisa ===
Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 343/330

Mapping: {{mapping| 1 0 -4 -3 -6 | 0 3 12 11 18 }}

Optimal tunings:
* WE: ~2 = 1202.6773{{c}}, ~10/7 = 632.7783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.6175{{c}}

{{Optimal ET sequence|legend=0| 17cee, 19 }}

Badness (Sintel): 1.81

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 81/80, 91/88, 147/143

Mapping: {{mapping| 1 0 -4 -3 -6 0 | 0 3 12 11 18 7 }}

Optimal tunings:
* WE: ~2 = 1203.6086{{c}}, ~10/7 = 633.1193{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.5346{{c}}

{{Optimal ET sequence|legend=0| 17cee, 19 }}

Badness (Sintel): 1.49

== Superpine ==
{{See also| No-sevens subgroup temperaments #Superpine }}

The superpine temperament is generated by 1/3 of a fourth, represented by [[~]][[35/32]], which resembles [[porcupine]], but it favors flat fifths instead of sharp ones. It may be described as {{nowrap| 36 & 43 }}; its ploidacot is omega-tricot. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent [[6/5]] – harmonics other than 3 all require the 15-tone mos ([[7L 8s]]) to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as [[11/10]] as in porcupine, which makes [[11/8]] high-[[complexity]] like the other harmonics, but in the 13-limit 5 generators up closely approximates [[13/8]]. [[43edo]] is a good tuning especially for the higher-limit extensions.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 1119744/1071875

{{Mapping|legend=1| 1 2 4 1 | 0 -3 -12 13 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3652{{c}}, ~35/32 = 167.1615{{c}}
: [[error map]]: {{val| -0.635 -4.709 +5.209 +3.639 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/32 = 167.2561{{c}}
: error map: {{val| 0.000 -3.723 +6.613 +5.503 }}

{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }}

[[Badness]] (Sintel): 3.46

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175, 864/847

Mapping: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }}

Optimal tunings:
* WE: ~2 = 1199.0522{{c}}, ~11/10 = 167.1904{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3382{{c}}

{{Optimal ET sequence|legend=0| 7, 36, 43 }}

Badness (Sintel): 1.90

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 144/143, 176/175

Mapping: {{mapping| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }}

Optimal tunings:
* WE: ~2 = 1199.4286{{c}}, ~11/10 = 167.3105{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3958{{c}}

{{Optimal ET sequence|legend=0| 7, 36, 43 }}

Badness (Sintel): 1.52

== Lithium ==
Lithium is named after the 3rd element for having a 3rd-octave period (and also for lithium's molar mass of 6.9 g/mol since 69edo supports it). Its ploidacot is triploid monocot. It supports a [[3L 6s]] scale and thus intuitively can be thought of as "tcherepnin meantone" in that context.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 3125/3087

{{Mapping|legend=1| 3 0 -12 -20 | 0 1 4 6 }}

: mapping generators: ~56/45, ~3

[[Optimal tuning]]s:
* [[WE]]: ~56/45 = 400.6744{{c}}, ~3/2 = 695.8474{{c}} {~15/14 = 105.5015{{c}})
: [[error map]]: {{val| +2.023 -4.084 -2.924 +4.910 }}
* [[CWE]]: ~56/45 = 400.0000{{c}}, ~3/2 = 695.1413{{c}} {~15/14 = 104.8587{{c}})
: error map: {{val| 0.000 -6.814 -5.748 +2.022 }}

{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}

[[Badness]] (Sintel): 1.75

== Squares ==
{{Main| Squares }}

Squares splits the [[6/1|6th harmonic]] into four subminor sixths of [[11/7]]~[[14/9]] (or splits a [[8/3|perfect eleventh]] into four supermajor thirds of [[9/7]]~[[14/11]]), and uses it for a generator. It may be described as {{nowrap| 14c & 17c }}; its ploidacot is beta-tetracot. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8-, 11-, and 14-note mos scales available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 2401/2400

{{Mapping|legend=1| 1 -1 -8 -3 | 0 4 16 9 }}

: mapping generators: ~2, ~14/9

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2488{{c}}, ~14/9 = 774.8640{{c}}
: [[error map]]: {{val| +1.249 -3.748 +1.520 +1.204 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 774.1560{{c}}
: error map: {{val| 0.000 -5.331 +0.183 -1.422 }}

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }}
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3/2 0 9/16 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

[[Algebraic generator]]: Sceptre2, the positive root of 9''x''2 + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.

{{Optimal ET sequence|legend=1| 14c, 17c, 31, 169b, 200b }}

[[Badness]] (Sintel): 1.16

Scales: [[skwares8]], [[skwares11]], [[skwares14]]

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 121/120

Mapping: {{mapping| 1 -1 -8 -3 -3 | 0 4 16 9 10 }}

Optimal tunings:
* WE: ~2 = 1201.6657{{c}}, ~11/7 = 775.1171{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.1754{{c}}

{{Optimal ET sequence|legend=0| 14c, 17c, 31, 130bee, 169beee }}

Badness (Sintel): 0.715

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 99/98, 121/120

Mapping: {{mapping| 1 -1 -8 -3 -3 5 | 0 4 16 9 10 -2 }}

Optimal tunings:
* WE: ~2 = 1199.8419{{c}}, ~11/7 = 774.3484{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4422{{c}}

{{Optimal ET sequence|legend=0| 14c, 17c, 31, 79cf }}

Badness (Sintel): 1.05

==== Squad ====
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 91/90, 99/98

Mapping: {{mapping| 1 -1 -8 -3 -3 -6 | 0 4 16 9 10 15 }}

Optimal tunings:
* WE: ~2 = 1202.0312{{c}}, ~11/7 = 775.5589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4140{{c}}

{{Optimal ET sequence|legend=0| 14cf, 17c, 31f }}

Badness (Sintel): 1.11

==== Agora ====
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 105/104, 121/120

Mapping: {{mapping| 1 -1 -8 -3 -3 -15 | 0 4 16 9 10 29 }}

Optimal tunings:
* WE: ~2 = 1202.3228{{c}}, ~11/7 = 775.2214{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8617{{c}}

{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 76e }}

Badness (Sintel): 1.01

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 105/104, 120/119, 121/119

Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 | 0 4 16 9 10 29 11 }}

Optimal tunings:
* WE: ~2 = 1201.4340{{c}}, ~11/7 = 774.7375{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8955{{c}}

{{Optimal ET sequence|legend=0| 14cf, 31 }}

Badness (Sintel): 1.15

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119

Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 -8 | 0 4 16 9 10 29 11 19 }}

Optimal tunings:
* WE: ~2 = 1201.2461{{c}}, ~11/7 = 774.5783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8479{{c}}

{{Optimal ET sequence|legend=0| 14cf, 31 }}

Badness (Sintel): 1.15

=== Cuboctahedra ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384, 1375/1372

Mapping: {{mapping| 1 -1 -8 -3 17 | 0 4 16 9 -21 }}

Optimal tunings:
* WE: ~2 = 1201.4436{{c}}, ~14/9 = 774.9386{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.0243{{c}}

{{Optimal ET sequence|legend=0| 31, 107b, 138b, 169be, 200be }}

Badness (Sintel): 1.88

== Jerome ==
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 51/20, or 139.316 cents. It may be described as {{nowrap| 17c & 26 }}; its ploidacot is pentacot. While the generator represents both 13/12 and 12/11, the CTE/CWE and Hieronymus generators are close to 13/12 in size.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 17280/16807

{{Mapping|legend=1| 1 1 0 2 | 0 5 20 7 }}

: mapping generators: ~2, ~54/49

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1640{{c}}, ~54/49 = 139.3624{{c}}
: [[error map]]: {{val| +0.164 -4.979 +0.934 +7.039 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~54/49 = 139.3528{{c}}
: error map: {{val| 0.000 -5.191 +0.741 +6.643 }}

{{Optimal ET sequence|legend=1| 17c, 26, 43 }}

[[Badness]] (Sintel): 2.75

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 864/847

Mapping: {{mapping| 1 1 0 2 3 | 0 5 20 7 4 }}

Optimal tunings:
* WE: ~2 = 1201.4436{{c}}, ~12/11 = 139.3714{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.4038{{c}}

{{Optimal ET sequence|legend=0| 17c, 26, 43 }}

Badness (Sintel): 1.58

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 99/98, 144/143

Mapping: {{mapping| 1 1 0 2 3 3 | 0 5 20 7 4 6 }}

Optimal tunings:
* WE: ~2 = 1199.8860{{c}}, ~13/12 = 139.3737{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3817{{c}}

{{Optimal ET sequence|legend=0| 17c, 26, 43 }}

Badness (Sintel): 1.21

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 81/80, 99/98, 144/143, 189/187

Mapping: {{mapping| 1 1 0 2 3 3 2 | 0 5 20 7 4 6 18 }}

Optimal tunings:
* WE: ~2 = 1199.8346{{c}}, ~13/12 = 139.3431{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3544{{c}}

{{Optimal ET sequence|legend=0| 17cg, 26, 43 }}

Badness (Sintel): 1.06

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143

Mapping: {{mapping| 1 1 0 2 3 3 2 1 | 0 5 20 7 4 6 18 28 }}

Optimal tunings:
* WE: ~2 = 1199.8891{{c}}, ~13/12 = 139.3001{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3080{{c}}

{{Optimal ET sequence|legend=0| 17cgh, 26, 43, 69 }}

Badness (Sintel): 1.11

== Meantritone ==
The meantritone temperament tempers out the [[mirkwai comma]] (16875/16807) and [[trimyna comma]] (50421/50000) in the 7-limit. In this temperament, the 6th harmonic is split into five generators of ~10/7; the ploidacot of this temperament is beta-pentacot. The name ''meantritone'' is a portmanteau of ''meantone'' and ''tritone'', the latter is a generator of this temperament.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 16875/16807

{{Mapping|legend=1| 1 -1 -8 -7 | 0 5 20 19 }}

: mapping generators: ~2, ~10/7

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.3832{{c}}, ~10/7 = 619.9478{{c}}
: [[error map]]: {{val| +1.383 -3.599 +1.576 +0.499 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.3176{{c}}
: error map: {{val| 0.000 -5.367 +0.038 -1.791 }}

{{Optimal ET sequence|legend=1| 29cd, 31, 188bcd, 219bbcd }}

[[Badness]] (Sintel): 2.08

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 2541/2500

Mapping: {{mapping| 1 -1 -8 -7 -11 | 0 5 20 19 28 }}

Optimal tunings:
* WE: ~2 = 1201.2054{{c}}, ~10/7 = 619.9752{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.4223{{c}}

{{Optimal ET sequence|legend=0| 29cde, 31 }}

Badness (Sintel): 1.42

== Injera ==
Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a ~15/14 semitone difference between a half-octave and a perfect fifth. Injera may be described as {{nowrap| 12 & 26 }}; its ploidacot is diploid monocot. It tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo]], which is two parallel [[19edo]]s, is an excellent tuning for injera.

[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 50/49, 81/80

{{Mapping|legend=1| 2 0 -8 -7 | 0 1 4 4 }}

: mapping generators: ~7/5, ~3

[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 600.6662{{c}}, ~3/2 = 695.1463{{c}} (~21/20 = 94.4801{{c}})
: [[error map]]: {{val| +1.332 -5.476 -5.729 +12.425 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 694.7712{{c}} (~21/20 = 94.7712{{c}})
: error map: {{val| 0.000 -7.184 -7.229 +10.259 }}

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]

{{Optimal ET sequence|legend=1| 12, 26, 38 }}

[[Badness]] (Sintel): 0.788

; Music
* [https://web.archive.org/web/20201127013520/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 ''Two Pairs of Socks''] by [[Igliashon Jones]] – in [[26edo]] tuning

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 81/80

Mapping: {{mapping| 2 0 -8 -7 -12 | 0 1 4 4 6 }}

Optimal tunings:
* WE: ~7/5 = 600.9350{{c}}, ~3/2 = 693.9198{{c}} (~21/20 = 92.9848{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.3539{{c}} (~21/20 = 93.3539{{c}})

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]

{{Optimal ET sequence|legend=0| 12, 26 }}

Badness (Sintel): 0.764

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 50/49, 78/77, 81/80

Mapping: {{mapping| 2 0 -8 -7 -12 -21 | 0 1 4 4 6 9 }}

Optimal tunings:
* WE: ~7/5 = 600.9982{{c}}, ~3/2 = 693.8249{{c}} (~21/20 = 92.8267{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.0992{{c}} (~21/20 = 93.0992{{c}})

Tuning ranges:
* 13-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]

{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}

Badness (Sintel): 0.891

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 50/49, 78/77, 81/80, 85/84

Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 | 0 1 4 4 6 9 1 }}

Optimal tunings:
* WE: ~7/5 = 601.1757{{c}}, ~3/2 = 693.8441{{c}} (~21/20 = 92.6684{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 692.8879{{c}} (~21/20 = 92.8879{{c}})

{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}

Badness (Sintel): 0.935

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84

Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 -1 | 0 1 4 4 6 9 1 3 }}

Optimal tunings:
* WE: ~7/5 = 601.4245{{c}}, ~3/2 = 693.9426{{c}} (~21/20 = 92.5181{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 692.7606{{c}} (~21/20 = 92.7606{{c}})

{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}

Badness (Sintel): 0.920

==== Enjera ====
Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 40/39, 45/44, 50/49

Mapping: {{mapping| 2 0 -8 -7 -12 -2 | 0 1 4 4 6 3 }}

Optimal tunings:
* WE: ~7/5 = 599.1863{{c}}, ~3/2 = 693.1791{{c}} (~21/20 = 93.9929{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.6809{{c}} (~21/20 = 93.6809{{c}})

{{Optimal ET sequence|legend=0| 10cdeef, 12f }}

Badness (Sintel): 1.10

=== Injerous ===
Subgroup: 2.3.5.7.11

Comma list: 33/32, 50/49, 55/54

Mapping: {{mapping| 2 0 -8 -7 10 | 0 1 4 4 -1 }}

Optimal tunings:
* WE: ~7/5 = 603.1682{{c}}, ~3/2 = 694.1945{{c}} (~21/20 = 91.0264{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 691.6107{{c}} (~21/20 = 91.6107{{c}})

{{Optimal ET sequence|legend=0| 12e, 14c, 26e, 40cee }}

Badness (Sintel): 1.28

=== Lahoh ===
Subgroup: 2.3.5.7.11

Comma list: 50/49, 56/55, 81/77

Mapping: {{mapping| 2 0 -8 -7 7 | 0 1 4 4 0 }}

Optimal tunings:
* WE: ~7/5 = 597.3179{{c}}, ~3/2 = 695.8759{{c}} (~21/20 = 98.5581{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 697.8757{{c}} (~21/20 = 97.8757{{c}})

{{Optimal ET sequence|legend=0| 10cd, 12 }}

Badness (Sintel): 1.42

=== Teff ===
{{Main| Teff }}

Teff, found and named by [[Mason Green]], is to injera what mohajira is to meantone; it splits the generator in halves in order to accommodate higher-limit intervals, creating a half-octave quartertone temperament. Its ploidacot is diploid alpha-dicot.

Subgroup: 2.3.5.7.11

Comma list: 50/49, 81/80, 864/847

Mapping: {{mapping| 2 1 -4 -3 8 | 0 2 8 8 -1 }}

: mapping generators: ~7/5, ~16/11

Optimal tunings:
* WE: ~7/5 = 600.2802{{c}}, ~16/11 = 647.7720{{c}} (~33/32 = 47.4918{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.5224{{c}} (~33/32 = 47.5224{{c}})

{{Optimal ET sequence|legend=0| 24d, 26, 50d }}

Badness (Sintel): 2.34

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 78/77, 81/80, 144/143

Mapping: {{mapping| 2 1 -4 -3 8 2 | 0 2 8 8 -1 5 }}

Optimal tunings:
* WE: ~7/5 = 600.3037{{c}}, ~16/11 = 647.7954{{c}} (~33/32 = 47.4917{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.5256{{c}} (~33/32 = 47.5256{{c}})

{{Optimal ET sequence|legend=0| 24d, 26, 50d }}

Badness (Sintel): 1.65

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 78/77, 81/80, 85/84, 144/143

Mapping: {{mapping| 2 1 -4 -3 8 2 6 | 0 2 8 8 -1 5 2 }}

Optimal tunings:
* WE: ~7/5 = 600.5123{{c}}, ~16/11 = 647.8970{{c}} (~34/33 = 47.3846{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.4314{{c}} (~34/33 = 47.4314{{c}})

{{Optimal ET sequence|legend=0| 24d, 26 }}

Badness (Sintel): 1.50

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143

Mapping: {{mapping| 2 1 -4 -3 8 2 6 2 | 0 2 8 8 -1 5 2 6 }}

Optimal tunings:
* WE: ~7/5 = 600.6308{{c}}, ~16/11 = 648.0424{{c}} (~34/33 = 47.4116{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.4715{{c}} (~34/33 = 47.4715{{c}})

{{Optimal ET sequence|legend=0| 24d, 26 }}

Badness (Sintel): 1.41

== Pombe ==
Pombe (named after the African millet beer) is a variant of [[#Teff]] by [[User:Kaiveran|Kaiveran Lugheidh]] that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Its ploidacot is diploid alpha-dicot, the same as teff. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 300125/294912

{{Mapping|legend=1| 2 1 -4 11 | 0 2 8 -5 }}

: mapping generators: ~735/512, ~35/24

[[Optimal tuning]]s:
* [[WE]]: ~735/512 = 601.0652{{c}}, ~35/24 = 648.9295{{c}} (~36/35 = 47.8642{{c}})
: [[error map]]: {{val| +2.130 -3.031 +0.861 -1.756 }}
* [[CWE]]: ~735/512 = 600.0000{{c}}, ~35/24 = 647.8628{{c}} (~36/35 = 47.8628{{c}})
: error map: {{val| 0.000 -6.229 -3.411 -8.140 }}

{{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}

[[Badness]] (Sintel): 2.94

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 245/242, 385/384

Mapping: {{mapping| 2 1 -4 11 8 | 0 2 8 -5 -1 }}

Optimal tunings:
* WE: ~99/70 = 600.7890{{c}}, ~16/11 = 648.7592{{c}} (~36/35 = 47.9701{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~16/11 = 647.9516{{c}} (~36/35 = 47.9516{{c}})

{{Optimal ET sequence|legend=0| 24, 26, 50 }}

Badness (Sintel): 1.72

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 144/143, 245/242

Mapping: {{mapping| 2 1 -4 11 8 2 | 0 2 8 -5 -1 5 }}

Optimal tunings:
* WE: ~99/70 = 600.6971{{c}}, ~16/11 = 648.6029{{c}} (~36/35 = 47.9058{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~16/11 = 647.8990{{c}} (~36/35 = 47.8990{{c}})

{{Optimal ET sequence|legend=0| 24, 26, 50 }}

Badness (Sintel): 1.28

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 144/143, 245/242, 273/272

Mapping: {{mapping| 2 1 -4 11 8 2 6 | 0 2 8 -5 -1 5 2 }}

Optimal tunings:
* WE: ~17/12 = 600.7610{{c}}, ~16/11 = 648.6638{{c}} (~36/35 = 47.9028{{c}})
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 647.8990{{c}} (~36/35 = 47.8990{{c}})

{{Optimal ET sequence|legend=0| 24, 26, 50 }}

Badness (Sintel): 1.08

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209

Mapping: {{mapping| 2 1 -4 11 8 2 6 2 | 0 2 8 -5 -1 5 2 6 }}

Optimal tunings:
* WE: ~17/12 = 600.8048{{c}}, ~16/11 = 648.7494{{c}} (~36/35 = 47.9446{{c}})
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 647.9425{{c}} (~36/35 = 47.9425{{c}})

{{Optimal ET sequence|legend=0| 24, 26, 50 }}

Badness (Sintel): 1.01

== Orphic ==
Orphic has a semi-octave period and four generators plus a period gives the 3rd harmonic; its ploidacot is diploid alpha-tetracot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 5898240/5764801

{{Mapping|legend=1| 2 1 -4 4 | 0 4 16 3 }}

: mapping generators: ~2401/1728, ~343/288

[[Optimal tuning]]s:
* [[WE]]: ~2401/1728 = 600.1767{{c}}, ~343/288 = 324.3015{{c}} (~7/6 = 275.8751{{c}})
: [[error map]]: {{val| +0.353 -4.572 +1.804 +4.785 }}
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~343/288 = 324.2285{{c}} (~7/6 = 275.7715{{c}})
: error map: {{val| 0.000 -5.041 +1.342 +3.860 }}

{{Optimal ET sequence|legend=1| 26, 48c, 74 }}

[[Badness]] (Sintel): 6.55

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 73728/73205

Mapping: {{mapping| 2 1 -4 4 8 | 0 4 16 3 -2 }}

Optimal tunings:
* WE: ~363/256 = 600.1011{{c}}, ~77/64 = 324.2923{{c}} (~7/6 = 275.8088{{c}})
* CWE: ~363/256 = 600.0000{{c}}, ~77/64 = 324.2463{{c}} (~7/6 = 275.7537{{c}})

{{Optimal ET sequence|legend=0| 26, 48c, 74 }}

Badness (Sintel): 3.36

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 144/143, 2200/2197

Mapping: {{mapping| 2 1 -4 4 8 2 | 0 4 16 3 -2 10 }}

Optimal tunings:
* WE: ~55/39 = 600.0540{{c}}, ~77/64 = 324.2551{{c}} (~7/6 = 275.7989{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~77/64 = 324.2307{{c}} (~7/6 = 275.7693{{c}})

{{Optimal ET sequence|legend=0| 26, 48c, 74 }}

Badness (Sintel): 2.21

== Cloudtone ==
The cloudtone temperament tempers out the [[cloudy comma]], 16807/16384 and the [[syntonic comma]], 81/80 in the 7-limit. It may be described as {{nowrap| 5 & 50 }}; its ploidacot is pentaploid monocot. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 16807/16384

{{Mapping|legend=1| 5 0 -20 14 | 0 1 4 0 }}

: mapping generators: ~8/7, ~3

[[Optimal tuning]]s:
* [[WE]]: ~8/7 = 240.4267{{c}}, ~3/2 = 696.9566{{c}} (~49/48 = 24.3235{{c}})
: [[error map]]: {{val| +2.133 -2.865 +1.513 -2.852 }}
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~3/2 = 696.1637{{c}} (~49/48 = 23.8373{{c}})
: error map: {{val| 0.000 -5.791 -1.659 -8.826 }}

{{Optimal ET sequence|legend=1| 5, 40c, 45, 50 }}

[[Badness]] (Sintel): 2.59

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384, 2401/2376

Mapping: {{mapping| 5 0 -20 14 41 | 0 1 4 0 -3 }}

Optimal tunings:
* WE: ~8/7 = 240.2740{{c}}, ~3/2 = 697.3317{{c}} (~56/55 = 23.4904{{c}})
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 696.6269{{c}} (~56/55 = 23.3731{{c}})

{{Optimal ET sequence|legend=0| 5, 45, 50 }}

Badness (Sintel): 2.33

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 144/143, 2401/2376

Mapping: {{mapping| 5 0 -20 14 41 -21 | 0 1 4 0 -3 5 }}

Optimal tunings:
* WE: ~8/7 = 240.2435{{c}}, ~3/2 = 696.8686{{c}} (~91/90 = 23.8618{{c}})
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 696.2653{{c}} (~91/90 = 23.7347{{c}})

{{Optimal ET sequence|legend=0| 5, 45f, 50 }}

Badness (Sintel): 2.02

== Subgroup extensions ==
=== Stützel (2.3.5.19) ===
[[Subgroup]]: 2.3.5.19

[[Comma list]]: 81/80, 96/95

{{Mapping|legend=2| 1 0 -4 9 | 0 1 4 -3 }}

{{Mapping|legend=3| 1 0 -4 0 0 0 0 9 | 0 1 4 0 0 0 0 -3 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5513{{c}}, ~3/2 = 697.6058{{c}}
: [[error map]]: {{val| -0.448 -4.798 +4.110 +6.977 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.8222{{c}}
: error map: {{val| 0.000 -4.133 +4.975 +9.020 }}

{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43, 98h }}

[[Badness]] (Sintel): 0.324

=== Hypnotone ===
Hypnotone is no-sevens [[#Flattone|flattone]].

[[Subgroup]]: 2.3.5.11

[[Comma list]]: 45/44, 81/80

{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}

{{Mapping|legend=3| 1 0 -4 0 -6 | 0 1 4 0 6 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.0621{{c}}, ~3/2 = 694.5448{{c}}
: [[error map]]: {{val| +2.062 -5.348 -8.135 +15.951 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.9085{{c}}
: error map: {{val| 0.000 -8.047 -10.680 +12.133 }}

{{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }}

[[Badness]] (Sintel): 0.326

==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13

Comma list: 45/44, 65/64, 81/80

Subgroup-val mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}

Gencom mapping: {{mapping| 1 0 -4 0 -6 10 | 0 1 4 0 6 -4 }}

Optimal tunings:
* WE: ~2 = 1202.6916{{c}}, ~3/2 = 694.4181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0870{{c}}

{{Optimal ET sequence|legend=0| 7, 12, 19, 26, 45f }}

Badness (Sintel): 0.561

=== Dequarter ===
[[Subgroup]]: 2.3.5.11

[[Comma list]]: 33/32, 55/54

{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}

{{Mapping|legend=3| 1 0 -4 0 5 | 0 1 4 0 -1 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1206.5832{{c}}, ~3/2 = 695.8763{{c}}
: [[error map]]: {{val| +6.583 +0.504 -2.809 -20.862 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 693.1206{{c}}
: error map: {{val| 0.000 -8.834 -13.831 -44.439 }}

{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }}

[[Badness]] (Sintel): 0.451

==== Dreamtone ====
Subgroup: 2.3.5.11.13

Comma list: 33/32, 55/54, 975/968

Subgroup-val mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}

Gencom mapping: {{mapping| 1 0 -4 0 5 21 | 0 1 4 0 -1 -11 }}

Optimal tunings:
* WE: ~2 = 1207.8248{{c}}, ~3/2 = 694.7806{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 690.1826{{c}}

{{Optimal ET sequence|legend=0| 7, 19eff, 26eff, 33ceeff, 40ceeff }}

Badness (Sintel): 1.40

== References ==
<references/>

[[Category:Temperament families]]
[[Category:Meantone family| ]] 
[[Category:Meantone| ]] 
[[Category:Rank 2]]
[[Category:Listen]]

Plücker coordinates

2026-05-06T17:27:55Z

Sintel: /* Projective distance */ Typo R^n

{{Expert|Wedgie}}
[[File:Plucker_embedding.png|thumb|600px|right|Schematic illustration of the Plücker embedding. Linear subspaces of <math>\mathbb{R}^n</math> (here lines) get mapped to points on a quadric surface in projective space.]]
{{Wikipedia|Plücker embedding}}

In [[exterior algebra]] applied to [[regular temperament theory]], '''Plücker coordinates''' (also known as the '''wedgie''') are a way to assign coordinates to abstract temperaments, by viewing them as elements of some projective space.

The usual way to write down an abstract temperament is via its mapping matrix, but Plücker coordinates give us a unique description that is useful for some calculations.
The definition here is given in terms of temperament matrices, but by duality, we can also embed interval spaces in the same way.
More specifically, the interval subspace spanned by the commas of some temperament can also be used to give unique coordinates to that temperament.
These two representations are related via the [[Hodge dual]].

== Definition ==
A temperament can be viewed as a point in what is called a Grassmannian variety, written as <math>\mathrm{Gr} (k, n)</math>.
This variety contains all possible k-dimensional subspaces of <math>\mathbb{R}^n</math>.
In musical terms, k represents the rank of the temperament (how many independent generators it has), and n is the number of primes we're considering in our [[just intonation subgroup]].

Let <math>M</math> be an element of <math>\mathrm{Gr} (k, n)</math>, spanned by basis vectors <math>m_1, \ldots, m_k</math>.
These basis vectors are the rows of the temperament mapping matrix.

The Plücker map takes a temperament and embeds it into a projective space by taking the wedge product of the basis vectors:
:<math>
\begin{align}
\iota: \mathrm{Gr} (k, n)
& \to \mathbf{P}\left(\Lambda^{k} \, \mathbb{R}^n \right) \\
\operatorname {span} (m_1, \ldots, m_k)
& \mapsto \left[ m_1 \wedge \ldots \wedge m_k \right] \, .
\end{align}
</math>

Here, <math>\Lambda^{k} \, \mathbb{R}^n</math> is the k-th exterior power (the subspace containing all k-vectors). This construction is independent of the basis we choose.
While the original space of temperaments has dimension <math>k(n-k)</math>, the space of Plücker coordinates is typically larger, with dimension <math>\binom{n}{k} - 1</math>.

== Examples ==
The space of lines through the origin is exactly projective space, so <math>\mathrm{Gr} (1, n) \cong \mathbf{P} (\mathbb{R}^n)</math>.
In 3 dimensions, a plane through the origin is completely defined by its normal, so we get that <math>\mathrm{Gr} (2, 3) \cong \mathrm{Gr} (1, 3) \cong \mathbf{P} (\mathbb{R}^3)</math>, the projective plane.

The simplest non-trivial case is <math>\mathrm{Gr} (2, 4)</math>.
An element <math>M</math> spanned by two lines <math>x, y</math>, can be represented as the matrix
:<math>
\begin{equation}
\begin{bmatrix}
x_{1} & x_{2} & x_{3} & x_{4} \\
y_{1} & y_{2} & y_{3} & y_{4}
\end{bmatrix} \, .
\end{equation}
</math>

These are not 'proper' coordinates, as doing row operations on this matrix preserves the row-span.

The projective coordinates can be calculated by taking the determinants of all <math>2 \times 2</math> sub-matrices
:<math>
p_{ij} =
\begin{vmatrix}
x_i & x_j \\
y_i & y_j
\end{vmatrix} \, ,
</math>

which finally gives us

:<math>
\begin{equation}
\iota (M) = \left[ x \wedge y \right] = \left[ p_{12} : p_{13} : p_{14} : p_{23} : p_{24} : p_{34} \right] \, .
\end{equation}
</math>

Note the use of colons to signify that these coordinates are homogeneous.

== Plücker relations ==

The coordinates must satisfy some algebraic relations called Plücker relations. Generally, the projective space is much 'larger' than the Grassmannian, and the image in the projective space is some quadric surface.

For the example above on <math>\mathrm{Gr} (2, 4)</math>, the Plücker relation is

:<math>
p_{12} p_{34} - p_{13} p_{24} + p_{14} p_{23} = 0 \, .
</math>

Note that in this case, there is only one such relation, but in higher dimensions there will be many.

== Rational points ==
A '''rational point''' <math>P</math> on <math>\mathrm{Gr}(k, n)</math> is a k-dimensional subspace such that <math>\mathcal{L} = P \cap \mathbb{Z}^n</math> is a rank k sublattice of <math>\mathbb{Z}^n</math>. Abstract temperaments correspond exactly to these rational points, although most have no practical musical use.

The same relations as above can be derived, where we represent P as integer matrix <math>M \in \mathbb{Z} ^ {k \times n}</math>, whose rows span <math>\mathcal{L}</math>.
The projective coordinates similarly have integer entries.
Because the Plücker coordinates are homogeneous, we can always put them in a canonical form by dividing all entries by their greatest common divisor (GCD) and ensuring the first element is non-negative.

An advantage of studying rational points is that we do not have to worry about [[torsion]].
The quotient group <math>\mathbb{Z}^n / \mathcal{L}</math> is a finitely generated abelian group.
When the Plücker coordinates are normalized (GCD = 1), we ensure that
<math>
\mathbb{Z}^n / \mathcal{L} \cong \mathbb{Z}^{n-k},
</math>
which is torsion-free.

== Height ==
A height function is a way to measure the 'arithmetic complexity' of a rational point. For example, the rational numbers <math>\frac{3}{2}</math> and <math>\frac{3001}{2001}</math> are close to eachother, but intuitively the second is much more complicated.

We can define the height of a rational point simply as the Euclidean norm on its Plücker coordinates <math>X = \iota (P)</math>.
:<math>
H(P) = \left\| X \right\| = \left\| m_1 \wedge \ldots \wedge m_n \right\| \\
</math>

In terms of the lattice defined by P, this definition is equivalent to the volume of the {{w|fundamental domain}}, also known as the lattice determinant.
It is easy to show that this does not depend on the basis we choose.

The height can be easily computed using the {{w|Gram matrix}}:
:<math>
\begin{align}
\mathrm{G}_{ij} &= \left\langle m_i, m_j \right\rangle \\
\sqrt{\det(\mathrm{G})} &= \left\| m_1 \wedge \ldots \wedge m_n \right\| = \left\| X \right\| \, .
\end{align}
</math>

In regular temperament theory, this height is usually known as simply the [[Tenney-Euclidean temperament_measures #TE complexity|complexity]].

== Projective distance ==
Given a temperament, we want to have some notion of distance, so that we can measure how well the temperament approximates JI. Since we are talking about linear subspaces (which all intersect at the origin), the only thing that is sensible to measure is the angle between them.

In Euclidean space, one usually takes advantage of the dot product to measure angles.
Given vectors <math>a, b \in \mathbb{R}^n</math>, we famously have

:<math>
\frac{a \cdot b}{\left\| a \right\| \left\| b \right\| } = \cos (\theta) \, .
</math>

In projective space, there is an analogous formula, using the wedge product instead.
Given some real point <math>j \in \mathbb{R}^n</math> with homogeneous coordinates <math>y</math>, and a linear subspace <math>P \in \mathrm{Gr} (k, n)</math> with Plücker coordinates <math>X</math>, we define the projective distance as

:<math>
d(P, j) = \frac{ \left\| X \wedge y \right\| }{\left\| X \right\| \left\| y \right\| } = \sin (\theta) \, .
</math>

Where we can take <math>j</math> to be the usual n-limit vector of log primes, so that <math> y = \left[ 1 : \log_2 (3) : \ldots : \log_2 (p_n) \right] </math>.
Unlike the dot product formula, this works for subspaces of any dimension.

Since for any decent temperament this angle will be extremely small, we can take <math>\sin (\theta) \approx \theta</math>.

== See also ==

* [[Wedgie supplement]] - Supplementary page going over additional information on wedgies
* [[Exterior algebra]] - exterior product, which produces wedgies
* [[Interior product]] - interior product, dual of the exterior product
* [[Hodge dual]] - acts on wedgies

[[Category:Exterior algebra]]

Argent comma

2026-05-06T00:54:48Z

Sintel: Redirected page to 5120/5103

#REDIRECT [[5120/5103]]

Talk:Logarithmic approximants

2026-05-06T00:53:50Z

Sintel:

{{WSArchiveLink}}

== Cleanup and notes ==

I cleaned up a lot of the formatting on this page. It is in a much better state now.

There's some genuinely cool stuff buried in here, and I haven't really taken the time to understand all the details.
I believe this material can be presented in much better ways. Not sure exactly how to go about this. A good start would maybe be to split off approximants into a more encyclopedic article rather than this essay format. The section on argent tuning is also of high interest, since it relates to what we now call the [[argent comma]].

– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 00:53, 6 May 2026 (UTC)

Logarithmic approximants

2026-05-05T17:10:39Z

Sintel: /* Sources and acknowledgements */ typo

A ''logarithmic approximant'' (or ''approximant'' for short) is an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:

* Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?
* Why are certain commas small, and roughly how small are they?
* Why does the 3-limit framework produce aesthetically pleasing scale structures?

The exact size, in cents, of an interval with frequency ratio ''r'' is

:<math>
J_c = 1200 \log_2{r} = 1200 \frac{\ln{r}}{\ln{2}}
</math>

where for just intervals r is rational and can be written as the ratio of two integers:

:<math>
r = \frac{n}{d}
</math>

When manipulating approximants it is convenient to work with a different logarithmic base, in which the interval is defined as

:<math>
J = \frac{1}{2} \ln{r}
</math>

This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e2 = 7.38906... This unit interval can conveniently be termed the dineper (dNp), being twice the size of the natural unit for logarithmic measurement, the Neper.

Comparing the two units of measurement we find

:1 dineper = 2400/ln(2) = 3462.468 cents

which is about 1.4 semitones short of three octaves.

The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus 3/2 is the perfect fifth. This can also be expressed by an explicit function: if bim(r) = (r-1)/(r+1), then r = bim(r). The inverse function can be written mib(v) = (1+v)/(1-v).

Three types of approximants are described here:

* Bimodular approximants (first order rational approximants)
* Padé approximants of order (1,2) (second order rational approximants)
* Quadratic approximants

= Bimodular approximants =

== Definition ==
The bimodular approximant of an interval with frequency ratio ''r = n/d'' is

:<math>
v = \frac{r-1}{r+1}
</math>

''v ''can thus be expressed as

:<math>
\begin{align}
v &= \frac{n-d}{n+d} \\
&= \text{(frequency difference) / (frequency sum)} \\
&= \frac{1}{2} \text{(frequency difference) / (mean frequency)}
\end{align}
</math>

''r'' can be retrieved from ''v'' using the inverse relation

:<math>
r = \frac{1+v}{1-v}
</math>

== Properties ==
When ''r'' is small, ''v'' provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.

Noting that the exact size (in dineper units) of the interval with frequency ratio ''r'' is

:<math>
J = \frac{1}{2} \ln{r}
</math>

the relationship between ''v'' and ''J'' can be expressed as

:<math>
v = \frac{r-1}{r+1} = \frac{e^{2J}-1}{e^{2J}+1} = \tanh{J} = J - \frac{1}{3}J^3 + \frac{2}{15}J^5 - \ldots
</math>

which shows that ''v'' ≈ ''J'' and provides an indication of the size and sign of the error involved in this approximation.

''J'' can be expressed in terms of ''v'' as

:<math>
J = \tanh^{-1}{v} = v + \frac{1}{3}v^3 + \frac{1}{5}v^5 - \ldots
</math>

The function ''v(r)'' is the order (1,1) [http://en.wikipedia.org/wiki/Pad%C3%A9_approximant Padé approximant] of the function ''J(r) =''½ ln ''r'' in the region of ''r'' = 1, which has the property of matching the function value and its first and second derivatives at this value of ''r''. The bimodular approximant function is thus accurate to second order in ''r'' - 1.

As an example, the size of the perfect fifth (in dNp units) is

:<math>
J = \frac{1}{2} \ln \left( \frac{3}{2} \right) = 0.20273\ldots
</math>

The bimodular approximant for this interval (''r'' = 3/2) is

:<math>
v = \frac{\frac{3}{2} - 1}{\frac{3}{2} + 1} = \frac{3 - 2}{3 + 2} = \frac{1}{5} = 0.2
</math>

and the Taylor series indicates that the error in this value is about

:<math>
-\frac{1}{3}v^3 = -0.00267 \ldots
</math>

The approximants of superparticular intervals are reciprocals of odd integers, as shown in Figure 1.

[[File:Low-order_superparticular_intervals.png|frame|none|Bimodular approximants for low-order superparticular intervals]]

If ''v''[''J''] denotes the bimodular approximant of an interval ''J'' with frequency ratio ''r'',

<math>
\begin{align}
v[-J] &= -v[J] \\
v[J_1 +J_2] &= \frac{v_1+v_2}{1+v_1 v_2}
\end{align}
</math>

This last result is equivalent to the identity expressing tanh(''J''1 + ''J''1) in terms of tanh(''J''1) and tanh(''J''2).

== Bimodular approximants and equal temperaments ==
While bimodular approximants have historically been used as a means of estimating the sizes of very small intervals, they remain reasonably accurate as the interval size is increased to an octave or more. And being easily computable, they provide a quick means of comparing the relative sizes of intervals. For example:

Two perfect fourths (''r'' = 4/3, ''v'' = 1/7) approximate a minor seventh (''r'' = 9/5, ''v'' = 2/7)

Three major thirds (''r'' = 5/4, ''v'' = 1/9) or two 7/5s (''v'' = 1/6) or five 8/7s (''v'' = 1/15) approximate an octave (''r'' = 2/1,'' v'' = 1/3)

Bimodular approximants (abbreviated to ‘approximants’ here) also provide simple explanations for the properties of certain equal temperaments.

Tuning the perfect fourth and perfect fifth in the ratio of their approximants (1/7 : 1/5 = 5 : 7) and adjusting their sum to a pure octave yields 12edo (considered as a 3-limit temperament). This is an example of the high accuracy typically obtainable from a tempering policy which takes two intervals which are similar in size and not too large, tunes them in their approximant ratio, and normalises their sum to a pure interval.

Aspects of 12edo considered as a 5-limit temperament can be explained by noting that it tunes the major third, major sixth and octave in the ratio of their approximants (1/9 : 1/4 : 1/3 = 4 : 9 : 12). The accuracy here is lower because the octave is of a size where the approximant has a significant error, and tuning the octave pure assigns the entire error to the smaller intervals.

Tuning the major third and perfect fifth in the in the ratio of their approximants (1/9 : 1/5) and tuning the fifth pure yields [[Carlos_Alpha|Carlos Alpha]].

Tuning the minor third and perfect fifth in the in the ratio of their approximants (1/11 : 1/5) and tuning the fifth pure yields [[Carlos_Beta|Carlos Beta]].

Tuning the minor third and major third in the ratio of their approximants (1/11 : 1/9) and adjusting their sum to a perfect fifth yields [[Carlos_Gamma|Carlos Gamma]] . This temperament has high accuracy because it conforms to the policy noted above.

Tuning the octave pure while preserving the ratios specified above yields, respectively, 31edo, 19edo and 34edo.

Tuning the intervals 9/7, 7/5 and 5/3 in the ratio of their approximants (1/8 : 1/6 : 1/4 = 3 : 4 : 6) and adjusting their sum to a perfect twelfth yields the equally tempered [[Bohlen–Pierce scale]].

Tuning the intervals 11/9, 9/7, 3/2 and 5/3 in the ratio of their approximants (1/10 : 1/8 : 1/5 : 1/4 = 4 : 5 : 8 : 10) and adjusting their sum to a major tenth yields [[88cET|88 cent equal temperament]].

Relationships of this sort can be identified in all equal temperaments.

== Bimodular commas ==
As a consequence of the near-rational interval relationships implied by approximants, any pair of source intervals can be used to define a comma.

Given two intervals ''J''1 and ''J''2 (with ''J''1 < ''J''2) and their approximants ''v''1 and ''v''2, we define the ''bimodular residue'' as

:<math>
b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}
</math>

and using the Taylor series expansion of ''J''(''v'') we find

:<math>
b_r(J_1,J_2) ≈ \frac{1}{3} (v_2^2 - v_1^2) = \frac{1}{3} (v_2 + v_1)(v_2 - v_1)
</math>

The bimodular comma is obtained from the bimodular residue by means of a rational multiplier which ensures that the result (in line with the usual convention applied to commas) is a linear combination of ''J''1 and ''J''2 with integer coefficients sharing no common factor:

:<math>
b(J_1,J_2) ≈ b_m(J_1,J_2) b_r(J_1,J_2)
</math>

where

:<math>
v_1 = \frac{j_1}{g_1}, v_2 = \frac{j_2}{g_2}
</math>

and (with rare exceptions)

:<math>
b_m(J_1,J_2) \approx \frac{\text{LCM}(j_1, j_2)}{\text{GCD}(g_1, g_2)}
</math>

The bimodular residue is accurately estimated by

:<math>
b_r(J_1,J_2) \approx \frac{1}{3} (J_1 + J_2) (J_2 - J_1)
</math>

and therefore

:<math>
b(J_1,J_2) \approx \frac{1}{3} (J_1 + J_2) (J_2 - J_1) b_m
</math>

===Examples===
If the source intervals are the perfect fourth (''f'' = 4/3'')'' and the perfect fifth (''F'' = 3/2), then ''v''1 = 1/7, ''v''2 = 1/5, and ''b'' is the Pythagorean comma:

:<math>
b(F,f) = b_r(F,f) = \frac{F}{\frac{1}{5}} - \frac{f}{\frac{1}{7}} = 5F - 7f
</math>

If the source intervals are the perfect fourth (''f'' = 4/3) and the minor seventh (''m''7 = 9/5), then ''v''1 = 1/7, ''v''2 = 2/7, ''b''r = 2/7 and ''b'' is the syntonic comma:

:<math>
b(m_7,f) = b_r(m_7,f) = \frac{2}{7} \left( \frac{m_7}{\frac{2}{7}} - \frac{f}{\frac{1}{7}} \right) = m_7 - 2f
</math>

For further examples of bimodular commas, and a fuller account of bimodular approximant theory, refer to [[:File:Bimod_Approx_2014-6-8.pdf|this paper]]. See also [[Don_Page_comma|Don Page comma]] (another name for this type of comma).

= Padé approximants of order (1,2) =

== Definition ==
In the section on bimodular approximants it was shown than an interval of logarithmic size ''J'' (measured in dineper units) is related to its bimodular approximant by

:<math>
J = \tanh^{-1}{v} = v + \frac{1}{3}v^3 + \frac{1}{5}v^5 - \ldots
</math>

where

:<math>
v = \frac{r-1}{r+1}
</math>

and ''r'' is the interval’s frequency ratio.

Another way to express this relationship is with a continued fraction:

:<math>
J = \tanh^{-1}{v} = \cfrac{v} {1-\cfrac{v^2}{3 - \cfrac{4v^2}{5 - \cfrac{9v^2}{7 - \ldots}}}}
</math>

The first convergent of this continued fraction is ''v'', the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is

:<math>
y = \cfrac{v}{1-\cfrac{v^2}{3}}
</math>

Values of this rational approximant for some simple 5-limit intervals are shown in the table below.

{| class="wikitable"
|-
| Interval ''J''
| (1,2) Padé approximant ''y''
|-
| Perfect twelfth = 3/1
| 6/11
|-
| Octave = 2/1
| 9/26
|-
| Major sixth = 5/3
| 12/47
|-
| Perfect fifth = 3/2
| 15/74
|-
| Perfect fourth = 4/3
| 21/146
|-
| Major third = 5/4
| 27/242
|}
The denominators of these fractions rapidly get large, so this type of approximant has limited usefulness. However, when combined with bimodular approximants it has occasional value in explaining apparent numerical coincidences and the smallness of the associated commas. For example:

(3/1) / (6/5) = 6.0257 ≈ (6/11) / (1/11) = 6 (kleisma)

(3/1) / (7/4) = 1.9632 ≈ (6/11) / (3/11) = 2 (septimal diesis = 49/48)

(2/1) / (7/6) = 4.4966 ≈ (9/26) / (1/13) = 9/2 (|-11 -9 0 9> comma)

(2/1) / (27/25) = 9.0065 ≈ (9/26) / (1/26) = 9 (ennealimma)

(5/3) / (49/45) = 5.9986 ≈ (12/47) / (2/47) = 6

(5/3) / (25/22) = 3.9960 ≈ (12/47) / (3/47) = 4

(5/3) / (26/21) = 2.3918 ≈ (12/47) / (5/47) = 12/5

(5/3) / (27/20) = 1.7022 ≈ (12/47) / (7/47) = 12/7

(3/2) / (20/17) = 2.4949 ≈ (15/74) / (6/74) = 5/2

= Quadratic approximants =

== Definition ==
The quadratic approximant ''q'' of an interval ''J'' with frequency ratio ''r'' = ''n''/''d'' is

:<math>
\begin{align}
q(r) &= \frac{1}{2} \left( r^{\frac{1}{2}} - r^{-\frac{1}{2}} \right) \\
&= \frac{1}{2} \left( e^{J} - e^{-J} \right) \\
&= \sinh{J} \\
&= J + \frac{1}{3!} J^3 + \frac{1}{5!} J^5 + \ldots
\end{align}
</math>

If this is compared with the expression for the bimodular approximant,

:<math>
v = \tanh{J} = J - \frac{1}{3}J^3 + \frac{2}{15}J^5 - \ldots
</math>

it is apparent that ''q'' is about twice as accurate as ''v'', with an error of opposite sign.

While ''v'' is the frequency difference divided by twice the arithmetic frequency mean, ''q'' is the frequency difference divided by twice the geometric frequency mean:

:<math>
q = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}
</math>

''r'' can be retrieved from ''q'' using

:<math>
\sqrt{r} = q + \sqrt{1+q^2}
</math>

The following are the quadratic approximants of some simple 5-limit intervals:

{| class="wikitable"
|-
| Interval ''J''
| Quadratic approximant ''q''
|-
| Perfect twelfth = 3/1
| 1/√3
|-
| Octave = 2/1
| 1/2√2
|-
| Minor seventh = 9/5
| 2/3√5
|-
| Pythagorean minor seventh = 16/9
| 7/24
|-
| Major sixth = 5/3
| 1/√15
|-
| Minor sixth = 8/5
| 3/4√10
|-
| Perfect fifth = 3/2
| 1/2√6
|-
| Perfect fourth = 4/3
| 1/4√3
|-
| Major third = 5/4
| 1/4√5
|-
| Minor third = 6/5
| 1/2√30
|-
| Pythagorean minor third = 32/27
| 5/24√6
|-
| Large tone = 9/8
| 1/12√2
|-
| Small tone = 10/9
| 1/6√10
|-
| Diatonic semitone = 16/15
| 1/8√15
|-
| Chroma = 25/24
| 1/20√6
|-
| Syntonic comma = 81/80
| 1/72√5
|}

Expressed in terms of the bimodular approximant, ''v = j/g'',

:<math>
q = \frac{v}{\sqrt{1-v^2}} = \frac{j}{\sqrt{g^2-j^2}}
</math>

Quadratic approximants of just intervals thus have the form ''q = j/√k'', where ''j'' and ''k'' are integers and ''j''2'' + k = g''2 is a perfect square.

The presence of a square root in the denominator of ''q'' (except where ''J'' is a double interval) means that quadratic approximants do not, on the whole, imply approximate rational ratios between just intervals or commas of the conventional type. Their interest stems from the fact that ratios involving integer square roots are expressible as repeating continued fractions.

== Properties ==
If ''v''[''J''] and ''q''[''J''] denote, respectively, the bimodular and quadratic approximants of an interval ''J'' with frequency ratio ''r'', and ''q''n denotes ''q''[''J''n] , then

:<math>
\begin{align}
v &= \tanh{J}, \; q = \sinh{J}, \; \frac{q}{v} = \cosh{J} \\
\sqrt{r} &= e^J = q(\frac{1}{v} + 1) \\
\frac{1}{\sqrt{r}} &= e^{-J} = q(\frac{1}{v} - 1) \\
\frac{1}{q^2} &= \frac{1}{v^2} - 1 \\

q[-J] &= -q[J] \\
q[J_2 + J_1] &= q_1 q_2 (\frac{1}{v_2} + \frac{1}{v_1}) \\
q[J_2 - J_1] &= q_1 q_2 (\frac{1}{v_2} - \frac{1}{v_1}) \\
\frac {q[J_2 + J_1]}{q[J_2 - J_1]} &= \frac{v_2+v_1}{v_2-v_1} \\
q[J_2 + J_1] q[J_2 - J_1] &= q_2^2 - q_1^2 \\
\end{align}
</math>

The last two expressions are rational for just intervals, and the last result is equivalent to the hyperbolic trigonometric identity

:<math>
\sinh{(J_2 + J_1)} \sinh{(J_2 - J_1)} = \sinh^2{J_2} - \sinh^2{J_1}
</math>

Where two quadratic approximants have the same square root in the denominator their ratio is rational. This seems to suggest a new source of approximate rational interval ratios, and therefore a new source of commas, but in this situation the approximants always represent the sum and difference of a pair of just intervals, and their ratio can be derived by an alternative route using the bimodular approximants of those intervals.

For example

:<math>
\frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6
</math>

where ''large tone'' = 9/8.

However, this can also be derived from bimodular approximants. Using

:<math>
\frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1}
</math>

with ''J''2 = ''F'' =3/2 and ''J''1 = ''f'' = 4/3 this gives

:<math>
\begin{align}
\frac{\text{octave}}{\text{large tone}} &\approx \frac{q[F+f]}{q[F-f]} \\
&= \frac{v[F] + v[f]}{v[F] - v[f]} \\
&= \frac{1/5 + 1/7}{1/5 - 1/7} = 6
\end{align}
</math>

The quadratic approximant ''q'' of a double interval 2''J'' (for example, the ditone) is rational, which suggests using ½ ''q''(''r''2) as a rational approximant of ''J'' (where ''J'' has frequency ratio ''r''):

:<math>
\frac{1}{2} q(r^2)
= \frac{1}{4} \left( r - \frac{1}{r} \right)
= \frac{1}{2} \sinh{2J}
= J + \frac{2}{3}J^3 + \frac{2}{15}J^5 + \ldots
</math>

However, this approximant is both less accurate and more complex than the corresponding bimodular approximant, and consequently of limited value.

The most interesting approximate interval ratios derivable from quadratic approximants are irrational.

== Relative sizes of intervals between 3 frequencies in arithmetic progression ==

=== Theorem ===
If three harmonics of a fundamental frequency form an arithmetic progression, then the ratio of the logarithmic sizes of the intervals formed between the lower and upper pairs of harmonics is close to the geometric mean of these intervals’ frequency ratios.

=== Remarks ===
If the harmonics have indices ''n - m, n'' and ''n + m'', the two intervals have reduced frequency ratios ''n/(n - m)'' and ''(n + m)/n''. It can be assumed that ''n'' and ''m'' have no common factor.

''m'' is the [[Superpartient|degree of epimoricity]] of the intervals. When ''m'' = 1 the intervals are adjacent superparticular intervals.

The geometric mean of the frequency ratios is the frequency ratio corresponding to the arithmetic mean of the intervals.

=== Proof ===
The ratio of the intervals as estimated from their quadratic approximants is

:<math>
\frac{m}{2\sqrt{n(n-m)}} / \frac{m}{2\sqrt{(n+m)n}} = \sqrt{\frac{n+m}{n-m}}
</math>

which is the geometric mean of their frequency ratios.

=== Examples ===
The ratio of the perfect fifth, ''F'' = 3/2, to the perfect fourth, ''f'' = 4/3, as estimated by their quadratic approximants (1/2√6 and 1/4√3) is √2, which is the frequency ratio of the arithmetic mean of these intervals (the half-octave).

:''F/f'' = 701.955/498.045 = 1.40942,

:√2 = 1.41421.

The ratio of the large tone, ''T'' = 9/8, to the small tone, ''t'' = 10/9, as estimated by their quadratic approximants (1/12√2 and 1/6√10) is √5/2, which is the frequency ratio of the mean tone.

:''T/t'' = 203.910/182.404 = 1.11790,

:√5/2 = 1.11803.

== Argent tuning ==
As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, 1.4000.

It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to

:Perfect fifth = 3/2 = 702.944 cents

:Perfect fourth = 4/3 = 497.056 cents

This fifth is wide by 0.989 cents, and the fourth narrow by the same amount. These errors are of about half the magnitude, and of opposite sign, as their counterparts in 12edo (where these intervals are tuned in the ratio of their bimodular approximants).

A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the [http://en.wikipedia.org/wiki/Silver_ratio silver ratio] (sometimes called the silver mean), ''δs'' = √2 + 1 = 2.4142. On this basis, and by analogy with [[Golden_Meantone|golden meantone]] temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘argent tuning' is proposed instead.

Argent tuning has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.

The continued fraction expansion of the silver ratio has a particularly simple form:

:<math>
\delta_s = \sqrt{2} + 1 = 2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ldots}}}
</math>

As a result, if two intervals ''L'' and ''s'' are tuned in the silver ratio, with ''s = L/δs'', subtracting twice the small interval ''s'' from the large interval ''L'' leaves a remainder of size ''s/δs'':

:<math>
L - 2s = (\delta_s - 2)s = \frac{s}{\delta_s}
</math>

(since <math> \tfrac{1}{\delta_s} = \sqrt{2} - 1 = \delta_s - 2 </math>) and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows. The names are assigned according to Pythagorean conventions (the limma being the Pythagorean semitone 256/243) followed by tempered and just sizes in cents:

{| class="wikitable"
|-
| Octave 1200.00 (1200.00)
| Perfect fourth 497.06 (498.04)
| Tone 205.89 (203.91)
| Limma 85.28 (90.22)
| Pythag. comma 35.32 (23.46)
|-
| Perfect 11th 1697.06 (1698.04)
| Perfect fifth 702.94 (701.96)
| Minor third 291.17 (294.13)
| Apotome 120.61 (113.69)
| 17-tone comma 49.96 (66.76)
|}
Thus for example:

:octave = 2×fourth + tone

:fourth = 2×tone + limma

:tone = 2×limma + Pythagorean comma

:perfect 11th (8/3) = 2×fifth + minor third

:fifth = 2×(minor third) + apotome

When picturing these relationships it makes most musical sense to place the small interval between the two larger ones, as in the ‘continued fraction jigsaw’ below.

The following relationships hold in the table, the first two being valid for the pure intervals as well as their tempered counterparts:

* Subtracting twice an interval from the interval on its left generates the interval on its right.
* An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.
* Adjacent horizontal pairs have ratio ''δs'' = √2 + 1.
* Adjacent vertical pairs have ratio √2.
* Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.

The regularity of this scheme, combined with the fact that the ratios between closely related intervals are of order 2, means that its intervals form orderly sequences in which successive terms are clearly differentiated but of comparable magnitude – highly desirable properties for the formation of musical scales.

In this fractal temperament, multiplying or dividing any interval by the factor ''δs'' = √2 + 1 produces another interval in the temperament. Any tempered interval ''J’'' can be split into three parts, two of equal size ''J’''/''δs'' and the other of size ''J’''/''δs''2.

A similar principle applies to multiplication and division by the factor √2, except that intervals in the top row of the table cannot be divided by √2 to yield another interval in the temperament. These properties means that the temperament would support compositional techniques based on novel types of intervallic augmentation and diminution.

Successive convergents of the silver ratio produce ratios involving [http://en.wikipedia.org/wiki/Pell_number Pell numbers].

:<math>
\sqrt{2} + 1 \approx 2, \; \frac{5}{2}, \; \frac{12}{5}, \; \frac{29}{12}, \; \frac{70}{29}, \; \ldots
</math>

Other approximations to the silver ratio are provided by ratios of consecutive half Pell-Lucas numbers, which are formed by adding consecutive Pell numbers

:<math>
\sqrt{2} + 1 \approx 3, \; \frac{7}{3}, \; \frac{17}{7}, \; \frac{41}{17}, \; \frac{99}{41}, \; \ldots
</math>

This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, 41edo, 70edo etc.).

The accuracy of the argent fifth means that the scheme produces workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur (the next column containing the 29-tone comma and ''minus'' the 41-tone comma).

Figure 2 is a ''continued fraction jigsaw'' showing the sizes of the octave (o), fourth (f), tone (T), limma (sp), Pythagorean comma (p) and 29-tone comma (p29) as tempered by 41edo - an approximation to argent tuning. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.

[[File:Continued_fraction_jigsaw_41edo.png|600px|thumb|none|Figure 2. Continued fraction jigsaw for 41edo]]

Figure 3 is a geometrical representation of argent tuning in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = large tone, mppp = Pythagorean apotome, p = Pythagorean comma.

[[File:Silver_temperament_graphic.png|600px|thumb|none|Figure 3. Geometrical representation of argent tuning]]

Argent tuning tunes the augmented fourth (tritone) and diminished fifth (double minor third) in the ratio 3/2√2, which is also the ratio of the quadratic approximants of 10/7 and 7/5:

:<math>
\frac{q[10/7]}{q[7/5]}= \frac{ \frac{3}{2\sqrt{70}} } { \frac{2} {2\sqrt{35}} } = \frac{3}{2\sqrt{2}}.
</math>

This means that in argent tuning the augmented fourth is very close to 10/7 and the diminished fifth is very close to 7/5. The discrepancy in each case is just 0.175 cents. Another way to express the first of these relationships is

:<math>
3 \left( \frac{1}{2\sqrt{6}} - \frac{1}{4\sqrt{3}} \right) \approx \frac{3}{2\sqrt{70}},
</math>

which after squaring both sides leads to √2 ≈ 99/70, a well-known approximation which can be confirmed by noting that 99/70 = √(2 + 1/4900).

As a consequence of these relationships the tempered diatonic semitone (85.281 cents) is close to 21/20 (84.467 cents), the tempered chromatic semitone (120.606 cents) is close to 15/14 (119.443 cents), and the tempered Pythagorean comma (35.325 cents) is close to 50/49 (34.976 cents).

If these 7-limit intervals are considered to be tempered to their 3-limit counterparts argent is an example of hemifamity temperament. Hemifamity (5120/5103) is the bimodular comma formed from 10/7 and 9/8

By the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem Gelfond-Schneider theorem] the frequency ratios of all argent intervals (''r'' = 2√2''a''+''b'', where'' a'' and ''b'' are integers) are transcendental, with the exception of octave multiples (''a'' = 0). The frequency ratio of the tempered perfect eleventh (8/3 = 2.6666...) is the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant Gelfond-Schneider constant] or Hilbert number, 2√2 = 2.665144...

==Golden temperaments==
It has been shown in an example above that the ratio of the large tone (''T'' = 9/8) to the small tone (''t'' = 10/9) is closely approximated by

:<math>
\frac{T}{t} = \frac{\sqrt{5}}{2}
</math>

It follows that

:<math>
\frac{T + \frac{t}{2}}{t} = \frac{\sqrt{5} + 1}{2} = \varphi
</math>

where ''ϕ'' = 1.61803... is the golden ratio.

If a Fibonacci sequence of intervals is formed from the pair of intervals ''T'' - ''t''/2 and ''t'', and extended in both directions, it can thus be expected that the ratios between successive intervals in this sequence will also be close to ''ϕ''. The sequence formed in this way is Sequence 1 in the following table.

{| class="wikitable"
|-
| Sequence 1:
| ''t''/2 - 3''c''
| 2''c''
| ''t''/2 ''- c''
| ''T - t''/2
| ''t''
| ''T + t''/2
| ''M + t''/2
| 2''M''
|-
| Sequence 2:
| ''magic''
| ''diesis''
| ''chroma''
| ''semitone''
| ''t''
| ''mp''
| ''f - c''
| ''m6p - c''
|-
| Difference:
| -3''σ''/2
| ''σ''
| -''σ''/2
| ''σ''/2
| 0
| ''σ''/2
| ''σ''/2
| ''σ''
|-
| Seq 1 ratios:
|
| 1.6120
| 1.6204
| 1.6171
| 1.6184
| 1.6179
| 1.6181
| 1.6180
|-
| Seq 2 ratios:
|
| 1.3865
| 1.7212
| 1.5810
| 1.6325
| 1.6125
| 1.6201
| 1.6172
|}
where ''f'' = 4/3, ''T'' = 9/8, ''t'' = 10/9, ''M'' = 5/4, ''magic'' = 3125/3072, ''diesis'' = 128/125, ''chroma'' = 25/24, ''semitone'' = 16/15, ''mp'' = 32/27, ''c'' = ''syntonic comma'' = 81/80, ''m6p'' = 128/81, ''σ'' = ''schisma'' = 32805/32768.

The ratios between successive intervals in Sequence 1 are shown in the row labelled ‘Seq 1 ratios’, and are indeed close to ''ϕ''.

Sequence 2 is another Fibonacci sequence of intervals which differ from those in Sequence 1 by small amounts of the order of one schisma (''σ''), as indicated by the row marked 'Difference' (which is itself a Fibonacci sequence).

The ratios of consecutive pairs of intervals in Sequence 2 are shown in the row labelled ‘Seq 2 ratios’. They approximate ''ϕ'' rather less accurately.

A suitable name for 5-limit tunings in which the intervals in either Sequence 1 or Sequence 2, or both, are tempered to exactly ''ϕ'' would be ‘golden temperaments’.

Tempering the Sequence 2 ratios to ''ϕ'' while tuning the octave pure and tempering out the syntonic comma yields [[Golden_Meantone|golden meantone]] temperament.

Tempering the Sequence 1 ratios to ''ϕ'' yields a range of temperaments which can be made extremely accurate by, for example, tuning the octave and fifth (and therefore all Pythagorean intervals) pure. In this temperament the errors in the intervals ''s, t'', ''M'' and ''m''=6/5 are all ±0.02106 cents.

Tempering out the schisma tunes Sequences 1 and 2 identically so that the ratios between consecutive intervals can be fixed at ''ϕ'' in both sequences. Normalised to a pure octave, the resulting temperament, ‘golden schismatic’, has a fifth of 701.791061 cents (error -0.163 cents) and a major third of 385.671509 cents (error -0.642 cents).

== Pythagorean triples of quadratic approximants ==
If the quadratic approximants ''q''1, q''2 and ''q''3 of a set of three intervals ''J''1, ''J''2 and ''J''3 satisfy

:<math>
q_1^2 + q_2^2 = q_3^2
</math>

they can be said to form a [http://en.wikipedia.org/wiki/Pythagorean_triple Pythagorean triple].

The following are three examples. In the first and third cases, their counterparts in 12edo, ''J''1', ''J''2' and ''J''3', are also Pythagorean triples:

{| class="wikitable"
|-
| | ''J''1
| | ''J''2
| | ''J''3
| | ''q''1
| | ''q''2
| | ''q''3
| | ''J''1'
| | ''J''2'
| | ''J''3'
|-
| | 6/5
| | 5/4
| | 4/3
| | 1/2√30
| | 1/4√5
| | 1/4√3
| | 3
| | 4
| | 5
|-
| | 4/3
| | 12/5
| | 5/2
| | 1/4√3
| | 7/4√15
| | 3/2√10
| |
| |
| |
|-
| | 8/5
| | 12/5
| | 8/3
| | 3/4√10
| | 7/4√15
| | 5/4√6
| | 8
| | 15
| | 17
|}

==A small 34edo comma==
As [[Gene_Ward_Smith|Gene Ward Smith]] has noted, the 5-limit comma |-433 -137 280> (''selenia'') is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using quadratic approximants.

It can be shown, using a suitable [[Comma-based_lattices|comma-based lattice]], that every comma tempered out by 34edo can be expressed as an integer linear combination of the [[Gammic_node|gammic comma]] |-29 -11 20> (4.769 cents) and the ''semisuper'' comma (''[[vishnuzma|vishnuzma]]'') |23 6 -14> (3.338 cents). In particular,

:''selenia'' = 7 ''gammic'' - 10 ''semisuper''

So to prove that ''selenia'' is small we must show that ''gammic/semisuper'' ≈ 10/7.

''Gammic'' and ''semisuper'' are both bimodular commas:

:''gammic'' = ''b''(6/5,5/4)

:''semisuper'' = ''b''(25/24,4/3)

Using a result given in the section on bimodular commas, the size of ''b''(''J''1,''J''2) can be estimated using

:<math>
b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 - J_1^2) b_m
</math>

Estimating ''J''2 and ''J''1 with their quadratic approximants we then have

:<math>
b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 - q_1^2) b_m
</math>

For ''gammic'':

:''J''₁= 6/5, ''J''₂= 5/4

:''v''₁ = 1/11, ''v''₂ = 1/9, ''b''m = 1

:''q''₁² = (1/4)(1/30), ''q''₂''² ='' (1/4)(1/20)

:''gammic'' = ''b''(''J''₁,''J''₂) ≈ (1/12) (1/30 - 1/20) = (1/12) (1/60)

For ''semisuper:''

:''J''₁= 25/24, ''J''₂= 4/3

:''v''₁ = 1/49, ''v''₂ = 1/7, ''b''m = 1/7

:''q''₁² = (1/4)(1/600), ''q''₂''² ='' (1/4)(1/12)

:''semisuper'' = ''b''(''J''₁,''J''₂) ≈ (1/12) (1/12 - 1/600)(1/7) = (1/12) (7/600)

Therefore

:''gammic/semisuper'' ≈ 10/7

as required.

To estimate the size of ''selenia'' we must quantify the error in this ratio. A more accurate analysis gives

:<math>
\begin{align}
b(J_1,J_2) &\approx \left( \frac{1}{3} \left(q_2^2 - q_1^2\right) - \frac{2}{15} \left( q_2^4 - q_1^4 \right) \right) b_m \\
&= \frac{1}{3} \left( q_2^2 - q_1^2 \right) \left( 1 - \frac{2}{5} \left(q_1^2 + q_2^2\right) \right) b_m
\end{align}
</math>

''b''(''J''1,''J''2) we should multiply them by

:<math>
f = 1 - \frac{2}{5} \left( q_1^2 + q_2^2 \right)
</math>

Thus a better estimate for ''gammic/semisuper'' is

:<math>
\frac{\text{gammic}}{\text{semisuper}} \approx \frac{10 f_{\text{gamma}}} {7 f_{\text{semisuper}}}
</math>

from which it follows that

:''selenia'' = 7 ''gammic'' - 10 ''semisuper''

:: ≈ 7 ''gammic'' (''f''''gammic'' - ''f''''semisuper'')/''f''''gammic''

Putting in the numbers:

:''f''''gammic'' = 1 - (2/5) (1/4) (1/30 + 1/20) = 1 - 1/120

:''f''''semisuper'' = 1 - (2/5)(1/4) (1/600 + 1/12) = 1 - (1/120) (51/50)

:''f''''gammic'' - ''f''''semisuper'' = 1/6000

Therefore

:''selenia'' ≈ 7 ''gammic'' (1/6000) (120/119) = ''gammic''/850 = 0.00561 cents

which is within 20% of the accurate value, 0.00476 cents. (The discrepancy is due to the influence of terms in ''q''6, which become significant when the ''f'' values are very similar.)

In summary, the reason ''selenia'' is small (compared to ''gammic'' and ''semisuper'') is because the quadratic approximants of ''gammic'' and ''semisuper'' are in the ratio 10/7. The reason it is ''very'' small (of order ''gammic''/1000 rather than ''gammic''/10) is because the fractional errors in those approximants are almost the same. That in turn is because the squares of the source intervals of these bimodular commas have nearly the same sum. Note that the quadratic approximants of three of these intervals form a Pythagorean triple:

:<math>
q \left( \frac{6}{5} \right) ^ 2 + q \left( \frac{5}{4} \right) ^ 2 = q \left( \frac{4}{3} \right) ^ 2
</math>

and (''q''(25/24))2 , being small in comparison to the other terms, compromises this equality only slightly.

= Sources and acknowledgements =
This article is based on original research by [[Martin_Gough|Martin Gough]]. See [[:File:Bimod_Approx_2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants.

Argent tuning is based on the continued fraction convergents of <math>\sqrt{2}</math>, which have been known since ancient times.
This application to tuning appears to have been first made by [[Erv Wilson]], who described it under the name '2-zig/2-zag' in a [http://anaphoria.com/meruthree.pdf note] dated December 1996, with a comment claiming it as his answer to [[Joseph Yasser]] "at about 1950".
The same construction was later arrived at independently by [[Graham Breed]] and Paul Hahn, who described it in posts ([https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12592.html#12599 #12599], [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12637.html#12670 #12670]) to the Yahoo tuning list on 10 and 12 August 2000.

Thanks to [[Gene_Ward_Smith|Gene Ward Smith]] for the Gelfond-Schneider result.

[[Category:Essays]]

Logarithmic approximants

2026-05-05T17:04:12Z

Sintel: Clarify contributions

A ''logarithmic approximant'' (or ''approximant'' for short) is an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:

* Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?
* Why are certain commas small, and roughly how small are they?
* Why does the 3-limit framework produce aesthetically pleasing scale structures?

The exact size, in cents, of an interval with frequency ratio ''r'' is

:<math>
J_c = 1200 \log_2{r} = 1200 \frac{\ln{r}}{\ln{2}}
</math>

where for just intervals r is rational and can be written as the ratio of two integers:

:<math>
r = \frac{n}{d}
</math>

When manipulating approximants it is convenient to work with a different logarithmic base, in which the interval is defined as

:<math>
J = \frac{1}{2} \ln{r}
</math>

This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e2 = 7.38906... This unit interval can conveniently be termed the dineper (dNp), being twice the size of the natural unit for logarithmic measurement, the Neper.

Comparing the two units of measurement we find

:1 dineper = 2400/ln(2) = 3462.468 cents

which is about 1.4 semitones short of three octaves.

The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus 3/2 is the perfect fifth. This can also be expressed by an explicit function: if bim(r) = (r-1)/(r+1), then r = bim(r). The inverse function can be written mib(v) = (1+v)/(1-v).

Three types of approximants are described here:

* Bimodular approximants (first order rational approximants)
* Padé approximants of order (1,2) (second order rational approximants)
* Quadratic approximants

= Bimodular approximants =

== Definition ==
The bimodular approximant of an interval with frequency ratio ''r = n/d'' is

:<math>
v = \frac{r-1}{r+1}
</math>

''v ''can thus be expressed as

:<math>
\begin{align}
v &= \frac{n-d}{n+d} \\
&= \text{(frequency difference) / (frequency sum)} \\
&= \frac{1}{2} \text{(frequency difference) / (mean frequency)}
\end{align}
</math>

''r'' can be retrieved from ''v'' using the inverse relation

:<math>
r = \frac{1+v}{1-v}
</math>

== Properties ==
When ''r'' is small, ''v'' provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.

Noting that the exact size (in dineper units) of the interval with frequency ratio ''r'' is

:<math>
J = \frac{1}{2} \ln{r}
</math>

the relationship between ''v'' and ''J'' can be expressed as

:<math>
v = \frac{r-1}{r+1} = \frac{e^{2J}-1}{e^{2J}+1} = \tanh{J} = J - \frac{1}{3}J^3 + \frac{2}{15}J^5 - \ldots
</math>

which shows that ''v'' ≈ ''J'' and provides an indication of the size and sign of the error involved in this approximation.

''J'' can be expressed in terms of ''v'' as

:<math>
J = \tanh^{-1}{v} = v + \frac{1}{3}v^3 + \frac{1}{5}v^5 - \ldots
</math>

The function ''v(r)'' is the order (1,1) [http://en.wikipedia.org/wiki/Pad%C3%A9_approximant Padé approximant] of the function ''J(r) =''½ ln ''r'' in the region of ''r'' = 1, which has the property of matching the function value and its first and second derivatives at this value of ''r''. The bimodular approximant function is thus accurate to second order in ''r'' - 1.

As an example, the size of the perfect fifth (in dNp units) is

:<math>
J = \frac{1}{2} \ln \left( \frac{3}{2} \right) = 0.20273\ldots
</math>

The bimodular approximant for this interval (''r'' = 3/2) is

:<math>
v = \frac{\frac{3}{2} - 1}{\frac{3}{2} + 1} = \frac{3 - 2}{3 + 2} = \frac{1}{5} = 0.2
</math>

and the Taylor series indicates that the error in this value is about

:<math>
-\frac{1}{3}v^3 = -0.00267 \ldots
</math>

The approximants of superparticular intervals are reciprocals of odd integers, as shown in Figure 1.

[[File:Low-order_superparticular_intervals.png|frame|none|Bimodular approximants for low-order superparticular intervals]]

If ''v''[''J''] denotes the bimodular approximant of an interval ''J'' with frequency ratio ''r'',

<math>
\begin{align}
v[-J] &= -v[J] \\
v[J_1 +J_2] &= \frac{v_1+v_2}{1+v_1 v_2}
\end{align}
</math>

This last result is equivalent to the identity expressing tanh(''J''1 + ''J''1) in terms of tanh(''J''1) and tanh(''J''2).

== Bimodular approximants and equal temperaments ==
While bimodular approximants have historically been used as a means of estimating the sizes of very small intervals, they remain reasonably accurate as the interval size is increased to an octave or more. And being easily computable, they provide a quick means of comparing the relative sizes of intervals. For example:

Two perfect fourths (''r'' = 4/3, ''v'' = 1/7) approximate a minor seventh (''r'' = 9/5, ''v'' = 2/7)

Three major thirds (''r'' = 5/4, ''v'' = 1/9) or two 7/5s (''v'' = 1/6) or five 8/7s (''v'' = 1/15) approximate an octave (''r'' = 2/1,'' v'' = 1/3)

Bimodular approximants (abbreviated to ‘approximants’ here) also provide simple explanations for the properties of certain equal temperaments.

Tuning the perfect fourth and perfect fifth in the ratio of their approximants (1/7 : 1/5 = 5 : 7) and adjusting their sum to a pure octave yields 12edo (considered as a 3-limit temperament). This is an example of the high accuracy typically obtainable from a tempering policy which takes two intervals which are similar in size and not too large, tunes them in their approximant ratio, and normalises their sum to a pure interval.

Aspects of 12edo considered as a 5-limit temperament can be explained by noting that it tunes the major third, major sixth and octave in the ratio of their approximants (1/9 : 1/4 : 1/3 = 4 : 9 : 12). The accuracy here is lower because the octave is of a size where the approximant has a significant error, and tuning the octave pure assigns the entire error to the smaller intervals.

Tuning the major third and perfect fifth in the in the ratio of their approximants (1/9 : 1/5) and tuning the fifth pure yields [[Carlos_Alpha|Carlos Alpha]].

Tuning the minor third and perfect fifth in the in the ratio of their approximants (1/11 : 1/5) and tuning the fifth pure yields [[Carlos_Beta|Carlos Beta]].

Tuning the minor third and major third in the ratio of their approximants (1/11 : 1/9) and adjusting their sum to a perfect fifth yields [[Carlos_Gamma|Carlos Gamma]] . This temperament has high accuracy because it conforms to the policy noted above.

Tuning the octave pure while preserving the ratios specified above yields, respectively, 31edo, 19edo and 34edo.

Tuning the intervals 9/7, 7/5 and 5/3 in the ratio of their approximants (1/8 : 1/6 : 1/4 = 3 : 4 : 6) and adjusting their sum to a perfect twelfth yields the equally tempered [[Bohlen–Pierce scale]].

Tuning the intervals 11/9, 9/7, 3/2 and 5/3 in the ratio of their approximants (1/10 : 1/8 : 1/5 : 1/4 = 4 : 5 : 8 : 10) and adjusting their sum to a major tenth yields [[88cET|88 cent equal temperament]].

Relationships of this sort can be identified in all equal temperaments.

== Bimodular commas ==
As a consequence of the near-rational interval relationships implied by approximants, any pair of source intervals can be used to define a comma.

Given two intervals ''J''1 and ''J''2 (with ''J''1 < ''J''2) and their approximants ''v''1 and ''v''2, we define the ''bimodular residue'' as

:<math>
b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}
</math>

and using the Taylor series expansion of ''J''(''v'') we find

:<math>
b_r(J_1,J_2) ≈ \frac{1}{3} (v_2^2 - v_1^2) = \frac{1}{3} (v_2 + v_1)(v_2 - v_1)
</math>

The bimodular comma is obtained from the bimodular residue by means of a rational multiplier which ensures that the result (in line with the usual convention applied to commas) is a linear combination of ''J''1 and ''J''2 with integer coefficients sharing no common factor:

:<math>
b(J_1,J_2) ≈ b_m(J_1,J_2) b_r(J_1,J_2)
</math>

where

:<math>
v_1 = \frac{j_1}{g_1}, v_2 = \frac{j_2}{g_2}
</math>

and (with rare exceptions)

:<math>
b_m(J_1,J_2) \approx \frac{\text{LCM}(j_1, j_2)}{\text{GCD}(g_1, g_2)}
</math>

The bimodular residue is accurately estimated by

:<math>
b_r(J_1,J_2) \approx \frac{1}{3} (J_1 + J_2) (J_2 - J_1)
</math>

and therefore

:<math>
b(J_1,J_2) \approx \frac{1}{3} (J_1 + J_2) (J_2 - J_1) b_m
</math>

===Examples===
If the source intervals are the perfect fourth (''f'' = 4/3'')'' and the perfect fifth (''F'' = 3/2), then ''v''1 = 1/7, ''v''2 = 1/5, and ''b'' is the Pythagorean comma:

:<math>
b(F,f) = b_r(F,f) = \frac{F}{\frac{1}{5}} - \frac{f}{\frac{1}{7}} = 5F - 7f
</math>

If the source intervals are the perfect fourth (''f'' = 4/3) and the minor seventh (''m''7 = 9/5), then ''v''1 = 1/7, ''v''2 = 2/7, ''b''r = 2/7 and ''b'' is the syntonic comma:

:<math>
b(m_7,f) = b_r(m_7,f) = \frac{2}{7} \left( \frac{m_7}{\frac{2}{7}} - \frac{f}{\frac{1}{7}} \right) = m_7 - 2f
</math>

For further examples of bimodular commas, and a fuller account of bimodular approximant theory, refer to [[:File:Bimod_Approx_2014-6-8.pdf|this paper]]. See also [[Don_Page_comma|Don Page comma]] (another name for this type of comma).

= Padé approximants of order (1,2) =

== Definition ==
In the section on bimodular approximants it was shown than an interval of logarithmic size ''J'' (measured in dineper units) is related to its bimodular approximant by

:<math>
J = \tanh^{-1}{v} = v + \frac{1}{3}v^3 + \frac{1}{5}v^5 - \ldots
</math>

where

:<math>
v = \frac{r-1}{r+1}
</math>

and ''r'' is the interval’s frequency ratio.

Another way to express this relationship is with a continued fraction:

:<math>
J = \tanh^{-1}{v} = \cfrac{v} {1-\cfrac{v^2}{3 - \cfrac{4v^2}{5 - \cfrac{9v^2}{7 - \ldots}}}}
</math>

The first convergent of this continued fraction is ''v'', the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is

:<math>
y = \cfrac{v}{1-\cfrac{v^2}{3}}
</math>

Values of this rational approximant for some simple 5-limit intervals are shown in the table below.

{| class="wikitable"
|-
| Interval ''J''
| (1,2) Padé approximant ''y''
|-
| Perfect twelfth = 3/1
| 6/11
|-
| Octave = 2/1
| 9/26
|-
| Major sixth = 5/3
| 12/47
|-
| Perfect fifth = 3/2
| 15/74
|-
| Perfect fourth = 4/3
| 21/146
|-
| Major third = 5/4
| 27/242
|}
The denominators of these fractions rapidly get large, so this type of approximant has limited usefulness. However, when combined with bimodular approximants it has occasional value in explaining apparent numerical coincidences and the smallness of the associated commas. For example:

(3/1) / (6/5) = 6.0257 ≈ (6/11) / (1/11) = 6 (kleisma)

(3/1) / (7/4) = 1.9632 ≈ (6/11) / (3/11) = 2 (septimal diesis = 49/48)

(2/1) / (7/6) = 4.4966 ≈ (9/26) / (1/13) = 9/2 (|-11 -9 0 9> comma)

(2/1) / (27/25) = 9.0065 ≈ (9/26) / (1/26) = 9 (ennealimma)

(5/3) / (49/45) = 5.9986 ≈ (12/47) / (2/47) = 6

(5/3) / (25/22) = 3.9960 ≈ (12/47) / (3/47) = 4

(5/3) / (26/21) = 2.3918 ≈ (12/47) / (5/47) = 12/5

(5/3) / (27/20) = 1.7022 ≈ (12/47) / (7/47) = 12/7

(3/2) / (20/17) = 2.4949 ≈ (15/74) / (6/74) = 5/2

= Quadratic approximants =

== Definition ==
The quadratic approximant ''q'' of an interval ''J'' with frequency ratio ''r'' = ''n''/''d'' is

:<math>
\begin{align}
q(r) &= \frac{1}{2} \left( r^{\frac{1}{2}} - r^{-\frac{1}{2}} \right) \\
&= \frac{1}{2} \left( e^{J} - e^{-J} \right) \\
&= \sinh{J} \\
&= J + \frac{1}{3!} J^3 + \frac{1}{5!} J^5 + \ldots
\end{align}
</math>

If this is compared with the expression for the bimodular approximant,

:<math>
v = \tanh{J} = J - \frac{1}{3}J^3 + \frac{2}{15}J^5 - \ldots
</math>

it is apparent that ''q'' is about twice as accurate as ''v'', with an error of opposite sign.

While ''v'' is the frequency difference divided by twice the arithmetic frequency mean, ''q'' is the frequency difference divided by twice the geometric frequency mean:

:<math>
q = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}
</math>

''r'' can be retrieved from ''q'' using

:<math>
\sqrt{r} = q + \sqrt{1+q^2}
</math>

The following are the quadratic approximants of some simple 5-limit intervals:

{| class="wikitable"
|-
| Interval ''J''
| Quadratic approximant ''q''
|-
| Perfect twelfth = 3/1
| 1/√3
|-
| Octave = 2/1
| 1/2√2
|-
| Minor seventh = 9/5
| 2/3√5
|-
| Pythagorean minor seventh = 16/9
| 7/24
|-
| Major sixth = 5/3
| 1/√15
|-
| Minor sixth = 8/5
| 3/4√10
|-
| Perfect fifth = 3/2
| 1/2√6
|-
| Perfect fourth = 4/3
| 1/4√3
|-
| Major third = 5/4
| 1/4√5
|-
| Minor third = 6/5
| 1/2√30
|-
| Pythagorean minor third = 32/27
| 5/24√6
|-
| Large tone = 9/8
| 1/12√2
|-
| Small tone = 10/9
| 1/6√10
|-
| Diatonic semitone = 16/15
| 1/8√15
|-
| Chroma = 25/24
| 1/20√6
|-
| Syntonic comma = 81/80
| 1/72√5
|}

Expressed in terms of the bimodular approximant, ''v = j/g'',

:<math>
q = \frac{v}{\sqrt{1-v^2}} = \frac{j}{\sqrt{g^2-j^2}}
</math>

Quadratic approximants of just intervals thus have the form ''q = j/√k'', where ''j'' and ''k'' are integers and ''j''2'' + k = g''2 is a perfect square.

The presence of a square root in the denominator of ''q'' (except where ''J'' is a double interval) means that quadratic approximants do not, on the whole, imply approximate rational ratios between just intervals or commas of the conventional type. Their interest stems from the fact that ratios involving integer square roots are expressible as repeating continued fractions.

== Properties ==
If ''v''[''J''] and ''q''[''J''] denote, respectively, the bimodular and quadratic approximants of an interval ''J'' with frequency ratio ''r'', and ''q''n denotes ''q''[''J''n] , then

:<math>
\begin{align}
v &= \tanh{J}, \; q = \sinh{J}, \; \frac{q}{v} = \cosh{J} \\
\sqrt{r} &= e^J = q(\frac{1}{v} + 1) \\
\frac{1}{\sqrt{r}} &= e^{-J} = q(\frac{1}{v} - 1) \\
\frac{1}{q^2} &= \frac{1}{v^2} - 1 \\

q[-J] &= -q[J] \\
q[J_2 + J_1] &= q_1 q_2 (\frac{1}{v_2} + \frac{1}{v_1}) \\
q[J_2 - J_1] &= q_1 q_2 (\frac{1}{v_2} - \frac{1}{v_1}) \\
\frac {q[J_2 + J_1]}{q[J_2 - J_1]} &= \frac{v_2+v_1}{v_2-v_1} \\
q[J_2 + J_1] q[J_2 - J_1] &= q_2^2 - q_1^2 \\
\end{align}
</math>

The last two expressions are rational for just intervals, and the last result is equivalent to the hyperbolic trigonometric identity

:<math>
\sinh{(J_2 + J_1)} \sinh{(J_2 - J_1)} = \sinh^2{J_2} - \sinh^2{J_1}
</math>

Where two quadratic approximants have the same square root in the denominator their ratio is rational. This seems to suggest a new source of approximate rational interval ratios, and therefore a new source of commas, but in this situation the approximants always represent the sum and difference of a pair of just intervals, and their ratio can be derived by an alternative route using the bimodular approximants of those intervals.

For example

:<math>
\frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6
</math>

where ''large tone'' = 9/8.

However, this can also be derived from bimodular approximants. Using

:<math>
\frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1}
</math>

with ''J''2 = ''F'' =3/2 and ''J''1 = ''f'' = 4/3 this gives

:<math>
\begin{align}
\frac{\text{octave}}{\text{large tone}} &\approx \frac{q[F+f]}{q[F-f]} \\
&= \frac{v[F] + v[f]}{v[F] - v[f]} \\
&= \frac{1/5 + 1/7}{1/5 - 1/7} = 6
\end{align}
</math>

The quadratic approximant ''q'' of a double interval 2''J'' (for example, the ditone) is rational, which suggests using ½ ''q''(''r''2) as a rational approximant of ''J'' (where ''J'' has frequency ratio ''r''):

:<math>
\frac{1}{2} q(r^2)
= \frac{1}{4} \left( r - \frac{1}{r} \right)
= \frac{1}{2} \sinh{2J}
= J + \frac{2}{3}J^3 + \frac{2}{15}J^5 + \ldots
</math>

However, this approximant is both less accurate and more complex than the corresponding bimodular approximant, and consequently of limited value.

The most interesting approximate interval ratios derivable from quadratic approximants are irrational.

== Relative sizes of intervals between 3 frequencies in arithmetic progression ==

=== Theorem ===
If three harmonics of a fundamental frequency form an arithmetic progression, then the ratio of the logarithmic sizes of the intervals formed between the lower and upper pairs of harmonics is close to the geometric mean of these intervals’ frequency ratios.

=== Remarks ===
If the harmonics have indices ''n - m, n'' and ''n + m'', the two intervals have reduced frequency ratios ''n/(n - m)'' and ''(n + m)/n''. It can be assumed that ''n'' and ''m'' have no common factor.

''m'' is the [[Superpartient|degree of epimoricity]] of the intervals. When ''m'' = 1 the intervals are adjacent superparticular intervals.

The geometric mean of the frequency ratios is the frequency ratio corresponding to the arithmetic mean of the intervals.

=== Proof ===
The ratio of the intervals as estimated from their quadratic approximants is

:<math>
\frac{m}{2\sqrt{n(n-m)}} / \frac{m}{2\sqrt{(n+m)n}} = \sqrt{\frac{n+m}{n-m}}
</math>

which is the geometric mean of their frequency ratios.

=== Examples ===
The ratio of the perfect fifth, ''F'' = 3/2, to the perfect fourth, ''f'' = 4/3, as estimated by their quadratic approximants (1/2√6 and 1/4√3) is √2, which is the frequency ratio of the arithmetic mean of these intervals (the half-octave).

:''F/f'' = 701.955/498.045 = 1.40942,

:√2 = 1.41421.

The ratio of the large tone, ''T'' = 9/8, to the small tone, ''t'' = 10/9, as estimated by their quadratic approximants (1/12√2 and 1/6√10) is √5/2, which is the frequency ratio of the mean tone.

:''T/t'' = 203.910/182.404 = 1.11790,

:√5/2 = 1.11803.

== Argent tuning ==
As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, 1.4000.

It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to

:Perfect fifth = 3/2 = 702.944 cents

:Perfect fourth = 4/3 = 497.056 cents

This fifth is wide by 0.989 cents, and the fourth narrow by the same amount. These errors are of about half the magnitude, and of opposite sign, as their counterparts in 12edo (where these intervals are tuned in the ratio of their bimodular approximants).

A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the [http://en.wikipedia.org/wiki/Silver_ratio silver ratio] (sometimes called the silver mean), ''δs'' = √2 + 1 = 2.4142. On this basis, and by analogy with [[Golden_Meantone|golden meantone]] temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘argent tuning' is proposed instead.

Argent tuning has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.

The continued fraction expansion of the silver ratio has a particularly simple form:

:<math>
\delta_s = \sqrt{2} + 1 = 2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ldots}}}
</math>

As a result, if two intervals ''L'' and ''s'' are tuned in the silver ratio, with ''s = L/δs'', subtracting twice the small interval ''s'' from the large interval ''L'' leaves a remainder of size ''s/δs'':

:<math>
L - 2s = (\delta_s - 2)s = \frac{s}{\delta_s}
</math>

(since <math> \tfrac{1}{\delta_s} = \sqrt{2} - 1 = \delta_s - 2 </math>) and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows. The names are assigned according to Pythagorean conventions (the limma being the Pythagorean semitone 256/243) followed by tempered and just sizes in cents:

{| class="wikitable"
|-
| Octave 1200.00 (1200.00)
| Perfect fourth 497.06 (498.04)
| Tone 205.89 (203.91)
| Limma 85.28 (90.22)
| Pythag. comma 35.32 (23.46)
|-
| Perfect 11th 1697.06 (1698.04)
| Perfect fifth 702.94 (701.96)
| Minor third 291.17 (294.13)
| Apotome 120.61 (113.69)
| 17-tone comma 49.96 (66.76)
|}
Thus for example:

:octave = 2×fourth + tone

:fourth = 2×tone + limma

:tone = 2×limma + Pythagorean comma

:perfect 11th (8/3) = 2×fifth + minor third

:fifth = 2×(minor third) + apotome

When picturing these relationships it makes most musical sense to place the small interval between the two larger ones, as in the ‘continued fraction jigsaw’ below.

The following relationships hold in the table, the first two being valid for the pure intervals as well as their tempered counterparts:

* Subtracting twice an interval from the interval on its left generates the interval on its right.
* An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.
* Adjacent horizontal pairs have ratio ''δs'' = √2 + 1.
* Adjacent vertical pairs have ratio √2.
* Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.

The regularity of this scheme, combined with the fact that the ratios between closely related intervals are of order 2, means that its intervals form orderly sequences in which successive terms are clearly differentiated but of comparable magnitude – highly desirable properties for the formation of musical scales.

In this fractal temperament, multiplying or dividing any interval by the factor ''δs'' = √2 + 1 produces another interval in the temperament. Any tempered interval ''J’'' can be split into three parts, two of equal size ''J’''/''δs'' and the other of size ''J’''/''δs''2.

A similar principle applies to multiplication and division by the factor √2, except that intervals in the top row of the table cannot be divided by √2 to yield another interval in the temperament. These properties means that the temperament would support compositional techniques based on novel types of intervallic augmentation and diminution.

Successive convergents of the silver ratio produce ratios involving [http://en.wikipedia.org/wiki/Pell_number Pell numbers].

:<math>
\sqrt{2} + 1 \approx 2, \; \frac{5}{2}, \; \frac{12}{5}, \; \frac{29}{12}, \; \frac{70}{29}, \; \ldots
</math>

Other approximations to the silver ratio are provided by ratios of consecutive half Pell-Lucas numbers, which are formed by adding consecutive Pell numbers

:<math>
\sqrt{2} + 1 \approx 3, \; \frac{7}{3}, \; \frac{17}{7}, \; \frac{41}{17}, \; \frac{99}{41}, \; \ldots
</math>

This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, 41edo, 70edo etc.).

The accuracy of the argent fifth means that the scheme produces workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur (the next column containing the 29-tone comma and ''minus'' the 41-tone comma).

Figure 2 is a ''continued fraction jigsaw'' showing the sizes of the octave (o), fourth (f), tone (T), limma (sp), Pythagorean comma (p) and 29-tone comma (p29) as tempered by 41edo - an approximation to argent tuning. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.

[[File:Continued_fraction_jigsaw_41edo.png|600px|thumb|none|Figure 2. Continued fraction jigsaw for 41edo]]

Figure 3 is a geometrical representation of argent tuning in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = large tone, mppp = Pythagorean apotome, p = Pythagorean comma.

[[File:Silver_temperament_graphic.png|600px|thumb|none|Figure 3. Geometrical representation of argent tuning]]

Argent tuning tunes the augmented fourth (tritone) and diminished fifth (double minor third) in the ratio 3/2√2, which is also the ratio of the quadratic approximants of 10/7 and 7/5:

:<math>
\frac{q[10/7]}{q[7/5]}= \frac{ \frac{3}{2\sqrt{70}} } { \frac{2} {2\sqrt{35}} } = \frac{3}{2\sqrt{2}}.
</math>

This means that in argent tuning the augmented fourth is very close to 10/7 and the diminished fifth is very close to 7/5. The discrepancy in each case is just 0.175 cents. Another way to express the first of these relationships is

:<math>
3 \left( \frac{1}{2\sqrt{6}} - \frac{1}{4\sqrt{3}} \right) \approx \frac{3}{2\sqrt{70}},
</math>

which after squaring both sides leads to √2 ≈ 99/70, a well-known approximation which can be confirmed by noting that 99/70 = √(2 + 1/4900).

As a consequence of these relationships the tempered diatonic semitone (85.281 cents) is close to 21/20 (84.467 cents), the tempered chromatic semitone (120.606 cents) is close to 15/14 (119.443 cents), and the tempered Pythagorean comma (35.325 cents) is close to 50/49 (34.976 cents).

If these 7-limit intervals are considered to be tempered to their 3-limit counterparts argent is an example of hemifamity temperament. Hemifamity (5120/5103) is the bimodular comma formed from 10/7 and 9/8

By the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem Gelfond-Schneider theorem] the frequency ratios of all argent intervals (''r'' = 2√2''a''+''b'', where'' a'' and ''b'' are integers) are transcendental, with the exception of octave multiples (''a'' = 0). The frequency ratio of the tempered perfect eleventh (8/3 = 2.6666...) is the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant Gelfond-Schneider constant] or Hilbert number, 2√2 = 2.665144...

==Golden temperaments==
It has been shown in an example above that the ratio of the large tone (''T'' = 9/8) to the small tone (''t'' = 10/9) is closely approximated by

:<math>
\frac{T}{t} = \frac{\sqrt{5}}{2}
</math>

It follows that

:<math>
\frac{T + \frac{t}{2}}{t} = \frac{\sqrt{5} + 1}{2} = \varphi
</math>

where ''ϕ'' = 1.61803... is the golden ratio.

If a Fibonacci sequence of intervals is formed from the pair of intervals ''T'' - ''t''/2 and ''t'', and extended in both directions, it can thus be expected that the ratios between successive intervals in this sequence will also be close to ''ϕ''. The sequence formed in this way is Sequence 1 in the following table.

{| class="wikitable"
|-
| Sequence 1:
| ''t''/2 - 3''c''
| 2''c''
| ''t''/2 ''- c''
| ''T - t''/2
| ''t''
| ''T + t''/2
| ''M + t''/2
| 2''M''
|-
| Sequence 2:
| ''magic''
| ''diesis''
| ''chroma''
| ''semitone''
| ''t''
| ''mp''
| ''f - c''
| ''m6p - c''
|-
| Difference:
| -3''σ''/2
| ''σ''
| -''σ''/2
| ''σ''/2
| 0
| ''σ''/2
| ''σ''/2
| ''σ''
|-
| Seq 1 ratios:
|
| 1.6120
| 1.6204
| 1.6171
| 1.6184
| 1.6179
| 1.6181
| 1.6180
|-
| Seq 2 ratios:
|
| 1.3865
| 1.7212
| 1.5810
| 1.6325
| 1.6125
| 1.6201
| 1.6172
|}
where ''f'' = 4/3, ''T'' = 9/8, ''t'' = 10/9, ''M'' = 5/4, ''magic'' = 3125/3072, ''diesis'' = 128/125, ''chroma'' = 25/24, ''semitone'' = 16/15, ''mp'' = 32/27, ''c'' = ''syntonic comma'' = 81/80, ''m6p'' = 128/81, ''σ'' = ''schisma'' = 32805/32768.

The ratios between successive intervals in Sequence 1 are shown in the row labelled ‘Seq 1 ratios’, and are indeed close to ''ϕ''.

Sequence 2 is another Fibonacci sequence of intervals which differ from those in Sequence 1 by small amounts of the order of one schisma (''σ''), as indicated by the row marked 'Difference' (which is itself a Fibonacci sequence).

The ratios of consecutive pairs of intervals in Sequence 2 are shown in the row labelled ‘Seq 2 ratios’. They approximate ''ϕ'' rather less accurately.

A suitable name for 5-limit tunings in which the intervals in either Sequence 1 or Sequence 2, or both, are tempered to exactly ''ϕ'' would be ‘golden temperaments’.

Tempering the Sequence 2 ratios to ''ϕ'' while tuning the octave pure and tempering out the syntonic comma yields [[Golden_Meantone|golden meantone]] temperament.

Tempering the Sequence 1 ratios to ''ϕ'' yields a range of temperaments which can be made extremely accurate by, for example, tuning the octave and fifth (and therefore all Pythagorean intervals) pure. In this temperament the errors in the intervals ''s, t'', ''M'' and ''m''=6/5 are all ±0.02106 cents.

Tempering out the schisma tunes Sequences 1 and 2 identically so that the ratios between consecutive intervals can be fixed at ''ϕ'' in both sequences. Normalised to a pure octave, the resulting temperament, ‘golden schismatic’, has a fifth of 701.791061 cents (error -0.163 cents) and a major third of 385.671509 cents (error -0.642 cents).

== Pythagorean triples of quadratic approximants ==
If the quadratic approximants ''q''1, q''2 and ''q''3 of a set of three intervals ''J''1, ''J''2 and ''J''3 satisfy

:<math>
q_1^2 + q_2^2 = q_3^2
</math>

they can be said to form a [http://en.wikipedia.org/wiki/Pythagorean_triple Pythagorean triple].

The following are three examples. In the first and third cases, their counterparts in 12edo, ''J''1', ''J''2' and ''J''3', are also Pythagorean triples:

{| class="wikitable"
|-
| | ''J''1
| | ''J''2
| | ''J''3
| | ''q''1
| | ''q''2
| | ''q''3
| | ''J''1'
| | ''J''2'
| | ''J''3'
|-
| | 6/5
| | 5/4
| | 4/3
| | 1/2√30
| | 1/4√5
| | 1/4√3
| | 3
| | 4
| | 5
|-
| | 4/3
| | 12/5
| | 5/2
| | 1/4√3
| | 7/4√15
| | 3/2√10
| |
| |
| |
|-
| | 8/5
| | 12/5
| | 8/3
| | 3/4√10
| | 7/4√15
| | 5/4√6
| | 8
| | 15
| | 17
|}

==A small 34edo comma==
As [[Gene_Ward_Smith|Gene Ward Smith]] has noted, the 5-limit comma |-433 -137 280> (''selenia'') is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using quadratic approximants.

It can be shown, using a suitable [[Comma-based_lattices|comma-based lattice]], that every comma tempered out by 34edo can be expressed as an integer linear combination of the [[Gammic_node|gammic comma]] |-29 -11 20> (4.769 cents) and the ''semisuper'' comma (''[[vishnuzma|vishnuzma]]'') |23 6 -14> (3.338 cents). In particular,

:''selenia'' = 7 ''gammic'' - 10 ''semisuper''

So to prove that ''selenia'' is small we must show that ''gammic/semisuper'' ≈ 10/7.

''Gammic'' and ''semisuper'' are both bimodular commas:

:''gammic'' = ''b''(6/5,5/4)

:''semisuper'' = ''b''(25/24,4/3)

Using a result given in the section on bimodular commas, the size of ''b''(''J''1,''J''2) can be estimated using

:<math>
b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 - J_1^2) b_m
</math>

Estimating ''J''2 and ''J''1 with their quadratic approximants we then have

:<math>
b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 - q_1^2) b_m
</math>

For ''gammic'':

:''J''₁= 6/5, ''J''₂= 5/4

:''v''₁ = 1/11, ''v''₂ = 1/9, ''b''m = 1

:''q''₁² = (1/4)(1/30), ''q''₂''² ='' (1/4)(1/20)

:''gammic'' = ''b''(''J''₁,''J''₂) ≈ (1/12) (1/30 - 1/20) = (1/12) (1/60)

For ''semisuper:''

:''J''₁= 25/24, ''J''₂= 4/3

:''v''₁ = 1/49, ''v''₂ = 1/7, ''b''m = 1/7

:''q''₁² = (1/4)(1/600), ''q''₂''² ='' (1/4)(1/12)

:''semisuper'' = ''b''(''J''₁,''J''₂) ≈ (1/12) (1/12 - 1/600)(1/7) = (1/12) (7/600)

Therefore

:''gammic/semisuper'' ≈ 10/7

as required.

To estimate the size of ''selenia'' we must quantify the error in this ratio. A more accurate analysis gives

:<math>
\begin{align}
b(J_1,J_2) &\approx \left( \frac{1}{3} \left(q_2^2 - q_1^2\right) - \frac{2}{15} \left( q_2^4 - q_1^4 \right) \right) b_m \\
&= \frac{1}{3} \left( q_2^2 - q_1^2 \right) \left( 1 - \frac{2}{5} \left(q_1^2 + q_2^2\right) \right) b_m
\end{align}
</math>

''b''(''J''1,''J''2) we should multiply them by

:<math>
f = 1 - \frac{2}{5} \left( q_1^2 + q_2^2 \right)
</math>

Thus a better estimate for ''gammic/semisuper'' is

:<math>
\frac{\text{gammic}}{\text{semisuper}} \approx \frac{10 f_{\text{gamma}}} {7 f_{\text{semisuper}}}
</math>

from which it follows that

:''selenia'' = 7 ''gammic'' - 10 ''semisuper''

:: ≈ 7 ''gammic'' (''f''''gammic'' - ''f''''semisuper'')/''f''''gammic''

Putting in the numbers:

:''f''''gammic'' = 1 - (2/5) (1/4) (1/30 + 1/20) = 1 - 1/120

:''f''''semisuper'' = 1 - (2/5)(1/4) (1/600 + 1/12) = 1 - (1/120) (51/50)

:''f''''gammic'' - ''f''''semisuper'' = 1/6000

Therefore

:''selenia'' ≈ 7 ''gammic'' (1/6000) (120/119) = ''gammic''/850 = 0.00561 cents

which is within 20% of the accurate value, 0.00476 cents. (The discrepancy is due to the influence of terms in ''q''6, which become significant when the ''f'' values are very similar.)

In summary, the reason ''selenia'' is small (compared to ''gammic'' and ''semisuper'') is because the quadratic approximants of ''gammic'' and ''semisuper'' are in the ratio 10/7. The reason it is ''very'' small (of order ''gammic''/1000 rather than ''gammic''/10) is because the fractional errors in those approximants are almost the same. That in turn is because the squares of the source intervals of these bimodular commas have nearly the same sum. Note that the quadratic approximants of three of these intervals form a Pythagorean triple:

:<math>
q \left( \frac{6}{5} \right) ^ 2 + q \left( \frac{5}{4} \right) ^ 2 = q \left( \frac{4}{3} \right) ^ 2
</math>

and (''q''(25/24))2 , being small in comparison to the other terms, compromises this equality only slightly.

= Sources and acknowledgements =
This article is based on original research by [[Martin_Gough|Martin Gough]]. See [[:File:Bimod_Approx_2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants.

Argent tuning is based on the continued fraction convergents of <math>sqrt{2}</math>, which have been known since ancient times.
This application to tuning appears to have been first made by [[Erv Wilson]], who described it under the name '2-zig/2-zag' in a [http://anaphoria.com/meruthree.pdf note] dated December 1996, with a comment claiming it as his answer to [[Joseph Yasser]] "at about 1950".
The same construction was later arrived at independently by [[Graham Breed]] and Paul Hahn, who described it in posts ([https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12592.html#12599 #12599], [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12637.html#12670 #12670]) to the Yahoo tuning list on 10 and 12 August 2000.

Thanks to [[Gene_Ward_Smith|Gene Ward Smith]] for the Gelfond-Schneider result.

[[Category:Essays]]

Logarithmic approximants

2026-05-05T16:51:14Z

Sintel: Cleanup formatting and fixing some errors.

A ''logarithmic approximant'' (or ''approximant'' for short) is an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:

* Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?
* Why are certain commas small, and roughly how small are they?
* Why does the 3-limit framework produce aesthetically pleasing scale structures?

The exact size, in cents, of an interval with frequency ratio ''r'' is

:<math>
J_c = 1200 \log_2{r} = 1200 \frac{\ln{r}}{\ln{2}}
</math>

where for just intervals r is rational and can be written as the ratio of two integers:

:<math>
r = \frac{n}{d}
</math>

When manipulating approximants it is convenient to work with a different logarithmic base, in which the interval is defined as

:<math>
J = \frac{1}{2} \ln{r}
</math>

This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e2 = 7.38906... This unit interval can conveniently be termed the dineper (dNp), being twice the size of the natural unit for logarithmic measurement, the Neper.

Comparing the two units of measurement we find

:1 dineper = 2400/ln(2) = 3462.468 cents

which is about 1.4 semitones short of three octaves.

The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus 3/2 is the perfect fifth. This can also be expressed by an explicit function: if bim(r) = (r-1)/(r+1), then r = bim(r). The inverse function can be written mib(v) = (1+v)/(1-v).

Three types of approximants are described here:

* Bimodular approximants (first order rational approximants)
* Padé approximants of order (1,2) (second order rational approximants)
* Quadratic approximants

= Bimodular approximants =

== Definition ==
The bimodular approximant of an interval with frequency ratio ''r = n/d'' is

:<math>
v = \frac{r-1}{r+1}
</math>

''v ''can thus be expressed as

:<math>
\begin{align}
v &= \frac{n-d}{n+d} \\
&= \text{(frequency difference) / (frequency sum)} \\
&= \frac{1}{2} \text{(frequency difference) / (mean frequency)}
\end{align}
</math>

''r'' can be retrieved from ''v'' using the inverse relation

:<math>
r = \frac{1+v}{1-v}
</math>

== Properties ==
When ''r'' is small, ''v'' provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.

Noting that the exact size (in dineper units) of the interval with frequency ratio ''r'' is

:<math>
J = \frac{1}{2} \ln{r}
</math>

the relationship between ''v'' and ''J'' can be expressed as

:<math>
v = \frac{r-1}{r+1} = \frac{e^{2J}-1}{e^{2J}+1} = \tanh{J} = J - \frac{1}{3}J^3 + \frac{2}{15}J^5 - \ldots
</math>

which shows that ''v'' ≈ ''J'' and provides an indication of the size and sign of the error involved in this approximation.

''J'' can be expressed in terms of ''v'' as

:<math>
J = \tanh^{-1}{v} = v + \frac{1}{3}v^3 + \frac{1}{5}v^5 - \ldots
</math>

The function ''v(r)'' is the order (1,1) [http://en.wikipedia.org/wiki/Pad%C3%A9_approximant Padé approximant] of the function ''J(r) =''½ ln ''r'' in the region of ''r'' = 1, which has the property of matching the function value and its first and second derivatives at this value of ''r''. The bimodular approximant function is thus accurate to second order in ''r'' - 1.

As an example, the size of the perfect fifth (in dNp units) is

:<math>
J = \frac{1}{2} \ln \left( \frac{3}{2} \right) = 0.20273\ldots
</math>

The bimodular approximant for this interval (''r'' = 3/2) is

:<math>
v = \frac{\frac{3}{2} - 1}{\frac{3}{2} + 1} = \frac{3 - 2}{3 + 2} = \frac{1}{5} = 0.2
</math>

and the Taylor series indicates that the error in this value is about

:<math>
-\frac{1}{3}v^3 = -0.00267 \ldots
</math>

The approximants of superparticular intervals are reciprocals of odd integers, as shown in Figure 1.

[[File:Low-order_superparticular_intervals.png|frame|none|Bimodular approximants for low-order superparticular intervals]]

If ''v''[''J''] denotes the bimodular approximant of an interval ''J'' with frequency ratio ''r'',

<math>
\begin{align}
v[-J] &= -v[J] \\
v[J_1 +J_2] &= \frac{v_1+v_2}{1+v_1 v_2}
\end{align}
</math>

This last result is equivalent to the identity expressing tanh(''J''1 + ''J''1) in terms of tanh(''J''1) and tanh(''J''2).

== Bimodular approximants and equal temperaments ==
While bimodular approximants have historically been used as a means of estimating the sizes of very small intervals, they remain reasonably accurate as the interval size is increased to an octave or more. And being easily computable, they provide a quick means of comparing the relative sizes of intervals. For example:

Two perfect fourths (''r'' = 4/3, ''v'' = 1/7) approximate a minor seventh (''r'' = 9/5, ''v'' = 2/7)

Three major thirds (''r'' = 5/4, ''v'' = 1/9) or two 7/5s (''v'' = 1/6) or five 8/7s (''v'' = 1/15) approximate an octave (''r'' = 2/1,'' v'' = 1/3)

Bimodular approximants (abbreviated to ‘approximants’ here) also provide simple explanations for the properties of certain equal temperaments.

Tuning the perfect fourth and perfect fifth in the ratio of their approximants (1/7 : 1/5 = 5 : 7) and adjusting their sum to a pure octave yields 12edo (considered as a 3-limit temperament). This is an example of the high accuracy typically obtainable from a tempering policy which takes two intervals which are similar in size and not too large, tunes them in their approximant ratio, and normalises their sum to a pure interval.

Aspects of 12edo considered as a 5-limit temperament can be explained by noting that it tunes the major third, major sixth and octave in the ratio of their approximants (1/9 : 1/4 : 1/3 = 4 : 9 : 12). The accuracy here is lower because the octave is of a size where the approximant has a significant error, and tuning the octave pure assigns the entire error to the smaller intervals.

Tuning the major third and perfect fifth in the in the ratio of their approximants (1/9 : 1/5) and tuning the fifth pure yields [[Carlos_Alpha|Carlos Alpha]].

Tuning the minor third and perfect fifth in the in the ratio of their approximants (1/11 : 1/5) and tuning the fifth pure yields [[Carlos_Beta|Carlos Beta]].

Tuning the minor third and major third in the ratio of their approximants (1/11 : 1/9) and adjusting their sum to a perfect fifth yields [[Carlos_Gamma|Carlos Gamma]] . This temperament has high accuracy because it conforms to the policy noted above.

Tuning the octave pure while preserving the ratios specified above yields, respectively, 31edo, 19edo and 34edo.

Tuning the intervals 9/7, 7/5 and 5/3 in the ratio of their approximants (1/8 : 1/6 : 1/4 = 3 : 4 : 6) and adjusting their sum to a perfect twelfth yields the equally tempered [[Bohlen–Pierce scale]].

Tuning the intervals 11/9, 9/7, 3/2 and 5/3 in the ratio of their approximants (1/10 : 1/8 : 1/5 : 1/4 = 4 : 5 : 8 : 10) and adjusting their sum to a major tenth yields [[88cET|88 cent equal temperament]].

Relationships of this sort can be identified in all equal temperaments.

== Bimodular commas ==
As a consequence of the near-rational interval relationships implied by approximants, any pair of source intervals can be used to define a comma.

Given two intervals ''J''1 and ''J''2 (with ''J''1 < ''J''2) and their approximants ''v''1 and ''v''2, we define the ''bimodular residue'' as

:<math>
b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}
</math>

and using the Taylor series expansion of ''J''(''v'') we find

:<math>
b_r(J_1,J_2) ≈ \frac{1}{3} (v_2^2 - v_1^2) = \frac{1}{3} (v_2 + v_1)(v_2 - v_1)
</math>

The bimodular comma is obtained from the bimodular residue by means of a rational multiplier which ensures that the result (in line with the usual convention applied to commas) is a linear combination of ''J''1 and ''J''2 with integer coefficients sharing no common factor:

:<math>
b(J_1,J_2) ≈ b_m(J_1,J_2) b_r(J_1,J_2)
</math>

where

:<math>
v_1 = \frac{j_1}{g_1}, v_2 = \frac{j_2}{g_2}
</math>

and (with rare exceptions)

:<math>
b_m(J_1,J_2) \approx \frac{\text{LCM}(j_1, j_2)}{\text{GCD}(g_1, g_2)}
</math>

The bimodular residue is accurately estimated by

:<math>
b_r(J_1,J_2) \approx \frac{1}{3} (J_1 + J_2) (J_2 - J_1)
</math>

and therefore

:<math>
b(J_1,J_2) \approx \frac{1}{3} (J_1 + J_2) (J_2 - J_1) b_m
</math>

===Examples===
If the source intervals are the perfect fourth (''f'' = 4/3'')'' and the perfect fifth (''F'' = 3/2), then ''v''1 = 1/7, ''v''2 = 1/5, and ''b'' is the Pythagorean comma:

:<math>
b(F,f) = b_r(F,f) = \frac{F}{\frac{1}{5}} - \frac{f}{\frac{1}{7}} = 5F - 7f
</math>

If the source intervals are the perfect fourth (''f'' = 4/3) and the minor seventh (''m''7 = 9/5), then ''v''1 = 1/7, ''v''2 = 2/7, ''b''r = 2/7 and ''b'' is the syntonic comma:

:<math>
b(m_7,f) = b_r(m_7,f) = \frac{2}{7} \left( \frac{m_7}{\frac{2}{7}} - \frac{f}{\frac{1}{7}} \right) = m_7 - 2f
</math>

For further examples of bimodular commas, and a fuller account of bimodular approximant theory, refer to [[:File:Bimod_Approx_2014-6-8.pdf|this paper]]. See also [[Don_Page_comma|Don Page comma]] (another name for this type of comma).

= Padé approximants of order (1,2) =

== Definition ==
In the section on bimodular approximants it was shown than an interval of logarithmic size ''J'' (measured in dineper units) is related to its bimodular approximant by

:<math>
J = \tanh^{-1}{v} = v + \frac{1}{3}v^3 + \frac{1}{5}v^5 - \ldots
</math>

where

:<math>
v = \frac{r-1}{r+1}
</math>

and ''r'' is the interval’s frequency ratio.

Another way to express this relationship is with a continued fraction:

:<math>
J = \tanh^{-1}{v} = \cfrac{v} {1-\cfrac{v^2}{3 - \cfrac{4v^2}{5 - \cfrac{9v^2}{7 - \ldots}}}}
</math>

The first convergent of this continued fraction is ''v'', the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is

:<math>
y = \cfrac{v}{1-\cfrac{v^2}{3}}
</math>

Values of this rational approximant for some simple 5-limit intervals are shown in the table below.

{| class="wikitable"
|-
| Interval ''J''
| (1,2) Padé approximant ''y''
|-
| Perfect twelfth = 3/1
| 6/11
|-
| Octave = 2/1
| 9/26
|-
| Major sixth = 5/3
| 12/47
|-
| Perfect fifth = 3/2
| 15/74
|-
| Perfect fourth = 4/3
| 21/146
|-
| Major third = 5/4
| 27/242
|}
The denominators of these fractions rapidly get large, so this type of approximant has limited usefulness. However, when combined with bimodular approximants it has occasional value in explaining apparent numerical coincidences and the smallness of the associated commas. For example:

(3/1) / (6/5) = 6.0257 ≈ (6/11) / (1/11) = 6 (kleisma)

(3/1) / (7/4) = 1.9632 ≈ (6/11) / (3/11) = 2 (septimal diesis = 49/48)

(2/1) / (7/6) = 4.4966 ≈ (9/26) / (1/13) = 9/2 (|-11 -9 0 9> comma)

(2/1) / (27/25) = 9.0065 ≈ (9/26) / (1/26) = 9 (ennealimma)

(5/3) / (49/45) = 5.9986 ≈ (12/47) / (2/47) = 6

(5/3) / (25/22) = 3.9960 ≈ (12/47) / (3/47) = 4

(5/3) / (26/21) = 2.3918 ≈ (12/47) / (5/47) = 12/5

(5/3) / (27/20) = 1.7022 ≈ (12/47) / (7/47) = 12/7

(3/2) / (20/17) = 2.4949 ≈ (15/74) / (6/74) = 5/2

= Quadratic approximants =

== Definition ==
The quadratic approximant ''q'' of an interval ''J'' with frequency ratio ''r'' = ''n''/''d'' is

:<math>
\begin{align}
q(r) &= \frac{1}{2} \left( r^{\frac{1}{2}} - r^{-\frac{1}{2}} \right) \\
&= \frac{1}{2} \left( e^{J} - e^{-J} \right) \\
&= \sinh{J} \\
&= J + \frac{1}{3!} J^3 + \frac{1}{5!} J^5 + \ldots
\end{align}
</math>

If this is compared with the expression for the bimodular approximant,

:<math>
v = \tanh{J} = J - \frac{1}{3}J^3 + \frac{2}{15}J^5 - \ldots
</math>

it is apparent that ''q'' is about twice as accurate as ''v'', with an error of opposite sign.

While ''v'' is the frequency difference divided by twice the arithmetic frequency mean, ''q'' is the frequency difference divided by twice the geometric frequency mean:

:<math>
q = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}
</math>

''r'' can be retrieved from ''q'' using

:<math>
\sqrt{r} = q + \sqrt{1+q^2}
</math>

The following are the quadratic approximants of some simple 5-limit intervals:

{| class="wikitable"
|-
| Interval ''J''
| Quadratic approximant ''q''
|-
| Perfect twelfth = 3/1
| 1/√3
|-
| Octave = 2/1
| 1/2√2
|-
| Minor seventh = 9/5
| 2/3√5
|-
| Pythagorean minor seventh = 16/9
| 7/24
|-
| Major sixth = 5/3
| 1/√15
|-
| Minor sixth = 8/5
| 3/4√10
|-
| Perfect fifth = 3/2
| 1/2√6
|-
| Perfect fourth = 4/3
| 1/4√3
|-
| Major third = 5/4
| 1/4√5
|-
| Minor third = 6/5
| 1/2√30
|-
| Pythagorean minor third = 32/27
| 5/24√6
|-
| Large tone = 9/8
| 1/12√2
|-
| Small tone = 10/9
| 1/6√10
|-
| Diatonic semitone = 16/15
| 1/8√15
|-
| Chroma = 25/24
| 1/20√6
|-
| Syntonic comma = 81/80
| 1/72√5
|}

Expressed in terms of the bimodular approximant, ''v = j/g'',

:<math>
q = \frac{v}{\sqrt{1-v^2}} = \frac{j}{\sqrt{g^2-j^2}}
</math>

Quadratic approximants of just intervals thus have the form ''q = j/√k'', where ''j'' and ''k'' are integers and ''j''2'' + k = g''2 is a perfect square.

The presence of a square root in the denominator of ''q'' (except where ''J'' is a double interval) means that quadratic approximants do not, on the whole, imply approximate rational ratios between just intervals or commas of the conventional type. Their interest stems from the fact that ratios involving integer square roots are expressible as repeating continued fractions.

== Properties ==
If ''v''[''J''] and ''q''[''J''] denote, respectively, the bimodular and quadratic approximants of an interval ''J'' with frequency ratio ''r'', and ''q''n denotes ''q''[''J''n] , then

:<math>
\begin{align}
v &= \tanh{J}, \; q = \sinh{J}, \; \frac{q}{v} = \cosh{J} \\
\sqrt{r} &= e^J = q(\frac{1}{v} + 1) \\
\frac{1}{\sqrt{r}} &= e^{-J} = q(\frac{1}{v} - 1) \\
\frac{1}{q^2} &= \frac{1}{v^2} - 1 \\

q[-J] &= -q[J] \\
q[J_2 + J_1] &= q_1 q_2 (\frac{1}{v_2} + \frac{1}{v_1}) \\
q[J_2 - J_1] &= q_1 q_2 (\frac{1}{v_2} - \frac{1}{v_1}) \\
\frac {q[J_2 + J_1]}{q[J_2 - J_1]} &= \frac{v_2+v_1}{v_2-v_1} \\
q[J_2 + J_1] q[J_2 - J_1] &= q_2^2 - q_1^2 \\
\end{align}
</math>

The last two expressions are rational for just intervals, and the last result is equivalent to the hyperbolic trigonometric identity

:<math>
\sinh{(J_2 + J_1)} \sinh{(J_2 - J_1)} = \sinh^2{J_2} - \sinh^2{J_1}
</math>

Where two quadratic approximants have the same square root in the denominator their ratio is rational. This seems to suggest a new source of approximate rational interval ratios, and therefore a new source of commas, but in this situation the approximants always represent the sum and difference of a pair of just intervals, and their ratio can be derived by an alternative route using the bimodular approximants of those intervals.

For example

:<math>
\frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6
</math>

where ''large tone'' = 9/8.

However, this can also be derived from bimodular approximants. Using

:<math>
\frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1}
</math>

with ''J''2 = ''F'' =3/2 and ''J''1 = ''f'' = 4/3 this gives

:<math>
\begin{align}
\frac{\text{octave}}{\text{large tone}} &\approx \frac{q[F+f]}{q[F-f]} \\
&= \frac{v[F] + v[f]}{v[F] - v[f]} \\
&= \frac{1/5 + 1/7}{1/5 - 1/7} = 6
\end{align}
</math>

The quadratic approximant ''q'' of a double interval 2''J'' (for example, the ditone) is rational, which suggests using ½ ''q''(''r''2) as a rational approximant of ''J'' (where ''J'' has frequency ratio ''r''):

:<math>
\frac{1}{2} q(r^2)
= \frac{1}{4} \left( r - \frac{1}{r} \right)
= \frac{1}{2} \sinh{2J}
= J + \frac{2}{3}J^3 + \frac{2}{15}J^5 + \ldots
</math>

However, this approximant is both less accurate and more complex than the corresponding bimodular approximant, and consequently of limited value.

The most interesting approximate interval ratios derivable from quadratic approximants are irrational.

== Relative sizes of intervals between 3 frequencies in arithmetic progression ==

=== Theorem ===
If three harmonics of a fundamental frequency form an arithmetic progression, then the ratio of the logarithmic sizes of the intervals formed between the lower and upper pairs of harmonics is close to the geometric mean of these intervals’ frequency ratios.

=== Remarks ===
If the harmonics have indices ''n - m, n'' and ''n + m'', the two intervals have reduced frequency ratios ''n/(n - m)'' and ''(n + m)/n''. It can be assumed that ''n'' and ''m'' have no common factor.

''m'' is the [[Superpartient|degree of epimoricity]] of the intervals. When ''m'' = 1 the intervals are adjacent superparticular intervals.

The geometric mean of the frequency ratios is the frequency ratio corresponding to the arithmetic mean of the intervals.

=== Proof ===
The ratio of the intervals as estimated from their quadratic approximants is

:<math>
\frac{m}{2\sqrt{n(n-m)}} / \frac{m}{2\sqrt{(n+m)n}} = \sqrt{\frac{n+m}{n-m}}
</math>

which is the geometric mean of their frequency ratios.

=== Examples ===
The ratio of the perfect fifth, ''F'' = 3/2, to the perfect fourth, ''f'' = 4/3, as estimated by their quadratic approximants (1/2√6 and 1/4√3) is √2, which is the frequency ratio of the arithmetic mean of these intervals (the half-octave).

:''F/f'' = 701.955/498.045 = 1.40942,

:√2 = 1.41421.

The ratio of the large tone, ''T'' = 9/8, to the small tone, ''t'' = 10/9, as estimated by their quadratic approximants (1/12√2 and 1/6√10) is √5/2, which is the frequency ratio of the mean tone.

:''T/t'' = 203.910/182.404 = 1.11790,

:√5/2 = 1.11803.

== Argent tuning ==
As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, 1.4000.

It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to

:Perfect fifth = 3/2 = 702.944 cents

:Perfect fourth = 4/3 = 497.056 cents

This fifth is wide by 0.989 cents, and the fourth narrow by the same amount. These errors are of about half the magnitude, and of opposite sign, as their counterparts in 12edo (where these intervals are tuned in the ratio of their bimodular approximants).

A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the [http://en.wikipedia.org/wiki/Silver_ratio silver ratio] (sometimes called the silver mean), ''δs'' = √2 + 1 = 2.4142. On this basis, and by analogy with [[Golden_Meantone|golden meantone]] temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘argent tuning' is proposed instead.

Argent tuning has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.

The continued fraction expansion of the silver ratio has a particularly simple form:

:<math>
\delta_s = \sqrt{2} + 1 = 2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ldots}}}
</math>

As a result, if two intervals ''L'' and ''s'' are tuned in the silver ratio, with ''s = L/δs'', subtracting twice the small interval ''s'' from the large interval ''L'' leaves a remainder of size ''s/δs'':

:<math>
L - 2s = (\delta_s - 2)s = \frac{s}{\delta_s}
</math>

(since <math> \tfrac{1}{\delta_s} = \sqrt{2} - 1 = \delta_s - 2 </math>) and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows. The names are assigned according to Pythagorean conventions (the limma being the Pythagorean semitone 256/243) followed by tempered and just sizes in cents:

{| class="wikitable"
|-
| Octave 1200.00 (1200.00)
| Perfect fourth 497.06 (498.04)
| Tone 205.89 (203.91)
| Limma 85.28 (90.22)
| Pythag. comma 35.32 (23.46)
|-
| Perfect 11th 1697.06 (1698.04)
| Perfect fifth 702.94 (701.96)
| Minor third 291.17 (294.13)
| Apotome 120.61 (113.69)
| 17-tone comma 49.96 (66.76)
|}
Thus for example:

:octave = 2×fourth + tone

:fourth = 2×tone + limma

:tone = 2×limma + Pythagorean comma

:perfect 11th (8/3) = 2×fifth + minor third

:fifth = 2×(minor third) + apotome

When picturing these relationships it makes most musical sense to place the small interval between the two larger ones, as in the ‘continued fraction jigsaw’ below.

The following relationships hold in the table, the first two being valid for the pure intervals as well as their tempered counterparts:

* Subtracting twice an interval from the interval on its left generates the interval on its right.
* An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.
* Adjacent horizontal pairs have ratio ''δs'' = √2 + 1.
* Adjacent vertical pairs have ratio √2.
* Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.

The regularity of this scheme, combined with the fact that the ratios between closely related intervals are of order 2, means that its intervals form orderly sequences in which successive terms are clearly differentiated but of comparable magnitude – highly desirable properties for the formation of musical scales.

In this fractal temperament, multiplying or dividing any interval by the factor ''δs'' = √2 + 1 produces another interval in the temperament. Any tempered interval ''J’'' can be split into three parts, two of equal size ''J’''/''δs'' and the other of size ''J’''/''δs''2.

A similar principle applies to multiplication and division by the factor √2, except that intervals in the top row of the table cannot be divided by √2 to yield another interval in the temperament. These properties means that the temperament would support compositional techniques based on novel types of intervallic augmentation and diminution.

Successive convergents of the silver ratio produce ratios involving [http://en.wikipedia.org/wiki/Pell_number Pell numbers].

:<math>
\sqrt{2} + 1 \approx 2, \; \frac{5}{2}, \; \frac{12}{5}, \; \frac{29}{12}, \; \frac{70}{29}, \; \ldots
</math>

Other approximations to the silver ratio are provided by ratios of consecutive half Pell-Lucas numbers, which are formed by adding consecutive Pell numbers

:<math>
\sqrt{2} + 1 \approx 3, \; \frac{7}{3}, \; \frac{17}{7}, \; \frac{41}{17}, \; \frac{99}{41}, \; \ldots
</math>

This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, 41edo, 70edo etc.).

The accuracy of the argent fifth means that the scheme produces workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur (the next column containing the 29-tone comma and ''minus'' the 41-tone comma).

Figure 2 is a ''continued fraction jigsaw'' showing the sizes of the octave (o), fourth (f), tone (T), limma (sp), Pythagorean comma (p) and 29-tone comma (p29) as tempered by 41edo - an approximation to argent tuning. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.

[[File:Continued_fraction_jigsaw_41edo.png|600px|thumb|none|Figure 2. Continued fraction jigsaw for 41edo]]

Figure 3 is a geometrical representation of argent tuning in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = large tone, mppp = Pythagorean apotome, p = Pythagorean comma.

[[File:Silver_temperament_graphic.png|600px|thumb|none|Figure 3. Geometrical representation of argent tuning]]

Argent tuning tunes the augmented fourth (tritone) and diminished fifth (double minor third) in the ratio 3/2√2, which is also the ratio of the quadratic approximants of 10/7 and 7/5:

:<math>
\frac{q[10/7]}{q[7/5]}= \frac{ \frac{3}{2\sqrt{70}} } { \frac{2} {2\sqrt{35}} } = \frac{3}{2\sqrt{2}}.
</math>

This means that in argent tuning the augmented fourth is very close to 10/7 and the diminished fifth is very close to 7/5. The discrepancy in each case is just 0.175 cents. Another way to express the first of these relationships is

:<math>
3 \left( \frac{1}{2\sqrt{6}} - \frac{1}{4\sqrt{3}} \right) \approx \frac{3}{2\sqrt{70}},
</math>

which after squaring both sides leads to √2 ≈ 99/70, a well-known approximation which can be confirmed by noting that 99/70 = √(2 + 1/4900).

As a consequence of these relationships the tempered diatonic semitone (85.281 cents) is close to 21/20 (84.467 cents), the tempered chromatic semitone (120.606 cents) is close to 15/14 (119.443 cents), and the tempered Pythagorean comma (35.325 cents) is close to 50/49 (34.976 cents).

If these 7-limit intervals are considered to be tempered to their 3-limit counterparts argent is an example of hemifamity temperament. Hemifamity (5120/5103) is the bimodular comma formed from 10/7 and 9/8

By the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem Gelfond-Schneider theorem] the frequency ratios of all argent intervals (''r'' = 2√2''a''+''b'', where'' a'' and ''b'' are integers) are transcendental, with the exception of octave multiples (''a'' = 0). The frequency ratio of the tempered perfect eleventh (8/3 = 2.6666...) is the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant Gelfond-Schneider constant] or Hilbert number, 2√2 = 2.665144...

==Golden temperaments==
It has been shown in an example above that the ratio of the large tone (''T'' = 9/8) to the small tone (''t'' = 10/9) is closely approximated by

:<math>
\frac{T}{t} = \frac{\sqrt{5}}{2}
</math>

It follows that

:<math>
\frac{T + \frac{t}{2}}{t} = \frac{\sqrt{5} + 1}{2} = \varphi
</math>

where ''ϕ'' = 1.61803... is the golden ratio.

If a Fibonacci sequence of intervals is formed from the pair of intervals ''T'' - ''t''/2 and ''t'', and extended in both directions, it can thus be expected that the ratios between successive intervals in this sequence will also be close to ''ϕ''. The sequence formed in this way is Sequence 1 in the following table.

{| class="wikitable"
|-
| Sequence 1:
| ''t''/2 - 3''c''
| 2''c''
| ''t''/2 ''- c''
| ''T - t''/2
| ''t''
| ''T + t''/2
| ''M + t''/2
| 2''M''
|-
| Sequence 2:
| ''magic''
| ''diesis''
| ''chroma''
| ''semitone''
| ''t''
| ''mp''
| ''f - c''
| ''m6p - c''
|-
| Difference:
| -3''σ''/2
| ''σ''
| -''σ''/2
| ''σ''/2
| 0
| ''σ''/2
| ''σ''/2
| ''σ''
|-
| Seq 1 ratios:
|
| 1.6120
| 1.6204
| 1.6171
| 1.6184
| 1.6179
| 1.6181
| 1.6180
|-
| Seq 2 ratios:
|
| 1.3865
| 1.7212
| 1.5810
| 1.6325
| 1.6125
| 1.6201
| 1.6172
|}
where ''f'' = 4/3, ''T'' = 9/8, ''t'' = 10/9, ''M'' = 5/4, ''magic'' = 3125/3072, ''diesis'' = 128/125, ''chroma'' = 25/24, ''semitone'' = 16/15, ''mp'' = 32/27, ''c'' = ''syntonic comma'' = 81/80, ''m6p'' = 128/81, ''σ'' = ''schisma'' = 32805/32768.

The ratios between successive intervals in Sequence 1 are shown in the row labelled ‘Seq 1 ratios’, and are indeed close to ''ϕ''.

Sequence 2 is another Fibonacci sequence of intervals which differ from those in Sequence 1 by small amounts of the order of one schisma (''σ''), as indicated by the row marked 'Difference' (which is itself a Fibonacci sequence).

The ratios of consecutive pairs of intervals in Sequence 2 are shown in the row labelled ‘Seq 2 ratios’. They approximate ''ϕ'' rather less accurately.

A suitable name for 5-limit tunings in which the intervals in either Sequence 1 or Sequence 2, or both, are tempered to exactly ''ϕ'' would be ‘golden temperaments’.

Tempering the Sequence 2 ratios to ''ϕ'' while tuning the octave pure and tempering out the syntonic comma yields [[Golden_Meantone|golden meantone]] temperament.

Tempering the Sequence 1 ratios to ''ϕ'' yields a range of temperaments which can be made extremely accurate by, for example, tuning the octave and fifth (and therefore all Pythagorean intervals) pure. In this temperament the errors in the intervals ''s, t'', ''M'' and ''m''=6/5 are all ±0.02106 cents.

Tempering out the schisma tunes Sequences 1 and 2 identically so that the ratios between consecutive intervals can be fixed at ''ϕ'' in both sequences. Normalised to a pure octave, the resulting temperament, ‘golden schismatic’, has a fifth of 701.791061 cents (error -0.163 cents) and a major third of 385.671509 cents (error -0.642 cents).

== Pythagorean triples of quadratic approximants ==
If the quadratic approximants ''q''1, q''2 and ''q''3 of a set of three intervals ''J''1, ''J''2 and ''J''3 satisfy

:<math>
q_1^2 + q_2^2 = q_3^2
</math>

they can be said to form a [http://en.wikipedia.org/wiki/Pythagorean_triple Pythagorean triple].

The following are three examples. In the first and third cases, their counterparts in 12edo, ''J''1', ''J''2' and ''J''3', are also Pythagorean triples:

{| class="wikitable"
|-
| | ''J''1
| | ''J''2
| | ''J''3
| | ''q''1
| | ''q''2
| | ''q''3
| | ''J''1'
| | ''J''2'
| | ''J''3'
|-
| | 6/5
| | 5/4
| | 4/3
| | 1/2√30
| | 1/4√5
| | 1/4√3
| | 3
| | 4
| | 5
|-
| | 4/3
| | 12/5
| | 5/2
| | 1/4√3
| | 7/4√15
| | 3/2√10
| |
| |
| |
|-
| | 8/5
| | 12/5
| | 8/3
| | 3/4√10
| | 7/4√15
| | 5/4√6
| | 8
| | 15
| | 17
|}

==A small 34edo comma==
As [[Gene_Ward_Smith|Gene Ward Smith]] has noted, the 5-limit comma |-433 -137 280> (''selenia'') is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using quadratic approximants.

It can be shown, using a suitable [[Comma-based_lattices|comma-based lattice]], that every comma tempered out by 34edo can be expressed as an integer linear combination of the [[Gammic_node|gammic comma]] |-29 -11 20> (4.769 cents) and the ''semisuper'' comma (''[[vishnuzma|vishnuzma]]'') |23 6 -14> (3.338 cents). In particular,

:''selenia'' = 7 ''gammic'' - 10 ''semisuper''

So to prove that ''selenia'' is small we must show that ''gammic/semisuper'' ≈ 10/7.

''Gammic'' and ''semisuper'' are both bimodular commas:

:''gammic'' = ''b''(6/5,5/4)

:''semisuper'' = ''b''(25/24,4/3)

Using a result given in the section on bimodular commas, the size of ''b''(''J''1,''J''2) can be estimated using

:<math>
b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 - J_1^2) b_m
</math>

Estimating ''J''2 and ''J''1 with their quadratic approximants we then have

:<math>
b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 - q_1^2) b_m
</math>

For ''gammic'':

:''J''₁= 6/5, ''J''₂= 5/4

:''v''₁ = 1/11, ''v''₂ = 1/9, ''b''m = 1

:''q''₁² = (1/4)(1/30), ''q''₂''² ='' (1/4)(1/20)

:''gammic'' = ''b''(''J''₁,''J''₂) ≈ (1/12) (1/30 - 1/20) = (1/12) (1/60)

For ''semisuper:''

:''J''₁= 25/24, ''J''₂= 4/3

:''v''₁ = 1/49, ''v''₂ = 1/7, ''b''m = 1/7

:''q''₁² = (1/4)(1/600), ''q''₂''² ='' (1/4)(1/12)

:''semisuper'' = ''b''(''J''₁,''J''₂) ≈ (1/12) (1/12 - 1/600)(1/7) = (1/12) (7/600)

Therefore

:''gammic/semisuper'' ≈ 10/7

as required.

To estimate the size of ''selenia'' we must quantify the error in this ratio. A more accurate analysis gives

:<math>
\begin{align}
b(J_1,J_2) &\approx \left( \frac{1}{3} \left(q_2^2 - q_1^2\right) - \frac{2}{15} \left( q_2^4 - q_1^4 \right) \right) b_m \\
&= \frac{1}{3} \left( q_2^2 - q_1^2 \right) \left( 1 - \frac{2}{5} \left(q_1^2 + q_2^2\right) \right) b_m
\end{align}
</math>

''b''(''J''1,''J''2) we should multiply them by

:<math>
f = 1 - \frac{2}{5} \left( q_1^2 + q_2^2 \right)
</math>

Thus a better estimate for ''gammic/semisuper'' is

:<math>
\frac{\text{gammic}}{\text{semisuper}} \approx \frac{10 f_{\text{gamma}}} {7 f_{\text{semisuper}}}
</math>

from which it follows that

:''selenia'' = 7 ''gammic'' - 10 ''semisuper''

:: ≈ 7 ''gammic'' (''f''''gammic'' - ''f''''semisuper'')/''f''''gammic''

Putting in the numbers:

:''f''''gammic'' = 1 - (2/5) (1/4) (1/30 + 1/20) = 1 - 1/120

:''f''''semisuper'' = 1 - (2/5)(1/4) (1/600 + 1/12) = 1 - (1/120) (51/50)

:''f''''gammic'' - ''f''''semisuper'' = 1/6000

Therefore

:''selenia'' ≈ 7 ''gammic'' (1/6000) (120/119) = ''gammic''/850 = 0.00561 cents

which is within 20% of the accurate value, 0.00476 cents. (The discrepancy is due to the influence of terms in ''q''6, which become significant when the ''f'' values are very similar.)

In summary, the reason ''selenia'' is small (compared to ''gammic'' and ''semisuper'') is because the quadratic approximants of ''gammic'' and ''semisuper'' are in the ratio 10/7. The reason it is ''very'' small (of order ''gammic''/1000 rather than ''gammic''/10) is because the fractional errors in those approximants are almost the same. That in turn is because the squares of the source intervals of these bimodular commas have nearly the same sum. Note that the quadratic approximants of three of these intervals form a Pythagorean triple:

:<math>
q \left( \frac{6}{5} \right) ^ 2 + q \left( \frac{5}{4} \right) ^ 2 = q \left( \frac{4}{3} \right) ^ 2
</math>

and (''q''(25/24))2 , being small in comparison to the other terms, compromises this equality only slightly.

= Sources and acknowledgements =
This article is based on original research by [[Martin_Gough|Martin Gough]]. See [[:File:Bimod_Approx_2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants.

The tuning referred to here as argent tuning appears to have been discovered 'about 1950' by Erv Wilson, who named it [http://anaphoria.com/meruthree.pdf 2-zig/2-zag]'. It was later rediscovered independently by [[Graham_Breed|Graham Breed]] and Paul Hahn, who described it in posts ([https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12592.html#12599 #12599], [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12637.html#12670 #12670]) to the Yahoo tuning list on 10 and 12 August 2000.

Thanks to [[Gene_Ward_Smith|Gene Ward Smith]] for the Gelfond-Schneider result.

[[Category:Essays]]

User:Sintel

2026-05-04T19:04:21Z

Sintel: update and link

PhD student computational biology who makes music in his spare time. I also like maths and coding. Currently working on [https://github.com/Sin-tel/tessera Tessera], a DAW for microtonal music.

I am interested in free pitch, JI, higher rank temperaments and medium-sized EDOs (e.g. 31, 34, 41). My intention is mostly to use familiar diatonic structures in spaces with more degrees of freedom.

Instruments I play: piano, clarinet, (bass) guitar, various cheap things.

My native language is Dutch, feel free to ask me if you need help translating some source.

I maintain a temperament calculator at: https://sintel.pythonanywhere.com/

Music links:

https://soundcloud.com/sintel-music

https://sintel.bandcamp.com/

Coding:

https://github.com/Sin-tel/
[[Category:User en-5]]
[[Category:User fr-2]]
[[Category:User nl-N]]

==Userspace pages==
{{Special:Prefixindex|prefix={{FULLPAGENAME}}/|hideredirects=1|stripprefix=1}}

Metallic harmony

2026-04-25T16:58:23Z

Sintel: {{idio}}

{{Distinguish|Metallic harmonic series}}

'''Metallic harmony'''{{idio}} is an approach of building harmony based on sevenths rather than thirds to produce consonant, resolved, sonorities. Specifically, metallic harmony treats [[7/4]] as the most consonant interval next to the octave. As a result, tunings that do not approximate 7/4 decently do not support metallic harmony. In addition, there must be an additional size of seventh/sixth that "clicks" with the 7/4 to form a triad. For example, [[13/7]] can be used as the upper interval, creating a 4:7:13 chord in [[just intonation]]. Other intervals such as [[12/7]] and [[19/11]] are notable possibilities. These seventh chords have a characteristic metallic and somewhat cold quality which earns them their name.

== Basic chord types ==
There are symmetrical and asymmetrical chords in metallic harmony. Asymmetric chords have a more rooted sound while symmetrical chords sound more ambiguous.

However, because metallic chords use only foreign intervals, they tend to sound exotic, or like metal. The beauty however is that they are capable of expressive harmony if used correctly. The two types of asymmetric triads soft and hard. Soft triads place the 7/4 on top of the chord while hard triads place it on the bottom. The names come from that if when the 7/4 is placed on top, the chord sounds smoother and mellow, on the bottom the chord has a rougher, [[JI]] crunch to it. Both are very nice chords but soft chords are more dissonant than hard chords regardless of what the names might suggest.

== EDOs that support metallic harmony ==
Some notable [[edo]]s for metallic harmony include {{EDOs|5, 10, 15, 26, 31, and 36}} and for more complex harmony [[21edo]] (or supersets of it like [[63edo]]) can be tried. [[10edo]] contains a 7 note MOS ([[3L 4s]]) that contains 3 hard and 3 soft metallic triads in addition to one symmetrical triad. In addition Metallic harmony can also be used to harmonize Mavila in [[16edo]]. However Mavila 7 only contains two hard and soft triads on degrees 1 4 and 2 6. Mavila 9 adds two more soft triads but there are still only two hard triads. Therefore, metallic harmony in 16 EDO doesn't work nearly as well in it would in 10 or [[20edo]].

[[Category:7-limit]]
[[Category:13-limit]]
[[Category:19-limit]]
[[Category:Harmony]]
[[Category:Metallic]]
[[Category:Mavila]]