Xenharmonic Wiki - User contributions [en]

12edo

2026-06-11T08:25:29Z

Eliora: /* Subsets and supersets */ fractional octave temperaments

{{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. Various [[12th-octave temperaments]] that augment 12edo exist, most prominent examples being [[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]]

384edo

2026-06-10T17:22:33Z

Eliora: +listen category

{{Infobox ET}}
{{ED intro}}

== Theory ==
384edo is [[consistent]] in the [[7-odd-limit]]. The equal temperament [[tempering out|tempers out]] the [[misty comma]] {{monzo| 26 -12 -3 }} and provides the [[optimal patent val]] for the 5-limit misty temperament. It also supports the 324 & 384 temperament which is an extension of misty that divides the octave into 12, reaches [[11/7]] and [[13/8]] within two steps, and is representable with a 36-note scale. It also tempers out the 5-limit tritriple comma {{monzo| 31 20 -27 }} in the 5-limit, and [[3136/3125]], [[5120/5103]], [[Landscape comma|250047/250000]], and the [[mistisma]] 458752/455625 in the 7-limit.

=== As a tuning standard ===
A step of 384edo is known as a '''pentamu''' (fifth MIDI-resolution unit, 5mu, 25 = 32 equal divisions of the [[12edo]] semitone). The internal data structure of the 5mu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and one other bit which is not used.

=== Prime harmonics ===
{{Harmonics in equal|384|intervals=prime}}

=== Subsets and supersets ===
Since 384 factors into {{factorization|384}}, 384edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, and 192 }}.

== See also ==
* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
** [[24edo]] (1mu tuning)
** [[48edo]] (2mu tuning)
** [[96edo]] (3mu tuning)
** [[192edo]] (4mu tuning)
** [[768edo]] (6mu tuning)
** [[1536edo]] (7mu tuning)
** [[3072edo]] (8mu tuning)
** [[6144edo]] (9mu tuning)
** [[12288edo]] (10mu tuning)
** [[24576edo]] (11mu tuning)
** [[49152edo]] (12mu tuning)
** [[98304edo]] (13mu tuning)
** [[196608edo]] (14mu tuning)

== Scales ==
* 19 13 19 13 19 13 19 13 19 13 19 13 19 13 19 13 19 13 19 13 19 13 19 13 - 324 & 384[24]
* 13 13 5 13 13 5 13 13 5 13 13 5 ... (12 times) ... 13 13 5 13 13 5 13 13 5 - 324 & 384[36]
== Music ==

; [[Eliora]]

* [https://www.youtube.com/watch?v=QR5SSfdAyLE ''Unmoveable''] (2026) - 324 & 384 temperament

== External links ==
* [http://tonalsoft.com/enc/number/5mu.aspx 5mu / pentamu] on [[Tonalsoft Encyclopedia]]

[[Category:Listen]]

384edo

2026-06-10T17:22:15Z

Eliora: expand, add music, note significance of the temp as it reaches 11/7 and 13/8 quickly

User:Eliora/324edo

2026-06-10T07:01:17Z

Eliora: Created page with "{{harmonics in equal|324}}"

User:Eliora/Proposed concept names

2026-06-10T06:59:10Z

Eliora:

== Anemoian (name later to be changed) ==
384p & 324p, 11-limit.

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 26411/26244, 250047/250000, 2097152/2096325

[[Mapping]]: {{mapping| 13 0 92 -355 | 0 5 12 14 -16 }}
: Mapping generators: ~1375/1296, ~243/200

[[Optimal tuning]]s:
* [[WE]]: ~1375/1296 = 99.997{{c}}, ~243/200 = 340.539{{c}}
: [[error map]]: {{val| -0.006 +0.038 -0.030 -0.013 }}
* [[CWE]]: ~1375/1296 = 100{{c}}, ~243/200 = 340.549{{c}}
: error map: {{val| 0.000, 0.788, 0.269, -1.146, -0.095 }}

{{Optimal ET sequence|legend=1| 60, 444, 384 }}

[[Badness]] (Sintel): 21.211

== Luminance (tuning) ==
https://www.wolframalpha.com/input?i=plot+sqrt%28%28RiemannSiegelZ%5B%282*pi*x%2Fln%282%29%29%5D%29%5E2%2B%28%28DivisorSum%5Bx%2C+%23+%26%5D-x%29%2Fx%29%5E2%29%2C+x+from+1+to+100

Luminance is a measure of an equal temperament based on both is [[abundancy index]] and [[zeta peak integer edo]] position. It is equal to sqrt(Z^2+A^2), where Z is the zeta value while A is the abundancy index.

Increasingly larger luminance values: {{EDOs|2, 3, 5, 7, 10, 12, 22, 24, 31, 41, 53, 87, ...}}
== Natrium ==
The natrium tempers out the {{monzo|403 -77 -121}} comma in the 5-limit, not only splitting the octave in 11, but using [[1125/1024]] as a generator, eleven of which plus one step of [[11edo]] make [[3/1]].
== Phosphorus ==

1125 & 2460, 23-limit.

1125 patent val branching tempers out the [[flashma]] and therefore is to be named white phosphorus, and 1125g val branching is to be named red phosphorus.
== Strontium ==

Described as the 1178 & 7334 temperament in the 19-limit.

== Cadmium ==
Described as the 624 & 4320 temperament upwards to the 23-limit.

In the 23-limit, the gen is mapped to [[70/69]].

== Thulium ==
https://sintel.pythonanywhere.com/result?subgroup=11&reduce=on&tenney=on&target=&edos=&commas=%5B-4%2C+2%2C+-11%2C+2%2C+6%3E%0D%0A%5B-21%2C+-14%2C+8%2C+10%2C+-1%3E%0D%0A%5B-25%2C+-12%2C+-3%2C+12%2C+5%3E%0D%0A%5B-17%2C+-16%2C+19%2C+8%2C+-7%3E%0D%0A%5B-29%2C+-10%2C+-14%2C+14%2C+11%3E%0D%0A%5B55%2C+-50%2C+-1%2C+7%2C+2%3E&submit_comma=submit

Subgroup: 2.3.5.7.11

Comma list: 781258401/781250000, 110341894140625/110336743047168, 3590222893590025814933504/3589489938459262943851245

Mapping: [{{val|69 0 4316 -2431 8769}}, {{val|0 1 -38 24 -78}}]

Mapping generators: ~100/99, ~3

{{Optimal ET sequence|legend=1| 759, 7797 }}

== Rutherfordium ==
Rutherfordium is described as the 624 & 4472 temperament in the 23-limit.

== Seaborgium ==
Named after the 106th element, most likely 2756 & 3498 in the 23-limit, but other options are likely.

== Kells ==
436 & 981 temperament.

== Meitnerium ==
981 & 3706 temperament.

== Copernicium ==
Named after the 112th element.

1904 & 3920 temperament.
== Tenessine ==

Described as the 234 & 1053 temperament, defined by tempering together the septimal ennealimma and the aluminium comma.

== Unpentennium ==
Described as the 795 & 3498 temperament and splits the octave into 159.

32edo

2026-06-08T22:09:54Z

Eliora: /* Subsets and supersets */ quick link

{{Infobox ET}}
{{ED intro}}

== Theory ==
32edo is generally the first power-of-2 edo which can be considered to handle [[limit|low-limit]] just intonation at all. It has unambiguous mappings for [[prime]]s up to the [[11-limit]], although [[6/5]] and Pythagorean intervals are especially poorly approximated if going by the [[patent val]] instead of using [[direct approximation|inconsistent approximations]]. Since 32edo is poor at approximating primes and it is a high power of 2, both traditional [[RTT]] and temperament-agnostic [[mos]] theory are of limited usefulness in the system (though it has an [[ultrasoft]] [[smitonic]] with {{nowrap|L/s {{=}} 5/4}}). 32edo's 5:2:1 [[blackdye]] scale {{nowrap|(1 5 2 5 1 5 2 5 1 5)}}, which is melodically comparable to [[31edo]]'s 5:2:1 [[diasem]], is notable for having 412.5¢ neogothic major thirds and 450¢ naiadics in place of the traditional 5-limit and Pythagorean major thirds in 5-limit blackdye, and the 75¢ semitone in place of 16/15. The 712.5¢ sharp fifth and the 675¢ flat fifth correspond to 3/2 and [[40/27]] in 5-limit blackdye, making 5:2:1 blackdye a [[dual-fifth]] scale.

=== As a tuning of other temperaments ===
While even advocates of less-common [[edo]]s can struggle to find something about 32edo worth noting, it does provide an excellent tuning for the [[sixix]] temperament, which [[tempering out|tempers out]] the [[5-limit]] sixix comma, [[3125/2916]], using its 9\32 generator of size 337.5 cents. [[Petr Pařízek]]'s preferred generator for sixix is (128/15)(1/11), which is 337.430 cents and which gives equal error to fifths and major thirds, so 32edo does sixix about as well as sixix can be done. It also can be used (with the 9\32 generator) to tune [[mohavila]], an 11-limit temperament which does not temper out sixix.

It also tempers out [[2048/2025]] in the 5-limit, and [[50/49]] with [[64/63]] in the [[7-limit]], which means it [[support]]s [[pajara]], with a very sharp fifth of 712.5 cents which could be experimented with by those with a penchant for fifths even sharper than the fifth of [[27edo]]; this fifth is in fact very close to the [[minimax tuning]] of the pajara extension [[Diaschismic family #Pajaro|pajaro]], using the 32f val. In the 11-limit it provides the [[optimal patent val]] for the {{nowrap| 15 & 17 }} temperament, tempering out [[55/54]], 64/63, and [[245/242]].

The sharp fifth of 32edo can be used to generate a very unequal [[archy]] (specifically [[oceanfront]]) [[5L 2s|diatonic scale]], with a [[diatonic semitone]] of 5 steps and a [[chromatic semitone]] of only 1. The diatonic [[major third]] (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450¢, approximating [[9/7]] and [[13/10]], and the minor third is 262.5¢, approximating [[7/6]]. Because of the unequalness of the scale, the minor second is reduced to a fifth-tone, but it still strongly resembles "normal" diatonic music, especially for darker [[mode]]s. In addition to the sharp fifth, there is an alternative [[mavila|mavila-like]] flat fifth of 675{{c}} (inherited from [[16edo]]), but it is much more inaccurate and discordant than the sharp fifth.

=== Odd harmonics ===
{{Harmonics in equal|32}}

=== Subsets and supersets ===
Since 32 is a power of two and factors as 25, 32edo contains subset edos {{EDOs| 2, 4, 8, and 16 }}.

See also [[32nd-octave temperaments]].

== Intervals ==
{| class="wikitable"
|-
! Degree
! Cents
! colspan="3" | [[Ups and downs notation]]
! 13-limit Ratios
! Other
|-
| 0
| 0.0
| P1
| perfect unison
| D
| 1/1
|-
| 1
| 37.5
| ^1, m2
| up unison, minor 2nd
| ^D, Eb
| 49/48, 50/49, 45/44
| 46/45, 52/51, 51/50
|-
| 2
| 75.0
| ^m2
| upminor 2nd
| ^Eb
| 22/21, 25/24
| 24/23, 23/22
|-
| 3
| 112.5
| ^^m2
| dupminor 2nd
| ^^Eb
| 16/15
| 49/46
|-
| 4
| 150.0
| vvM2
| dudmajor 2nd
| vvE
| 12/11, 49/45
| 25/23
|-
| 5
| 187.5
| A1, vM2
| aug 1sn, downmajor 2nd
| D#, vE
| 10/9, 39/35
| 19/17
|-
| 6
| 225.0
| M2
| major 2nd
| E
| 8/7, 25/22
| 57/50
|-
| 7
| 262.5
| m3
| minor 3rd
| F
| 7/6, 64/55
| 57/49
|-
| 8
| 300.0
| ^m3
| upminor 3rd
| ^F
| 6/5, 32/27
| 19/16
|-
| 9
| 337.5
| ^^m3
| dupminor 3rd
| ^^F
| 11/9, 39/32, 63/52
| 17/14, 28/23
|-
| 10
| 375.0
| vvM3
| dudmajor 3rd
| vvF#
| 5/4, 26/21, 56/45, 96/77
| 36/29
|-
| 11
| 412.5
| vM3
| downmajor 3rd
| vF#
| 14/11, 33/26, 80/63
| 19/15
|-
| 12
| 450.0
| M3
| major 3rd
| F#
| 13/10, 35/27, 64/49
| 22/17, 57/44
|-
| 13
| 487.5
| P4
| perfect 4th
| G
| 4/3, 33/25, 160/121
| 45/34, 85/64
|-
| 14
| 525.0
| ^4
| up 4th
| ^G
| 27/20, 110/81
| 19/14, 23/17
|-
| 15
| 562.5
| ^^4, ^d5
| dup 4th, updim 5th
| ^^G, ^Ab
| 18/13, 11/8
|-
| 16
| 600.0
| vvA4, ^^d5
| dudaug 4th, dupdim 5th
| vvG#, ^^Ab
| 7/5, 10/7, 99/70, 140/99
| 17/12, 12/17
|-
| 17
| 637.5
| vA4, vv5
| downaug 4th, dud 5th
| vG#, vvA
| 13/9, 16/11
|-
| 18
| 675.0
| v5
| down 5th
| vA
| 40/27, 81/55
| 28/19, 34/23
|-
| 19
| 712.5
| P5
| perfect 5th
| A
| 3/2, 50/33, 121/80
| 68/45, 128/85
|-
| 20
| 750.0
| m6
| minor 6th
| Bb
| 20/13, 54/35, 49/32
| 17/11, 88/57
|-
| 21
| 787.5
| ^m6
| upminor 6th
| ^Bb
| 11/7, 52/33, 63/40
| 30/19
|-
| 22
| 825.0
| ^^m6
| dupminor 6th
| ^^Bb
| 8/5, 21/13, 45/28, 77/48
| 29/18
|-
| 23
| 862.5
| vvM6
| dudmajor 6th
| vvB
| 18/11, 64/39, 104/63
| 28/17, 23/14
|-
| 24
| 900.0
| vM6
| downmajor 6th
| vB
| 5/3, 27/16
| 32/19
|-
| 25
| 937.5
| M6
| major 6th
| B
| 12/7, 55/32
| 98/57
|-
| 26
| 975.0
| m7
| minor 7th
| C
| 7/4, 44/25
| 100/57
|-
| 27
| 1012.5
| ^m7
| upminor 7th
| ^C
| 9/5, 70/39
| 34/19
|-
| 28
| 1050.0
| ^^m7
| dupminor 7th
| ^^C
| 11/6, 90/49
| 46/25
|-
| 29
| 1087.5
| vvM7
| dudmajor 7th
| vvC#
| 15/8
| 92/49
|-
| 30
| 1125.0
| vM7
| downmajor 7th
| vC#
| 21/11, 48/25
| 23/12, 44/23
|-
| 31
| 1162.5
| M7, v8
| major 7th, down 8ve
| C#, vD
| 96/49, 49/25, 88/45
| 45/23, 51/26, 100/51
|-
| 32
| 1200.0
| P8
| 8ve
| D
| 2/1
|
|}

== Notation ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5-szg}}

If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using triple arrows.

=== Kite's ups and downs notation ===
32edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
{{Ups and downs sharpness}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as [[25edo #Sagittal notation|25edo]], and is a subset of the notation for [[64edo #Second-best fifth notation|64b-edo]].

==== Evo flavor ====
<imagemap>
File:32-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 407 0 567 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 407 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:32-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:32-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 367 0 527 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 3067 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:32-EDO_Revo_Sagittal.svg]]
</imagemap>

== Approximation to JI ==
{{Q-odd-limit intervals|32}}

== 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| 51 -32 }}
| {{Mapping| 32 51 }}
| -3.327
| 3.32
| 8.87
|-
| 2.3.7
| 64/63, 46118408/43046721
| {{Mapping| 32 51 90 }}
| -2.950
| 2.76
| 7.38
|- style="border-top: double;"
| 2.3.5
| 648/625, 20480/19683
| {{Mapping| 32 51 75 }} (32c)
| -5.965
| 4.61
| 12.3
|-
| 2.3.5.7
| 64/63, 245/243, 392/375
| {{Mapping| 32 51 75 90 }} (32c)
| -5.027
| 4.31
| 11.5
|- style="border-top: double;"
| 2.3.5
| 2048/2025, 3125/2916
| {{Mapping| 32 51 74 }} (32)
| +0.177
| 4.72
| 12.6
|-
| 2.3.5.7
| 50/49, 64/63, 3125/2916
| {{Mapping| 32 51 75 90 }} (32)
| -1.008
| 4.15
| 11.1
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Associated ratio*
! Temperaments
|-
| 1
| 1\32
| 37.5
| 49/48
| [[Slender]] (32)
|-
| 1
| 9\32
| 262.5
| 7/6
| [[Septimin]] (32f)
|-
| 1
| 9\32
| 337.5
| 6/5
| [[Sixix]] (32f)
|-
| 1
| 13\32
| 487.5
| 4/3
| [[Superpyth]] (32c, 7-limit) / [[ultrapyth]] (32) / [[quasiultra]] (32)
|-
| 1
| 15\32
| 562.5
| 7/5
| [[Progress]] (32cf)
|-
| 2
| 13\32
| 487.5
| 4/3
| [[Pajara]] (32, 7-limit)
|-
| 8
| 14\33 (1\32)
| 487.5 (37.5)
| 4/3 (36/35)
| [[Octonion]] (32cf)
|-
| 16
| 14\33 (1\32)
| 487.5 (37.5)
| 4/3 (45/44)
| [[Sedecic]] (32)
|}
<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

== Delta-rational harmony ==
The tables below show chords that approximate 3-integer-limit [[delta-rational]] chords with least-squares error less than 0.0015.

{| class="wikitable mw-collapsible mw-collapsed sortable"
|+ style="font-size: 105%; white-space: nowrap;" | Fully delta-rational triads
|-
! Steps
! Delta signature
! Least-squares error
|-
| 0,1,2
| +1+1
| 0.00023
|-
| 0,1,3
| +1+2
| 0.00051
|-
| 0,1,4
| +1+3
| 0.00083
|-
| 0,2,3
| +2+1
| 0.00041
|-
| 0,2,4
| +1+1
| 0.00092
|-
| 0,3,4
| +3+1
| 0.00060
|-
| 0,3,11
| +1+3
| 0.00014
|-
| 0,4,11
| +1+2
| 0.00087
|-
| 0,5,8
| +3+2
| 0.00076
|-
| 0,6,16
| +1+2
| 0.00076
|-
| 0,8,26
| +1+3
| 0.00016
|-
| 0,9,23
| +1+2
| 0.00000
|-
| 0,12,17
| +2+1
| 0.00004
|-
| 0,13,20
| +3+2
| 0.00008
|-
| 0,15,21
| +2+1
| 0.00007
|-
| 0,18,27
| +3+2
| 0.00000
|-
| 0,22,30
| +2+1
| 0.00030
|-
| 0,25,31
| +3+1
| 0.00062
|}

{| class="wikitable mw-collapsible mw-collapsed sortable"
|+ style="font-size: 105%; white-space: nowrap;" | Partially delta-rational tetrads
|-
! Steps
! Delta signature
! Least-squares error
|-
| 0,1,2,3
| +1+?+1
| 0.00056
|-
| 0,1,2,4
| +1+?+2
| 0.00100
|-
| 0,1,3,4
| +1+?+1
| 0.00085
|-
| 0,1,16,17
| +2+?+3
| 0.00091
|-
| 0,1,16,18
| +1+?+3
| 0.00093
|-
| 0,1,17,18
| +2+?+3
| 0.00058
|-
| 0,1,17,19
| +1+?+3
| 0.00051
|-
| 0,1,18,19
| +2+?+3
| 0.00025
|-
| 0,1,18,20
| +1+?+3
| 0.00009
|-
| 0,1,19,20
| +2+?+3
| 0.00010
|-
| 0,1,19,21
| +1+?+3
| 0.00034
|-
| 0,1,20,21
| +2+?+3
| 0.00045
|-
| 0,1,20,22
| +1+?+3
| 0.00078
|-
| 0,1,21,22
| +2+?+3
| 0.00081
|-
| 0,1,30,31
| +1+?+2
| 0.00076
|-
| 0,2,3,4
| +2+?+1
| 0.00082
|-
| 0,2,6,11
| +1+?+3
| 0.00077
|-
| 0,2,7,12
| +1+?+3
| 0.00009
|-
| 0,2,8,13
| +1+?+3
| 0.00097
|-
| 0,2,12,13
| +3+?+2
| 0.00072
|-
| 0,2,12,15
| +1+?+2
| 0.00060
|-
| 0,2,13,14
| +3+?+2
| 0.00032
|-
| 0,2,13,16
| +1+?+2
| 0.00018
|-
| 0,2,14,15
| +3+?+2
| 0.00009
|-
| 0,2,14,17
| +1+?+2
| 0.00097
|-
| 0,2,15,16
| +3+?+2
| 0.00050
|-
| 0,2,16,17
| +3+?+2
| 0.00093
|-
| 0,2,17,21
| +1+?+3
| 0.00061
|-
| 0,2,18,20
| +2+?+3
| 0.00050
|-
| 0,2,18,22
| +1+?+3
| 0.00025
|-
| 0,2,19,21
| +2+?+3
| 0.00020
|-
| 0,2,20,22
| +2+?+3
| 0.00091
|-
| 0,3,4,8
| +2+?+3
| 0.00098
|-
| 0,3,5,9
| +2+?+3
| 0.00007
|-
| 0,3,7,12
| +1+?+2
| 0.00048
|-
| 0,3,8,13
| +1+?+2
| 0.00071
|-
| 0,3,9,16
| +1+?+3
| 0.00074
|-
| 0,3,10,17
| +1+?+3
| 0.00057
|-
| 0,3,17,23
| +1+?+3
| 0.00026
|-
| 0,3,18,19
| +2+?+1
| 0.00082
|-
| 0,3,18,21
| +2+?+3
| 0.00075
|-
| 0,3,18,22
| +1+?+2
| 0.00025
|-
| 0,3,19,20
| +2+?+1
| 0.00035
|-
| 0,3,19,21
| +1+?+1
| 0.00019
|-
| 0,3,19,22
| +2+?+3
| 0.00030
|-
| 0,3,19,23
| +1+?+2
| 0.00094
|-
| 0,3,20,21
| +2+?+1
| 0.00013
|-
| 0,3,20,22
| +1+?+1
| 0.00066
|-
| 0,3,21,22
| +2+?+1
| 0.00063
|-
| 0,3,26,31
| +1+?+3
| 0.00016
|-
| 0,4,5,12
| +1+?+2
| 0.00059
|-
| 0,4,5,15
| +1+?+3
| 0.00060
|-
| 0,4,8,13
| +2+?+3
| 0.00013
|-
| 0,4,11,20
| +1+?+3
| 0.00049
|-
| 0,4,12,18
| +1+?+2
| 0.00042
|-
| 0,4,13,14
| +3+?+1
| 0.00079
|-
| 0,4,13,16
| +1+?+1
| 0.00088
|-
| 0,4,14,15
| +3+?+1
| 0.00035
|-
| 0,4,14,16
| +3+?+2
| 0.00024
|-
| 0,4,14,17
| +1+?+1
| 0.00024
|-
| 0,4,15,16
| +3+?+1
| 0.00009
|-
| 0,4,15,17
| +3+?+2
| 0.00060
|-
| 0,4,16,17
| +3+?+1
| 0.00055
|-
| 0,4,17,25
| +1+?+3
| 0.00058
|-
| 0,4,19,23
| +2+?+3
| 0.00040
|-
| 0,4,21,26
| +1+?+2
| 0.00030
|-
| 0,4,23,30
| +1+?+3
| 0.00062
|-
| 0,5,6,9
| +3+?+2
| 0.00013
|-
| 0,5,7,19
| +1+?+3
| 0.00069
|-
| 0,5,9,17
| +1+?+2
| 0.00047
|-
| 0,5,10,16
| +2+?+3
| 0.00038
|-
| 0,5,11,13
| +2+?+1
| 0.00067
|-
| 0,5,11,15
| +1+?+1
| 0.00027
|-
| 0,5,11,22
| +1+?+3
| 0.00052
|-
| 0,5,12,14
| +2+?+1
| 0.00015
|-
| 0,5,13,15
| +2+?+1
| 0.00099
|-
| 0,5,15,22
| +1+?+2
| 0.00090
|-
| 0,5,16,26
| +1+?+3
| 0.00034
|-
| 0,5,19,24
| +2+?+3
| 0.00051
|-
| 0,5,23,29
| +1+?+2
| 0.00015
|-
| 0,5,24,25
| +3+?+1
| 0.00090
|-
| 0,5,24,27
| +1+?+1
| 0.00085
|-
| 0,5,25,26
| +3+?+1
| 0.00034
|-
| 0,5,25,27
| +3+?+2
| 0.00011
|-
| 0,5,25,28
| +1+?+1
| 0.00058
|-
| 0,5,26,27
| +3+?+1
| 0.00023
|-
| 0,5,26,28
| +3+?+2
| 0.00096
|-
| 0,5,27,28
| +3+?+1
| 0.00081
|-
| 0,6,9,14
| +1+?+1
| 0.00013
|-
| 0,6,11,18
| +2+?+3
| 0.00020
|-
| 0,6,12,21
| +1+?+2
| 0.00064
|-
| 0,6,15,18
| +3+?+2
| 0.00025
|-
| 0,6,18,26
| +1+?+2
| 0.00075
|-
| 0,6,19,25
| +2+?+3
| 0.00062
|-
| 0,6,20,22
| +2+?+1
| 0.00074
|-
| 0,6,20,24
| +1+?+1
| 0.00046
|-
| 0,6,20,31
| +1+?+3
| 0.00043
|-
| 0,6,21,23
| +2+?+1
| 0.00025
|-
| 0,6,24,31
| +1+?+2
| 0.00091
|-
| 0,7,8,12
| +3+?+2
| 0.00097
|-
| 0,7,8,14
| +1+?+1
| 0.00076
|-
| 0,7,8,24
| +1+?+3
| 0.00043
|-
| 0,7,9,11
| +3+?+1
| 0.00053
|-
| 0,7,9,12
| +2+?+1
| 0.00018
|-
| 0,7,9,13
| +3+?+2
| 0.00054
|-
| 0,7,9,20
| +1+?+2
| 0.00020
|-
| 0,7,10,12
| +3+?+1
| 0.00028
|-
| 0,7,12,20
| +2+?+3
| 0.00010
|-
| 0,7,14,24
| +1+?+2
| 0.00004
|-
| 0,7,15,29
| +1+?+3
| 0.00028
|-
| 0,7,17,22
| +1+?+1
| 0.00091
|-
| 0,7,19,26
| +2+?+3
| 0.00073
|-
| 0,7,22,25
| +3+?+2
| 0.00065
|-
| 0,7,23,26
| +3+?+2
| 0.00086
|-
| 0,7,27,31
| +1+?+1
| 0.00074
|-
| 0,7,28,30
| +2+?+1
| 0.00044
|-
| 0,7,29,31
| +2+?+1
| 0.00074
|-
| 0,8,11,23
| +1+?+2
| 0.00070
|-
| 0,8,11,28
| +1+?+3
| 0.00080
|-
| 0,8,13,22
| +2+?+3
| 0.00070
|-
| 0,8,14,20
| +1+?+1
| 0.00072
|-
| 0,8,15,19
| +3+?+2
| 0.00057
|-
| 0,8,16,18
| +3+?+1
| 0.00031
|-
| 0,8,16,19
| +2+?+1
| 0.00023
|-
| 0,8,16,27
| +1+?+2
| 0.00085
|-
| 0,8,17,19
| +3+?+1
| 0.00063
|-
| 0,8,19,27
| +2+?+3
| 0.00084
|-
| 0,8,23,28
| +1+?+1
| 0.00055
|-
| 0,9,10,15
| +3+?+2
| 0.00092
|-
| 0,9,11,30
| +1+?+3
| 0.00012
|-
| 0,9,13,20
| +1+?+1
| 0.00100
|-
| 0,9,13,26
| +1+?+2
| 0.00021
|-
| 0,9,17,29
| +1+?+2
| 0.00062
|-
| 0,9,19,28
| +2+?+3
| 0.00096
|-
| 0,9,20,26
| +1+?+1
| 0.00070
|-
| 0,9,21,25
| +3+?+2
| 0.00055
|-
| 0,9,22,24
| +3+?+1
| 0.00031
|-
| 0,9,22,25
| +2+?+1
| 0.00034
|-
| 0,9,23,25
| +3+?+1
| 0.00077
|-
| 0,10,13,17
| +2+?+1
| 0.00066
|-
| 0,10,14,25
| +2+?+3
| 0.00076
|-
| 0,10,16,21
| +3+?+2
| 0.00034
|-
| 0,10,18,25
| +1+?+1
| 0.00004
|-
| 0,10,27,29
| +3+?+1
| 0.00080
|-
| 0,10,27,30
| +2+?+1
| 0.00029
|-
| 0,10,27,31
| +3+?+2
| 0.00077
|-
| 0,10,28,30
| +3+?+1
| 0.00040
|-
| 0,11,12,18
| +3+?+2
| 0.00040
|-
| 0,11,12,28
| +1+?+2
| 0.00038
|-
| 0,11,13,16
| +3+?+1
| 0.00049
|-
| 0,11,14,17
| +3+?+1
| 0.00085
|-
| 0,11,14,26
| +2+?+3
| 0.00077
|-
| 0,11,16,24
| +1+?+1
| 0.00085
|-
| 0,11,18,22
| +2+?+1
| 0.00057
|-
| 0,11,21,26
| +3+?+2
| 0.00058
|-
| 0,11,23,30
| +1+?+1
| 0.00023
|-
| 0,12,15,24
| +1+?+1
| 0.00060
|-
| 0,12,18,21
| +3+?+1
| 0.00014
|-
| 0,12,21,29
| +1+?+1
| 0.00078
|-
| 0,12,23,27
| +2+?+1
| 0.00036
|-
| 0,12,25,30
| +3+?+2
| 0.00084
|-
| 0,13,16,21
| +2+?+1
| 0.00057
|-
| 0,13,19,28
| +1+?+1
| 0.00023
|-
| 0,13,22,25
| +3+?+1
| 0.00019
|-
| 0,13,27,31
| +2+?+1
| 0.00012
|-
| 0,14,15,30
| +2+?+3
| 0.00004
|-
| 0,14,17,24
| +3+?+2
| 0.00028
|-
| 0,14,20,25
| +2+?+1
| 0.00048
|-
| 0,14,26,29
| +3+?+1
| 0.00012
|-
| 0,15,16,20
| +3+?+1
| 0.00002
|-
| 0,15,24,29
| +2+?+1
| 0.00028
|-
| 0,16,20,31
| +1+?+1
| 0.00042
|-
| 0,16,24,31
| +3+?+2
| 0.00051
|-
| 0,17,21,29
| +3+?+2
| 0.00090
|-
| 0,17,22,28
| +2+?+1
| 0.00062
|-
| 0,17,23,27
| +3+?+1
| 0.00039
|-
| 0,18,25,31
| +2+?+1
| 0.00007
|-
| 0,18,26,30
| +3+?+1
| 0.00001
|-
| 0,19,21,30
| +3+?+2
| 0.00014
|-
| 0,20,21,26
| +3+?+1
| 0.00032
|-
| 0,21,24,29
| +3+?+1
| 0.00026
|}

== Octave stretch or compression ==
Whether [[octave stretch]], shrink or neither is advised for 32edo depends on which [[val]]s one wishes to use.

For 32, pure-octaves, or slight compression (~0.5{{c}}), works well.

For 32f, moderate compression (~2{{c}}) works well. This is close to [[zpi|133zpi]] (32.07edo).

For 32c or 32cf, substantial compression (3-4{{c}}) is well suited.

For 32be, substantial ''stretch'' works (~5{{c}}). This is close to [[zpi|132zpi]] (31.86edo).

The graph shows [[zeta]] near 32edo.

[[File:plot32.png|alt=plot32.png|plot32.png]]

== Instruments ==
* [[Lumatone mapping for 32edo]]

== Music ==
=== Modern renderings ===
; {{W|Koji Kondo}}
* [https://www.youtube.com/shorts/OUlNwN-bAsc "Lost Woods" from ''The Legend of Zelda: Ocarina of Time OST''] (1998) – covered by [[Bryan Deister]] (2025)

=== 21st century ===
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=yMokW3-0vIs ''Beyond the Grid''] (2024)

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/nTQfjPjeee8 ''32edo improv''] (2025)
* ''Licorice Hearted'' (2026)
** [https://www.youtube.com/shorts/zFgw-AfGEcQ short 1] · [https://www.youtube.com/shorts/ocgMIf4xopo short 2]

; [[groundfault]]
* "Winter's Mortal Hope", from ''A New Dusk'' (2024) – [https://groundfco.bandcamp.com/track/winters-mortal-hope-32edo Bandcamp] | [https://www.youtube.com/watch?v=1bnEO8vGvbo&t=1357 YouTube (22:37–26:00)]

; [[Claudi Meneghin]]
* ''Canon on Twinkle Twinkle Little Star''
** [https://www.youtube.com/watch?v=y2G6Fs2HMUs for organ] (2023) · [https://www.youtube.com/watch?v=JWRGLa59ZwY for baroque oboe & viola] (2024)

; [[Petr Pařízek]]
* [https://web.archive.org/web/20201127014118/http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg ''Sixix'']

; [[Billy Stiltner]]
* [https://billystiltner.bandcamp.com/album/1332 ''1332''] (2019)

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127013223/http://micro.soonlabel.com/32edo/32-32-32-nothing-less-will-do.mp3 ''32 32 32 Nothing Less Will Do'']

; [[Stephen Weigel]]
* [https://www.youtube.com/watch?v=00kH3CqSgMY "Zinnia Riplet"], featured in [https://spectropolrecords.bandcamp.com/album/possible-worlds-vol-4 ''Possible Worlds Vol. 4''] (2019) of Spectropol Records
* [https://soundcloud.com/overtoneshock/admins-hot-tub-32-edo ''Admin's Hot Tub''] (2019)

[[Category:Listen]]
[[Category:Sixix]]
{{Todo|add scales list}}

1789edo

2026-06-05T05:59:36Z

Eliora:

{{Infobox ET}}
{{ED intro}}

== Theory ==
1789edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. Otherwise, it is excellent in approximating harmonics [[5/1|5]], [[9/1|9]], [[11/1|11]], [[13/1|13]] and [[21/1|21]], making it suitable for a 2.9.5.21.11.13 [[subgroup]] interpretation.

For higher harmonics, 1789edo can be adapted for use with the 2.9.5.21.11.13.29.31.47.59.61 subgroup. Perhaps the most notable fact about 1789edo is that it [[tempering out|tempers out]] the jacobin comma ([[6656/6655]]), and it is also consistent on the subgroup 2.5.11.13 of the comma, which is quite appropriate for edo's number. Although there are temperaments which are better suited for tempering this comma, 1789edo is unique in that its number is the hallmark year of the French Revolution, thus making the tempering of the jacobin comma on topic.

1789bd val, {{Val|1789 '''2836''' 4154 '''5023'''}} is better tuned than the patent val, and it tempers out 67108864/66976875, 48828125/48771072, 96889010407/96855122250.

=== Odd harmonics ===
{{Harmonics in equal|1789}}

=== Jacobin temperaments ===
{{Main| The Jacobins }}

Since 1789edo tempers out the jacobin comma and it is defined by stacking three 11/8s to reach 13/10, one can use that as a generator. The resulting temperament is {{nowrap|37 & 1789}}, called onzonic. Name "onzonic" comes from the French word for eleven, ''onze''.

1789edo supports the 2.5.11.13.19 subgroup temperament called ''estates general'' defined as {{nowrap|1789 & 3125}}. This is referencing the fact that Estates General were called by Louis XVI on 5th May 1789, written as 05/05, and 3125 is 5 to the 5th power and also provides an optimal patent val for tempering out the jacobin comma, contuing the lore.

=== Other ===
1789edo can be used for the finite "French decimal" temperament—that is, where all the interval targets in just intonation are expressed as terminating decimals. For example, [[5/4]], [[25/16]], [[128/125]], [[32/25]], 625/512, etc.

Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[ed5/4]] temperaments—more exactly those which are divisors of 576, and that includes all from [[2ed5/4]] to [[9ed5/4]], skipping [[7ed5/4]]. One such scale which stands for [[4ed5/4]], is a tuning for the [[hemiluna]] temperament in the 1789bd val in the 13-limit.

1789edo has an essentially perfect [[9/8]], a very common interval. 1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9/8s over the octave in this case. It is defined as a {{nowrap|460 & 1789}} temperament.

Since 1789edo has a very precise 31/29, it supports tricesimoprimal miracloid—a version of secor with 31/29 as the generator and a flat, meantone-esque fifth of about 692.23 cents. Using the maximal evenness method, we find a {{nowrap|52 & 1789}} temperament. Best subgroup for it is 2.5.7.11.19.29.31, since both 52edo and 1789edo support it well, and the comma basis is 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688.

On the patent val in the 7-limit, 1789edo supports {{nowrap|99 & 373}} temperament called maviloid. In addition, it also tempers out [[2401/2400]].

=== Subsets and supersets ===
1789edo is the 278th [[prime edo]]. [[3578edo]], which doubles it, is consistent in the [[21-odd-limit]].

== Table of selected intervals ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Selected intervals in 1789edo
|-
! Step
! Eliora's naming system
! JI approximation or other interpretations*
|-
| 0
| Unison
| 1/1
|-
| 25
| Oquatonic comma
| {{monzo| 65 -28 }}
|-
| 35
| Pythagorean comma
| [[531441/524288]]
|-
| 36
|
| 145/143
|-
| 61
| Lesser diesis
| [[128/125]]
|-
| 74
|
| 319/310
|-
| 122
|
| 65/62
|-
| 125
| Sextilimeans generator
| 16807/16000
|-
| 172
| Tricesimoprimal Miracle semitone
| [[31/29]]
|-
| 226
|
| 440/403
|-
| 290
| Jacobin minor interval
| 160/143, 649/580
|-
| 338
| Minor sqrt(13/10)
|
|-
| 339
| Major sqrt(13/10)
| {{monzo| -69 0 0 0 20 }}
|-
| 387
| Jacobin major interval
| 754/649
|-
| 523
| Breedsmic neutral third
| 49/40, 60/49
|-
| 576
| Major third
| [[5/4]]
|-
| 677
| Jacobin naiadic
| [[13/10]]
|-
| 750
| Sextilimeans fourth
|
|-
| 777
| Maviloid generator
| 875/648
|-
| 822
| Jacobin superfourth, Mongolian fourth
| [[11/8]]
|-
| 1032
| Secor fifth, Tricesimoprimal Miracle fifth
| (31/29)6
|-
| 1039
| Sextilimeans fifth
|
|-
| 1046
| Minor fifth
| [[3/2]]**
|-
| 1047
| Major fifth
| [[3/2]]**
|-
| 1213
| Classical minor sixth
| [[8/5]]
|-
| 1444
| Harmonic seventh
| [[7/4]]
|-
| 1535
| 29th harmonic
| [[29/16]]
|-
| 1579
| 59th harmonic
| [[59/32]]
|-
| 1707
| 31st harmonic
| [[31/16]]
|-
| 1789
| Octave
| 2/1
|}
<nowiki />* Based on the 2.5.11.13.29.31 subgroup where applicable

<nowiki />** 1046\1789 as 3/2 is the patent val, 1047\1789 as 3/2 is the 1789b val

== 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]] (%)
|-
| 2.9
| {{monzo| -5671 1789 }}
| {{mapping| 1789 5671 }}
| −0.00044
| 0.00044
| 0.06
|-
| 2.9.5
| {{monzo| -70 36 -19 }}, {{monzo| 129 -7 -46 }}
| {{mapping| 1789 5671 4154 }}
| −0.00710
| 0.00942
| 1.40
|-
| 2.9.5.7
| 420175/419904, {{monzo| 34 2 -21 3 }}, {{monzo| -55 15 2 1 }}
| {{mapping| 1789 5671 4154 5022 }}
| +0.01606
| 0.04093
| 6.10
|- style="border-top: double;"
| 2.5.11.13
| 6656/6655, {{monzo| 43 -18 5 -5 }}, {{monzo| -38 -32 10 21 }}
| {{mapping| 1789 4154 6189 6620}}
| −0.00490
| 0.01405
| 2.09
|-
| 2.5.11.13.29
| 6656/6655, 371293/371200, {{monzo| -18 -6 -1 3 5 }}, {{monzo| 34 -20 5 0 -1 }}
| {{mapping| 1789 4154 6189 6620 8691 }}
| −0.00591
| 0.01272
| 1.90
|-
| 2.5.11.13.29.31
| 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321
| {{mapping| 1789 4154 6189 6620 8691 8863 }}
| −0.00363
| 0.01268
| 1.89
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Associated ratio*
! Temperament
|-
| 1
| 35\1789
| 23.48
| 531441/524288
| [[Commatose]]
|-
| "
| 125\1789
| 83.85
| 16807/16000
| [[Sextilimeans]]
|-
| "
| 144\1789
| 96.59
| 200/189
| [[Hemiluna]] (1789bd)
|-
| "
| 172\1789
| 115.37
| 31/29
| [[Tricesimoprimal miracloid]]
|-
| "
| 377\1789
| 252.88
| 53094899/45875200
| [[Double bastille]]
|-
| "
| 576\1789
| 386.36
| 5/4
| [[French decimal]]
|-
| "
| 754\1789
| 505.76
| {{monzo| 104 0 57 0 -14 5 }}
| [[Pure bastille]]
|-
| "
| 777\1789
| 521.18
| 875/648
| [[Maviloid]]
|-
| "
| 778\1789
| 521.86
| 80275/59392
| [[Estates general]]
|-
| "
| 822\1789
| 551.37
| 11/8
| [[Onzonic]]
|-
| "
| 865\1789
| 580.21
| 6875/4914
| [[Eternal revolutionary]] (1789bd)
|}
<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

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=1zrnsGODQSg ''Etude la (R)evolution''] (2022)

[[Category:Jacobin]]
[[Category:Listen]]

{{Todo| review | clarify }}

Pure bastille

2026-05-29T19:29:03Z

Eliora: fix lettering

#REDIRECT[[The Jacobins #Pure bastille]]

No-threes subgroup temperaments

2026-05-29T19:24:49Z

Eliora: /* Bastille */ again link

{{Technical data page}}
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 3.

== Overview by mapping of 5 ==
Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].

* For no-fives, see [[#No-threes-or-fives subgroup temperaments]].
* French decimal and trader have a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.
* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
* Insect has a ~55/32 generator, three of which give the ~5/1.
* Frostburn has a ~28/25 generator, four of which give the ~8/5.
Others have a more complex mapping of 5.

== 2.5.7 temperaments ==

Temperaments discussed elsewhere include
* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus|Hemimean clan]]
* Mercy ([[823543/819200]]) → [[Quince clan #Mercy|Quince clan]]
* Llywelyn a.k.a. shoe ([[4194304/4117715]]) → [[Llywelynsmic clan #Llywelyn a.k.a. shoe|Llywelynsmic clan]]

=== Frostburn ===
{{See also| Magic family #Quadrimage | Subgroup temperaments #Baldy }}

[[Subgroup]]: 2.5.7

[[Comma list]]: 78125/76832

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

: Sval mapping generators: ~2, ~28/25

[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389

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

[[Badness]] (Sintel): 0.886

==== 2.5.7.11 ====
Subgroup: 2.5.7.11

Comma list: 245/242, 625/616

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

: Sval mapping generators: ~2, ~28/25

Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745

{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}

Badness (Sintel): 0.463

=== Mabilic ===
{{See also| Chromatic pairs #Mabilic }}{{Main|Mabilic and trismegistus}}Given below is the no-three version of [[Mavila family#Armodue|armodue]], [[Mabila family#Semabila|semabila]], and [[Magic family#Trismegistus|trismegistus]]. It is the 7 & 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma.

[[Subgroup]]: 2.5.7

[[Comma list]]: 1071875/1048576

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

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

: [[gencom]]: [2 175/128; 1071875/1048576]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 527.236

{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305bc }}

[[Tp tuning #T2 tuning|RMS error]]: 0.7729 cents

=== Rainy ===
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]] (and [[valentine]]). Rainy is notable theoretically as it equates ([[2/1]])/([[5/4]])3 (128/125, the lesser diesis) with ([[2/1]])/([[8/7]])5 (the 2.7-subgroup [[cloudy comma]], which is similar to the 2.5-subgroup lesser diesis in that tempering it out tunes the 8/7 about 8.8{{cent}} sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7{{cent}}). By tempering out their difference, stacked 5s and stacked 7s become easier to navigate, using the general-purpose diesis to simplify clusters. (Note that this analysis assumes a [[lattice]]-based conceptualization of [[JI]] which is often called "stacking-based"; see [[taxonomies of xen approaches]].)

A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.

[[Subgroup]]: 2.5.7

[[Comma list]]: [[2100875/2097152]]

[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]

[[Gencom]]: [2 256/245; 2100875/2097152]

[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]

Optimal tuning ([[POTE]]): ~256/245 = 77.205

{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents

=== French decimal ===
Conceived upon the fact that 1789edo has an excellent 5/4, and uses it as the generator. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, a 1525 & 1789 temperament is obtained.

Subgroup: 2.5.7

Comma basis: {{monzo|372 -159 -1}}

Sval mapping: [{{val|1 2 54}}, {{val|0 1 -159}}]

Optimal tuning (CTE): ~5/4 = 386.360

{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...

[[Badness]] (Sintel): 148.6

==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11

Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}

Sval mapping: [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]

Optimal tuning (CTE): ~5/4 = 386.361

{{Optimal ET sequence|legend=0|264, 733}}, ...

Badness (Sintel): 52.150

==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13

Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625

Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]

Optimal tuning (CTE): ~5/4 = 386.361

{{Optimal ET sequence|legend=0|1525, 1789}}, ...

Badness (Sintel): 10.518

=== Bastille ===
{{Main| Bastille }}

Described as the 2.5.7 subgroup 1407 & 1789 temperament, and named after an [[wikipedia:Storming of the Bastille|eponymous historical event]] which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Double bastille|double bastille]].

Subgroup: 2.5.7

Comma list: {{Monzo|1426 -596 -15}}

Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]

Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243

{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...

[[Badness]] (Sintel): 7224.3

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

Augment is related to [[augmented]].

[[Subgroup]]: 2.5.7.11

[[Comma list]]: 56/55, 128/125

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

{{Mapping|legend=3| 3 0 7 9 11| 0 0 0 -1 -1 }}

: [[gencom]]: [5/4 8/7; 56/55 128/125]

[[Optimal tuning]] ([[POTE]]): ~5/4 = 1\3, ~8/7 = 228.275

{{Optimal ET sequence|legend=1| 3, 6, 9, 15, 21 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.422 cents

=== Ostara ===
'''Ostara''' is a temperament that is derived from 93 & 524 solar calendar leap rule scale. It was initially defined by taking the 2.7.13.17.19 subgroup, but it can also be intepreted in general no-threes 19-limit.

Ostara can also refer to a collection of temperaments which temper out 16807/16796.

[[Subgroup]]: 2.5.7.11

[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200

[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000¢, ~5120/3773 = 529.003¢
* [[CWE]]: ~2 = 1200.000¢, ~5120/3773 = 529.004¢

{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}

[[Badness]] (Sintel): 11.731

==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13

Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125

Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]

Optimal tunings:
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢

{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}

Badness (Sintel): 3.415

==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17

Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]

Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251

Optimal tunings:
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢

{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}

Badness (Sintel): 1.985

==== 2.5.7.11.13.17.19 subgroup ====
Subgroup: 2.5.7.11.13.17.19

Sval Mapping: [{{val| 1 1 20 -49 35 42 21 }}, {{val| 0 3 -39 119 -71 -86 -38 }}]

Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875

Optimal tunings:
* CTE: ~2 = 1200.000¢, ~19/14 = 529.006¢
* CWE: ~2 = 1200.000¢, ~19/14 = 529.005¢

{{Optimal ET sequence|legend=0| 93, 338, 431 }}

Badness (Sintel): 1.285

=== Tricesimoprimal miracloid ===
{{See also|Tricesimoprimal miracloid/Eliora's approach|l1=Eliora's approach to tricesimoprimal miracloid}}
Described as the 52 & 1789 temperament in the 2.5.7.11.19.29.31 subgroup, with harmonics specifically selected for 52edo and 1789edo. Its generator is [[31/29]], which is also close to the secor. Since it is conceived as the temperament in the above specific subgroup, it makes no sense to name it for smaller subgroups. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable [[well temperament]] for 52edo.

Subgroup: 2.5.7.11.19.29.31

Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688

Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]

Optimal tuning (CTE): ~58/31 = 1084.628

{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...

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

Huntington may be described as the 10 & 27 temperament in the 2.5.7.13 subgroup.

[[Subgroup]]: 2.5.7.13

[[Comma list]]: [[640/637]], [[10985/10976]]

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

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

: [[gencom]]: [2 16/13; 640/637 10985/10976]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/13 = 357.002

{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3452 cents

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

Silver can be described as the 10 & 27 temperament in the 2.5.7.13.17 subgroup.

[[Subgroup]]: 2.5.7.13.17

[[Comma list]]: [[170/169]], [[640/637]], [[5525/5488]]

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

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

: [[gencom]]: [2 13/8; 170/169 640/637 5525/5488]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/8 = 842.711

{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents

=== Pakkanen ===
[[Subgroup]]: 2.5.7.11

[[Comma list]]: 625/616

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

: mapping generators: ~2, ~5, ~11

[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653

{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}

[[Badness]] (Sintel): 0.573

=== No-threes naiad ===
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}

This temperament can be described as the 21 & 29 & 37 temperament in no-threes subgroups. It expands [[Subgroup temperaments #Tridec|tridec]] and [[Subgroup temperaments #Naiadec|naiadec]].

[[Subgroup]]: 2.5.7.11

[[Comma list]]: 5021863/5000000

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

: mapping generators: ~2, ~5, ~100/77

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.080¢, ~5 = 2786.820¢, ~100/77 = 454.618¢
* [[CWE]]: ~2 = 1200.000¢, ~5 = 2786.740¢, ~100/77 = 454.590¢

{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}

[[Badness]] (Sintel): 1.862

==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13

Comma list: 847/845, 1001/1000

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

Optimal tunings:
* WE: ~2 = 1200.034¢, ~5 = 2786.678¢, ~13/10 = 454.569¢
* CWE: ~2 = 1200.000¢, ~5 = 2786.646¢, ~13/10 = 454.557¢

{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}

Badness (Sintel): 0.179

==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17

Comma list: 170/169, 221/220, 847/845

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

Optimal tunings:
* WE: ~2 = 1200.407¢, ~5 = 2787.484¢, ~13/10 = 455.036¢
* CWE: ~2 = 1200.000¢, ~5 = 2787.107¢, ~13/10 = 454.906¢

{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 50, 58, 66g, 87g }}

Badness (Sintel): 0.438

== Higher 2.5 temperaments ==

Temperaments discussed elsewhere include:
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]

=== Movila ===
This temperament has a structure very similar to [[mavila]] but is somewhat more accurate because the generator is a flat [[11/8]] rather than a sharp [[4/3]]. The major third is still ~[[5/4]], but the minor third is now ~[[64/55]] instead of ~[[6/5]].

[[Subgroup]]: 2.5.11

[[Comma list]]: 1331/1280

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

[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[11/8]] = 529.846

{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}

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

Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, is related to [[wizard]].

[[Subgroup]]: 2.5.11

[[Comma list]]: [[15625/15488]]

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

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

: [[gencom]]: [125/88 5/4; 15625/15488]

[[Optimal tuning]] ([[POTE]]): ~125/88 = 1\2, ~5/4 = 383.768

{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3997

=== Insect ===
[[Subgroup]]: 2.5.11

[[Comma list]]: 33275/32768

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

: Mapping generators, ~2, ~[[55/32]]

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[55/32]] = 928.032

{{Optimal ET sequence|legend=1|9, 13, 22, 97e, 119e, 141e, 163e, 304ceee}}

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

Sephiroth is the no-7 restriction of [[muggles]].

[[Subgroup]]: 2.5.11.13.17

[[Comma list]]: 65/64, 170/169, 221/220

{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}

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

: [[gencom]]: [2 5/4; 65/64 170/169 221/220]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 372.236

{{Optimal ET sequence|legend=1| 10, 13, 16, 29 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.774 cents

=== Trader ===
[[Subgroup]]: 2.5.13

[[Comma list]]: [[26/25]]

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

: Mapping generators, ~2, ~[[5/4]]

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[5/4]] = 407.079

{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}

=== Superquintal ===
[[Subgroup]]: 2.5.13

[[Comma list]]: 64000000/62748517

{{Mapping|legend=2|1 5 6|0 -7 -6}}

: Mapping generators, ~2, ~13/10

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281

{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}

=== Vengeance ===
{{Main| Vengeance }}

Another lower-error replica of mavila, with the fifth being ~[[25/17]] instead of ~[[3/2]].

[[Subgroup]]: 2.5.17

[[Comma list]]: 78608/78125

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[34/25]] = 529.095

{{Optimal ET sequence|legend=1|7g, 9, 25, 34, 93, 127, 288, 415}}

== No-threes-or-fives subgroup temperaments ==
Temperaments discussed elsewhere include
* Orgone → [[Orgonia #Orgone|Orgonia]]
* Berylic → [[4th-octave temperaments #Berylic|4th-octave temperaments]]
* 21-23-commatic → [[21st-octave temperaments #21-23-commatic|21st-octave temperaments]]
* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic|31st-octave temperaments]]
* 37-11-commatic (rank-1) → [[37th-octave temperaments #37-11-commatic (rank-1)|37th-octave temperaments]]
* etc.

=== Amaranthine ===
{{See also| No-fives subgroup temperaments #Chrysanthemum }}

Amaranthine is the high-accuracy 2.7.11 subgroup strong restriction of [[Gamelismic clan#11-limit 3|undecimal mothra]].

[[Subgroup]]: 2.7.11

[[Comma list]]: 5767168/5764801

{{Mapping|legend=2| 1 2 -3 | 0 1 8 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~7/4 = 968.913

{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d }}

Badness (Sintel): 0.031

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

[[Subgroup]]: 2.7.11.13

[[Comma list]]: 343/338, 847/832

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

{{Mapping|legend=3| 1 0 0 1 3 1| 0 0 0 4 1 6 }}

: [[gencom]]: [2 11/8; 343/338 847/832]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 540.099

{{Optimal ET sequence|legend=1| 5, 7, 9, 11, 20 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.282 cents

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

Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup.

[[Subgroup]]: 2.7.11.13

[[Comma list]]: [[1573/1568]], [[15488/15379]]

{{Mapping|legend=2| 1 0 1 3 | 0 8 7 2 }}

{{Mapping|legend=3| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}

: [[gencom]]: [2 14/11; 1573/1568 15488/15379]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/11 = 421.309

{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 225, 319cd, 413bcd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.4043 cents

=== Voltage ===
Voltage is the 3 & 7 temperament in the 2.7.13 subgroup.

[[Subgroup]]: 2.7.13

[[Comma list]]: [[28672/28561]]

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

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

: [[gencom]]: [2, 16/13; 28672/28561]

[[Optimal tuning]]:
* [[POTE]]: ~2 = 1\1, ~16/13 = 357.677
* [[TOP tuning|POTT]]: ~2 = 1\1, ~16/13 = 357.794 (1200 - 300 log2(7))

{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.1423 cents

=== Ultrakleismic ===
[[Subgroup]]: 2.7.17

[[Comma list]]: 4913/4802

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

: Mapping generators, ~2, ~[[17/14]]

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 324.446

{{Optimal ET sequence|legend=1|4, 7, 11, 26, 37}}

=== Counterultrakleismic ===
[[Subgroup]]: 2.7.17

[[Comma list]]: 2024782584832/2015993900449

{{Mapping|legend=2|1 0 1|0 10 11}}

: Mapping generators, ~2, ~[[17/14]]

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 336.858

{{Optimal ET sequence|legend=1|7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g}}

=== Shipwreck ===

[[Subgroup]]: 2.7.53

[[Comma list]]: 1048576/1042139

[[Gencom]]: [2 64/53; 1048576/1042139]

[[Mapping]]: [{{val|1 0 6}}, {{val|0 3 -1}}]]

[[POTE generator]]: ~64/53 = 323.034

{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}

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

Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, is generated by ~13/11. Two generator steps give ~11/8 and three generator steps give ~13/8.

[[Subgroup]]: 2.11.13

[[Comma list]]: [[1352/1331]]

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

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

: [[gencom]]: [2 13/11; 1352/1331]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/11 = 279.318

{{Optimal ET sequence|legend=1| 13, 30, 43, 73, 116 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.8449 cents

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

Blackbirds is a fairly straightforward temperament. It simply equates ~13/11 to 1/4 of the octave with a generator for prime 11 or 13.

[[Subgroup]]: 2.11.13

[[Comma list]]: [[29282/28561]]

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

{{Mapping|legend=3| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}

: [[gencom]]: [13/11 11/8; 29282/28561]

[[Optimal tuning]] ([[POTE]]): ~13/11 = 1\4, ~11/8 = 546.660

{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 44, 68, 112c, 180bc }}

[[Tp tuning #T2 tuning|RMS error]]: 0.8685 cents

=== Bluebirds ===
{{Distinguish| Bluebird }}
{{See also| Chromatic pairs #Bluebirds }}

[[Subgroup]]: 2.11.13

[[Comma list]]: [[265837/262144]]

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

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

: [[gencom]]: [2 143/128; 265837/262144]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~143/128 = 182.368

{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125c, 204bc, 329bc }}

[[Tp tuning #T2 tuning|RMS error]]: 0.4444 cents

=== Yamablu ===
Yamablu, with a generator of ~17/13, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The [[Kite's Method of Naming Rank-2 Scales using Mode Numbers|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].

[[Subgroup]]: 2.11.13.17.19

[[Comma list]]: 209/208, 2057/2048, 83521/83486

[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]

Optimal tuning ([[POTE]]): ~17/13 = 462.9606

{{Optimal ET sequence|legend=1| 13, 44, 57, 70}}

[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents

=== Mavericks ===

[[Subgroup]]: 2.13.19

[[Comma list]]: 47525504/47045881

[[Mapping]]: [{{val|1 1 2}}, {{val|0 6 5}}]

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/19 = 539.886

{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}

=== Yer (rank 3) ===
[[Subgroup]]: 2.11.13.17.19

[[Comma list]]: 209/208, 2057/2048

[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}

[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638

{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}

[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]

Bastille

2026-05-29T19:22:53Z

Eliora: and link to what this actually is named after

'''Bastille''' is a rank-2 temperament associated with the {{monzo|1426 0 -596 -15}} comma in the 2.5.7 subgroup, that is no-threes subgroup. It is extracted from a sequence of [[mos scale]]s supported by [[1407edo]] and [[1789edo]]. It derives its name from the fact that the [[wikipedia:storming of the Bastille|storming of the Bastille]] happened on 14 July 1789, which in date format is written as 14/07/1789.

For technical information see [[No-threes subgroup temperaments#Bastille]].

== Theory ==
Since 1407edo and 1789edo have large relative errors on harmonics outside of the 2.5.7 subgroup, there initially was no single temperament named bastille, and the term used to refer to merely a sequence of mos scales to accommodate many possible vals that 1407edo and 1789edo support. However, the temperament can be firmly defined in the 2.5.7 subgroup.

The defining characteristic of Bastille is that the generator is a "meantone" flat fifth of around 694.5 cents. Fifteen stacked generators lead to [[8/5]]. Given the precision, such a characteristic is notable for edos in the thousands to support.

=== Extensions ===
The [[The Jacobins #Pure bastille|1407eff & 1789 temperament]], named ''pure bastille'', tempers out the [[jacobin comma]], which is thematically relevant to the naming reason as it combines two French Revolution-associated historical terms in one temperament. However, from a just intonation perspective, such temperament may be too complex, even for edos in the thousands. On the other hand, taking patent vals of 1407edo and 1789edo in the full 13-limit produces a temperament whose fifth is mapped to [[112/75]], a relatively simple interval on this scale. Comma basis for this version of bastille is {384912/384475, 41765625/41746432, 4501875/4499456, 540078539/540000000}. Since 112/75 is called marvelous fifth, the most straightforward name for this temperament is ''[[marvelous bastille]]''.

=== MOS scales of Bastille ===
* [[1L 1s]]
* [[2L 1s]]
* [[2L 3s]]
* [[5L 2s]] - minisoft diatonic
* [[7L 5s]] - minihard m-chromatic
* [[7L 12s]] - ultrasoft m-enharmonic - can be used as a well-temperament for [[19edo]].

== Related pages ==
* [[Bastille comma]]

[[Category:Rank 2]]

Bastille

2026-05-29T19:20:45Z

Eliora: /* Extensions */ slight formality rewrite

'''Bastille''' is a rank-2 temperament associated with the {{monzo|1426 0 -596 -15}} comma in the 2.5.7 subgroup, that is no-threes subgroup. It is extracted from a sequence of [[mos scale]]s supported by [[1407edo]] and [[1789edo]]. It derives its name from the fact that the storming of Bastille happened on 14 July 1789, which in date format is written as 14/07/1789.

For technical information see [[No-threes subgroup temperaments#Bastille]].

== Theory ==
Since 1407edo and 1789edo have large relative errors on harmonics outside of the 2.5.7 subgroup, there initially was no single temperament named bastille, and the term used to refer to merely a sequence of mos scales to accommodate many possible vals that 1407edo and 1789edo support. However, the temperament can be firmly defined in the 2.5.7 subgroup.

The defining characteristic of Bastille is that the generator is a "meantone" flat fifth of around 694.5 cents. Fifteen stacked generators lead to [[8/5]]. Given the precision, such a characteristic is notable for edos in the thousands to support.

=== Extensions ===
The [[The Jacobins #Pure bastille|1407eff & 1789 temperament]], named ''pure bastille'', tempers out the [[jacobin comma]], which is thematically relevant to the naming reason as it combines two French Revolution-associated historical terms in one temperament. However, from a just intonation perspective, such temperament may be too complex, even for edos in the thousands. On the other hand, taking patent vals of 1407edo and 1789edo in the full 13-limit produces a temperament whose fifth is mapped to [[112/75]], a relatively simple interval on this scale. Comma basis for this version of bastille is {384912/384475, 41765625/41746432, 4501875/4499456, 540078539/540000000}. Since 112/75 is called marvelous fifth, the most straightforward name for this temperament is ''[[marvelous bastille]]''.

=== MOS scales of Bastille ===
* [[1L 1s]]
* [[2L 1s]]
* [[2L 3s]]
* [[5L 2s]] - minisoft diatonic
* [[7L 5s]] - minihard m-chromatic
* [[7L 12s]] - ultrasoft m-enharmonic - can be used as a well-temperament for [[19edo]].

== Related pages ==
* [[Bastille comma]]

[[Category:Rank 2]]

496edo

2026-05-10T21:30:02Z

Eliora: That "compound scale" is not really a temperament so moving accordingly

{{Infobox ET}}
{{ED intro}}

== Theory ==
496edo is [[enfactoring|enfactored]] in the 11-limit, with the same tuning as [[248edo]], but the [[patent val]]s differ on the mapping for 13. In the 13-limit patent val, it tempers out [[4225/4224]].

496edo is good with the 2.3.11.19 [[subgroup]]. For higher limits, it has good approximations of [[31/1|31]], [[37/1|37]], and [[47/1|47]]. In the 2.3.11.19 subgroup, it tempers out 131072/131043.

=== Odd harmonics ===
{{Harmonics in equal|496}}

=== Subsets and supersets ===
496 is the 3rd {{w|perfect number}}, factoring into {{factorization|496}}. Its nontrivial divisors are {{EDOs| 2, 4, 8, 16, 31, 62, 124, 248 }}, the most notable being 31.

== Scales ==
Since 496edo is enfactored 248edo, in the 11-limit it can represent a [[compound scale|compound]] of two chains of 11-limit [[bischismic]] temperaments, interlaced.

User:Eliora/377edo

2026-05-02T00:22:50Z

Eliora: Created page with "{{Infobox ET}} {{ED intro}} 377edo is consistent only in the 3-limit and represents lower harmonics poorly. Nonetheless, 2.7/5.9.13.17 subgroup is represented quite well. Some commas 377edo tempers out in this subgroup are 2000033/2000000, 4303125/4302592, and 3955078125/3954653486. === Odd harmonics === {{harmonics in equal|377}} === Subsets and supersets === Since 377 factors as {{Factorization|377}}, 377edo contains 13edo and 29edo as its subsets. Given..."

{{Infobox ET}}
{{ED intro}}

377edo is consistent only in the 3-limit and represents lower harmonics poorly. Nonetheless, 2.7/5.9.13.17 subgroup is represented quite well. Some commas 377edo tempers out in this subgroup are 2000033/2000000, 4303125/4302592, and 3955078125/3954653486.

=== Odd harmonics ===

{{harmonics in equal|377}}

=== Subsets and supersets ===

Since 377 factors as {{Factorization|377}}, 377edo contains [[13edo]] and [[29edo]] as its subsets.

Given the fact that this equal division does not support any obvious temperaments, the best way to use it is as a polychromatic tuning that is a superposition of 13edo and 29edo.

User:Eliora/3200edo

2026-04-21T22:08:02Z

Eliora: Created page with "{{Infobox ET|3200}} {{ED intro|3200}} 3200edo is enfactored in the 11-limit, sharing the tuning with 1600edo. === Odd harmonics === {{harmonics in equal|3200}}"

{{Infobox ET|3200}}
{{ED intro|3200}}

3200edo is [[enfactoring|enfactored]] in the 11-limit, sharing the tuning with [[1600edo]].

=== Odd harmonics ===

{{harmonics in equal|3200}}

Factor 9 grid

2026-04-01T22:44:53Z

Eliora: /* Theory */

'''Factor 9 grid''' is a type of musical scale which was first proposed for esoteric reasons as a supposed replacement to [[12edo]]. The scale also became notable in music theory when esoteric properties of the scale were subsequently refuted by the famous YouTuber, bassist, and composer [[Adam Neely]].

== Theory ==
The scale is an isoharmonic sequence consisting of the following frequencies (in Hz): 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, and their octave equivalents. This sequence forms an arithmetic progression with a constant difference of 9, which gives rise to the name "Factor 9 grid". It hence is identical to the [[14ado]] scale spanning the 14th through 28th harmonics, and since 14ado is a [[23-limit]] just intonation system, Factor 9 grid correspondingly is a part of of [[23-limit]] just intonation.

More precisely, the "Factor 9 grid" refers to a specific mode of [[14ado]] whose tonic is placed on the step corresponding to 432 Hz and its octave equivalents, such as 216 Hz or 864 Hz, which is the 11th step of 14ado itself. It is this particular modal alignment that is commonly associated with "A = 432 Hz" conspiracy theories, where the emphasis is placed on organizing the scale around 432 Hz as a tonic. Proponents of the "Factor 9 grid" manly present it as the more consonant or acoustically "healthier" alternative to the prevailing [[12edo|12-tone equal temperament]], often accompanied by references to the symbolic or "sacred" significance of the number 12.

However, descriptions of the scale, as they are presented in the video, contain several internal inconsistencies. The underlying structure of the grid, as mentioned above, corresponds to [[14ado]], which by definition contains 14 distinct steps per octave rather than 12, thus conflicting with the initial claims that the scale is 12-note or is a replacement for 12edo.

Furthermore, in the cited material, the sequence appears to omit the frequency 243 Hz (and its octave equivalents), despite it being a member of arithmetic progression that constitutes the scale. This omission appears to be motivated by an attempt to align the number of pitches with the 12-note framework of standard Western notation, as the presentation maps the resulting tones onto conventional note names. However, the rationale for excluding specifically 243 Hz and its octave displacement, as opposed to any other member of the sequence, is not explicitly addressed. No criteria are provided for why this particular step is removed while the remaining tones are retained, leaving the adjustment unexplained within the context of the scale’s stated arithmetic construction.

Additionally, the accompanying table distinguishes between G♯ and A♭ as separate pitches, resulting in a 13-note scale rather than either 12 or 14. This is not explained in the video as well, and further complicates the stated aim of aligning the system with a 12-note framework, particularly in light of the emphasis placed on the number 12 in the associated commentary.

== Intervals ==
The 9/8 interval was skipped by the original video for unknown reasons.
{| class="wikitable"
|+ Factor 9 Grid on 432 Hz
|-
! Frequency (Hz) !! Note !! Interval
|-
| 432 || A || [[1/1]]
|-
| 450 || A#/Bb || [[25/24]]
|-
| 468 || B || [[13/12]]
|-
| 486 || - || [[9/8]]
|-
| 504 || C || [[7/6]]
|-
| 540 || C#/Db || [[5/4]]
|-
| 576 || D || [[4/3]]
|-
| 612 || D#/Eb || [[17/12]]
|-
| 648 || E || [[3/2]]
|-
| 684 || F || [[19/12]]
|-
| 720 || F#/Gb || [[5/3]]
|-
| 756 || G || [[7/4]]
|-
| 792 || G# || [[11/6]]
|-
| 828 || Ab || [[23/12]]
|-
| 864 || A || [[2/1]]
|}

== Attempts at representing the scale through regular temperament theory ==
Within regular temperament theory, the Factor 9 grid admits close approximations through equal divisions of just intervals, and is well-supported by any tuning system that either represents the [[23-limit]] intervals well, or the direct intervals mentioned in the scale.

Though, in particular, it is worth noting that it can be represented by 666 equal divisions of the 15/14 interval (666ed15/14), which directly models its harmonic structure, as well as by 666edo, which supports a related regular temperament with similar properties, which is a very ironic aspect of the scale.

The scale, often justified by proponents through numerological arguments emphasizing purity and the special status of certain numbers, is located in close correspondence with equal temperament systems involving the number [[wikipedia:666 (number)|666]], which is a value widely regarded in popular numerological traditions, and Western popular culture as a whole as [[wikipedia:the Number of the Beast|the Number of the Beast]] or satanic. Thus, within the proponents’ own framework, where numerical symbolism is treated as musically or metaphysically significant, this association with 666 directly contradicts the stated rationale for privileging the scale over conventional tuning systems.

[[Eliora]] when trying to represent the Factor 9 grid via a rank-2 temperament did not realize that x31eq.com resource takes the first interval in the number series as the equivalence interval, and assumed that it is represented by [[666edo]] instead. "q666" in the temperament finder, when 14:15:16:... typed out, stands for 666ed15/14 as opposed to 666ed2.

This being said, 666edo, more specifically, the 495 & 666 23-limit temperament, regular temperament preserves more properties of the "Factor 9 Grid" than the corrected version, since it's period-9 and its period minus reduced generator interval also maps to the smallest interval in the scale, [[28/27]]. So in a limited way, 666edo does ironically represent the Factor 9 grid well. The corrected version would be with 495ed15/14 & 666ed15/14, which corresponds to a 4973 & 6691 temperament in the 23-limit, however that temperament has no structure resembling the initial factor-9 grid other than closely approximating its constituent intervals.

The 495 & 666 temperament is given the name ''enneasoteric'' by Eliora, since music was already composed in the 666edo scale approximating Factor 9 grid, while the name "factor 9 grid" to avoid ambiguity will be retained with the scale only.

=== Enneasoteric (495 & 666) ===
[[Subgroup]]: 2.3.5.7.11.13.17.19.23

[[Comma list]]: 442/441, 715/714, 2300/2299, 3060/3059, 3179/3174, 9025/9009, 57375/57344

[[Mapping]] [[generators]]: ~250/231, ~336/323

[[Optimal tuning]] ([[CTE]]): ~336/323 = 70.270

{{Optimal ET sequence|legend=1|171f, 495, 666}}.

== Mysticism and reality ==
The motivation behind the scale, as with many esoteric just intonation proposals, is the claim that the irrational pitch relationships of equal temperament produce acoustically unpleasant effects which, in turn, are said to propagate into subconscious perception and negatively affect human well-being.

However, claims that specific tuning systems (such as just intonation or particular frequency standards like 432 Hz) have direct effects on public health, social cohesion, or global conditions are not supported by empirical evidence. While differences in tuning can influence perceived consonance, timbre, and listener preference, these effects operate at the level of auditory perception and musical aesthetics rather than large-scale societal outcomes.

Furthermore, from a mathematical perspective, it is not possible to simultaneously achieve the exact rational interval relationships of just intonation and the structural evenness of equal temperament. The irrationality inherent to equal divisions of the octave has been recognized since antiquity, most commonly through proofs such as the irrationality of √2. For example, if there were an exact just intonation fraction corresponding to the 600-cent tritone, its numerator and denominator would be required to satisfy mutually incompatible conditions — [[wikipedia:Square root of 2#Proof by infinite descent|being both even and coprime]]. Similarly, if a stack of pure fifths (3/2) were to close exactly at the octave, the resulting comma {{Monzo|-X Y}} would have to equal 1. In this number, numerator X must be a power of 2 and the denominator Y a power of 3, thus implying the existence of an even power of 3, which is not possible.

More broadly, while differences in tuning systems can affect perceived consonance, beating, and timbral character, these effects remain within the domain of auditory perception and musical aesthetics. There is no established physiological or cognitive mechanism by which specific frequency ratios could influence complex outcomes such as public health, social stability, or quality of life at scale. Such phenomena are determined by a wide range of economic, environmental, and social factors, none of which have demonstrated dependence on musical tuning standards.

Accordingly, claims that particular tuning systems produce measurable improvements in human well-being would require empirical verification under controlled conditions. In the absence of such evidence, assertions of this kind remain unsupported, particularly when presented within numerological or occultist frameworks rather than scientific methodology.

== External links ==
* [https://www.youtube.com/watch?v=FY74AFQl2qQ Sonic Geometry: The Language of Frequency and Form] - the original video
* [https://www.youtube.com/watch?v=ghUs-84NAAU Testing 432Hz Frequencies and Temperaments] - refuting by Adam Neely

[[Category:23-limit]]

Factor 9 grid

2026-03-31T03:29:29Z

Eliora: /* Mysticism and reality */

'''Factor 9 grid''' is a type of musical scale which was first proposed for esoteric reasons as a supposed replacement to [[12edo]]. The scale also became notable in music theory when esoteric properties of the scale were subsequently refuted by the famous YouTuber, bassist, and composer [[Adam Neely]].

== Theory ==
The scale is an isoharmonic sequence consisting of the following frequencies (in Hz): 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, and their octave equivalents. This sequence forms an arithmetic progression with a constant difference of 9, which gives rise to the name "Factor 9 grid". It hence is identical to the [[14ado]] scale spanning the 14th through 28th harmonics, and since 14ado is a [[23-limit]] just intonation system, Factor 9 grid correspondingly is a part of of [[23-limit]] just intonation.

More precisely, the "Factor 9 grid" refers to a specific mode of [[14ado]] whose tonic is placed on the step corresponding to 432 Hz and its octave equivalents, such as 216 Hz or 864 Hz, which is the 11th step of 14ado itself. It is this particular modal alignment that is commonly associated with "A = 432 Hz" conspiracy theories, where the emphasis is placed on organizing the scale around 432 Hz as a tonic.

Proponents of the "Factor 9 grid" manly present it as the more consonant or acoustically "healthier" alternative to the prevailing [[12edo|12-tone equal temperament]], often accompanied by references to the symbolic or "sacred1 significance of the number 12. However, descriptions of the scale, as they are presented in the video, contain several internal inconsistencies. The underlying structure of the grid, as mentioned above, corresponds to [[14ado]], which by definition contains 14 distinct steps per octave rather than 12, thus conflicting with the initial claims that the scale is 12-note or is a replacement for 12edo.

Furthermore, in the cited material, the sequence appears to omit the frequency 243 Hz (and its octave equivalents), despite it being a member of arithmetic progression that constitutes the scale. This omission appears to be motivated by an attempt to align the number of pitches with the 12-note framework of standard Western notation, as the presentation maps the resulting tones onto conventional note names. However, the rationale for excluding specifically 243 Hz and its octave displacement, as opposed to any other member of the sequence, is not explicitly addressed. No criteria are provided for why this particular step is removed while the remaining tones are retained, leaving the adjustment unexplained within the context of the scale’s stated arithmetic construction.

Additionally, the accompanying table distinguishes between G♯ and A♭ as separate pitches, resulting in a 13-note scale rather than either 12 or 14. This is not explained in the video as well, and further complicates the stated aim of aligning the system with a 12-note framework, particularly in light of the emphasis placed on the number 12 in the associated commentary.

== Intervals ==
The 9/8 interval was skipped by the original video for unknown reasons.
{| class="wikitable"
|+ Factor 9 Grid on 432 Hz
|-
! Frequency (Hz) !! Note !! Interval
|-
| 432 || A || [[1/1]]
|-
| 450 || A#/Bb || [[25/24]]
|-
| 468 || B || [[13/12]]
|-
| 486 || - || [[9/8]]
|-
| 504 || C || [[7/6]]
|-
| 540 || C#/Db || [[5/4]]
|-
| 576 || D || [[4/3]]
|-
| 612 || D#/Eb || [[17/12]]
|-
| 648 || E || [[3/2]]
|-
| 684 || F || [[19/12]]
|-
| 720 || F#/Gb || [[5/3]]
|-
| 756 || G || [[7/4]]
|-
| 792 || G# || [[11/6]]
|-
| 828 || Ab || [[23/12]]
|-
| 864 || A || [[2/1]]
|}

== Attempts at representing the scale through regular temperament theory ==
Within regular temperament theory, the Factor 9 grid admits close approximations through equal divisions of just intervals, and is well-supported by any tuning system that either represents the [[23-limit]] intervals well, or the direct intervals mentioned in the scale.

Though, in particular, it is worth noting that it can be represented by 666 equal divisions of the 15/14 interval (666ed15/14), which directly models its harmonic structure, as well as by 666edo, which supports a related regular temperament with similar properties, which is a very ironic aspect of the scale.

The scale, often justified by proponents through numerological arguments emphasizing purity and the special status of certain numbers, is located in close correspondence with equal temperament systems involving the number [[wikipedia:666 (number)|666]], which is a value widely regarded in popular numerological traditions, and Western popular culture as a whole as [[wikipedia:the Number of the Beast|the Number of the Beast]] or satanic. Thus, within the proponents’ own framework, where numerical symbolism is treated as musically or metaphysically significant, this association with 666 directly contradicts the stated rationale for privileging the scale over conventional tuning systems.

[[Eliora]] when trying to represent the Factor 9 grid via a rank-2 temperament did not realize that x31eq.com resource takes the first interval in the number series as the equivalence interval, and assumed that it is represented by [[666edo]] instead. "q666" in the temperament finder, when 14:15:16:... typed out, stands for 666ed15/14 as opposed to 666ed2.

This being said, 666edo, more specifically, the 495 & 666 23-limit temperament, regular temperament preserves more properties of the "Factor 9 Grid" than the corrected version, since it's period-9 and its period minus reduced generator interval also maps to the smallest interval in the scale, [[28/27]]. So in a limited way, 666edo does ironically represent the Factor 9 grid well. The corrected version would be with 495ed15/14 & 666ed15/14, which corresponds to a 4973 & 6691 temperament in the 23-limit, however that temperament has no structure resembling the initial factor-9 grid other than closely approximating its constituent intervals.

The 495 & 666 temperament is given the name ''enneasoteric'' by Eliora, since music was already composed in the 666edo scale approximating Factor 9 grid, while the name "factor 9 grid" to avoid ambiguity will be retained with the scale only.

=== Enneasoteric (495 & 666) ===
[[Subgroup]]: 2.3.5.7.11.13.17.19.23

[[Comma list]]: 442/441, 715/714, 2300/2299, 3060/3059, 3179/3174, 9025/9009, 57375/57344

[[Mapping]] [[generators]]: ~250/231, ~336/323

[[Optimal tuning]] ([[CTE]]): ~336/323 = 70.270

{{Optimal ET sequence|legend=1|171f, 495, 666}}.

== Mysticism and reality ==
The motivation behind the scale, as with many esoteric just intonation proposals, is the claim that the irrational pitch relationships of equal temperament produce acoustically unpleasant effects which, in turn, are said to propagate into subconscious perception and negatively affect human well-being.

However, claims that specific tuning systems (such as just intonation or particular frequency standards like 432 Hz) have direct effects on public health, social cohesion, or global conditions are not supported by empirical evidence. While differences in tuning can influence perceived consonance, timbre, and listener preference, these effects operate at the level of auditory perception and musical aesthetics rather than large-scale societal outcomes.

Furthermore, from a mathematical perspective, it is not possible to simultaneously achieve the exact rational interval relationships of just intonation and the structural evenness of equal temperament. The irrationality inherent to equal divisions of the octave has been recognized since antiquity, most commonly through proofs such as the irrationality of √2. For example, if there were an exact just intonation fraction corresponding to the 600-cent tritone, its numerator and denominator would be required to satisfy mutually incompatible conditions — [[wikipedia:Square root of 2#Proof by infinite descent|being both even and coprime]]. Similarly, if a stack of pure fifths (3/2) were to close exactly at the octave, the resulting comma {{Monzo|-X Y}} would have to equal 1. In this number, numerator X must be a power of 2 and the denominator Y a power of 3, thus implying the existence of an even power of 3, which is not possible.

More broadly, while differences in tuning systems can affect perceived consonance, beating, and timbral character, these effects remain within the domain of auditory perception and musical aesthetics. There is no established physiological or cognitive mechanism by which specific frequency ratios could influence complex outcomes such as public health, social stability, or quality of life at scale. Such phenomena are determined by a wide range of economic, environmental, and social factors, none of which have demonstrated dependence on musical tuning standards.

Accordingly, claims that particular tuning systems produce measurable improvements in human well-being would require empirical verification under controlled conditions. In the absence of such evidence, assertions of this kind remain unsupported, particularly when presented within numerological or occultist frameworks rather than scientific methodology.

== External links ==
* [https://www.youtube.com/watch?v=FY74AFQl2qQ Sonic Geometry: The Language of Frequency and Form] - the original video
* [https://www.youtube.com/watch?v=ghUs-84NAAU Testing 432Hz Frequencies and Temperaments] - refuting by Adam Neely

[[Category:23-limit]]

Factor 9 grid

2026-03-31T03:28:39Z

Eliora: /* Mysticism and reality */

'''Factor 9 grid''' is a type of musical scale which was first proposed for esoteric reasons as a supposed replacement to [[12edo]]. The scale also became notable in music theory when esoteric properties of the scale were subsequently refuted by the famous YouTuber, bassist, and composer [[Adam Neely]].

== Theory ==
The scale is an isoharmonic sequence consisting of the following frequencies (in Hz): 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, and their octave equivalents. This sequence forms an arithmetic progression with a constant difference of 9, which gives rise to the name "Factor 9 grid". It hence is identical to the [[14ado]] scale spanning the 14th through 28th harmonics, and since 14ado is a [[23-limit]] just intonation system, Factor 9 grid correspondingly is a part of of [[23-limit]] just intonation.

More precisely, the "Factor 9 grid" refers to a specific mode of [[14ado]] whose tonic is placed on the step corresponding to 432 Hz and its octave equivalents, such as 216 Hz or 864 Hz, which is the 11th step of 14ado itself. It is this particular modal alignment that is commonly associated with "A = 432 Hz" conspiracy theories, where the emphasis is placed on organizing the scale around 432 Hz as a tonic.

Proponents of the "Factor 9 grid" manly present it as the more consonant or acoustically "healthier" alternative to the prevailing [[12edo|12-tone equal temperament]], often accompanied by references to the symbolic or "sacred1 significance of the number 12. However, descriptions of the scale, as they are presented in the video, contain several internal inconsistencies. The underlying structure of the grid, as mentioned above, corresponds to [[14ado]], which by definition contains 14 distinct steps per octave rather than 12, thus conflicting with the initial claims that the scale is 12-note or is a replacement for 12edo.

Furthermore, in the cited material, the sequence appears to omit the frequency 243 Hz (and its octave equivalents), despite it being a member of arithmetic progression that constitutes the scale. This omission appears to be motivated by an attempt to align the number of pitches with the 12-note framework of standard Western notation, as the presentation maps the resulting tones onto conventional note names. However, the rationale for excluding specifically 243 Hz and its octave displacement, as opposed to any other member of the sequence, is not explicitly addressed. No criteria are provided for why this particular step is removed while the remaining tones are retained, leaving the adjustment unexplained within the context of the scale’s stated arithmetic construction.

Additionally, the accompanying table distinguishes between G♯ and A♭ as separate pitches, resulting in a 13-note scale rather than either 12 or 14. This is not explained in the video as well, and further complicates the stated aim of aligning the system with a 12-note framework, particularly in light of the emphasis placed on the number 12 in the associated commentary.

== Intervals ==
The 9/8 interval was skipped by the original video for unknown reasons.
{| class="wikitable"
|+ Factor 9 Grid on 432 Hz
|-
! Frequency (Hz) !! Note !! Interval
|-
| 432 || A || [[1/1]]
|-
| 450 || A#/Bb || [[25/24]]
|-
| 468 || B || [[13/12]]
|-
| 486 || - || [[9/8]]
|-
| 504 || C || [[7/6]]
|-
| 540 || C#/Db || [[5/4]]
|-
| 576 || D || [[4/3]]
|-
| 612 || D#/Eb || [[17/12]]
|-
| 648 || E || [[3/2]]
|-
| 684 || F || [[19/12]]
|-
| 720 || F#/Gb || [[5/3]]
|-
| 756 || G || [[7/4]]
|-
| 792 || G# || [[11/6]]
|-
| 828 || Ab || [[23/12]]
|-
| 864 || A || [[2/1]]
|}

== Attempts at representing the scale through regular temperament theory ==
Within regular temperament theory, the Factor 9 grid admits close approximations through equal divisions of just intervals, and is well-supported by any tuning system that either represents the [[23-limit]] intervals well, or the direct intervals mentioned in the scale.

Though, in particular, it is worth noting that it can be represented by 666 equal divisions of the 15/14 interval (666ed15/14), which directly models its harmonic structure, as well as by 666edo, which supports a related regular temperament with similar properties, which is a very ironic aspect of the scale.

The scale, often justified by proponents through numerological arguments emphasizing purity and the special status of certain numbers, is located in close correspondence with equal temperament systems involving the number [[wikipedia:666 (number)|666]], which is a value widely regarded in popular numerological traditions, and Western popular culture as a whole as [[wikipedia:the Number of the Beast|the Number of the Beast]] or satanic. Thus, within the proponents’ own framework, where numerical symbolism is treated as musically or metaphysically significant, this association with 666 directly contradicts the stated rationale for privileging the scale over conventional tuning systems.

[[Eliora]] when trying to represent the Factor 9 grid via a rank-2 temperament did not realize that x31eq.com resource takes the first interval in the number series as the equivalence interval, and assumed that it is represented by [[666edo]] instead. "q666" in the temperament finder, when 14:15:16:... typed out, stands for 666ed15/14 as opposed to 666ed2.

This being said, 666edo, more specifically, the 495 & 666 23-limit temperament, regular temperament preserves more properties of the "Factor 9 Grid" than the corrected version, since it's period-9 and its period minus reduced generator interval also maps to the smallest interval in the scale, [[28/27]]. So in a limited way, 666edo does ironically represent the Factor 9 grid well. The corrected version would be with 495ed15/14 & 666ed15/14, which corresponds to a 4973 & 6691 temperament in the 23-limit, however that temperament has no structure resembling the initial factor-9 grid other than closely approximating its constituent intervals.

The 495 & 666 temperament is given the name ''enneasoteric'' by Eliora, since music was already composed in the 666edo scale approximating Factor 9 grid, while the name "factor 9 grid" to avoid ambiguity will be retained with the scale only.

=== Enneasoteric (495 & 666) ===
[[Subgroup]]: 2.3.5.7.11.13.17.19.23

[[Comma list]]: 442/441, 715/714, 2300/2299, 3060/3059, 3179/3174, 9025/9009, 57375/57344

[[Mapping]] [[generators]]: ~250/231, ~336/323

[[Optimal tuning]] ([[CTE]]): ~336/323 = 70.270

{{Optimal ET sequence|legend=1|171f, 495, 666}}.

== Mysticism and reality ==
The motivation behind the scale, as with many esoteric just intonation proposals, is the claim that the irrational pitch relationships of equal temperament produce acoustically unpleasant effects which, in turn, are said to propagate into subconscious perception and negatively affect human well-being.

However, claims that specific tuning systems (such as just intonation or particular frequency standards like 432 Hz) have direct effects on public health, social cohesion, or global conditions are not supported by empirical evidence. While differences in tuning can influence perceived consonance, timbre, and listener preference, these effects operate at the level of auditory perception and musical aesthetics rather than large-scale societal outcomes.

Furthermore, from a mathematical perspective, it is not possible to simultaneously achieve the exact rational interval relationships of just intonation and the structural evenness of equal temperament. The irrationality inherent to equal divisions of the octave has been recognized since antiquity, most commonly through proofs such as the irrationality of √2. For example, if there were an exact just intonation fraction corresponding to the 600-cent tritone, its numerator and denominator would be required to satisfy mutually incompatible conditions — [[wikipedia:Square root of 2#Proof by infinite descent|being both even and coprime]]. Similarly, if a stack of pure fifths (3/2) were to close exactly at the octave, the resulting comma {{Monzo|-X Y}} would have to equal 1. In this number, numerator X must be a power of 2 and the denominator Y a power of 3, thus implying the existence of an even power of 3, which is not possible.

More broadly, while differences in tuning systems can affect perceived consonance, beating, and timbral character, these effects remain within the domain of auditory perception and musical aesthetics. There is no established physiological or cognitive mechanism by which specific frequency ratios could influence complex outcomes such as public health, social stability, or quality of life at scale. Such phenomena are determined by a wide range of economic, environmental, and social factors, none of which have demonstrated dependence on musical tuning standards.

Accordingly, claims that particular tuning systems produce measurable improvements in human well-being would require empirical verification under controlled conditions. In the absence of such evidence, assertions of this kind remain unsupported, particularly when presented within numerological or occultist frameworks rather than scientific methodology.

== External links ==
* [https://www.youtube.com/watch?v=FY74AFQl2qQ Sonic Geometry: The Language of Frequency and Form] - the original video
* [https://www.youtube.com/watch?v=ghUs-84NAAU Testing 432Hz Frequencies and Temperaments] - refuting by Adam Neely

[[Category:23-limit]]

Factor 9 grid

2026-03-31T03:17:28Z

Eliora: /* Attempts at representing the scale through regular temperament theory */ do the same, expand

'''Factor 9 grid''' is a type of musical scale which was first proposed for esoteric reasons as a supposed replacement to [[12edo]]. The scale also became notable in music theory when esoteric properties of the scale were subsequently refuted by the famous YouTuber, bassist, and composer [[Adam Neely]].

== Theory ==
The scale is an isoharmonic sequence consisting of the following frequencies (in Hz): 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, and their octave equivalents. This sequence forms an arithmetic progression with a constant difference of 9, which gives rise to the name "Factor 9 grid". It hence is identical to the [[14ado]] scale spanning the 14th through 28th harmonics, and since 14ado is a [[23-limit]] just intonation system, Factor 9 grid correspondingly is a part of of [[23-limit]] just intonation.

More precisely, the "Factor 9 grid" refers to a specific mode of [[14ado]] whose tonic is placed on the step corresponding to 432 Hz and its octave equivalents, such as 216 Hz or 864 Hz, which is the 11th step of 14ado itself. It is this particular modal alignment that is commonly associated with "A = 432 Hz" conspiracy theories, where the emphasis is placed on organizing the scale around 432 Hz as a tonic.

Proponents of the "Factor 9 grid" manly present it as the more consonant or acoustically "healthier" alternative to the prevailing [[12edo|12-tone equal temperament]], often accompanied by references to the symbolic or "sacred1 significance of the number 12. However, descriptions of the scale, as they are presented in the video, contain several internal inconsistencies. The underlying structure of the grid, as mentioned above, corresponds to [[14ado]], which by definition contains 14 distinct steps per octave rather than 12, thus conflicting with the initial claims that the scale is 12-note or is a replacement for 12edo.

Furthermore, in the cited material, the sequence appears to omit the frequency 243 Hz (and its octave equivalents), despite it being a member of arithmetic progression that constitutes the scale. This omission appears to be motivated by an attempt to align the number of pitches with the 12-note framework of standard Western notation, as the presentation maps the resulting tones onto conventional note names. However, the rationale for excluding specifically 243 Hz and its octave displacement, as opposed to any other member of the sequence, is not explicitly addressed. No criteria are provided for why this particular step is removed while the remaining tones are retained, leaving the adjustment unexplained within the context of the scale’s stated arithmetic construction.

Additionally, the accompanying table distinguishes between G♯ and A♭ as separate pitches, resulting in a 13-note scale rather than either 12 or 14. This is not explained in the video as well, and further complicates the stated aim of aligning the system with a 12-note framework, particularly in light of the emphasis placed on the number 12 in the associated commentary.

== Intervals ==
The 9/8 interval was skipped by the original video for unknown reasons.
{| class="wikitable"
|+ Factor 9 Grid on 432 Hz
|-
! Frequency (Hz) !! Note !! Interval
|-
| 432 || A || [[1/1]]
|-
| 450 || A#/Bb || [[25/24]]
|-
| 468 || B || [[13/12]]
|-
| 486 || - || [[9/8]]
|-
| 504 || C || [[7/6]]
|-
| 540 || C#/Db || [[5/4]]
|-
| 576 || D || [[4/3]]
|-
| 612 || D#/Eb || [[17/12]]
|-
| 648 || E || [[3/2]]
|-
| 684 || F || [[19/12]]
|-
| 720 || F#/Gb || [[5/3]]
|-
| 756 || G || [[7/4]]
|-
| 792 || G# || [[11/6]]
|-
| 828 || Ab || [[23/12]]
|-
| 864 || A || [[2/1]]
|}

== Attempts at representing the scale through regular temperament theory ==
Within regular temperament theory, the Factor 9 grid admits close approximations through equal divisions of just intervals, and is well-supported by any tuning system that either represents the [[23-limit]] intervals well, or the direct intervals mentioned in the scale.

Though, in particular, it is worth noting that it can be represented by 666 equal divisions of the 15/14 interval (666ed15/14), which directly models its harmonic structure, as well as by 666edo, which supports a related regular temperament with similar properties, which is a very ironic aspect of the scale.

The scale, often justified by proponents through numerological arguments emphasizing purity and the special status of certain numbers, is located in close correspondence with equal temperament systems involving the number [[wikipedia:666 (number)|666]], which is a value widely regarded in popular numerological traditions, and Western popular culture as a whole as [[wikipedia:the Number of the Beast|the Number of the Beast]] or satanic. Thus, within the proponents’ own framework, where numerical symbolism is treated as musically or metaphysically significant, this association with 666 directly contradicts the stated rationale for privileging the scale over conventional tuning systems.

[[Eliora]] when trying to represent the Factor 9 grid via a rank-2 temperament did not realize that x31eq.com resource takes the first interval in the number series as the equivalence interval, and assumed that it is represented by [[666edo]] instead. "q666" in the temperament finder, when 14:15:16:... typed out, stands for 666ed15/14 as opposed to 666ed2.

This being said, 666edo, more specifically, the 495 & 666 23-limit temperament, regular temperament preserves more properties of the "Factor 9 Grid" than the corrected version, since it's period-9 and its period minus reduced generator interval also maps to the smallest interval in the scale, [[28/27]]. So in a limited way, 666edo does ironically represent the Factor 9 grid well. The corrected version would be with 495ed15/14 & 666ed15/14, which corresponds to a 4973 & 6691 temperament in the 23-limit, however that temperament has no structure resembling the initial factor-9 grid other than closely approximating its constituent intervals.

The 495 & 666 temperament is given the name ''enneasoteric'' by Eliora, since music was already composed in the 666edo scale approximating Factor 9 grid, while the name "factor 9 grid" to avoid ambiguity will be retained with the scale only.

=== Enneasoteric (495 & 666) ===
[[Subgroup]]: 2.3.5.7.11.13.17.19.23

[[Comma list]]: 442/441, 715/714, 2300/2299, 3060/3059, 3179/3174, 9025/9009, 57375/57344

[[Mapping]] [[generators]]: ~250/231, ~336/323

[[Optimal tuning]] ([[CTE]]): ~336/323 = 70.270

{{Optimal ET sequence|legend=1|171f, 495, 666}}.

== Mysticism and reality ==
The motivation behind the scale, or any kind of esoteric just intonation proposal, is that the irrational pitch standard of equal temperament has an acoustically unpleasant effect, that subsequently allegedly spreads to the subconscious levels of the mind and results in deterioration of the quality of human life.

However, it is not possible to have both the rationality of just intonation and evenness of equal temperament at the same time. The irrationality of equal temperament has been known since ancient times by various names, the most common being the proof by infinite descent of the square root of two. If there was an exact JI interval fraction which corresponds to the 600 cent tritone, it's numerator and denominator would have to be both even and coprime, which cannot happen. Another example is that if a stack of fifths (3/2) were to close at the octave, the resulting comma {{Monzo|-X Y}} would be equal to 1, while having X be a power of 2, and Y be a power of 3. This would imply the existence of an even power of 3, which is nonsense.

If there is credible scientific evidence that playing just intonation music through public loudspeakers, in soundtracks, or other public venues will increase peace and quality of human life, it must be presented properly, without occultist context.

== External links ==
* [https://www.youtube.com/watch?v=FY74AFQl2qQ Sonic Geometry: The Language of Frequency and Form] - the original video
* [https://www.youtube.com/watch?v=ghUs-84NAAU Testing 432Hz Frequencies and Temperaments] - refuting by Adam Neely

[[Category:23-limit]]

Factor 9 grid

2026-03-31T03:04:59Z

Eliora: /* Theory */

'''Factor 9 grid''' is a type of musical scale which was first proposed for esoteric reasons as a supposed replacement to [[12edo]]. The scale also became notable in music theory when esoteric properties of the scale were subsequently refuted by the famous YouTuber, bassist, and composer [[Adam Neely]].

== Theory ==
The scale is an isoharmonic sequence consisting of the following frequencies (in Hz): 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, and their octave equivalents. This sequence forms an arithmetic progression with a constant difference of 9, which gives rise to the name "Factor 9 grid". It hence is identical to the [[14ado]] scale spanning the 14th through 28th harmonics, and since 14ado is a [[23-limit]] just intonation system, Factor 9 grid correspondingly is a part of of [[23-limit]] just intonation.

More precisely, the "Factor 9 grid" refers to a specific mode of [[14ado]] whose tonic is placed on the step corresponding to 432 Hz and its octave equivalents, such as 216 Hz or 864 Hz, which is the 11th step of 14ado itself. It is this particular modal alignment that is commonly associated with "A = 432 Hz" conspiracy theories, where the emphasis is placed on organizing the scale around 432 Hz as a tonic.

Proponents of the "Factor 9 grid" manly present it as the more consonant or acoustically "healthier" alternative to the prevailing [[12edo|12-tone equal temperament]], often accompanied by references to the symbolic or "sacred1 significance of the number 12. However, descriptions of the scale, as they are presented in the video, contain several internal inconsistencies. The underlying structure of the grid, as mentioned above, corresponds to [[14ado]], which by definition contains 14 distinct steps per octave rather than 12, thus conflicting with the initial claims that the scale is 12-note or is a replacement for 12edo.

Furthermore, in the cited material, the sequence appears to omit the frequency 243 Hz (and its octave equivalents), despite it being a member of arithmetic progression that constitutes the scale. This omission appears to be motivated by an attempt to align the number of pitches with the 12-note framework of standard Western notation, as the presentation maps the resulting tones onto conventional note names. However, the rationale for excluding specifically 243 Hz and its octave displacement, as opposed to any other member of the sequence, is not explicitly addressed. No criteria are provided for why this particular step is removed while the remaining tones are retained, leaving the adjustment unexplained within the context of the scale’s stated arithmetic construction.

Additionally, the accompanying table distinguishes between G♯ and A♭ as separate pitches, resulting in a 13-note scale rather than either 12 or 14. This is not explained in the video as well, and further complicates the stated aim of aligning the system with a 12-note framework, particularly in light of the emphasis placed on the number 12 in the associated commentary.

== Intervals ==
The 9/8 interval was skipped by the original video for unknown reasons.
{| class="wikitable"
|+ Factor 9 Grid on 432 Hz
|-
! Frequency (Hz) !! Note !! Interval
|-
| 432 || A || [[1/1]]
|-
| 450 || A#/Bb || [[25/24]]
|-
| 468 || B || [[13/12]]
|-
| 486 || - || [[9/8]]
|-
| 504 || C || [[7/6]]
|-
| 540 || C#/Db || [[5/4]]
|-
| 576 || D || [[4/3]]
|-
| 612 || D#/Eb || [[17/12]]
|-
| 648 || E || [[3/2]]
|-
| 684 || F || [[19/12]]
|-
| 720 || F#/Gb || [[5/3]]
|-
| 756 || G || [[7/4]]
|-
| 792 || G# || [[11/6]]
|-
| 828 || Ab || [[23/12]]
|-
| 864 || A || [[2/1]]
|}

== Attempts at representing the scale through regular temperament theory ==
In an extremely ironic twist, this allegedly divine and enlightening scale is well-approximated by an evil satanic temperament of 666 equal divisions of the 15/14 interval. However, [[Eliora]] when trying to represent the Factor 9 grid via a rank-2 temperament did not realize that x31eq.com resource takes the first interval in the number series as the equivalence interval, and assumed that it is represented by [[666edo]] instead. "q666" in the temperament finder, when 14:15:16:... typed out, stands for 666ed15/14 as opposed to 666ed2.

This being said, this temperament preserves more properties of the "Factor 9 Grid" than the corrected version, since it's period-9 and its period minus reduced generator interval also maps to the smallest interval in the Factor 9 Grid system, [[28/27]]. So in the end, 666edo does ironically represent the Factor 9 grid well. The corrected version would be with 495ed15/14 & 666ed15/14, which corresponds to a 4973 & 6691 temperament in the 23-limit, that has no structure resembling the initial factor-9 grid other than being made of two good edos in the 23-limit.

The 495 & 666 temperament is given the name ''enneasoteric'' by Eliora, since music was already composed in the 666edo scale approximating Factor 9 grid, while the name "factor 9 grid" to avoid ambiguity will be retained with the scale only.

=== Enneasoteric (495 & 666) ===
[[Subgroup]]: 2.3.5.7.11.13.17.19.23

[[Comma list]]: 442/441, 715/714, 2300/2299, 3060/3059, 3179/3174, 9025/9009, 57375/57344

[[Mapping]] [[generators]]: ~250/231, ~336/323

[[Optimal tuning]] ([[CTE]]): ~336/323 = 70.270

{{Optimal ET sequence|legend=1|171f, 495, 666}}.

== Mysticism and reality ==
The motivation behind the scale, or any kind of esoteric just intonation proposal, is that the irrational pitch standard of equal temperament has an acoustically unpleasant effect, that subsequently allegedly spreads to the subconscious levels of the mind and results in deterioration of the quality of human life.

However, it is not possible to have both the rationality of just intonation and evenness of equal temperament at the same time. The irrationality of equal temperament has been known since ancient times by various names, the most common being the proof by infinite descent of the square root of two. If there was an exact JI interval fraction which corresponds to the 600 cent tritone, it's numerator and denominator would have to be both even and coprime, which cannot happen. Another example is that if a stack of fifths (3/2) were to close at the octave, the resulting comma {{Monzo|-X Y}} would be equal to 1, while having X be a power of 2, and Y be a power of 3. This would imply the existence of an even power of 3, which is nonsense.

If there is credible scientific evidence that playing just intonation music through public loudspeakers, in soundtracks, or other public venues will increase peace and quality of human life, it must be presented properly, without occultist context.

== External links ==
* [https://www.youtube.com/watch?v=FY74AFQl2qQ Sonic Geometry: The Language of Frequency and Form] - the original video
* [https://www.youtube.com/watch?v=ghUs-84NAAU Testing 432Hz Frequencies and Temperaments] - refuting by Adam Neely

[[Category:23-limit]]

Factor 9 grid

2026-03-31T03:04:40Z

Eliora: /* Theory */ streamline the writing, make it more formal, read like a proper reading of a source not a claim about F9G

'''Factor 9 grid''' is a type of musical scale which was first proposed for esoteric reasons as a supposed replacement to [[12edo]]. The scale also became notable in music theory when esoteric properties of the scale were subsequently refuted by the famous YouTuber, bassist, and composer [[Adam Neely]].

== Theory ==
The scale is an isoharmonic sequence consisting of the following frequencies (in Hz): 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, and their octave equivalents. This sequence forms an arithmetic progression with a constant difference of 9, which gives rise to the name "Factor 9 grid". It hence is identical to the [[14ado]] scale spanning the 14th through 28th harmonics, and since 14ado is a [[23-limit]] just intonation system, Factor 9 grid correspondingly is a part of of [[23-limit]] just intonation.

More precisely, the "Factor 9 grid" refers to a specific mode of [[14ado]] whose tonic is placed on the step corresponding to 432 Hz and its octave equivalents, such as 216 Hz or 864 Hz, which is the 11th step of 14ado itself. It is this particular modal alignment that is commonly associated with "A = 432 Hz" conspiracy theories, where the emphasis is placed on organizing the scale around 432 Hz as a tonic.

Proponents of the "Factor 9 grid" manly present it as the more consonant or acoustically "healthier" alternative to the prevailing [[12edo|12-tone equal temperament]], often accompanied by references to the symbolic or "sacred” significance of the number 12. However, descriptions of the scale, as they are presented in the video, contain several internal inconsistencies. The underlying structure of the grid, as mentioned above, corresponds to [[14ado]], which by definition contains 14 distinct steps per octave rather than 12, thus conflicting with the initial claims that the scale is 12-note or is a replacement for 12edo.

Furthermore, in the cited material, the sequence appears to omit the frequency 243 Hz (and its octave equivalents), despite it being a member of arithmetic progression that constitutes the scale. This omission appears to be motivated by an attempt to align the number of pitches with the 12-note framework of standard Western notation, as the presentation maps the resulting tones onto conventional note names. However, the rationale for excluding specifically 243 Hz and its octave displacement, as opposed to any other member of the sequence, is not explicitly addressed. No criteria are provided for why this particular step is removed while the remaining tones are retained, leaving the adjustment unexplained within the context of the scale’s stated arithmetic construction.

Additionally, the accompanying table distinguishes between G♯ and A♭ as separate pitches, resulting in a 13-note scale rather than either 12 or 14. This is not explained in the video as well, and further complicates the stated aim of aligning the system with a 12-note framework, particularly in light of the emphasis placed on the number 12 in the associated commentary.

== Intervals ==
The 9/8 interval was skipped by the original video for unknown reasons.
{| class="wikitable"
|+ Factor 9 Grid on 432 Hz
|-
! Frequency (Hz) !! Note !! Interval
|-
| 432 || A || [[1/1]]
|-
| 450 || A#/Bb || [[25/24]]
|-
| 468 || B || [[13/12]]
|-
| 486 || - || [[9/8]]
|-
| 504 || C || [[7/6]]
|-
| 540 || C#/Db || [[5/4]]
|-
| 576 || D || [[4/3]]
|-
| 612 || D#/Eb || [[17/12]]
|-
| 648 || E || [[3/2]]
|-
| 684 || F || [[19/12]]
|-
| 720 || F#/Gb || [[5/3]]
|-
| 756 || G || [[7/4]]
|-
| 792 || G# || [[11/6]]
|-
| 828 || Ab || [[23/12]]
|-
| 864 || A || [[2/1]]
|}

== Attempts at representing the scale through regular temperament theory ==
In an extremely ironic twist, this allegedly divine and enlightening scale is well-approximated by an evil satanic temperament of 666 equal divisions of the 15/14 interval. However, [[Eliora]] when trying to represent the Factor 9 grid via a rank-2 temperament did not realize that x31eq.com resource takes the first interval in the number series as the equivalence interval, and assumed that it is represented by [[666edo]] instead. "q666" in the temperament finder, when 14:15:16:... typed out, stands for 666ed15/14 as opposed to 666ed2.

This being said, this temperament preserves more properties of the "Factor 9 Grid" than the corrected version, since it's period-9 and its period minus reduced generator interval also maps to the smallest interval in the Factor 9 Grid system, [[28/27]]. So in the end, 666edo does ironically represent the Factor 9 grid well. The corrected version would be with 495ed15/14 & 666ed15/14, which corresponds to a 4973 & 6691 temperament in the 23-limit, that has no structure resembling the initial factor-9 grid other than being made of two good edos in the 23-limit.

The 495 & 666 temperament is given the name ''enneasoteric'' by Eliora, since music was already composed in the 666edo scale approximating Factor 9 grid, while the name "factor 9 grid" to avoid ambiguity will be retained with the scale only.

=== Enneasoteric (495 & 666) ===
[[Subgroup]]: 2.3.5.7.11.13.17.19.23

[[Comma list]]: 442/441, 715/714, 2300/2299, 3060/3059, 3179/3174, 9025/9009, 57375/57344

[[Mapping]] [[generators]]: ~250/231, ~336/323

[[Optimal tuning]] ([[CTE]]): ~336/323 = 70.270

{{Optimal ET sequence|legend=1|171f, 495, 666}}.

== Mysticism and reality ==
The motivation behind the scale, or any kind of esoteric just intonation proposal, is that the irrational pitch standard of equal temperament has an acoustically unpleasant effect, that subsequently allegedly spreads to the subconscious levels of the mind and results in deterioration of the quality of human life.

However, it is not possible to have both the rationality of just intonation and evenness of equal temperament at the same time. The irrationality of equal temperament has been known since ancient times by various names, the most common being the proof by infinite descent of the square root of two. If there was an exact JI interval fraction which corresponds to the 600 cent tritone, it's numerator and denominator would have to be both even and coprime, which cannot happen. Another example is that if a stack of fifths (3/2) were to close at the octave, the resulting comma {{Monzo|-X Y}} would be equal to 1, while having X be a power of 2, and Y be a power of 3. This would imply the existence of an even power of 3, which is nonsense.

If there is credible scientific evidence that playing just intonation music through public loudspeakers, in soundtracks, or other public venues will increase peace and quality of human life, it must be presented properly, without occultist context.

== External links ==
* [https://www.youtube.com/watch?v=FY74AFQl2qQ Sonic Geometry: The Language of Frequency and Form] - the original video
* [https://www.youtube.com/watch?v=ghUs-84NAAU Testing 432Hz Frequencies and Temperaments] - refuting by Adam Neely

[[Category:23-limit]]

319edo

2026-03-22T16:58:34Z

Eliora: /* Theory */

{{Infobox ET}}
{{ED intro}}

== Theory ==
319edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway its steps. Nonetheless, it provides the optimal patent val for 5-limit [[mystery]] temperament, which tempers out the [[29-comma]], despite poor 5-limit harmonic approximation.

The full 11-limit [[patent val]] is nonetheless a reasonable interpretation since the lower harmonics all tend sharp. Using this val, it [[tempering out|tempers out]] [[6144/6125]], [[10976/10935]], and {{monzo| 9 -10 9 -5 }} in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], 15488/15435, 59290/59049, and [[65536/65219]] in the 11-limit. It [[support]]s [[mystery]] in the 5-limit and [[protolangwidge]].

If we instead adopt the 2.9.… [[subgroup]] interpretation, then 2.9.15.21 is a good subgroup to start with.

=== Odd harmonics ===
{{Harmonics in equal|319}}

=== Subsets and supersets ===
Since 319 factors into 11 × 29, 319edo has [[11edo]] and [[29edo]] as its subsets. [[638edo]], which doubles it, gives a good correction to the harmonics 3, 5, and 7.

== 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]] (%)
|-
| 2.9
| {{monzo| -1011 319 }}
| {{mapping| 319 1011 }}
| +0.1223
| 0.1223
| 3.25
|-
| 2.9.15
| {{monzo| -51 5 9 }}, {{monzo| -16 26 -17 }}
| {{mapping| 319 1011 1246 }}
| +0.1869
| 0.1353
| 3.60
|}

29th-octave temperaments

2026-03-22T16:55:41Z

Eliora: /* Copper */ less charged language

{{Infobox fractional-octave|29}}
[[29edo]] is notable for being the first equal division to have a more precise [[3/2]] than [[12edo]], and the first tuning to be consistent in the [[15-odd-limit]]. 29th-octave temperaments occur naturally when temperament-merging edos whose greatest common divisor is 29.

== Mystery (5-limit) ==
: ''Main article: [[Mystery]] and for higher-limit versions see [[Hemifamity temperaments #Mystery]]''

The mystery temperament in the 5-limit is described by tempering out the comma {{monzo| 46 -29 }}, where a circle of 29 fifths closes on 17 octaves, and it is supported by small multiples of 29edo.

Subgroup: 2.3.5

[[Comma]]: {{monzo| 46 -29 }}

[[Mapping]]: [{{val| 29 46 0 }}, {{val| 0 0 1 }}]

Mapping generators: ~531441/524288, ~5

[[POTE generator]]: ~5/4 = 387.408

{{Optimal ET sequence|legend=1| 29, 58, 87, 232, 319 }}

[[Badness]]: 1.020556

== Copper ==
Copper temperament is derived from a 5-limit comma called [[copper comma]], because it is constructed the same way towards 29edo as [[Kirnberger's atom]] is towards 12edo. A fifth of each of these tunings is modified by a tiny amount, then a circle of these fifths is set to close eventually at the octave.

It is worth noting that despite 29edo's fifth being closer to 3/2 than 12edo's, copper has a higher TE error than [[atomic]] and hence is not a [[very high accuracy temperaments|very high accuracy temperament]].

Subgroup: 2.3.5

Comma list: {{monzo|-481 261 29}}

Mapping: [{{val|29 0 481}}, {{val|0 1 -9}}]

Mapping generators: ~{{monzo|-199 12 108}} = 1\29, ~3/2 = 701.905

Optimal tuning (CTE): ~3/2 = 701.905

[[Support]]ing [[ET]]s: {{EDOs|29, 754, 783, 812, 1566, 1537, 2320, 3103, 3132}}, ...

{{Navbox fractional-octave}}

[[Category:29edo]]
{{Todo| review }}

80th-octave temperaments

2026-03-05T16:45:41Z

Eliora: /* Mercury */

{{Technical data page}}
{{Infobox fractional-octave|80}}

This page describes 80th-octave temperaments. [[80edo]] is extremely accurate for the [[17/1|17th harmonic]], a relationship which is so far seen in all documented temperaments on this page. In addition, it is the first equal division to be consistent in the [[19-limit]].

Temperaments discussed elsewhere include

* ''Octogintic'', → [[Parkleiness temperaments#Octogintic|Parkleiness temperaments]]

== Tetraicosic ==
Tetraicosic is described as 1600 & 2320, and named after the fact that 4 × 20 = 80.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: {{monzo| -52 17 12 -1 }}, {{monzo| 25 42 -8 -26 }}

[[Mapping]]: [{{val| 80 1 343 -27 }}, {{val| 0 4 -5 8 }}]

Mapping generators: ~31637227888/31381059609, ~7381125/5619712

[[Optimal tuning]] ([[CTE]]): ~7381125/5619712 = 471.7339

{{Optimal ET sequence|legend=1| 720, 1600, 2320, 3920 }}

[[Badness]]: 1.49

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 928760463360/928426965851, {{monzo| 49 -13 -14 -1 2 }}

Mapping: [{{val| 80 1 343 -27 434 }}, {{val| 0 4 -5 8 -5 }}]

Optimal tuning (CTE): ~2278125/1734656 = 471.7342

{{Optimal ET sequence|legend=1| 720, 1600, 2320, 3920 }}

Badness: 0.178

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 9801/9800, 4100625/4100096, 14236560/14235529, 143327232/143286143

Mapping: [{{val| 80 1 343 -27 434 13 }}, {{val| 0 4 -5 8 -5 9 }}]

Optimal tuning (CTE): ~130/99 = 471.7322

{{Optimal ET sequence|legend=1| 720, 1600, 2320, 3920 }}

Badness: 0.0741

=== 17-limit ===
Subgroup 2.3.5.7.11.13.17

Comma list: 9801/9800, 14400/14399, 373527/373490, 1812608/1812525, 4685824/4685625

Mapping: [{{val| 80 1 343 -27 434 13 327}}, {{val| 0 4 -5 8 -5 9 0}}]

Optimal tuning (CTE): ~130/99 = 471.7...

{{Optimal ET sequence|legend=1| 720, 1600, 2320, 3920 }}

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 9801/9800, 10830/10839, 12636/12635, 23409/23408, 373527/373490, 32133332/32131125

Mapping: [{{val| 80 1 343 -27 434 13 327 -69}}, {{val| 0 4 -5 8 -5 9 0 13}}]

Optimal tuning (CTE): ~67473/51376 = 471.732

{{Optimal ET sequence|legend=1| 720, 1600, 2320, 3920 }}

== Mercury ==
: ''Not to be confused with [[mercury meantone]] and [[mercurial comma|mercurial]].''

Named after the 80th element, defined as the 320 & 2000 temperament.
=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 9801/9800, 67392/67375, 1399680/1399489, {{monzo|14 -8 -8 5 2 -1}}

{{mapping|legend=1| 80 3 161 -23 252 197 | 0 5 1 10 1 4 }}

: mapping generators: ~9295/9216, ~2304/1859

Optimal tuning (CTE): ~2304/1859 = 371.385

{{Optimal ET sequence|legend=1|320, 2000, 4320}}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 9801/9800, 12376/12375, 67392/67375, 4230144/4229225, 494534656/494515125

{{mapping|legend=1| 80 3 161 -23 252 197 327 | 0 5 1 10 1 4 0 }}

: mapping generators: ~459/455, ~1859/1500

Optimal tuning (CTE): ~2275/1836 = 371.386

{{Optimal ET sequence|legend=1|320, 2000, 4320}}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 9801/9800, 12376/12375, 67392/67375, 392445/392392, 401408/401375, 1549184/1549125

Mapping: [{{val|80 3 161 -23 252 197 327 117}}, {{val|0 5 1 10 1 4 0 9}}]

: mapping generators: ~459/455, ~2275/1836

Optimal tuning (CTE): ~2275/1836 = 371.386

{{Optimal ET sequence|legend=1|320, 2000, 4320}}

== Octodeca ==
Octodeca can be described as the 80 & 1920 temperament.

Subgroup: 2.3.5.7

Comma list: {{monzo|21 60 -50}}, 184528125/184473632

{{mapping|legend=1|80 2 36 -25|0 5 6 10}}

: mapping generators: ~1071875/1062882 = 1\80, ~7476806640625/6025163444928 = 374.386

Optimal tuning (CTE): ~7476806640625/6025163444928 = 374.386

[[Support]]ing [[ET]]s: {{EDOs|80, 1920, 2000}}
=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 5832/5831, 9801/9800, 89376/89375, 123201/123200, 392445/392392, 653184/653125

{{mapping|legend=1|80 2 36 -25 -127 -321 -327 -240|0 5 6 10 6 -1 0 4}}

: mapping generators: ~12000/12103 = 1\80, ~1625/1309 = 374.386

Optimal tuning (CTE): ~1625/1309 = 374.386

=== 23-limit ===

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 5832/5831, 8625/8624, 9801/9800, 10626/10625, 89376/89375, 95013/95000, 123201/123200

{{mapping|legend=1|80 2 36 -25 -127 -321 -327 -240 -287|0 5 6 10 6 -1 0 4 3}}

: mapping generators: ~12000/12103 = 1\80, ~920/741 = 374.387

Optimal tuning (CTE): ~920/741 = 374.387

{{Navbox fractional-octave}}

[[Category:80edo]]]

User:Eliora/2025edo

2025-12-30T21:33:11Z

Eliora: Created page with "{{Infobox ET|2025edo}} {{ED intro|2025edo}} == Theory == {{harmonics in equal|2025}} == Music == * [https://www.youtube.com/watch?v=jTyp85G_Fsw Rumbling Air]"

{{Infobox ET|2025edo}}
{{ED intro|2025edo}}

== Theory ==
{{harmonics in equal|2025}}

== Music ==

* [https://www.youtube.com/watch?v=jTyp85G_Fsw Rumbling Air]

261edo

2025-12-21T22:35:14Z

Eliora: expand, also remove links to temperaments where 261 isn't explicitly listed in val lists - clarify those?

{{Infobox ET}}
{{ED intro}}

261edo is [[consistent]] in the [[7-odd-limit]] and shares its harmonic 3 with [[29edo]], though with a significant error. Like other small EDO multiples of 29, it tunes the [[mystery]] temperament. A comma basis for the 7-limit is {10976/10935, 15625/15552, 2097152/2083725}. It is also [[enfactoring|enfactored]] in the 2.5.11 subgroup, inheriting it from [[87edo]] which it triples.

261edo's patent val supports and is part of the [[optimal ET sequence]] for the [[diatessic]] temperament in the 11- and 13-limit, [[novemkleismic]] in the 7-limit and the 11-limit. 261ccdee val, {{val|261 414 '''605''' '''732''' '''904'''}} supports the [[superkleismic]] temperament.

=== Odd harmonics ===
{{Harmonics in equal|261}}

=== Subsets and supersets ===
Since 261 factors as {{Factorization|261}}, 261edo has subset edos {{EDOs|1, 3, 9, 29, 87}}.

[[783edo]], which divides each edostep in three, provides a strong correction for harmonics 3 and 7.

[[Category:Novemkleismic]]
[[Category:Diatessic]]

User:Eliora/899edo

2025-12-21T22:22:34Z

Eliora: Created page with "{{Infobox ET|899edo}} {{ED intro|899edo}} === Prime harmonics === {{harmonics in equal|899}} === Subsets and supersets === Since 899 factors as {{Factorization|899}}, 899edo contains 29edo and 31edo."

{{Infobox ET|899edo}}
{{ED intro|899edo}}

=== Prime harmonics ===
{{harmonics in equal|899}}

=== Subsets and supersets ===

Since 899 factors as {{Factorization|899}}, 899edo contains [[29edo]] and [[31edo]].

2024edo

2025-12-18T17:58:42Z

Eliora: /* Theory */ duplicate text removed

{{Infobox ET}}
{{ED intro}}
== Theory ==
2024edo is [[Enfactoring|enfactored]] in the 13-limit, with the same tuning as [[1012edo]], which is also a [[zeta]] edo. It corrects 1012edo's mapping for 17, being a strong 2.3.7.11.17.19 subgroup temperament. A comma basis for the said subgroup is {23409/23408, 117649/117612, 323456/323433, 131072/131043, 26042368/26040609}. In the said subgroup, it tunes the {{nowrap|629 & 2024}} temperament, which has [[19/17]] as a generator and reaches [[17/16]] in 13 steps.

It has two suitable mappings for [[5/1|5th harmonic]], one which derives from 1012edo, and other in the 2024c val. In the 2024c val, it [[tempering out|tempers out]] the [[wizma]], 420175/419904 in the 7-limit, as well as [[3025/3024]], [[4225/4224]] and [[10648/10647]] in the 13-limit. If the sharp and flat mappings of 5/4 are combined, then 2024edo is a good 2.3.25 [[subgroup]] tuning. In the 2.3.25.7.11 subgroup, it tempers out [[4375/4374]] and tunes a messed-up version of the [[heimdall]] temperament, which reaches 7th harmonic in 2 second generators instead of 4, and 11th harmonic in 6 second generators instead of 12, taking half as much.

=== Prime harmonics ===
{{Harmonics in equal|2024}}

=== Subsets and supersets ===
Since 2024 factors into {{factorization|2024}}, 2024edo has subset edos {{EDOs| 2, 4, 8, 11, 22, 23, 44, 46, 88, 92, 184, 253, 506, and 1012 }}.

== Music ==
; [[Eliora]]
* ''[https://youtu.be/6wO9uONbJhY Piano Juice]'' (2024) - palladium[92] (from [[506edo]]), ruthenium[88] (from [[1012edo]]), {{nowrap|629 & 2024[25]}}, major-arcana[88] (from [[506edo]])

[[Category:Listen]]

3125/2916

2025-11-06T23:50:58Z

Eliora: fronting, emphasis

{{Infobox Interval
| Name = sixix comma
| Color name = y51, quinyo unison
| Comma = yes
}}
'''3125/2916''', or the '''sixix comma''' (119.839 [[cent]]s in size) is the comma which defines the [[sixix]] temperament, a temperament available in [[25edo]], [[32edo]] and [[43edo]].

If treated as a melodic interval rather than a comma, then it is very well approximated by one step of [[10edo]]. The small difference between a stack of 10 3125/2916's and the octave, {{monzo| 21 60 -50 }}, is called the [[neon comma]] and it realizes the [[neon]] 5-limit temperament.

== See also ==
* [[Gallery of just intervals]]
* [[Large comma]]

[[Category:Sixix]]
[[Category:Commas with unknown etymology]]

696edo

2025-10-11T23:55:35Z

Eliora: /* Regular temperament properties */

{{Infobox ET}}
{{ED intro}}

696edo is a strong 7-limit tuning, but unfortunately it is consistent only up to the [[9-odd-limit]]. In the 5-limit, it tempers out the schisma, and in the 7-limit, the landscape comma. It supports the [[magnesium]] temperament which divides the octave in 12, as well as [[chromium]] temperament that divides it in 24.

Nonetheless despite inconsistency, it is a valuable xenharmonic system in higher limits. It provides the [[optimal patent val]] for the [[octant]] temperament in the 13-limit, even if its approximation of 13 is almost half a step off. Likewise, 696edo tunes [[altierran]] and [[house]] temperaments in the 11-limit. In the higher limits, it may be used as a 2.3.5.7.17.31 subgroup tuning.

The 696cc val is also very close to the [[POTE]] tuning for the [[witcher]] temperament, while 696f tunes [[semiterm]] and the inaccurate 696d tunes [[pontic]].

=== Prime harmonics ===
{{Harmonics in equal|696}}

=== Subsets and supersets ===

Since 696 factors as {{Factorization|696}}, 696edo has subset edos {{EDOs|1, 2, 3, 4, 6, 8, 12, 24, 29, 58, 87, 116, 174, 232, 348}}.

== 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|-1103 696}}
|{{mapping|696 1103}}
|0.072829
|0.073
|4.22
|-
|2.3.5
|32805/32768, {{monzo|52 80 -77}}
|{{mapping|696 1103 1616}}
|0.060798
|0.064
|3.71
|-
|2.3.5.7
|32805/32768, 250047/250000, {{monzo|22 10 -3 -11}}
|{{mapping|696 1103 1616 1954}}
|0.072061
|0.035
|2.06
|-
|2.3.5.7.11
|9801/9800, 32805/32768, 46656/46585, 250047/250000
|{{mapping|696 1103 1616 1954 2408}}
|0.004896
|0.089
|5.15
|-
|2.3.5.7.11.13
|729/728, 1575/1573, 4096/4095, 67392/67375, 250047/250000
|{{mapping|696 1103 1616 1954 2408 2576}}
| -0.034283
|0.119
|6.92
|}

696edo

2025-10-11T23:55:05Z

Eliora:

{{Infobox ET}}
{{ED intro}}

696edo is a strong 7-limit tuning, but unfortunately it is consistent only up to the [[9-odd-limit]]. In the 5-limit, it tempers out the schisma, and in the 7-limit, the landscape comma. It supports the [[magnesium]] temperament which divides the octave in 12, as well as [[chromium]] temperament that divides it in 24.

Nonetheless despite inconsistency, it is a valuable xenharmonic system in higher limits. It provides the [[optimal patent val]] for the [[octant]] temperament in the 13-limit, even if its approximation of 13 is almost half a step off. Likewise, 696edo tunes [[altierran]] and [[house]] temperaments in the 11-limit. In the higher limits, it may be used as a 2.3.5.7.17.31 subgroup tuning.

The 696cc val is also very close to the [[POTE]] tuning for the [[witcher]] temperament, while 696f tunes [[semiterm]] and the inaccurate 696d tunes [[pontic]].

=== Prime harmonics ===
{{Harmonics in equal|696}}

=== Subsets and supersets ===

Since 696 factors as {{Factorization|696}}, 696edo has subset edos {{EDOs|1, 2, 3, 4, 6, 8, 12, 24, 29, 58, 87, 116, 174, 232, 348}}.

== 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|-1103 696}}
|{{mapping|696 1103}}
|0.072829
|0.073
|4.22
|-
|2.3.5
|32805/32768, 52 80 -77
|{{mapping|696 1103 1616}}
|0.060798
|0.064
|3.71
|-
|2.3.5.7
|32805/32768, 250047/250000, 22 10 -3 -11
|{{mapping|696 1103 1616 1954}}
|0.072061
|0.035
|2.06
|-
|2.3.5.7.11
|9801/9800, 32805/32768, 46656/46585, 250047/250000
|{{mapping|696 1103 1616 1954 2408}}
|0.004896
|0.089
|5.15
|-
|2.3.5.7.11.13
|729/728, 1575/1573, 4096/4095, 67392/67375, 250047/250000
|{{mapping|696 1103 1616 1954 2408 2576}}
| -0.034283
|0.119
|6.92
|}

Barium

2025-08-22T22:41:52Z

Eliora: /* Theory */

'''Barium''' is a [[rank-2 temperament]] defined in the [[5-limit]] by [[tempering out]] the comma which sets 56 syntonic commas equal to the octave. Extensions exist in the 7-limit and the 11-limit. It is named after the 56th chemical element.

For technical data see: [[56th-octave temperaments#Barium]]
== Theory ==
An octave is equal to <math>\frac{1}{\log_{2}{\frac{81}{80}}} \approx 55.79763</math> syntonic commas, which when rounded to the closest integer yields 56. The associated comma in the 5-limit is {{monzo|-225 224 -56}}, and therefore is tempered if and only if the EDO divides 56. The comma is about 4 cents wide, but since each 81/80 is flattened by only about 0.07 cents as a consequence, barium is a very precise microtemperament.

Because the period is set to 81/80, interval stacking scheme works the same way as in [[meantone]], with the only difference being that the resulting intervals are represented in different 56ths of the octave. When the interval 3/2 is stacked 4 times, it also mirrors the pattern in every 1/56th of the octave, reaching [[5/4]] in 4 steps just as meantone would.

In the 7-limit, the reduced generator of barium is equal to the [[126/125]], a comma which together with the syntonic comma completes the basis for the [[septimal meantone]]. As such, barium can be interpreted this way as an "unfolding" of the septimal meantone into the fractional-octave temperament where one comma (81/80) is the period and the other (126/125) is the generator. Reading directly from the mapping, 7/4 is attained in 5 stacked intervals.

Barium in the 7-limit also tempers out the [[akjaysma]], meaning that 40 periods are set to [[105/64]].

== Music ==

Although full gamuts for barium start at 224 notes per octave, it is possible to use what is effectively a subset of barium temperament of much smaller gamut to produce the good major thirds of quarter-comma meantone while still getting good fifths, although any stacking of fifths will rapidly increase the size of gamut needed.

* [https://www.youtube.com/watch?v=Hmjx4wvLG7Q Uccellini - «Aria Sopra La Bergamasca» (1642), arranged for Organ, tuned into Adaptive Just Intonation] rendered by [[Claudi Meneghin]] (2024)

[[Category:Barium| ]] 
[[Category:Rank-2 temperaments]]

298edo

2025-08-14T05:25:28Z

Eliora: /* Theory */ +barton

{{Infobox ET}}
{{ED intro}}

== Theory ==
298edo is [[enfactoring|enfactored]] in the [[5-limit]] and only [[consistent]] in the [[5-odd-limit]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo—it retains the mapping for [[harmonic]]s [[2/1|2]], [[3/1|3]], [[5/1|5]], and [[17/1|17]] but differs on the mapping for [[7/4|7]], [[11/8|11]], [[13/8|13]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 [[subgroup]], 298edo supports [[emka]] and is a strong tuning for [[barton]]. In the full 13-limit, 298edo supports an unnamed {{nowrap|77 & 298}} temperament with [[13/8]] as its generator.

Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the {{nowrap|31 & 298d}} variant and the {{nowrap|118 & 298d}} variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val—for example, it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].

The 298cd val, {{val| 298 472 '''691''' '''836''' 1031 }} supports [[miracle]].

In higher limits, 298edo has excellent representation of the 2.5.11.17.23.43.53.59 subgroup, with all the harmonics having errors of less than 10 r¢. A comma basis for the 2.5.11.17.23.43.53.59 subgroup is {1376/1375, 3128/3127, 4301/4300, 25075/25069, 38743/38720, 58351/58300, 973360/972961}.

=== Odd harmonics ===
{{Harmonics in equal|298}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 6144/6125, 78732/78125, 3796875/3764768
| {{mapping| 298 472 692 837 }} (298)
| +0.0275
| 0.5022
| 12.5
|-
| 2.3.5.7.11
| 243/242, 1375/1372, 6144/6125, 72171/71680
| {{mapping| 298 472 692 837 1031 }} (298)
| +0.0012
| 0.4523
| 11.2
|-
| 2.3.5.7.11
| 243/242, 1029/1024, 3136/3125, 9801/9800
| {{mapping| 298 472 692 836 1031 }} (298d)
| +0.2882
| 0.4439
| 11.0
|-
| 2.3.5.7.11.13
| 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925
| {{mapping| 298 472 692 837 1031 1103 }} (298)
| −0.0478
| 0.4271
| 10.6
|-
| 2.3.5.7.11.13.17
| 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925
| {{mapping| 298 472 692 837 1031 1103 1218 }} (298)
| −0.0320
| 0.3974
| 9.87
|}

=== Rank-2 temperaments ===
Note: 5-limit temperaments supported by 149et are not listed.
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Associated ratio*
! Temperaments
|-
| 1
| 113\298
| 455.033
| 13/10
| [[Petrtri]]
|-
| 1
| 137\298
| 551.67
| 11/8
| [[Emka]]
|-
| 2
| 39\298
| 157.04
| 35/32
| [[Bison]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
The [[concoctic]] scale for 298edo is a scale produced by a generator of 105 steps (paraconcoctic), and the associated rank-2 temperament is {{nowrap|105 & 298}}.

[[Category:Bison]]
[[Category:Emka]]

768edo

2025-08-13T05:02:35Z

Eliora: /* Theory */ it's actually an 11-limit OPV for bezique

{{Infobox ET}}
{{ED intro}}

== Theory ==
768edo is [[consistent]] in the [[7-odd-limit]]. The equal temperament [[tempering out|tempers out]] the [[mutt comma]] {{monzo| -44 -3 21 }} and the 5-limit [[bicommatic]] comma {{monzo| -37 38 -10 }} in the 5-limit, and [[horwell comma|65625/65536]], [[250047/250000]], [[mitonismic temperaments|5250987/5242880]], {{monzo| -12 -5 11 -2 }}, {{monzo| 7 18 -2 -11 }}, and {{monzo| -36 8 4 5 }} in the 7-limit.

It provides the 11-limit [[optimal patent val]] for the 32nd-octave [[bezique]] temperament.

=== As a tuning standard ===
A step of 768edo is known as a '''hexamu''' (sixth MIDI-resolution unit, 6mu, 26 = 64 equal divisions of the [[12edo]] semitone). The internal data structure of the 6mu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and all six of the remaining bits used for the tuning data.

=== Odd harmonics ===
{{Harmonics in equal|768|intervals=prime}}

=== Subsets and supersets ===
Since 768 factors into {{factorization|768}}, 768edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, and 384 }}.

== See also ==

* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
** [[24edo]] (1mu tuning)
** [[48edo]] (2mu tuning)
** [[96edo]] (3mu tuning)
** [[192edo]] (4mu tuning)
** [[384edo]] (5mu tuning)
** [[1536edo]] (7mu tuning)
** [[3072edo]] (8mu tuning)
** [[6144edo]] (9mu tuning)
** [[12288edo]] (10mu tuning)
** [[24576edo]] (11mu tuning)
** [[49152edo]] (12mu tuning)
** [[98304edo]] (13mu tuning)
** [[196608edo]] (14mu tuning)

== External links ==
* [http://tonalsoft.com/enc/number/6mu.aspx 6mu / hexamu / 768-edo] on [[Tonalsoft Encyclopedia]]

Warped diatonic

2025-08-12T04:14:52Z

Eliora: fix pagename mid-page

A '''warped diatonic scale''' is a scale (excluding the diatonic scale itself) that contains relatively long substrings of the [[5L 2s]] diatonic scale ("LLsLLLsLLsLLLs...") in its sequence of large and small steps, and such that the sizes of those steps are similar to those of the diatonic scale (namely, in the ballpark of 200 and 100 cents).

Such scales may mislead a diatonic-conditioned listener into assigning the intervals to diatonic scale categories, but the categorization will be violated when either (1) the part of the scale that doesn't agree with 5L2s is reached, or (2) the harmonic nature of the intervals is drastically different than what's expected from the diatonic scale.

Combinatorically, most [[distributionally_even|distributionally even]] scales with more L steps than s steps do have significantly long substrings of the 5L2s diatonic scale in them. So, when searching for distributionally even warped diatonics, we can use the simple figure of merit that {{nowrap|''x''L ''y''s}} is a good warped diatonic when {{nowrap|''x'' > ''y''}} and {{nowrap|2''x'' + ''y''}} is in the vicinity of 12. If {{nowrap|2''x'' + ''y''}} is much less than 12, the steps will be too large to be recognized as the diatonic scale; conversely if {{nowrap|2''x'' + ''y''}} is much more than 12, the steps will be too small.

The scales at the top and bottom of this table are questionable as "warped diatonics", but the ones near {{nowrap|2''x'' + ''y'' {{=}} 12}} are good examples.

{| class="wikitable"
! {{nowrap|2''x'' + ''y''}}
! Formula
! Temperaments
! 5L 2s substrings
! Comments
|-
| 4
| ([[2edo]])
|
| (SS, SSS)
|
|-
| 5
| [[2L 1s]]
|
| LLs
|
|-
| 6
| ([[3edo]])
|
| (SS, SSS)
|
|-
| 7
| [[3L 1s]]
|
| LLLs
|
|-
| 8
| ([[4edo]])
|
| (SS, SSS)
|
|-
| 8
| [[3L 2s]]
| Sensi, Squares, Petrtri, A-Team, Father
| LsLLsL
|
|-
| 9
| [[4L 1s]]
| Bug, superpelog
| LLLsLL, LLsLLL
|
|-
| 9
| [[3L 3s]]
| Augmented
| Ls
|
|-
| 10
| [[4L 2s]]
| Decimal, lemba
| LLsLLsLL
| "Remove one L"
|-
| 10
| ([[5edo]])
|
| (SS, SSS)
| "Remove two s's"
|-
| 11
| [[4L 3s]]
| Orgone, keemun, sixix
| LsLLsL
| "Replace one L with s"
|-
| 11
| [[5L 1s]]
| Machine, gorgo
| LLLsLL, LLsLLL
| "Remove one s"
|-
| '''12'''
| '''[[5L 2s]]'''
| '''Ordinary diatonic'''
| (infinitely long)
|
|-
| 13
| [[5L 3s]]
| Father, A-Team, Petrtri
| LsLLsLL, LLsLLsL
| "Add one s"
|-
| 13
| [[6L 1s]]
| Glacial, leantone, tetracot
| LLLsLL, LLsLLL
| "Replace one s with L"
|-
| 14
| [[5L 4s]]
| Superpelog, godzilla
| LsLLsL
| "Add two s's"
|-
| 14
| [[6L 2s]]
| Hedgehog
| LLsLLLsLL
| "Add one L"
|-
| 15
| [[6L 3s]]
| Triforce
| LLsLLsLL
|
|-
| 15
| [[7L 1s]]
| Porcupine
| LLLsLL, LLsLLL
|
|-
| 16
| [[6L 4s]]
| Lemba, antikythera
| LsLLsL
|
|-
| 16
| [[7L 2s]]
| Mavila
| LLsLLLsLL
|
|-
| 17
| [[7L 3s]]
| Dicot
| LLsLLLsLLs
|
|-
| 17
| [[8L 1s]]
| Bleu, Tsaharuk, Quanharuk, bohpier
| LLLsLL, LLsLLL
|
|-
| 18
| [[8L 2s]]
| Octokaidecal
| LLLsLL, LLsLLL
|
|-
| 19
| [[8L 3s]]
| Sensi
| LLLsLLsLLLs
|
|-
| 19
| [[9L 1s]]
| Negri, Twothirdtonic
| LLLsLL, LLsLLL
|
|-
| 20
| [[9L 2s]]
| Casablanca
| LLLsLL, LLsLLL
|
|}

=== Rank 3 ===
{| class="wikitable"
! {{nowrap|3''x'' + 2''y'' + ''z''}}
! Formula
! Temperaments/names
! Max variety
|-
| 13
| 2L 3m 2s
|
| 3
|-
| 14
| 3L 1m 3s
|
| 3
|-
| 15
| 2L 4m 1s
|
| 4
|-
| 16
| 4L 1m 2s
|
| 4
|-
| 17
| 4L 2m 1s
|
| 4
|-
| 17
| 2L 4m 3s
|
| 4
|-
| 18
| 5L 1m 1s
|
| 3
|-
| 18
| 4L 1s 3m
|
| 4
|-
| 18
| 3L 3m 3s
|
| 3
|-
| 19
| 5L 1m 2s
|
| 4
|-
| 19
| 4L 2m 3s
|
| 4
|-
| 19
| 2L 5m 3s
|
| 4
|}

=== Warped antidiatonic ===
The scales at the top and bottom of this table are questionable as "warped antidiatonics", but the ones near {{nowrap|2''x'' + ''y'' {{=}} 9}} are good examples.
{| class="wikitable"
! {{nowrap|2''x'' + ''y''}}
! Formula
! Temperaments/names
! 2L 5s substrings
! Comments
|-
| 5
| [[5edo]]
|
| (ss, sss)
|
|-
| 6
| [[1L 4s]]
| Slendric, Gorgo, Machine
| sLsss
|
|-
| 7
| [[1L 5s]]
| Glacial, leantone, tetracot
| ssLsss
| "Remove one L"
|-
| 7
| [[2L 3s]]
| '''Ordinary pentatonic'''
| LssLs
| "Remove two s’s"
|-
| 8
| [[1L 6s]]
| Porcupine
|
| "Replace one L with s"
|-
| 8
| [[2L 4s]]
| Hedgehog
| LssLss
| "Remove one s"
|-
| 9
| [[2L 5s]]
| '''Ordinary antidiatonic'''
| (infinitely long)
|
|-
| 10
| [[2L 6s]]
| Twothirdtonic, srutal/Pajara, shrutar
| ssLsssLs
| "Add one s"
|-
| 10
| [[3L 4s]]
| '''Ordinary neutral diatonic'''
|
| "Replace one s with L"
|-
| 11
| [[2L 7s]]
| Score
| ssLsssLss
| "Add two s's"
|-
| 11
| [[3L 5s]]
| Sensi
| LsssLssL
| "Add one L"
|-
| 12
| [[3L 6s]]
| August, Augene
| LssLss
|
|-
| 12
| [[4L 4s]]
| Diminished
| Ls
|
|-
| 13
| [[3L 7s]]
| Magic
| LssLsssLss
|
|-
| 13
| [[4L 5s]]
| Orwell
|
|
|}

== Example: Godzilla[9] ==
Godzilla[9] in [[19edo|19edo]] has steps of {{nowrap|3 3 1 3 1 3 1 3 1}}. That corresponds to LLsLsLsLs where L is 189.5 cents and s is 63.2 cents. These step sizes are well within the range where they are "recognized" as diatonic scale steps. If you play 0 3 6 7 10 in 19edo (with no drone or harmony, just the melody) it will sound like "do re mi fa sol", leading you to believe that "sol" is a 3/2 interval above "do", but in fact it's not—it's closer to 10/7. The "real" 3/2 is another small step above that, in the melodic position of a flatted "la".

Here's a summary of all the contradicted expectations in godzilla[9], of which there are many:

<ul>
<li>{{nowrap|2L + s}}, which is expected to be 4/3, is really more like 9/7.</li>
<li>{{nowrap|2L + 2s}}, which is expected to be a "dissonant tritone" like 10/7, is actually the "real" 4/3. In "ti do re mi fa", the ti-fa interval is a tempered 4/3.</li>
<li>{{nowrap|3L + s}}, which is expected to be 3/2, is really more like 10/7.</li>
<li>{{nowrap|3L + 2s}}, which is expected to be an 8/5, is actually the "real" 3/2. In the "minor" scale "la ti do re mi fa", it's the "fa" rather than the "mi" that forms a 3/2 with the root "la".</li>
<li>{{nowrap|4L + 2s}}, which is expected to be a "minor seventh" ({{nowrap|16/9 ~ 9/5}}), is actually 5/3. So in the "natural minor" scale "la ti do re mi fa sol", the outer la-sol interval is the easily recognizable consonance 5/3, but that's the "wrong" consonance for a diatonically-conditioned listener.</li>
<li>{{nowrap|L + 2s}}, which is not found in the ordinary diatonic scale but is familiar from the melodic and harmonic minor scales, is expected to be something close to a "major third" (in 12edo it's 400 cents, in meantone it's a tempered 9/7). In godzilla[9] it's the just minor third 6/5.</li>
<li>Finally, although there are 7 or even 8 consecutive notes of the scale that sound melodically familiar, the way the scale closes at the 2/1 is very unexpected and jarring for a diatonically-conditioned listener. When you get to {{nowrap|5L + 3s}} that sounds melodically like it ought to be at least an "octave", probably something larger like an "augmented octave". But the real 2/1 is actually one small step beyond that, at {{nowrap|5L + 4s}}.</li></ul>

[[Category:todo:link]]
[[Category:Diatonic]]
[[Category:Lists of scales]]

Mercurial comma

2025-08-12T03:35:53Z

Eliora: expand and clarify that a mere stack of 19/17s and 15/14s in a diatonic way is not a "temperament" and this comma like any others has a full rank temperament to it

{{Infobox Interval
| Ratio = 557122275/556583944
| Name = mercurial comma
| Color name = 19o517u5rryy-3, quinnosu-abiruyo negative 3rd
| Comma = yes
}}

'''557122275/556583944''', or the '''mercurial comma''', is an unnoticeable comma measuring 1.673649 [[cent]]s. It represents the amount by which five justly tuned [[19/17]]s and two [[15/14]]s exceed an octave, thus being of importance in [[5L 2s]] diatonic scale.

== Etymology ==
As the ratio between these intervals is very close to phi, a diatonic scale consisting of a 19/17 as the tone and 15/14 as the semitone represents a strong harmonic entropy minimum adjacent to the [[golden meantone]] sequence. This is particularly well suited to stringed instruments that are normally tuned with slight octave stretches due to the inharmonicity of their partials. Hence, the mercurial comma is named after the chemical element mercury due to its adjacency to gold, since mercury meantone is adjacent to golden meantone.

It is worth noting that per principles of regular temperament theory, a temperament produced by the mercurial comma is a rank-7 19-limit temperament, which would naturally have the name mercurial.

== Temperaments ==
Tempering this comma out in the full 19-limit produces the rank-19 '''mercurial''' temperament.

Subgroup:

Comma list: 557122275/556583944

[[Mapping]]: 
{| class="right-all"
|-
| [⟨ || 1 || 1 || 2 || 4 || 3 || -3 || 4 || 5 ||],
|-
| ⟨ || 0 || 1 || 0 || 1 || 0 || 0 || 0 || 0 ||],
|-
| ⟨ || 0 || 0 || 1 || 1 || 0 || 0 || 0 || 0 || ],
|-
| ⟨ || 0 || 0 || 0 || -5 || 0 || 0 || 0 || -2 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ]],
|-
| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ]],
|-
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || 0 || ]],
|-
| ⟨ || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || ]],
|}

: mapping generators: ~2, ~3, ~5, ~5415/4046, ~11, ~13, ~17

The interval 5415/4046 is within the range of meantone fourths. From this, 5415/4046 can be derived as the mercurial meantone fourth and respectively its inverse, 8092/5415, mercurial meantone fifth. The comma that separates each from the just versions is [[16245/16184]].

Despite sharing the name, mercurial temperament is unrelated to the [[Mercury|80th-octave temperament]] by the same name, though as a curiosity, the [[2960edo]], more specifically, the 2960dh val, is the unique tuning supporting them both.

[[Category:Meantone]]
[[Category:Mercurial]]
[[Category:Commas named after elements]]

768edo

2025-08-08T01:02:04Z

Eliora: /* Theory */ +fact it supports bezique

{{Infobox ET}}
{{ED intro}}

== Theory ==
768edo is [[consistent]] in the [[7-odd-limit]]. The equal temperament [[tempering out|tempers out]] the [[mutt comma]] {{monzo| -44 -3 21 }} and the 5-limit [[bicommatic]] comma {{monzo| -37 38 -10 }} in the 5-limit, and [[horwell comma|65625/65536]], [[250047/250000]], [[mitonismic temperaments|5250987/5242880]], {{monzo| -12 -5 11 -2 }}, {{monzo| 7 18 -2 -11 }}, and {{monzo| -36 8 4 5 }} in the 7-limit.

It is a strong tuning and a member of the optimal ET sequence for the 32nd-octave [[bezique]] temperament.

=== As a tuning standard ===
A step of 768edo is known as a '''hexamu''' (sixth MIDI-resolution unit, 6mu, 26 = 64 equal divisions of the [[12edo]] semitone). The internal data structure of the 6mu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and all six of the remaining bits used for the tuning data.

=== Odd harmonics ===
{{Harmonics in equal|768|intervals=prime}}

=== Subsets and supersets ===
Since 768 factors into {{factorization|768}}, 768edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, and 384 }}.

== See also ==

* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
** [[24edo]] (1mu tuning)
** [[48edo]] (2mu tuning)
** [[96edo]] (3mu tuning)
** [[192edo]] (4mu tuning)
** [[384edo]] (5mu tuning)
** [[1536edo]] (7mu tuning)
** [[3072edo]] (8mu tuning)
** [[6144edo]] (9mu tuning)
** [[12288edo]] (10mu tuning)
** [[24576edo]] (11mu tuning)
** [[49152edo]] (12mu tuning)
** [[98304edo]] (13mu tuning)
** [[196608edo]] (14mu tuning)

== External links ==
* [http://tonalsoft.com/enc/number/6mu.aspx 6mu / hexamu / 768-edo] on [[Tonalsoft Encyclopedia]]

Gregorian leap day

2025-08-07T20:37:21Z

Eliora: Correct as requested, 97 -> 97g

{{Novelty}}

'''Gregorian leap day''' is a [[rank-2 temperament]] which is produced by temperament-merging 97edo, which has the cardinality of leap years in Gregorian calendar's cycle, and 400edo, the whole duration of the cycle.

400 is the number of years in the Gregorian calendar's leap cycle. They are not spread evenly, but if they were, this would produce a scale with a 33\400 generator which is associated to [[18/17]] and [[55/52]], three of which make [[19/16]]. The optimal tuning is very close to 18/17, which makes it very similar to [[Galilei's tuning]]. Gregorian leap day has mos of size 12, 13, 25, 37, 49, 61, 73, and 97. [[1L 11s]] mos of this temperament is a barely noticeable circulating temperament for [[12edo]].

In the 7-limit, temperament reaches [[15/8]] in 11 generators, entirely contained within the 12-tone well temperament, and also [[7/5]] in 18 generators.

== Temperament data ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 67108864/66976875, {{monzo| -13 3 -17 17 }}

{{Mapping|legend=1| 1 10 -7 8 | 0 -102 113 131 }}

: mapping generators: ~2, 160000/151263

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, 160000/151263 = 98.9982

{{Optimal ET sequence|legend=1| 97, 303, 400, 1297, 1697c }}

[[Badness]]: 1.10

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 166698/166375, 422576/421875, 67108864/66976875

Mapping: {{mapping| 1 10 -7 8 16 | 0 -102 113 131 -152 }}

Optimal tuning (CTE): ~2 = 1\1, 69120/65219 = 98.9994

Optimal ET sequence: {{Optimal ET sequence| 97, 303, 400 }}

Badness: 0.251

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 4096/4095, 105644/105625, 166698/166375

Mapping: {{mapping| 1 10 -7 8 16 7 | 0 -102 113 131 -152 -40 }}

Optimal tuning (CTE): ~2 = 1\1, 55/52 = 98.9992

Optimal ET sequence: {{Optimal ET sequence| 97, 303, 400 }}

Badness: 0.126

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 4096/4095, 11016/11011, 14400/14399, 93639/93500

Mapping: {{mapping| 1 10 -7 8 16 7 -12 | 0 -102 113 131 -152 -40 195 }}

Optimal tuning (CTE): ~2 = 1\1, 18/17 = 98.9993

Optimal ET sequence: {{Optimal ET sequence| 97g, 303g, 400 }}

Badness: 0.0776

=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 2926/2925, 4096/4095, 6175/6174, 11016/11011, 14400/14399

Mapping: {{mapping| 1 10 -7 8 16 7 -12 4 | 0 -102 113 131 -152 -40 195 3 }}

Optimal tuning (CTE): ~2 = 1\1, 18/17 = 98.9993

Optimal ET sequence: {{Optimal ET sequence| 97g, 303g, 400 }}

Badness: 0.0541

[[Category:Gregorian leap day| ]] 
[[Category:Rank-2 temperaments]]

10th-octave temperaments

2025-08-04T18:13:25Z

Eliora: /* Neon */ explain what exactly makes this notable

{{Technical data page}}
{{Infobox fractional-octave|10}}
[[10edo]] is notable for having close approximations of [[15/14]] to one step, and [[13/8]] to 7 steps. 10th-octave temperaments naturally occur between any equal divisions of the octave whose greatest common divisor is 10.

Temperaments discussed elsewhere include: [[Quintosec family #Decoid|decoid]], [[Ragismic microtemperaments #Deca|deca]], [[Quintile family #Decile|decile]], [[Metric microtemperaments #Decimetra|decimetra]], [[Stearnsmic clan #Decistearn|decistearn]], [[Vishnuzmic family #Decavish|decavish]], and [[Kalismic temperaments #Linus|linus]].

== Neon ==
Neon tempers out {{monzo| 21 60 -50 }} in the 5-limit, equating [[3125/2916]] with one step of 10edo. Neon extensions discussed elsewhere include [[deca]], [[calcium]], and [[zinc]].

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| 21 60 -50 }}

{{Mapping|legend=1| 10 4 9 | 0 5 6 }}

: Mapping generators: ~3125/2916, {{monzo| 10 29 -24 }}

[[Optimal tuning]] ([[CTE]]): ~3125/2916 = 1\10, {{monzo| 10 29 -24 }} = 284.3888

{{Optimal ET sequence|legend=1| 80, 190, 270, 460, 730, 1730, 2460, 3190, 5650 }}

[[Badness]]: 0.206596

{{Navbox fractional-octave}}

[[Category:10edo]]

User talk:Eliora/2592edo

2025-08-04T02:54:43Z

Eliora: /* Notability */

== Notability ==

I don't think this page meets the [[notability guidelines]]. I understand that this edo is the optimal patent val of one particular temperament (whose notability is either tenuous or insufficiently explained) and has a lot of divisors. Unfortunately, the first aspect is most likely the case of a very large number of edos in the thousands, and the second aspect, as far as I know, isn't a particularly notable aspect, especially in edos of that size (for the vast majority of musicians, should I specify). I believe it's sufficient that this edo figures in the optimal patent val list of windrose temperament, but otherwise I'll consider deleting this page per [[XW:NG]]. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 04:29, 3 August 2025 (UTC)

: +1 Delete – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 15:29, 3 August 2025 (UTC)

:: A compromise: I ask that the page be reinstated and then contents moved to [[User:Eliora/2592edo]], so I can work on my niche edos without clogging up the main namespace if the ideas aren't notable enough. [[User:Eliora|Eliora]] ([[User talk:Eliora|talk]]) 02:54, 4 August 2025 (UTC)

56edo

2025-08-02T22:56:34Z

Eliora: /* Subsets and supersets */

{{Infobox ET}}
{{ED intro}}

== Theory ==
56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo.

56edo can be used to tune [[hemithirds]], [[superkleismic]], [[sycamore]] and [[keen]] temperaments, and using {{val| 56 89 130 158 }} (56d) as the equal temperament val, for [[pajara]]. It provides the [[optimal patent val]] for 7-, 11- and 13-limit [[sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for 11-limit pajara.

=== Prime harmonics ===
{{Harmonics in equal|56}}

=== Subsets and supersets ===
Since 56 factors into {{nowrap|23 × 7}}, 56edo has subset edos {{EDOs| 2, 4, 7, 8, 14, 28 }}.

One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the unrounded value being 55.7976. [[Barium]] temperament realizes this proximity through regular temperament theory, and is supported by notable edos like [[224edo]], [[1848edo]], and [[2520edo]], which is a highly composite edo.

== Intervals ==
{| class="wikitable center-all right-2 left-3"
|-
! #
! Cents
! Approximate ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 21.4
| ''[[49/48]]'', [[55/54]], [[56/55]], [[64/63]]
| {{UDnote|step=1}}
|-
| 2
| 42.9
| ''[[28/27]]'', [[40/39]], [[45/44]], [[50/49]], ''[[81/80]]''
| {{UDnote|step=2}}
|-
| 3
| 64.3
| [[25/24]], ''[[36/35]]'', ''[[33/32]]''
| {{UDnote|step=3}}
|-
| 4
| 85.7
| [[19/18]], [[20/19]], [[21/20]], [[22/21]]
| {{UDnote|step=4}}
|-
| 5
| 107.1
| [[16/15]], [[17/16]], [[18/17]]
| {{UDnote|step=5}}
|-
| 6
| 128.6
| [[15/14]], [[13/12]], [[14/13]]
| {{UDnote|step=6}}
|-
| 7
| 150.0
| [[12/11]]
| {{UDnote|step=7}}
|-
| 8
| 171.4
| ''[[10/9]]'', [[11/10]], [[21/19]]
| {{UDnote|step=8}}
|-
| 9
| 192.9
| [[19/17]], [[28/25]]
| {{UDnote|step=9}}
|-
| 10
| 214.3
| [[9/8]], [[17/15]]
| {{UDnote|step=10}}
|-
| 11
| 235.7
| [[8/7]]
| {{UDnote|step=11}}
|-
| 12
| 257.1
| [[7/6]]
| {{UDnote|step=12}}
|-
| 13
| 278.6
| [[13/11]], [[20/17]]
| {{UDnote|step=13}}
|-
| 14
| 300.0
| [[19/16]], [[25/21]]
| {{UDnote|step=14}}
|-
| 15
| 321.4
| [[6/5]]
| {{UDnote|step=15}}
|-
| 16
| 342.9
| [[11/9]], [[17/14]]
| {{UDnote|step=16}}
|-
| 17
| 364.3
| [[16/13]], [[21/17]], [[26/21]]
| {{UDnote|step=17}}
|-
| 18
| 385.7
| [[5/4]]
| {{UDnote|step=18}}
|-
| 19
| 407.1
| [[14/11]], [[19/12]], [[24/19]]
| {{UDnote|step=19}}
|-
| 20
| 428.6
| [[32/25]], [[33/26]]
| {{UDnote|step=20}}
|-
| 21
| 450.0
| ''[[9/7]]'', [[13/10]]
| {{UDnote|step=21}}
|-
| 22
| 471.4
| [[17/13]], [[21/16]]
| {{UDnote|step=22}}
|-
| 23
| 492.9
| [[4/3]]
| {{UDnote|step=23}}
|-
| 24
| 514.3
| [[35/26]]
| {{UDnote|step=24}}
|-
| 25
| 535.7
| [[15/11]], [[19/14]], [[26/19]], ''[[27/20]]''
| {{UDnote|step=25}}
|-
| 26
| 557.1
| [[11/8]]
| {{UDnote|step=26}}
|-
| 27
| 578.6
| [[7/5]]
| {{UDnote|step=27}}
|-
| 28
| 600.0
| [[17/12]], [[24/17]]
| {{UDnote|step=28}}
|-
| …
| …
| …
| …
|}
<nowiki/>* The following table assumes the 19-limit [[patent val]]; other approaches are possible. Inconsistent intervals are marked in ''italics''.

== Notation ==

=== Ups and downs notation ===

56edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
{{Sharpness-sharp7a}}

Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{Sharpness-sharp7}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as [[63edo#Sagittal notation|63-EDO]].

==== Evo flavor ====
<imagemap>
File:56-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 575 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
rect 120 80 220 106 [[81/80]]
rect 220 80 340 106 [[33/32]]
default [[File:56-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:56-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
rect 120 80 220 106 [[81/80]]
rect 220 80 340 106 [[33/32]]
default [[File:56-EDO_Revo_Sagittal.svg]]
</imagemap>

== Approximation to JI ==
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 276
| steps = 56.0083399588546
| step size = 21.4253805929895
| tempered height = 6.063216
| pure height = 6.023344
| integral = 0.931117
| gap = 14.804703
| octave = 1199.82131320741
| consistent = 8
| distinct = 8
}}

== 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| 89 -56 }}
| {{mapping| 56 89 }}
| −1.64
| 1.63
| 7.64
|-
| 2.3.5
| 2048/2025, 1953125/1889568
| {{mapping| 56 89 130 }}
| −1.01
| 1.61
| 7.50
|-
| 2.3.5.7
| 686/675, 875/864, 1029/1024
| {{mapping| 56 89 130 157 }}
| −0.352
| 1.80
| 8.38
|-
| 2.3.5.7.11
| 100/99, 245/242, 385/384, 686/675
| {{mapping| 56 89 130 157 194 }}
| −0.618
| 1.69
| 7.90
|-
| 2.3.5.7.11.13
| 91/90, 100/99, 169/168, 245/242, 385/384
| {{mapping| 56 89 130 157 194 207 }}
| −0.299
| 1.70
| 7.95
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Associated ratio*
! Temperament
|-
| 1
| 3\56
| 64.29
| 25/24
| [[Sycamore]]
|-
| 1
| 9\56
| 192.86
| 28/25
| [[Hemithirds]]
|-
| 1
| 11\56
| 235.71
| 8/7
| [[Slendric]]
|-
| 1
| 15\56
| 321.43
| 6/5
| [[Superkleismic]]
|-
| 1
| 25\56
| 535.71
| 15/11
| [[Maquila]] (56d) / [[maquiloid]] (56)
|-
| 2
| 11\56
| 235.71
| 8/7
| [[Echidnic]]
|-
| 2
| 23\56 (5\56)
| 492.86 (107.14)
| 4/3 (17/16)
| [[Keen]] / keenic
|-
| 4
| 23\56 (5\56)
| 492.86 (107.14)
| 4/3 (17/16)
| [[Bidia]] (7-limit)
|-
| 7
| 23\56 (1\56)
| 492.86 (21.43)
| 4/3 (250/243)
| [[Sevond]]
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* [[Supra7]]
* [[Supra12]]
* Subsets of [[echidnic]][16] (6u8d):
** Frankincense (this is the original/default tuning): 364.3 - 492.9 - 707.1 - 835.7 - 1200.0
** Quasi-[[equipentatonic]]: 257.1 - 492.9 - 707.1 - 964.3 - 1200.0
** Sakura-like scale containing [[phi]]: 107.1 - 492.9 - 707.1 - 835.7 - 1200.0
* Subsets of [[sevond]][14]
** Evened minor pentatonic (approximated from [[72edo]]): 321.4 - 492.9 - 685.7 - 1028.6 - 1200.0

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/o0imqFPDh9k ''56edo''] (2023)

; [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] (2025)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=xWKa59qDkXQ ''Prelude & Fugue in Pajara''] (2020) – in pajara, 56edo tuning
* [https://www.youtube.com/watch?v=3oO1SIVWBgI ''Mirror Canon in F''] (2020)
* [https://www.youtube.com/watch?v=s1h083BRWXU ''Canon 3-in-1 on a Ground''] (2020)

== See also ==
* [[Lumatone mapping for 56edo]]

[[Category:Hemithirds]]
[[Category:Keen]]
[[Category:Listen]]
[[Category:Pajara]]
[[Category:Superkleismic]]
[[Category:Sycamore]]

EDO

2025-08-02T22:54:09Z

Eliora: /* 2000…9999 */

{{Todo|discuss title}}
{{interwiki
| de = EDO
| en = EDO
| es = EDOs
| ja = オクターブ平均律
| ko = EDO (Korean)
| ro = DEO
}}
An '''equal division of the octave''' ('''EDO''', ''EE-dee-oh''; '''edo''', ''EE-doh'') is a [[Tuning system|tuning]] obtained by dividing the [[octave]] in a certain number of [[Equal-step tuning|equal steps]]. This means that the [[interval]] between any two consecutive pitches is identical.

A tuning with ''n'' equal divisions of the octave is usually called "''n''-edo" ("''n''-EDO"). For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO).

An equal (pitch) division of the octave is the most common type of [[EPD|equal (pitch) division]], which is a kind of [[equal-step tuning]]. Therefore, it is also a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.

== History ==
Tuning theorists first used the term "equal temperament" for edos designed to approximate [[Low-complexity JI|low-complexity just intervals]]. The same term is still used today for all rank-1 [[Regular temperament|temperaments]]. For example, [[15edo]] can be referred to as 15-tone equal temperament (15-TET, 15-tET, 15tet, etc.), or more simply 15 equal temperament (15-ET, 15et, etc.).

The acronym "EDO" was coined by [[Daniel Anthony Stearns]] in 1999, originally standing for "equidistant divisions of the octave"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_65#65 Yahoo! Tuning Group | ''Where F + f = O'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_117#117 Yahoo! Tuning Group | ''f + F and WFS/MOS'']</ref>. More recently, the {{w|anacronym}} "edo", spelled in lowercase and pronounced as a regular word, has also become common.

With the development of [[Edonoi|equal divisions of non-octave intervals (edonoi)]], some people started writing "ed2" ("ED2"), especially when naming a specific tuning.

== Calculating the step size ==
To find the step size of ''n''-edo in terms of [[cent]]s, divide 1200 by ''n''. The size ''s'' of ''k'' steps of ''n''-edo (''k''\''n'') is

<math>\displaystyle s = 1200 \cdot k/n</math>

To find the step size of ''n''-edo in terms of [[frequency ratio]], take the ''n''-th root of 2. For example, the step of 12edo is 21/12 (≈ 1.059). So the ratio ''c'' of the ''k'' steps of ''n''-edo is

<math>\displaystyle c = 2^{k/n}</math>

In particular, when ''k'' is 0, ''c'' is simply 1, because any number to the 0th power is 1. And when {{nowrap|''k'' {{=}} ''n''}}, ''c'' is simply 2, because any number to the 1st power is itself.

== Properties ==
EDO scales are straightforward to work with due to their uniform step size.
Some musicians find the consistency bland, while others appreciate the stable foundation it provides for composition.
The only property shared by all of them is the equality of their step-sizes; otherwise, their individual properties are as different as can be.
Lower-numbered EDOs, especially 5 to 24, possess very strong and unique "characters", which some composers find inspiring.

== Practical advantages ==
=== Fretted instruments ===
If you are a [[guitar]]ist (or a player of some other fretted string instrument, like a bass guitar, Appalachian dulcimer, [[ukulele]], banjo, mandolin, sitar, saz, pipa, or zhong ruan), an EDO will provide you with the simplest possible fretboard layout, as all of the frets will go straight across the fretboard, regardless of how you want to tune the open strings.
Fret crowding can become an issue with smaller divisions, especially high up the neck.
For these cases, [[ed4|equal divisions of the double octave]] or higher multiples offer a compromise solution.

=== Free modulation ===
EDOs allow for modulation to every single key in the tuning, without any alteration in harmonic properties, thus making transposition totally seamless.
This also makes them somewhat easier to learn, as you do not have to memorize the harmonic and melodic variations that appear in various keys (which you would have to learn in JI, an unequal regular temperament, or a well-temperament, especially with smaller numbers of tones).
For those accustomed to the "equality" of 12-TET, the equality of the alternative EDOs can be reassuringly familiar.

== Approaches to exploring EDOs ==
If you are interested in exploring the unique merits and challenges of each EDO, irrespective of any desire to approximate just intonation (or any other a priori musical goal), starting at the bottom and working your way up can be a most illuminating exercise.

If you're a classically trained musician and you'd like to start with some EDOs that have some relationship to common-practice tonal music, starting with reasonably-low EDOs that give a good approximation to the perfect fifth ([[3/2]]) can be rewarding.
These include {{EDOs| 12, 17, 19, 22, 26, 27, 29, 31, 39, 41, 43, 45, 46, 49, 50, and 53 }}.
All of these can be notated with some variant on the [[Circle-of-fifths notation|A–G "circle of fifths" notation]], while other EDOs, including {{EDOs| 24, 34, 36, 38, 44, 48, or 51}} involve multiple such circles.

Some EDOs, such as {{EDOs| 26, 27, 32, 33, or 37 }} have fifths which are reasonably good but quite audibly not just. Other EDOs, such as {{EDOs| 11, 13, 14, 15, 16, 18, 20, 21, 23, or 25 }}, are of interest to the avid seeker of totally unusual sounds that have next-to-no connection with the common practice.

If your interest lies in the nuanced approximation of just intonation through EDOs, then delving into EDOs characterized by a strong [[The Riemann zeta function and tuning#Zeta EDO lists|zeta peak]] could be especially captivating. Such EDOs, including {{EDOs| 12, 19, 22, 27, 31, 34, 41, 46, 53, 58, 60, 65, 68, 72, 77, 80, 84, 87, 94, and 99 }}, offer rich avenues for exploration in the quest for harmonic purity and transparent [[temperament]]s.

EDOs with a less pronounced, yet still noteworthy [[The Riemann zeta function and tuning#Local zeta edos|zeta peak]]—specifically {{EDOs| 10, 14, 15, 16, 17, 21, 24, 26, 29, 32, 36, 37, 38, 39, 43, 45, 48, 49, 50, 56, 62, 63, 82, 89, and 96 }}—present a unique palette for harmony explorers. Although these systems may lack the harmonic precision found in EDOs with more prominent zeta peaks, they strike an intriguing balance between consonance and more distant harmonic textures.

EDOs can be further subdivided and classified according to the size of the fifth, such as with [[Margo Schulter]]'s [[gentle region]] or the distinction between negative, positive, doubly negative and doubly positive of [[R. H. M. Bosanquet]]. [[Kite Giedraitis]] has proposed these six categories, based on the size of the fifth. From narrowest to widest:

* '''Superflat''' EDOs ({{EDOs| 9, 11, 13b, 16, 18b, and 23 }}) have a fifth narrower than four-sevenths of an octave ({{nowrap|4\7 {{=}} 685.714{{c}}}})
* '''Perfect''' EDOs ({{EDOs| 7, 14, 21, 28, and 35 }}) have a fifth equal to {{nowrap|4\7 {{=}} 685.714{{c}}}}
* '''Diatonic''' EDOs ({{EDOs| 12, 17, 19, 22, 24, etc. }}) have a fifth between 685.714{{c}} and 720{{c}}
* '''Pentatonic''' EDOs ({{EDOs| 5, 10, 15, 20, 25, and 30 }}) have a fifth of three-fifths of an octave ({{nowrap|3\5 {{=}} 720{{c}}}})
* '''Supersharp''' EDOs ({{EDOs| 8, 13, and 18 }}) have a fifth wider than 720{{c}}
* '''Trivial''' EDOs ({{EDOs| 1, 2, 3, 4, and 6 }}) have a fifth about 100{{c}} from just, and are contained in 12edo

== Structural properties ==
You will quickly find that the ''factorization'' of the total number of notes in each EDO has consequences for its structure and the way it relates to other EDOs. For example, {{nowrap|6 {{=}} 2 x 3}}, so 6edo contains all of the intervals in both 2edo and 3edo. On the other hand, 7 is a prime number, so no 7edo intervals are redundant with those of smaller EDOs. See [[Prime EDO]] and [[Highly composite EDO]] for more details.

The [[MOS]] paradigm is a fascinating way of thinking about building sub-scales of EDOs and relating them to non-EDO scales, as well as finding common melodic patterns between multiple EDOs.

=== Adding EDOs ===
Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). This really amounts to the consideration of adding the associated [[vals]], which are the mappings to primes larger than 2. An EDO is defined by a certain number of steps equating to 2; if we have more steps equating to 3, we get a 3-limit val, and so forth. So, for example, 12edo can be written {{val| 12 }}, saying that twelve steps maps to 2, but the 3-limit val for 12 is {{val| 12 19 }}, telling us that 19 steps maps to 3, and the 5-limit val is {{val| 12 19 28 }}, telling us that 28 steps maps to 5.

If we add 12 and 19 we get another good division, {{nowrap| 12 + 19 {{=}} 31 }}. We can understand why this works if we look at it as adding vals; {{val| 12 19 28 }} + {{val| 19 30 44 }} = {{val| 31 49 72 }}. The relative error in terms of [[relative cent]]s is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is {{nowrap|[-1.955 13.686]}} (the same as absolute cents) and the error of 19edo is {{nowrap|[-11.429 -11.663]}}, and this sums to {{nowrap|[-13.384 2.023]}}. In relative cents the error of the fifth for 31edo is not much increased from 19edo, and on converting to absolute cents we find it is even better, and the error of the major third is much smaller due to the cancellation. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 31edo) can have a much smaller error due to the cancellation. 24edo's flat fifth and 29edo's sharp fifth can be added to form 53edo.

We may also look at addition of EDOs in terms of MOS; if ''a''\''n'' is a generator for an ''n''-edo MOS, and ''b''\''m'' for an ''m''-edo MOS, where both of these are generators for the same linear temperament, then the mediant, {{nowrap|(''a'' + ''b'')\(''n'' + ''m'')}}, will be a generator for a MOS for the same temperament, this time in {{nowrap|(''n'' + ''m'')}}-edo. A visual way of putting this is that through this addition of ''n'' and ''m'', one becomes the accidentals or black keys, and the other the naturals or white keys. The choice of accidental/natural or black keys/white keys is a question of emphasis on the part of the composer or designer. Furthermore, one may add more than two numbers, hierarchically expanding the possibilities to double flats and sharps and beyond. This can be useful in designing keyboards and systems of notation.

=== Scale size considerations ===
EDOs with fewer than 12 divisions have steps exceeding 100 cents.
Of these, 1, 2, 3, 4, and 6 divide 12 and so are already available.
{{EDOs| 5, 7, and 9 }} have arguably been used in various musical traditions worldwide.

When using EDOs to tune scales or [[regular temperament]]s, the size becomes less conceptually important since not all notes need to be used.
Some of the EDOs which can be used to tune various temperaments are listed on the [[optimal patent val]] page. Tuning a scale in just intonation by one of these EDOs can be regarded as automatically tempering it to the corresponding regular temperament.

To practically tune large edos through software tuning, one may take advantage of [[MIDI]] channels. See [[Tuning per channel]].

All of these tools are also applicable to equal divisions of other ([[nonoctave]]) intervals as well.

== EDOs versus Equal Temperaments ==
See [[EDO vs ET]].

== Individual pages for EDOs ==
=== 0…999 ===
{| class="wikitable center-all mw-collapsible"
|+ style="font-size: 105%; white-space: nowrap;" | 0…99
|-
| [[0edo|0]]
| [[1edo|1]]
| [[2edo|2]]
| [[3edo|3]]
| [[4edo|4]]
| [[5edo|5]]
| [[6edo|6]]
| [[7edo|7]]
| [[8edo|8]]
| [[9edo|9]]
|-
| [[10edo|10]]
| [[11edo|11]]
| [[12edo|12]]
| [[13edo|13]]
| [[14edo|14]]
| [[15edo|15]]
| [[16edo|16]]
| [[17edo|17]]
| [[18edo|18]]
| [[19edo|19]]
|-
| [[20edo|20]]
| [[21edo|21]]
| [[22edo|22]]
| [[23edo|23]]
| [[24edo|24]]
| [[25edo|25]]
| [[26edo|26]]
| [[27edo|27]]
| [[28edo|28]]
| [[29edo|29]]
|-
| [[30edo|30]]
| [[31edo|31]]
| [[32edo|32]]
| [[33edo|33]]
| [[34edo|34]]
| [[35edo|35]]
| [[36edo|36]]
| [[37edo|37]]
| [[38edo|38]]
| [[39edo|39]]
|-
| [[40edo|40]]
| [[41edo|41]]
| [[42edo|42]]
| [[43edo|43]]
| [[44edo|44]]
| [[45edo|45]]
| [[46edo|46]]
| [[47edo|47]]
| [[48edo|48]]
| [[49edo|49]]
|-
| [[50edo|50]]
| [[51edo|51]]
| [[52edo|52]]
| [[53edo|53]]
| [[54edo|54]]
| [[55edo|55]]
| [[56edo|56]]
| [[57edo|57]]
| [[58edo|58]]
| [[59edo|59]]
|-
| [[60edo|60]]
| [[61edo|61]]
| [[62edo|62]]
| [[63edo|63]]
| [[64edo|64]]
| [[65edo|65]]
| [[66edo|66]]
| [[67edo|67]]
| [[68edo|68]]
| [[69edo|69]]
|-
| [[70edo|70]]
| [[71edo|71]]
| [[72edo|72]]
| [[73edo|73]]
| [[74edo|74]]
| [[75edo|75]]
| [[76edo|76]]
| [[77edo|77]]
| [[78edo|78]]
| [[79edo|79]]
|-
| [[80edo|80]]
| [[81edo|81]]
| [[82edo|82]]
| [[83edo|83]]
| [[84edo|84]]
| [[85edo|85]]
| [[86edo|86]]
| [[87edo|87]]
| [[88edo|88]]
| [[89edo|89]]
|-
| [[90edo|90]]
| [[91edo|91]]
| [[92edo|92]]
| [[93edo|93]]
| [[94edo|94]]
| [[95edo|95]]
| [[96edo|96]]
| [[97edo|97]]
| [[98edo|98]]
| [[99edo|99]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 100…199
|-
| [[100edo|100]]
| [[101edo|101]]
| [[102edo|102]]
| [[103edo|103]]
| [[104edo|104]]
| [[105edo|105]]
| [[106edo|106]]
| [[107edo|107]]
| [[108edo|108]]
| [[109edo|109]]
|-
| [[110edo|110]]
| [[111edo|111]]
| [[112edo|112]]
| [[113edo|113]]
| [[114edo|114]]
| [[115edo|115]]
| [[116edo|116]]
| [[117edo|117]]
| [[118edo|118]]
| [[119edo|119]]
|-
| [[120edo|120]]
| [[121edo|121]]
| [[122edo|122]]
| [[123edo|123]]
| [[124edo|124]]
| [[125edo|125]]
| [[126edo|126]]
| [[127edo|127]]
| [[128edo|128]]
| [[129edo|129]]
|-
| [[130edo|130]]
| [[131edo|131]]
| [[132edo|132]]
| [[133edo|133]]
| [[134edo|134]]
| [[135edo|135]]
| [[136edo|136]]
| [[137edo|137]]
| [[138edo|138]]
| [[139edo|139]]
|-
| [[140edo|140]]
| [[141edo|141]]
| [[142edo|142]]
| [[143edo|143]]
| [[144edo|144]]
| [[145edo|145]]
| [[146edo|146]]
| [[147edo|147]]
| [[148edo|148]]
| [[149edo|149]]
|-
| [[150edo|150]]
| [[151edo|151]]
| [[152edo|152]]
| [[153edo|153]]
| [[154edo|154]]
| [[155edo|155]]
| [[156edo|156]]
| [[157edo|157]]
| [[158edo|158]]
| [[159edo|159]]
|-
| [[160edo|160]]
| [[161edo|161]]
| [[162edo|162]]
| [[163edo|163]]
| [[164edo|164]]
| [[165edo|165]]
| [[166edo|166]]
| [[167edo|167]]
| [[168edo|168]]
| [[169edo|169]]
|-
| [[170edo|170]]
| [[171edo|171]]
| [[172edo|172]]
| [[173edo|173]]
| [[174edo|174]]
| [[175edo|175]]
| [[176edo|176]]
| [[177edo|177]]
| [[178edo|178]]
| [[179edo|179]]
|-
| [[180edo|180]]
| [[181edo|181]]
| [[182edo|182]]
| [[183edo|183]]
| [[184edo|184]]
| [[185edo|185]]
| [[186edo|186]]
| [[187edo|187]]
| [[188edo|188]]
| [[189edo|189]]
|-
| [[190edo|190]]
| [[191edo|191]]
| [[192edo|192]]
| [[193edo|193]]
| [[194edo|194]]
| [[195edo|195]]
| [[196edo|196]]
| [[197edo|197]]
| [[198edo|198]]
| [[199edo|199]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 200…299
|-
| [[200edo|200]]
| [[201edo|201]]
| [[202edo|202]]
| [[203edo|203]]
| [[204edo|204]]
| [[205edo|205]]
| [[206edo|206]]
| [[207edo|207]]
| [[208edo|208]]
| [[209edo|209]]
|-
| [[210edo|210]]
| [[211edo|211]]
| [[212edo|212]]
| [[213edo|213]]
| [[214edo|214]]
| [[215edo|215]]
| [[216edo|216]]
| [[217edo|217]]
| [[218edo|218]]
| [[219edo|219]]
|-
| [[220edo|220]]
| [[221edo|221]]
| [[222edo|222]]
| [[223edo|223]]
| [[224edo|224]]
| [[225edo|225]]
| [[226edo|226]]
| [[227edo|227]]
| [[228edo|228]]
| [[229edo|229]]
|-
| [[230edo|230]]
| [[231edo|231]]
| [[232edo|232]]
| [[233edo|233]]
| [[234edo|234]]
| [[235edo|235]]
| [[236edo|236]]
| [[237edo|237]]
| [[238edo|238]]
| [[239edo|239]]
|-
| [[240edo|240]]
| [[241edo|241]]
| [[242edo|242]]
| [[243edo|243]]
| [[244edo|244]]
| [[245edo|245]]
| [[246edo|246]]
| [[247edo|247]]
| [[248edo|248]]
| [[249edo|249]]
|-
| [[250edo|250]]
| [[251edo|251]]
| [[252edo|252]]
| [[253edo|253]]
| [[254edo|254]]
| [[255edo|255]]
| [[256edo|256]]
| [[257edo|257]]
| [[258edo|258]]
| [[259edo|259]]
|-
| [[260edo|260]]
| [[261edo|261]]
| [[262edo|262]]
| [[263edo|263]]
| [[264edo|264]]
| [[265edo|265]]
| [[266edo|266]]
| [[267edo|267]]
| [[268edo|268]]
| [[269edo|269]]
|-
| [[270edo|270]]
| [[271edo|271]]
| [[272edo|272]]
| [[273edo|273]]
| [[274edo|274]]
| [[275edo|275]]
| [[276edo|276]]
| [[277edo|277]]
| [[278edo|278]]
| [[279edo|279]]
|-
| [[280edo|280]]
| [[281edo|281]]
| [[282edo|282]]
| [[283edo|283]]
| [[284edo|284]]
| [[285edo|285]]
| [[286edo|286]]
| [[287edo|287]]
| [[288edo|288]]
| [[289edo|289]]
|-
| [[290edo|290]]
| [[291edo|291]]
| [[292edo|292]]
| [[293edo|293]]
| [[294edo|294]]
| [[295edo|295]]
| [[296edo|296]]
| [[297edo|297]]
| [[298edo|298]]
| [[299edo|299]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 300…399
|-
| [[300edo|300]]
| [[301edo|301]]
| [[302edo|302]]
| [[303edo|303]]
| [[304edo|304]]
| [[305edo|305]]
| [[306edo|306]]
| [[307edo|307]]
| [[308edo|308]]
| [[309edo|309]]
|-
| [[310edo|310]]
| [[311edo|311]]
| [[312edo|312]]
| [[313edo|313]]
| [[314edo|314]]
| [[315edo|315]]
| [[316edo|316]]
| [[317edo|317]]
| [[318edo|318]]
| [[319edo|319]]
|-
| [[320edo|320]]
| [[321edo|321]]
| [[322edo|322]]
| [[323edo|323]]
| [[324edo|324]]
| [[325edo|325]]
| [[326edo|326]]
| [[327edo|327]]
| [[328edo|328]]
| [[329edo|329]]
|-
| [[330edo|330]]
| [[331edo|331]]
| [[332edo|332]]
| [[333edo|333]]
| [[334edo|334]]
| [[335edo|335]]
| [[336edo|336]]
| [[337edo|337]]
| [[338edo|338]]
| [[339edo|339]]
|-
| [[340edo|340]]
| [[341edo|341]]
| [[342edo|342]]
| [[343edo|343]]
| [[344edo|344]]
| [[345edo|345]]
| [[346edo|346]]
| [[347edo|347]]
| [[348edo|348]]
| [[349edo|349]]
|-
| [[350edo|350]]
| [[351edo|351]]
| [[352edo|352]]
| [[353edo|353]]
| [[354edo|354]]
| [[355edo|355]]
| [[356edo|356]]
| [[357edo|357]]
| [[358edo|358]]
| [[359edo|359]]
|-
| [[360edo|360]]
| [[361edo|361]]
| [[362edo|362]]
| [[363edo|363]]
| [[364edo|364]]
| [[365edo|365]]
| [[366edo|366]]
| [[367edo|367]]
| [[368edo|368]]
| [[369edo|369]]
|-
| [[370edo|370]]
| [[371edo|371]]
| [[372edo|372]]
| [[373edo|373]]
| [[374edo|374]]
| [[375edo|375]]
| [[376edo|376]]
| [[377edo|377]]
| [[378edo|378]]
| [[379edo|379]]
|-
| [[380edo|380]]
| [[381edo|381]]
| [[382edo|382]]
| [[383edo|383]]
| [[384edo|384]]
| [[385edo|385]]
| [[386edo|386]]
| [[387edo|387]]
| [[388edo|388]]
| [[389edo|389]]
|-
| [[390edo|390]]
| [[391edo|391]]
| [[392edo|392]]
| [[393edo|393]]
| [[394edo|394]]
| [[395edo|395]]
| [[396edo|396]]
| [[397edo|397]]
| [[398edo|398]]
| [[399edo|399]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 400…499
|-
| [[400edo|400]]
| [[401edo|401]]
| [[402edo|402]]
| [[403edo|403]]
| [[404edo|404]]
| [[405edo|405]]
| [[406edo|406]]
| [[407edo|407]]
| [[408edo|408]]
| [[409edo|409]]
|-
| [[410edo|410]]
| [[411edo|411]]
| [[412edo|412]]
| [[413edo|413]]
| [[414edo|414]]
| [[415edo|415]]
| [[416edo|416]]
| [[417edo|417]]
| [[418edo|418]]
| [[419edo|419]]
|-
| [[420edo|420]]
| [[421edo|421]]
| [[422edo|422]]
| [[423edo|423]]
| [[424edo|424]]
| [[425edo|425]]
| [[426edo|426]]
| [[427edo|427]]
| [[428edo|428]]
| [[429edo|429]]
|-
| [[430edo|430]]
| [[431edo|431]]
| [[432edo|432]]
| [[433edo|433]]
| [[434edo|434]]
| [[435edo|435]]
| [[436edo|436]]
| [[437edo|437]]
| [[438edo|438]]
| [[439edo|439]]
|-
| [[440edo|440]]
| [[441edo|441]]
| [[442edo|442]]
| [[443edo|443]]
| [[444edo|444]]
| [[445edo|445]]
| [[446edo|446]]
| [[447edo|447]]
| [[448edo|448]]
| [[449edo|449]]
|-
| [[450edo|450]]
| [[451edo|451]]
| [[452edo|452]]
| [[453edo|453]]
| [[454edo|454]]
| [[455edo|455]]
| [[456edo|456]]
| [[457edo|457]]
| [[458edo|458]]
| [[459edo|459]]
|-
| [[460edo|460]]
| [[461edo|461]]
| [[462edo|462]]
| [[463edo|463]]
| [[464edo|464]]
| [[465edo|465]]
| [[466edo|466]]
| [[467edo|467]]
| [[468edo|468]]
| [[469edo|469]]
|-
| [[470edo|470]]
| [[471edo|471]]
| [[472edo|472]]
| [[473edo|473]]
| [[474edo|474]]
| [[475edo|475]]
| [[476edo|476]]
| [[477edo|477]]
| [[478edo|478]]
| [[479edo|479]]
|-
| [[480edo|480]]
| [[481edo|481]]
| [[482edo|482]]
| [[483edo|483]]
| [[484edo|484]]
| [[485edo|485]]
| [[486edo|486]]
| [[487edo|487]]
| [[488edo|488]]
| [[489edo|489]]
|-
| [[490edo|490]]
| [[491edo|491]]
| [[492edo|492]]
| [[493edo|493]]
| [[494edo|494]]
| [[495edo|495]]
| [[496edo|496]]
| [[497edo|497]]
| [[498edo|498]]
| [[499edo|499]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 500…599
|-
| [[500edo|500]]
| [[501edo|501]]
| [[502edo|502]]
| [[503edo|503]]
| [[504edo|504]]
| [[505edo|505]]
| [[506edo|506]]
| [[507edo|507]]
| [[508edo|508]]
| [[509edo|509]]
|-
| [[510edo|510]]
| [[511edo|511]]
| [[512edo|512]]
| [[513edo|513]]
| [[514edo|514]]
| [[515edo|515]]
| [[516edo|516]]
| [[517edo|517]]
| [[518edo|518]]
| [[519edo|519]]
|-
| [[520edo|520]]
| [[521edo|521]]
| [[522edo|522]]
| [[523edo|523]]
| [[524edo|524]]
| [[525edo|525]]
| [[526edo|526]]
| [[527edo|527]]
| [[528edo|528]]
| [[529edo|529]]
|-
| [[530edo|530]]
| [[531edo|531]]
| [[532edo|532]]
| [[533edo|533]]
| [[534edo|534]]
| [[535edo|535]]
| [[536edo|536]]
| [[537edo|537]]
| [[538edo|538]]
| [[539edo|539]]
|-
| [[540edo|540]]
| [[541edo|541]]
| [[542edo|542]]
| [[543edo|543]]
| [[544edo|544]]
| [[545edo|545]]
| [[546edo|546]]
| [[547edo|547]]
| [[548edo|548]]
| [[549edo|549]]
|-
| [[550edo|550]]
| [[551edo|551]]
| [[552edo|552]]
| [[553edo|553]]
| [[554edo|554]]
| [[555edo|555]]
| [[556edo|556]]
| [[557edo|557]]
| [[558edo|558]]
| [[559edo|559]]
|-
| [[560edo|560]]
| [[561edo|561]]
| [[562edo|562]]
| [[563edo|563]]
| [[564edo|564]]
| [[565edo|565]]
| [[566edo|566]]
| [[567edo|567]]
| [[568edo|568]]
| [[569edo|569]]
|-
| [[570edo|570]]
| [[571edo|571]]
| [[572edo|572]]
| [[573edo|573]]
| [[574edo|574]]
| [[575edo|575]]
| [[576edo|576]]
| [[577edo|577]]
| [[578edo|578]]
| [[579edo|579]]
|-
| [[580edo|580]]
| [[581edo|581]]
| [[582edo|582]]
| [[583edo|583]]
| [[584edo|584]]
| [[585edo|585]]
| [[586edo|586]]
| [[587edo|587]]
| [[588edo|588]]
| [[589edo|589]]
|-
| [[590edo|590]]
| [[591edo|591]]
| [[592edo|592]]
| [[593edo|593]]
| [[594edo|594]]
| [[595edo|595]]
| [[596edo|596]]
| [[597edo|597]]
| [[598edo|598]]
| [[599edo|599]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 600…699
|-
| [[600edo|600]]
| [[601edo|601]]
| [[602edo|602]]
| [[603edo|603]]
| [[604edo|604]]
| [[605edo|605]]
| [[606edo|606]]
| [[607edo|607]]
| [[608edo|608]]
| [[609edo|609]]
|-
| [[610edo|610]]
| [[611edo|611]]
| [[612edo|612]]
| [[613edo|613]]
| [[614edo|614]]
| [[615edo|615]]
| [[616edo|616]]
| [[617edo|617]]
| [[618edo|618]]
| [[619edo|619]]
|-
| [[620edo|620]]
| [[621edo|621]]
| [[622edo|622]]
| [[623edo|623]]
| [[624edo|624]]
| [[625edo|625]]
| [[626edo|626]]
| [[627edo|627]]
| [[628edo|628]]
| [[629edo|629]]
|-
| [[630edo|630]]
| [[631edo|631]]
| [[632edo|632]]
| [[633edo|633]]
| [[634edo|634]]
| [[635edo|635]]
| [[636edo|636]]
| [[637edo|637]]
| [[638edo|638]]
| [[639edo|639]]
|-
| [[640edo|640]]
| [[641edo|641]]
| [[642edo|642]]
| [[643edo|643]]
| [[644edo|644]]
| [[645edo|645]]
| [[646edo|646]]
| [[647edo|647]]
| [[648edo|648]]
| [[649edo|649]]
|-
| [[650edo|650]]
| [[651edo|651]]
| [[652edo|652]]
| [[653edo|653]]
| [[654edo|654]]
| [[655edo|655]]
| [[656edo|656]]
| [[657edo|657]]
| [[658edo|658]]
| [[659edo|659]]
|-
| [[660edo|660]]
| [[661edo|661]]
| [[662edo|662]]
| [[663edo|663]]
| [[664edo|664]]
| [[665edo|665]]
| [[666edo|666]]
| [[667edo|667]]
| [[668edo|668]]
| [[669edo|669]]
|-
| [[670edo|670]]
| [[671edo|671]]
| [[672edo|672]]
| [[673edo|673]]
| [[674edo|674]]
| [[675edo|675]]
| [[676edo|676]]
| [[677edo|677]]
| [[678edo|678]]
| [[679edo|679]]
|-
| [[680edo|680]]
| [[681edo|681]]
| [[682edo|682]]
| [[683edo|683]]
| [[684edo|684]]
| [[685edo|685]]
| [[686edo|686]]
| [[687edo|687]]
| [[688edo|688]]
| [[689edo|689]]
|-
| [[690edo|690]]
| [[691edo|691]]
| [[692edo|692]]
| [[693edo|693]]
| [[694edo|694]]
| [[695edo|695]]
| [[696edo|696]]
| [[697edo|697]]
| [[698edo|698]]
| [[699edo|699]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 700…799
|-
| [[700edo|700]]
| [[701edo|701]]
| [[702edo|702]]
| [[703edo|703]]
| [[704edo|704]]
| [[705edo|705]]
| [[706edo|706]]
| [[707edo|707]]
| [[708edo|708]]
| [[709edo|709]]
|-
| [[710edo|710]]
| [[711edo|711]]
| [[712edo|712]]
| [[713edo|713]]
| [[714edo|714]]
| [[715edo|715]]
| [[716edo|716]]
| [[717edo|717]]
| [[718edo|718]]
| [[719edo|719]]
|-
| [[720edo|720]]
| [[721edo|721]]
| [[722edo|722]]
| [[723edo|723]]
| [[724edo|724]]
| [[725edo|725]]
| [[726edo|726]]
| [[727edo|727]]
| [[728edo|728]]
| [[729edo|729]]
|-
| [[730edo|730]]
| [[731edo|731]]
| [[732edo|732]]
| [[733edo|733]]
| [[734edo|734]]
| [[735edo|735]]
| [[736edo|736]]
| [[737edo|737]]
| [[738edo|738]]
| [[739edo|739]]
|-
| [[740edo|740]]
| [[741edo|741]]
| [[742edo|742]]
| [[743edo|743]]
| [[744edo|744]]
| [[745edo|745]]
| [[746edo|746]]
| [[747edo|747]]
| [[748edo|748]]
| [[749edo|749]]
|-
| [[750edo|750]]
| [[751edo|751]]
| [[752edo|752]]
| [[753edo|753]]
| [[754edo|754]]
| [[755edo|755]]
| [[756edo|756]]
| [[757edo|757]]
| [[758edo|758]]
| [[759edo|759]]
|-
| [[760edo|760]]
| [[761edo|761]]
| [[762edo|762]]
| [[763edo|763]]
| [[764edo|764]]
| [[765edo|765]]
| [[766edo|766]]
| [[767edo|767]]
| [[768edo|768]]
| [[769edo|769]]
|-
| [[770edo|770]]
| [[771edo|771]]
| [[772edo|772]]
| [[773edo|773]]
| [[774edo|774]]
| [[775edo|775]]
| [[776edo|776]]
| [[777edo|777]]
| [[778edo|778]]
| [[779edo|779]]
|-
| [[780edo|780]]
| [[781edo|781]]
| [[782edo|782]]
| [[783edo|783]]
| [[784edo|784]]
| [[785edo|785]]
| [[786edo|786]]
| [[787edo|787]]
| [[788edo|788]]
| [[789edo|789]]
|-
| [[790edo|790]]
| [[791edo|791]]
| [[792edo|792]]
| [[793edo|793]]
| [[794edo|794]]
| [[795edo|795]]
| [[796edo|796]]
| [[797edo|797]]
| [[798edo|798]]
| [[799edo|799]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 800…899
|-
| [[800edo|800]]
| [[801edo|801]]
| [[802edo|802]]
| [[803edo|803]]
| [[804edo|804]]
| [[805edo|805]]
| [[806edo|806]]
| [[807edo|807]]
| [[808edo|808]]
| [[809edo|809]]
|-
| [[810edo|810]]
| [[811edo|811]]
| [[812edo|812]]
| [[813edo|813]]
| [[814edo|814]]
| [[815edo|815]]
| [[816edo|816]]
| [[817edo|817]]
| [[818edo|818]]
| [[819edo|819]]
|-
| [[820edo|820]]
| [[821edo|821]]
| [[822edo|822]]
| [[823edo|823]]
| [[824edo|824]]
| [[825edo|825]]
| [[826edo|826]]
| [[827edo|827]]
| [[828edo|828]]
| [[829edo|829]]
|-
| [[830edo|830]]
| [[831edo|831]]
| [[832edo|832]]
| [[833edo|833]]
| [[834edo|834]]
| [[835edo|835]]
| [[836edo|836]]
| [[837edo|837]]
| [[838edo|838]]
| [[839edo|839]]
|-
| [[840edo|840]]
| [[841edo|841]]
| [[842edo|842]]
| [[843edo|843]]
| [[844edo|844]]
| [[845edo|845]]
| [[846edo|846]]
| [[847edo|847]]
| [[848edo|848]]
| [[849edo|849]]
|-
| [[850edo|850]]
| [[851edo|851]]
| [[852edo|852]]
| [[853edo|853]]
| [[854edo|854]]
| [[855edo|855]]
| [[856edo|856]]
| [[857edo|857]]
| [[858edo|858]]
| [[859edo|859]]
|-
| [[860edo|860]]
| [[861edo|861]]
| [[862edo|862]]
| [[863edo|863]]
| [[864edo|864]]
| [[865edo|865]]
| [[866edo|866]]
| [[867edo|867]]
| [[868edo|868]]
| [[869edo|869]]
|-
| [[870edo|870]]
| [[871edo|871]]
| [[872edo|872]]
| [[873edo|873]]
| [[874edo|874]]
| [[875edo|875]]
| [[876edo|876]]
| [[877edo|877]]
| [[878edo|878]]
| [[879edo|879]]
|-
| [[880edo|880]]
| [[881edo|881]]
| [[882edo|882]]
| [[883edo|883]]
| [[884edo|884]]
| [[885edo|885]]
| [[886edo|886]]
| [[887edo|887]]
| [[888edo|888]]
| [[889edo|889]]
|-
| [[890edo|890]]
| [[891edo|891]]
| [[892edo|892]]
| [[893edo|893]]
| [[894edo|894]]
| [[895edo|895]]
| [[896edo|896]]
| [[897edo|897]]
| [[898edo|898]]
| [[899edo|899]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 900…999
|-
| [[900edo|900]]
| [[901edo|901]]
| [[902edo|902]]
| [[903edo|903]]
| [[904edo|904]]
| [[905edo|905]]
| [[906edo|906]]
| [[907edo|907]]
| [[908edo|908]]
| [[909edo|909]]
|-
| [[910edo|910]]
| [[911edo|911]]
| [[912edo|912]]
| [[913edo|913]]
| [[914edo|914]]
| [[915edo|915]]
| [[916edo|916]]
| [[917edo|917]]
| [[918edo|918]]
| [[919edo|919]]
|-
| [[920edo|920]]
| [[921edo|921]]
| [[922edo|922]]
| [[923edo|923]]
| [[924edo|924]]
| [[925edo|925]]
| [[926edo|926]]
| [[927edo|927]]
| [[928edo|928]]
| [[929edo|929]]
|-
| [[930edo|930]]
| [[931edo|931]]
| [[932edo|932]]
| [[933edo|933]]
| [[934edo|934]]
| [[935edo|935]]
| [[936edo|936]]
| [[937edo|937]]
| [[938edo|938]]
| [[939edo|939]]
|-
| [[940edo|940]]
| [[941edo|941]]
| [[942edo|942]]
| [[943edo|943]]
| [[944edo|944]]
| [[945edo|945]]
| [[946edo|946]]
| [[947edo|947]]
| [[948edo|948]]
| [[949edo|949]]
|-
| [[950edo|950]]
| [[951edo|951]]
| [[952edo|952]]
| [[953edo|953]]
| [[954edo|954]]
| [[955edo|955]]
| [[956edo|956]]
| [[957edo|957]]
| [[958edo|958]]
| [[959edo|959]]
|-
| [[960edo|960]]
| [[961edo|961]]
| [[962edo|962]]
| [[963edo|963]]
| [[964edo|964]]
| [[965edo|965]]
| [[966edo|966]]
| [[967edo|967]]
| [[968edo|968]]
| [[969edo|969]]
|-
| [[970edo|970]]
| [[971edo|971]]
| [[972edo|972]]
| [[973edo|973]]
| [[974edo|974]]
| [[975edo|975]]
| [[976edo|976]]
| [[977edo|977]]
| [[978edo|978]]
| [[979edo|979]]
|-
| [[980edo|980]]
| [[981edo|981]]
| [[982edo|982]]
| [[983edo|983]]
| [[984edo|984]]
| [[985edo|985]]
| [[986edo|986]]
| [[987edo|987]]
| [[988edo|988]]
| [[989edo|989]]
|-
| [[990edo|990]]
| [[991edo|991]]
| [[992edo|992]]
| [[993edo|993]]
| [[994edo|994]]
| [[995edo|995]]
| [[996edo|996]]
| [[997edo|997]]
| [[998edo|998]]
| [[999edo|999]]
|}

=== 1000…1999 ===
{{EDOs
| 1000, 1001, 1012, 1015, 1019, 1051, 1053, 1059, 1063, 1065, 1080, 1092, 1106, 1125, 1131, 1147, 1152, 1166, 1171, 1178, 1193, 1200, 1210, 1224, 1230, 1236, 1240, 1244, 1260, 1272, 1277, 1289, 1308, 1312, 1323, 1330, 1337, 1342, 1361, 1376, 1381, 1395, 1407, 1419, 1429, 1440, 1448, 1489, 1506, 1517, 1520, 1525, 1536, 1547, 1553, 1554, 1559, 1578, 1583, 1590, 1600, 1609, 1612, 1619, 1637, 1641, 1643, 1650, 1665, 1672, 1700, 1730, 1759, 1776, 1778, 1783, 1789, 1793, 1794, 1802, 1803, 1817, 1848, 1861, 1879, 1880, 1889, 1911, 1920, 1944, 1955, 1957, 1983, 1984 }}

=== 2000…9999 ===
{{EDOs
| 2000, 2016, 2019, 2022, 2023, 2024, 2025, 2029, 2048, 2053, 2072, 2081, 2100, 2101, 2113, 2118, 2129, 2153, 2190, 2200, 2207, 2242, 2243, 2320, 2444, 2460, 2477, 2513, 2520, 2544, 2549, 2554, 2592, 2619, 2684, 2711, 2730, 2777, 2809, 2814, 2819, 2897, 2901, 2912, 2960, 2964, 3041, 3071, 3072, 3079, 3080, 3125, 3178, 3361, 3395, 3422, 3476, 3498, 3558, 3566, 3578, 3600, 3643, 3684, 3696, 3776, 3889, 3920, 4004, 4007, 4079, 4096, 4172, 4190, 4231, 4296, 4320, 4327, 4349, 4380, 4501, 4650, 4973, 5040, 5280, 5544, 5585, 5809, 5902, 5941, 6079, 6349, 6380, 6650, 6664, 6691, 7033, 7315, 7980, 8103, 8192, 8269, 8404, 8539, 8736, 8745, 9539
}}

=== 10000 and up ===
{{EDOs
| 10009, 10459, 10600, 10729, 11664, 12276, 12348, 12500, 13382, 14124, 14348, 14618, 14842, 15601, 15900, 16218, 16384, 16625, 16808, 17461, 18355, 20203, 20567, 25281, 25282, 28000, 28742, 30103, 30631, 31867, 31920, 32436, 33616, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 99693, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 571611, 762148, 1714833, 1905370, 2547047, 2667518, 2901533, 3159811, 4191814, 6000000, 11358058, 402653184, 5407372813
}}

== Non-integer EDO ==
A non-integer EDO can be defined as using a non-integer divisor to divide the octave. Typically, non-integer EDOs are understood as ''not'' containing the exact octave, so that they remain [[equal tuning]]s. If the exact octave is retained and if the generator resets itself at each period, then this results in a [[MOS scale]] with only 1 small step.

All fractional EDOs are integer equal divisions of another integer interval. For example, (25/2)edo is equivalent to 25ed4. In general:

<math>\displaystyle (p/q) \text{edo} = p \text{-ed} 2^q</math>

for integers ''p'' and ''q''. Many irrational EDOs cannot be converted to integer equal divisions of another integer interval, so they are things of their own.

Non-integer EDOs can be written in decimal form, such as 12.1edo. This is often meant to be approximate, used in the context of [[octave stretch]] of an integer EDO, rather than as a fractional EDO.

== Scale tree ==

The scale tree, or Stern-Brocot tree, provides a visual map of the world of EDOs, based on fifth size.

[[File:The_Scale_Tree.png|alt=The Scale Tree.png|800x1023px|The Scale Tree.png]]

The regular EDOs, up to 72edo:

[[File:Scale_Tree_close-up.png|alt=Scale Tree close-up.png|Scale Tree close-up.png]]

== Pergens ==
{{See also| Pergen #Pergens and EDOs }}

[[Pergen]]s provide a JI-agnostic way to name the rank-2 scales of an EDO. This table lists every possible period/generator combination for EDOs 5–24, and for each coprime combination, the simplest pergen that it can represent. Non-coprime combinations such as {{nowrap|P = 6\12}}, {{nowrap|G = 4\12}} are marked as "-".

{| class="wikitable"
|-
! EDO
! Period
! colspan="11" | Generator in EDO steps
|-
!
! in EDO steps
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
|-
! 5
! 5 = P8
| P4/2
| P5
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 6
! 6 = P8
| P4/2
| -
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/2
| P5
|
|
|
|
|
|
|
|
|
|
|-
! 7
! 7 = P8
| P4/3
| P5/2
| P5
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 8
! 8 = P8
| P4/3
| -
| P5
|
|
|
|
|
|
|
|
|-
! 4 = P8/2
| P5
| -
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 9
! 9 = P8
| P4/4
| P4/2
| -
| P5
|
|
|
|
|
|
|
|-
! 3 = P8/3
| P5
|
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 10
! 10 = P8
| P4/4
| -
| P5/2
| -
|
|
|
|
|
|
|
|-
! 5 = P8/2
| P5
| P4/2
|
|
|
|
|
|
|
|
|
|-
! 11
! 11 = P8
| P4/5
| P5/3
| P5/2
| P11/4
| P5
|
|
|
|
|
|
|-
! rowspan="4" | 12
! 12 = P8
| P4/5
| -
| -
| -
| P5
|
|
|
|
|
|
|-
! 6 = P8/2
| P5
| -
| -
|
|
|
|
|
|
|
|
|-
! 4 = P8/3
| P5
| -
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/4
| P5
|
|
|
|
|
|
|
|
|
|
|-
! 13b
! 13 = P8
| P4/6
| P4/3
| P4/2
| P12/5
| P12/4
| P5
|
|
|
|
|
|-
! rowspan="2" | 14
! 14 = P8
| P4/6
| -
| P4/2
| -
| P11/4
| -
|
|
|
|
|
|-
! 7 = P8/2
| P5
| P4/3
| P4/2
|
|
|
|
|
|
|
|
|-
! rowspan="3" | 15
! 15 = P8
| P4/6
| P4/3
| -
| P12/6
| -
| -
| P11/3
|
|
|
|
|-
! 5 = P8/3
| P5
| P4/2
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/5
| P4/3
|
|
|
|
|
|
|
|
|
|
|-
! rowspan="3" | 16
! 16 = P8
| P4/7
| -
| P5/3
| -
| P12/5
| -
| P5
|
|
|
|
|-
! 8 = P8/2
| P5
| -
| P5/3
| -
|
|
|
|
|
|
|
|-
! 4 = P8/4
| P5
| -
|
|
|
|
|
|
|
|
|
|-
! 17
! 17 = P8
| P4/7
| P5/5
| P11/8
| P11/6
| P5/2
| P11/4
| P5
| P11/3
|
|
|
|-
! rowspan="4" | 18b
! 18 = P8
| P4/8
| -
| -
| -
| P5/2
| -
| P12/4
| -
|
|
|
|-
! 9 = P8/2
| P5
| P4/4
| -
| P4/2
|
|
|
|
|
|
|
|-
! 6 = P8/3
| P5/2
| -
| -
|
|
|
|
|
|
|
|
|-
! 3 = P8/6
| P5
|
|
|
|
|
|
|
|
|
|
|-
! 19
! 19 = P8
| P4/8
| P4/4
| P11/9
| P4/2
| P12/6
| P12/5
| ccP5/7
| P5
| P11/3
|
|
|-
! rowspan="4" | 20
! 20 = P8
| P4/8
| -
| P5/4
| -
| -
| -
| P11/4
| -
| c3P5/8
|
|
|-
! 10 = P8/2
| M2/4
| -
| P5/4
| -
| -
|
|
|
|
|
|
|-
! 5 = P8/4
| P4/2
| P5
|
|
|
|
|
|
|
|
|
|-
! 4 = P8/5
| P5/4
| -
|
|
|
|
|
|
|
|
|
|-
! rowspan="3" | 21
! 21 = P8
| P4/9
| P5/6
| -
| P5/3
| P11/6
| -
| -
| c3P4/9
| -
| P11/3
|
|-
! 7 = P8/3
| P5/2
| P5
| P4/3
|
|
|
|
|
|
|
|
|-
! 3 = P8/7
| P5/3
|
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 22
! 22 = P8
| P4/9
| -
| P4/3
| -
| P12/7
| -
| P12/5
| -
| P5
| -
|
|-
! 11 = P8/2
| M2/4
| P5
| P4/3
| P12/5
| P12/7
|
|
|
|
|
|
|-
! 23
! 23 = P8
| P4/10
| P4/5
| P11/11
| P12/9
| P4/2
| P12/6
| ccP4/8
| ccP4/7
| P12/4
| P5
| P11/3
|-
! rowspan="6" | 24
! 24 = P8
| P4/10
| -
| -
| -
| P4/2
| -
| P5/2
| -
| -
| -
| c4P5/10
|-
! 12 = P8/2
| M2/4
| -
| -
| -
| P4/2
|
|
|
|
|
|
|-
! 8 = P8/3
| P5/2
| -
| P4/2
|
|
|
|
|
|
|
|
|-
! 6 = P8/4
| P4/2
| -
| -
|
|
|
|
|
|
|
|
|-
! 4 = P8/6
| P4/2
| -
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/8
| P5
|
|
|
|
|
|
|
|
|
|
|-
!
!
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
|}

== See also ==
Related topics
* [[Equal-step tuning]]
* [[Highly composite equal division]]
* [[List of rank one temperaments by step size]]
* [[Prime equal division]]

Technical data
* [[Absolute errors of small EDOs]]
* [[Consistency limits of small EDOs]]
* [[Distinct EDO Scales]]
* [[Minimal consistent EDOs]]
* [[Monotonicity levels of small EDOs]]
* [[Relative errors of small EDOs]]

Opinions
* [[Collection of EDO impressions]]

Other
* [[:Category:Equal divisions of the octave]]

== External links ==
* [[Ivor Darreg]], [https://www.webcitation.org/5xZz8RtQB Teen Tunes]

== Notes ==
<references />

[[Category:Equal divisions of the octave| ]] 
[[Category:Acronyms]]
[[Category:Lists of scales]]

EDO

2025-08-02T22:51:22Z

Eliora: /* 1000…1999 */

{{Todo|discuss title}}
{{interwiki
| de = EDO
| en = EDO
| es = EDOs
| ja = オクターブ平均律
| ko = EDO (Korean)
| ro = DEO
}}
An '''equal division of the octave''' ('''EDO''', ''EE-dee-oh''; '''edo''', ''EE-doh'') is a [[Tuning system|tuning]] obtained by dividing the [[octave]] in a certain number of [[Equal-step tuning|equal steps]]. This means that the [[interval]] between any two consecutive pitches is identical.

A tuning with ''n'' equal divisions of the octave is usually called "''n''-edo" ("''n''-EDO"). For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO).

An equal (pitch) division of the octave is the most common type of [[EPD|equal (pitch) division]], which is a kind of [[equal-step tuning]]. Therefore, it is also a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.

== History ==
Tuning theorists first used the term "equal temperament" for edos designed to approximate [[Low-complexity JI|low-complexity just intervals]]. The same term is still used today for all rank-1 [[Regular temperament|temperaments]]. For example, [[15edo]] can be referred to as 15-tone equal temperament (15-TET, 15-tET, 15tet, etc.), or more simply 15 equal temperament (15-ET, 15et, etc.).

The acronym "EDO" was coined by [[Daniel Anthony Stearns]] in 1999, originally standing for "equidistant divisions of the octave"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_65#65 Yahoo! Tuning Group | ''Where F + f = O'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_117#117 Yahoo! Tuning Group | ''f + F and WFS/MOS'']</ref>. More recently, the {{w|anacronym}} "edo", spelled in lowercase and pronounced as a regular word, has also become common.

With the development of [[Edonoi|equal divisions of non-octave intervals (edonoi)]], some people started writing "ed2" ("ED2"), especially when naming a specific tuning.

== Calculating the step size ==
To find the step size of ''n''-edo in terms of [[cent]]s, divide 1200 by ''n''. The size ''s'' of ''k'' steps of ''n''-edo (''k''\''n'') is

<math>\displaystyle s = 1200 \cdot k/n</math>

To find the step size of ''n''-edo in terms of [[frequency ratio]], take the ''n''-th root of 2. For example, the step of 12edo is 21/12 (≈ 1.059). So the ratio ''c'' of the ''k'' steps of ''n''-edo is

<math>\displaystyle c = 2^{k/n}</math>

In particular, when ''k'' is 0, ''c'' is simply 1, because any number to the 0th power is 1. And when {{nowrap|''k'' {{=}} ''n''}}, ''c'' is simply 2, because any number to the 1st power is itself.

== Properties ==
EDO scales are straightforward to work with due to their uniform step size.
Some musicians find the consistency bland, while others appreciate the stable foundation it provides for composition.
The only property shared by all of them is the equality of their step-sizes; otherwise, their individual properties are as different as can be.
Lower-numbered EDOs, especially 5 to 24, possess very strong and unique "characters", which some composers find inspiring.

== Practical advantages ==
=== Fretted instruments ===
If you are a [[guitar]]ist (or a player of some other fretted string instrument, like a bass guitar, Appalachian dulcimer, [[ukulele]], banjo, mandolin, sitar, saz, pipa, or zhong ruan), an EDO will provide you with the simplest possible fretboard layout, as all of the frets will go straight across the fretboard, regardless of how you want to tune the open strings.
Fret crowding can become an issue with smaller divisions, especially high up the neck.
For these cases, [[ed4|equal divisions of the double octave]] or higher multiples offer a compromise solution.

=== Free modulation ===
EDOs allow for modulation to every single key in the tuning, without any alteration in harmonic properties, thus making transposition totally seamless.
This also makes them somewhat easier to learn, as you do not have to memorize the harmonic and melodic variations that appear in various keys (which you would have to learn in JI, an unequal regular temperament, or a well-temperament, especially with smaller numbers of tones).
For those accustomed to the "equality" of 12-TET, the equality of the alternative EDOs can be reassuringly familiar.

== Approaches to exploring EDOs ==
If you are interested in exploring the unique merits and challenges of each EDO, irrespective of any desire to approximate just intonation (or any other a priori musical goal), starting at the bottom and working your way up can be a most illuminating exercise.

If you're a classically trained musician and you'd like to start with some EDOs that have some relationship to common-practice tonal music, starting with reasonably-low EDOs that give a good approximation to the perfect fifth ([[3/2]]) can be rewarding.
These include {{EDOs| 12, 17, 19, 22, 26, 27, 29, 31, 39, 41, 43, 45, 46, 49, 50, and 53 }}.
All of these can be notated with some variant on the [[Circle-of-fifths notation|A–G "circle of fifths" notation]], while other EDOs, including {{EDOs| 24, 34, 36, 38, 44, 48, or 51}} involve multiple such circles.

Some EDOs, such as {{EDOs| 26, 27, 32, 33, or 37 }} have fifths which are reasonably good but quite audibly not just. Other EDOs, such as {{EDOs| 11, 13, 14, 15, 16, 18, 20, 21, 23, or 25 }}, are of interest to the avid seeker of totally unusual sounds that have next-to-no connection with the common practice.

If your interest lies in the nuanced approximation of just intonation through EDOs, then delving into EDOs characterized by a strong [[The Riemann zeta function and tuning#Zeta EDO lists|zeta peak]] could be especially captivating. Such EDOs, including {{EDOs| 12, 19, 22, 27, 31, 34, 41, 46, 53, 58, 60, 65, 68, 72, 77, 80, 84, 87, 94, and 99 }}, offer rich avenues for exploration in the quest for harmonic purity and transparent [[temperament]]s.

EDOs with a less pronounced, yet still noteworthy [[The Riemann zeta function and tuning#Local zeta edos|zeta peak]]—specifically {{EDOs| 10, 14, 15, 16, 17, 21, 24, 26, 29, 32, 36, 37, 38, 39, 43, 45, 48, 49, 50, 56, 62, 63, 82, 89, and 96 }}—present a unique palette for harmony explorers. Although these systems may lack the harmonic precision found in EDOs with more prominent zeta peaks, they strike an intriguing balance between consonance and more distant harmonic textures.

EDOs can be further subdivided and classified according to the size of the fifth, such as with [[Margo Schulter]]'s [[gentle region]] or the distinction between negative, positive, doubly negative and doubly positive of [[R. H. M. Bosanquet]]. [[Kite Giedraitis]] has proposed these six categories, based on the size of the fifth. From narrowest to widest:

* '''Superflat''' EDOs ({{EDOs| 9, 11, 13b, 16, 18b, and 23 }}) have a fifth narrower than four-sevenths of an octave ({{nowrap|4\7 {{=}} 685.714{{c}}}})
* '''Perfect''' EDOs ({{EDOs| 7, 14, 21, 28, and 35 }}) have a fifth equal to {{nowrap|4\7 {{=}} 685.714{{c}}}}
* '''Diatonic''' EDOs ({{EDOs| 12, 17, 19, 22, 24, etc. }}) have a fifth between 685.714{{c}} and 720{{c}}
* '''Pentatonic''' EDOs ({{EDOs| 5, 10, 15, 20, 25, and 30 }}) have a fifth of three-fifths of an octave ({{nowrap|3\5 {{=}} 720{{c}}}})
* '''Supersharp''' EDOs ({{EDOs| 8, 13, and 18 }}) have a fifth wider than 720{{c}}
* '''Trivial''' EDOs ({{EDOs| 1, 2, 3, 4, and 6 }}) have a fifth about 100{{c}} from just, and are contained in 12edo

== Structural properties ==
You will quickly find that the ''factorization'' of the total number of notes in each EDO has consequences for its structure and the way it relates to other EDOs. For example, {{nowrap|6 {{=}} 2 x 3}}, so 6edo contains all of the intervals in both 2edo and 3edo. On the other hand, 7 is a prime number, so no 7edo intervals are redundant with those of smaller EDOs. See [[Prime EDO]] and [[Highly composite EDO]] for more details.

The [[MOS]] paradigm is a fascinating way of thinking about building sub-scales of EDOs and relating them to non-EDO scales, as well as finding common melodic patterns between multiple EDOs.

=== Adding EDOs ===
Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). This really amounts to the consideration of adding the associated [[vals]], which are the mappings to primes larger than 2. An EDO is defined by a certain number of steps equating to 2; if we have more steps equating to 3, we get a 3-limit val, and so forth. So, for example, 12edo can be written {{val| 12 }}, saying that twelve steps maps to 2, but the 3-limit val for 12 is {{val| 12 19 }}, telling us that 19 steps maps to 3, and the 5-limit val is {{val| 12 19 28 }}, telling us that 28 steps maps to 5.

If we add 12 and 19 we get another good division, {{nowrap| 12 + 19 {{=}} 31 }}. We can understand why this works if we look at it as adding vals; {{val| 12 19 28 }} + {{val| 19 30 44 }} = {{val| 31 49 72 }}. The relative error in terms of [[relative cent]]s is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is {{nowrap|[-1.955 13.686]}} (the same as absolute cents) and the error of 19edo is {{nowrap|[-11.429 -11.663]}}, and this sums to {{nowrap|[-13.384 2.023]}}. In relative cents the error of the fifth for 31edo is not much increased from 19edo, and on converting to absolute cents we find it is even better, and the error of the major third is much smaller due to the cancellation. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 31edo) can have a much smaller error due to the cancellation. 24edo's flat fifth and 29edo's sharp fifth can be added to form 53edo.

We may also look at addition of EDOs in terms of MOS; if ''a''\''n'' is a generator for an ''n''-edo MOS, and ''b''\''m'' for an ''m''-edo MOS, where both of these are generators for the same linear temperament, then the mediant, {{nowrap|(''a'' + ''b'')\(''n'' + ''m'')}}, will be a generator for a MOS for the same temperament, this time in {{nowrap|(''n'' + ''m'')}}-edo. A visual way of putting this is that through this addition of ''n'' and ''m'', one becomes the accidentals or black keys, and the other the naturals or white keys. The choice of accidental/natural or black keys/white keys is a question of emphasis on the part of the composer or designer. Furthermore, one may add more than two numbers, hierarchically expanding the possibilities to double flats and sharps and beyond. This can be useful in designing keyboards and systems of notation.

=== Scale size considerations ===
EDOs with fewer than 12 divisions have steps exceeding 100 cents.
Of these, 1, 2, 3, 4, and 6 divide 12 and so are already available.
{{EDOs| 5, 7, and 9 }} have arguably been used in various musical traditions worldwide.

When using EDOs to tune scales or [[regular temperament]]s, the size becomes less conceptually important since not all notes need to be used.
Some of the EDOs which can be used to tune various temperaments are listed on the [[optimal patent val]] page. Tuning a scale in just intonation by one of these EDOs can be regarded as automatically tempering it to the corresponding regular temperament.

To practically tune large edos through software tuning, one may take advantage of [[MIDI]] channels. See [[Tuning per channel]].

All of these tools are also applicable to equal divisions of other ([[nonoctave]]) intervals as well.

== EDOs versus Equal Temperaments ==
See [[EDO vs ET]].

== Individual pages for EDOs ==
=== 0…999 ===
{| class="wikitable center-all mw-collapsible"
|+ style="font-size: 105%; white-space: nowrap;" | 0…99
|-
| [[0edo|0]]
| [[1edo|1]]
| [[2edo|2]]
| [[3edo|3]]
| [[4edo|4]]
| [[5edo|5]]
| [[6edo|6]]
| [[7edo|7]]
| [[8edo|8]]
| [[9edo|9]]
|-
| [[10edo|10]]
| [[11edo|11]]
| [[12edo|12]]
| [[13edo|13]]
| [[14edo|14]]
| [[15edo|15]]
| [[16edo|16]]
| [[17edo|17]]
| [[18edo|18]]
| [[19edo|19]]
|-
| [[20edo|20]]
| [[21edo|21]]
| [[22edo|22]]
| [[23edo|23]]
| [[24edo|24]]
| [[25edo|25]]
| [[26edo|26]]
| [[27edo|27]]
| [[28edo|28]]
| [[29edo|29]]
|-
| [[30edo|30]]
| [[31edo|31]]
| [[32edo|32]]
| [[33edo|33]]
| [[34edo|34]]
| [[35edo|35]]
| [[36edo|36]]
| [[37edo|37]]
| [[38edo|38]]
| [[39edo|39]]
|-
| [[40edo|40]]
| [[41edo|41]]
| [[42edo|42]]
| [[43edo|43]]
| [[44edo|44]]
| [[45edo|45]]
| [[46edo|46]]
| [[47edo|47]]
| [[48edo|48]]
| [[49edo|49]]
|-
| [[50edo|50]]
| [[51edo|51]]
| [[52edo|52]]
| [[53edo|53]]
| [[54edo|54]]
| [[55edo|55]]
| [[56edo|56]]
| [[57edo|57]]
| [[58edo|58]]
| [[59edo|59]]
|-
| [[60edo|60]]
| [[61edo|61]]
| [[62edo|62]]
| [[63edo|63]]
| [[64edo|64]]
| [[65edo|65]]
| [[66edo|66]]
| [[67edo|67]]
| [[68edo|68]]
| [[69edo|69]]
|-
| [[70edo|70]]
| [[71edo|71]]
| [[72edo|72]]
| [[73edo|73]]
| [[74edo|74]]
| [[75edo|75]]
| [[76edo|76]]
| [[77edo|77]]
| [[78edo|78]]
| [[79edo|79]]
|-
| [[80edo|80]]
| [[81edo|81]]
| [[82edo|82]]
| [[83edo|83]]
| [[84edo|84]]
| [[85edo|85]]
| [[86edo|86]]
| [[87edo|87]]
| [[88edo|88]]
| [[89edo|89]]
|-
| [[90edo|90]]
| [[91edo|91]]
| [[92edo|92]]
| [[93edo|93]]
| [[94edo|94]]
| [[95edo|95]]
| [[96edo|96]]
| [[97edo|97]]
| [[98edo|98]]
| [[99edo|99]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 100…199
|-
| [[100edo|100]]
| [[101edo|101]]
| [[102edo|102]]
| [[103edo|103]]
| [[104edo|104]]
| [[105edo|105]]
| [[106edo|106]]
| [[107edo|107]]
| [[108edo|108]]
| [[109edo|109]]
|-
| [[110edo|110]]
| [[111edo|111]]
| [[112edo|112]]
| [[113edo|113]]
| [[114edo|114]]
| [[115edo|115]]
| [[116edo|116]]
| [[117edo|117]]
| [[118edo|118]]
| [[119edo|119]]
|-
| [[120edo|120]]
| [[121edo|121]]
| [[122edo|122]]
| [[123edo|123]]
| [[124edo|124]]
| [[125edo|125]]
| [[126edo|126]]
| [[127edo|127]]
| [[128edo|128]]
| [[129edo|129]]
|-
| [[130edo|130]]
| [[131edo|131]]
| [[132edo|132]]
| [[133edo|133]]
| [[134edo|134]]
| [[135edo|135]]
| [[136edo|136]]
| [[137edo|137]]
| [[138edo|138]]
| [[139edo|139]]
|-
| [[140edo|140]]
| [[141edo|141]]
| [[142edo|142]]
| [[143edo|143]]
| [[144edo|144]]
| [[145edo|145]]
| [[146edo|146]]
| [[147edo|147]]
| [[148edo|148]]
| [[149edo|149]]
|-
| [[150edo|150]]
| [[151edo|151]]
| [[152edo|152]]
| [[153edo|153]]
| [[154edo|154]]
| [[155edo|155]]
| [[156edo|156]]
| [[157edo|157]]
| [[158edo|158]]
| [[159edo|159]]
|-
| [[160edo|160]]
| [[161edo|161]]
| [[162edo|162]]
| [[163edo|163]]
| [[164edo|164]]
| [[165edo|165]]
| [[166edo|166]]
| [[167edo|167]]
| [[168edo|168]]
| [[169edo|169]]
|-
| [[170edo|170]]
| [[171edo|171]]
| [[172edo|172]]
| [[173edo|173]]
| [[174edo|174]]
| [[175edo|175]]
| [[176edo|176]]
| [[177edo|177]]
| [[178edo|178]]
| [[179edo|179]]
|-
| [[180edo|180]]
| [[181edo|181]]
| [[182edo|182]]
| [[183edo|183]]
| [[184edo|184]]
| [[185edo|185]]
| [[186edo|186]]
| [[187edo|187]]
| [[188edo|188]]
| [[189edo|189]]
|-
| [[190edo|190]]
| [[191edo|191]]
| [[192edo|192]]
| [[193edo|193]]
| [[194edo|194]]
| [[195edo|195]]
| [[196edo|196]]
| [[197edo|197]]
| [[198edo|198]]
| [[199edo|199]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 200…299
|-
| [[200edo|200]]
| [[201edo|201]]
| [[202edo|202]]
| [[203edo|203]]
| [[204edo|204]]
| [[205edo|205]]
| [[206edo|206]]
| [[207edo|207]]
| [[208edo|208]]
| [[209edo|209]]
|-
| [[210edo|210]]
| [[211edo|211]]
| [[212edo|212]]
| [[213edo|213]]
| [[214edo|214]]
| [[215edo|215]]
| [[216edo|216]]
| [[217edo|217]]
| [[218edo|218]]
| [[219edo|219]]
|-
| [[220edo|220]]
| [[221edo|221]]
| [[222edo|222]]
| [[223edo|223]]
| [[224edo|224]]
| [[225edo|225]]
| [[226edo|226]]
| [[227edo|227]]
| [[228edo|228]]
| [[229edo|229]]
|-
| [[230edo|230]]
| [[231edo|231]]
| [[232edo|232]]
| [[233edo|233]]
| [[234edo|234]]
| [[235edo|235]]
| [[236edo|236]]
| [[237edo|237]]
| [[238edo|238]]
| [[239edo|239]]
|-
| [[240edo|240]]
| [[241edo|241]]
| [[242edo|242]]
| [[243edo|243]]
| [[244edo|244]]
| [[245edo|245]]
| [[246edo|246]]
| [[247edo|247]]
| [[248edo|248]]
| [[249edo|249]]
|-
| [[250edo|250]]
| [[251edo|251]]
| [[252edo|252]]
| [[253edo|253]]
| [[254edo|254]]
| [[255edo|255]]
| [[256edo|256]]
| [[257edo|257]]
| [[258edo|258]]
| [[259edo|259]]
|-
| [[260edo|260]]
| [[261edo|261]]
| [[262edo|262]]
| [[263edo|263]]
| [[264edo|264]]
| [[265edo|265]]
| [[266edo|266]]
| [[267edo|267]]
| [[268edo|268]]
| [[269edo|269]]
|-
| [[270edo|270]]
| [[271edo|271]]
| [[272edo|272]]
| [[273edo|273]]
| [[274edo|274]]
| [[275edo|275]]
| [[276edo|276]]
| [[277edo|277]]
| [[278edo|278]]
| [[279edo|279]]
|-
| [[280edo|280]]
| [[281edo|281]]
| [[282edo|282]]
| [[283edo|283]]
| [[284edo|284]]
| [[285edo|285]]
| [[286edo|286]]
| [[287edo|287]]
| [[288edo|288]]
| [[289edo|289]]
|-
| [[290edo|290]]
| [[291edo|291]]
| [[292edo|292]]
| [[293edo|293]]
| [[294edo|294]]
| [[295edo|295]]
| [[296edo|296]]
| [[297edo|297]]
| [[298edo|298]]
| [[299edo|299]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 300…399
|-
| [[300edo|300]]
| [[301edo|301]]
| [[302edo|302]]
| [[303edo|303]]
| [[304edo|304]]
| [[305edo|305]]
| [[306edo|306]]
| [[307edo|307]]
| [[308edo|308]]
| [[309edo|309]]
|-
| [[310edo|310]]
| [[311edo|311]]
| [[312edo|312]]
| [[313edo|313]]
| [[314edo|314]]
| [[315edo|315]]
| [[316edo|316]]
| [[317edo|317]]
| [[318edo|318]]
| [[319edo|319]]
|-
| [[320edo|320]]
| [[321edo|321]]
| [[322edo|322]]
| [[323edo|323]]
| [[324edo|324]]
| [[325edo|325]]
| [[326edo|326]]
| [[327edo|327]]
| [[328edo|328]]
| [[329edo|329]]
|-
| [[330edo|330]]
| [[331edo|331]]
| [[332edo|332]]
| [[333edo|333]]
| [[334edo|334]]
| [[335edo|335]]
| [[336edo|336]]
| [[337edo|337]]
| [[338edo|338]]
| [[339edo|339]]
|-
| [[340edo|340]]
| [[341edo|341]]
| [[342edo|342]]
| [[343edo|343]]
| [[344edo|344]]
| [[345edo|345]]
| [[346edo|346]]
| [[347edo|347]]
| [[348edo|348]]
| [[349edo|349]]
|-
| [[350edo|350]]
| [[351edo|351]]
| [[352edo|352]]
| [[353edo|353]]
| [[354edo|354]]
| [[355edo|355]]
| [[356edo|356]]
| [[357edo|357]]
| [[358edo|358]]
| [[359edo|359]]
|-
| [[360edo|360]]
| [[361edo|361]]
| [[362edo|362]]
| [[363edo|363]]
| [[364edo|364]]
| [[365edo|365]]
| [[366edo|366]]
| [[367edo|367]]
| [[368edo|368]]
| [[369edo|369]]
|-
| [[370edo|370]]
| [[371edo|371]]
| [[372edo|372]]
| [[373edo|373]]
| [[374edo|374]]
| [[375edo|375]]
| [[376edo|376]]
| [[377edo|377]]
| [[378edo|378]]
| [[379edo|379]]
|-
| [[380edo|380]]
| [[381edo|381]]
| [[382edo|382]]
| [[383edo|383]]
| [[384edo|384]]
| [[385edo|385]]
| [[386edo|386]]
| [[387edo|387]]
| [[388edo|388]]
| [[389edo|389]]
|-
| [[390edo|390]]
| [[391edo|391]]
| [[392edo|392]]
| [[393edo|393]]
| [[394edo|394]]
| [[395edo|395]]
| [[396edo|396]]
| [[397edo|397]]
| [[398edo|398]]
| [[399edo|399]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 400…499
|-
| [[400edo|400]]
| [[401edo|401]]
| [[402edo|402]]
| [[403edo|403]]
| [[404edo|404]]
| [[405edo|405]]
| [[406edo|406]]
| [[407edo|407]]
| [[408edo|408]]
| [[409edo|409]]
|-
| [[410edo|410]]
| [[411edo|411]]
| [[412edo|412]]
| [[413edo|413]]
| [[414edo|414]]
| [[415edo|415]]
| [[416edo|416]]
| [[417edo|417]]
| [[418edo|418]]
| [[419edo|419]]
|-
| [[420edo|420]]
| [[421edo|421]]
| [[422edo|422]]
| [[423edo|423]]
| [[424edo|424]]
| [[425edo|425]]
| [[426edo|426]]
| [[427edo|427]]
| [[428edo|428]]
| [[429edo|429]]
|-
| [[430edo|430]]
| [[431edo|431]]
| [[432edo|432]]
| [[433edo|433]]
| [[434edo|434]]
| [[435edo|435]]
| [[436edo|436]]
| [[437edo|437]]
| [[438edo|438]]
| [[439edo|439]]
|-
| [[440edo|440]]
| [[441edo|441]]
| [[442edo|442]]
| [[443edo|443]]
| [[444edo|444]]
| [[445edo|445]]
| [[446edo|446]]
| [[447edo|447]]
| [[448edo|448]]
| [[449edo|449]]
|-
| [[450edo|450]]
| [[451edo|451]]
| [[452edo|452]]
| [[453edo|453]]
| [[454edo|454]]
| [[455edo|455]]
| [[456edo|456]]
| [[457edo|457]]
| [[458edo|458]]
| [[459edo|459]]
|-
| [[460edo|460]]
| [[461edo|461]]
| [[462edo|462]]
| [[463edo|463]]
| [[464edo|464]]
| [[465edo|465]]
| [[466edo|466]]
| [[467edo|467]]
| [[468edo|468]]
| [[469edo|469]]
|-
| [[470edo|470]]
| [[471edo|471]]
| [[472edo|472]]
| [[473edo|473]]
| [[474edo|474]]
| [[475edo|475]]
| [[476edo|476]]
| [[477edo|477]]
| [[478edo|478]]
| [[479edo|479]]
|-
| [[480edo|480]]
| [[481edo|481]]
| [[482edo|482]]
| [[483edo|483]]
| [[484edo|484]]
| [[485edo|485]]
| [[486edo|486]]
| [[487edo|487]]
| [[488edo|488]]
| [[489edo|489]]
|-
| [[490edo|490]]
| [[491edo|491]]
| [[492edo|492]]
| [[493edo|493]]
| [[494edo|494]]
| [[495edo|495]]
| [[496edo|496]]
| [[497edo|497]]
| [[498edo|498]]
| [[499edo|499]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 500…599
|-
| [[500edo|500]]
| [[501edo|501]]
| [[502edo|502]]
| [[503edo|503]]
| [[504edo|504]]
| [[505edo|505]]
| [[506edo|506]]
| [[507edo|507]]
| [[508edo|508]]
| [[509edo|509]]
|-
| [[510edo|510]]
| [[511edo|511]]
| [[512edo|512]]
| [[513edo|513]]
| [[514edo|514]]
| [[515edo|515]]
| [[516edo|516]]
| [[517edo|517]]
| [[518edo|518]]
| [[519edo|519]]
|-
| [[520edo|520]]
| [[521edo|521]]
| [[522edo|522]]
| [[523edo|523]]
| [[524edo|524]]
| [[525edo|525]]
| [[526edo|526]]
| [[527edo|527]]
| [[528edo|528]]
| [[529edo|529]]
|-
| [[530edo|530]]
| [[531edo|531]]
| [[532edo|532]]
| [[533edo|533]]
| [[534edo|534]]
| [[535edo|535]]
| [[536edo|536]]
| [[537edo|537]]
| [[538edo|538]]
| [[539edo|539]]
|-
| [[540edo|540]]
| [[541edo|541]]
| [[542edo|542]]
| [[543edo|543]]
| [[544edo|544]]
| [[545edo|545]]
| [[546edo|546]]
| [[547edo|547]]
| [[548edo|548]]
| [[549edo|549]]
|-
| [[550edo|550]]
| [[551edo|551]]
| [[552edo|552]]
| [[553edo|553]]
| [[554edo|554]]
| [[555edo|555]]
| [[556edo|556]]
| [[557edo|557]]
| [[558edo|558]]
| [[559edo|559]]
|-
| [[560edo|560]]
| [[561edo|561]]
| [[562edo|562]]
| [[563edo|563]]
| [[564edo|564]]
| [[565edo|565]]
| [[566edo|566]]
| [[567edo|567]]
| [[568edo|568]]
| [[569edo|569]]
|-
| [[570edo|570]]
| [[571edo|571]]
| [[572edo|572]]
| [[573edo|573]]
| [[574edo|574]]
| [[575edo|575]]
| [[576edo|576]]
| [[577edo|577]]
| [[578edo|578]]
| [[579edo|579]]
|-
| [[580edo|580]]
| [[581edo|581]]
| [[582edo|582]]
| [[583edo|583]]
| [[584edo|584]]
| [[585edo|585]]
| [[586edo|586]]
| [[587edo|587]]
| [[588edo|588]]
| [[589edo|589]]
|-
| [[590edo|590]]
| [[591edo|591]]
| [[592edo|592]]
| [[593edo|593]]
| [[594edo|594]]
| [[595edo|595]]
| [[596edo|596]]
| [[597edo|597]]
| [[598edo|598]]
| [[599edo|599]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 600…699
|-
| [[600edo|600]]
| [[601edo|601]]
| [[602edo|602]]
| [[603edo|603]]
| [[604edo|604]]
| [[605edo|605]]
| [[606edo|606]]
| [[607edo|607]]
| [[608edo|608]]
| [[609edo|609]]
|-
| [[610edo|610]]
| [[611edo|611]]
| [[612edo|612]]
| [[613edo|613]]
| [[614edo|614]]
| [[615edo|615]]
| [[616edo|616]]
| [[617edo|617]]
| [[618edo|618]]
| [[619edo|619]]
|-
| [[620edo|620]]
| [[621edo|621]]
| [[622edo|622]]
| [[623edo|623]]
| [[624edo|624]]
| [[625edo|625]]
| [[626edo|626]]
| [[627edo|627]]
| [[628edo|628]]
| [[629edo|629]]
|-
| [[630edo|630]]
| [[631edo|631]]
| [[632edo|632]]
| [[633edo|633]]
| [[634edo|634]]
| [[635edo|635]]
| [[636edo|636]]
| [[637edo|637]]
| [[638edo|638]]
| [[639edo|639]]
|-
| [[640edo|640]]
| [[641edo|641]]
| [[642edo|642]]
| [[643edo|643]]
| [[644edo|644]]
| [[645edo|645]]
| [[646edo|646]]
| [[647edo|647]]
| [[648edo|648]]
| [[649edo|649]]
|-
| [[650edo|650]]
| [[651edo|651]]
| [[652edo|652]]
| [[653edo|653]]
| [[654edo|654]]
| [[655edo|655]]
| [[656edo|656]]
| [[657edo|657]]
| [[658edo|658]]
| [[659edo|659]]
|-
| [[660edo|660]]
| [[661edo|661]]
| [[662edo|662]]
| [[663edo|663]]
| [[664edo|664]]
| [[665edo|665]]
| [[666edo|666]]
| [[667edo|667]]
| [[668edo|668]]
| [[669edo|669]]
|-
| [[670edo|670]]
| [[671edo|671]]
| [[672edo|672]]
| [[673edo|673]]
| [[674edo|674]]
| [[675edo|675]]
| [[676edo|676]]
| [[677edo|677]]
| [[678edo|678]]
| [[679edo|679]]
|-
| [[680edo|680]]
| [[681edo|681]]
| [[682edo|682]]
| [[683edo|683]]
| [[684edo|684]]
| [[685edo|685]]
| [[686edo|686]]
| [[687edo|687]]
| [[688edo|688]]
| [[689edo|689]]
|-
| [[690edo|690]]
| [[691edo|691]]
| [[692edo|692]]
| [[693edo|693]]
| [[694edo|694]]
| [[695edo|695]]
| [[696edo|696]]
| [[697edo|697]]
| [[698edo|698]]
| [[699edo|699]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 700…799
|-
| [[700edo|700]]
| [[701edo|701]]
| [[702edo|702]]
| [[703edo|703]]
| [[704edo|704]]
| [[705edo|705]]
| [[706edo|706]]
| [[707edo|707]]
| [[708edo|708]]
| [[709edo|709]]
|-
| [[710edo|710]]
| [[711edo|711]]
| [[712edo|712]]
| [[713edo|713]]
| [[714edo|714]]
| [[715edo|715]]
| [[716edo|716]]
| [[717edo|717]]
| [[718edo|718]]
| [[719edo|719]]
|-
| [[720edo|720]]
| [[721edo|721]]
| [[722edo|722]]
| [[723edo|723]]
| [[724edo|724]]
| [[725edo|725]]
| [[726edo|726]]
| [[727edo|727]]
| [[728edo|728]]
| [[729edo|729]]
|-
| [[730edo|730]]
| [[731edo|731]]
| [[732edo|732]]
| [[733edo|733]]
| [[734edo|734]]
| [[735edo|735]]
| [[736edo|736]]
| [[737edo|737]]
| [[738edo|738]]
| [[739edo|739]]
|-
| [[740edo|740]]
| [[741edo|741]]
| [[742edo|742]]
| [[743edo|743]]
| [[744edo|744]]
| [[745edo|745]]
| [[746edo|746]]
| [[747edo|747]]
| [[748edo|748]]
| [[749edo|749]]
|-
| [[750edo|750]]
| [[751edo|751]]
| [[752edo|752]]
| [[753edo|753]]
| [[754edo|754]]
| [[755edo|755]]
| [[756edo|756]]
| [[757edo|757]]
| [[758edo|758]]
| [[759edo|759]]
|-
| [[760edo|760]]
| [[761edo|761]]
| [[762edo|762]]
| [[763edo|763]]
| [[764edo|764]]
| [[765edo|765]]
| [[766edo|766]]
| [[767edo|767]]
| [[768edo|768]]
| [[769edo|769]]
|-
| [[770edo|770]]
| [[771edo|771]]
| [[772edo|772]]
| [[773edo|773]]
| [[774edo|774]]
| [[775edo|775]]
| [[776edo|776]]
| [[777edo|777]]
| [[778edo|778]]
| [[779edo|779]]
|-
| [[780edo|780]]
| [[781edo|781]]
| [[782edo|782]]
| [[783edo|783]]
| [[784edo|784]]
| [[785edo|785]]
| [[786edo|786]]
| [[787edo|787]]
| [[788edo|788]]
| [[789edo|789]]
|-
| [[790edo|790]]
| [[791edo|791]]
| [[792edo|792]]
| [[793edo|793]]
| [[794edo|794]]
| [[795edo|795]]
| [[796edo|796]]
| [[797edo|797]]
| [[798edo|798]]
| [[799edo|799]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 800…899
|-
| [[800edo|800]]
| [[801edo|801]]
| [[802edo|802]]
| [[803edo|803]]
| [[804edo|804]]
| [[805edo|805]]
| [[806edo|806]]
| [[807edo|807]]
| [[808edo|808]]
| [[809edo|809]]
|-
| [[810edo|810]]
| [[811edo|811]]
| [[812edo|812]]
| [[813edo|813]]
| [[814edo|814]]
| [[815edo|815]]
| [[816edo|816]]
| [[817edo|817]]
| [[818edo|818]]
| [[819edo|819]]
|-
| [[820edo|820]]
| [[821edo|821]]
| [[822edo|822]]
| [[823edo|823]]
| [[824edo|824]]
| [[825edo|825]]
| [[826edo|826]]
| [[827edo|827]]
| [[828edo|828]]
| [[829edo|829]]
|-
| [[830edo|830]]
| [[831edo|831]]
| [[832edo|832]]
| [[833edo|833]]
| [[834edo|834]]
| [[835edo|835]]
| [[836edo|836]]
| [[837edo|837]]
| [[838edo|838]]
| [[839edo|839]]
|-
| [[840edo|840]]
| [[841edo|841]]
| [[842edo|842]]
| [[843edo|843]]
| [[844edo|844]]
| [[845edo|845]]
| [[846edo|846]]
| [[847edo|847]]
| [[848edo|848]]
| [[849edo|849]]
|-
| [[850edo|850]]
| [[851edo|851]]
| [[852edo|852]]
| [[853edo|853]]
| [[854edo|854]]
| [[855edo|855]]
| [[856edo|856]]
| [[857edo|857]]
| [[858edo|858]]
| [[859edo|859]]
|-
| [[860edo|860]]
| [[861edo|861]]
| [[862edo|862]]
| [[863edo|863]]
| [[864edo|864]]
| [[865edo|865]]
| [[866edo|866]]
| [[867edo|867]]
| [[868edo|868]]
| [[869edo|869]]
|-
| [[870edo|870]]
| [[871edo|871]]
| [[872edo|872]]
| [[873edo|873]]
| [[874edo|874]]
| [[875edo|875]]
| [[876edo|876]]
| [[877edo|877]]
| [[878edo|878]]
| [[879edo|879]]
|-
| [[880edo|880]]
| [[881edo|881]]
| [[882edo|882]]
| [[883edo|883]]
| [[884edo|884]]
| [[885edo|885]]
| [[886edo|886]]
| [[887edo|887]]
| [[888edo|888]]
| [[889edo|889]]
|-
| [[890edo|890]]
| [[891edo|891]]
| [[892edo|892]]
| [[893edo|893]]
| [[894edo|894]]
| [[895edo|895]]
| [[896edo|896]]
| [[897edo|897]]
| [[898edo|898]]
| [[899edo|899]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 900…999
|-
| [[900edo|900]]
| [[901edo|901]]
| [[902edo|902]]
| [[903edo|903]]
| [[904edo|904]]
| [[905edo|905]]
| [[906edo|906]]
| [[907edo|907]]
| [[908edo|908]]
| [[909edo|909]]
|-
| [[910edo|910]]
| [[911edo|911]]
| [[912edo|912]]
| [[913edo|913]]
| [[914edo|914]]
| [[915edo|915]]
| [[916edo|916]]
| [[917edo|917]]
| [[918edo|918]]
| [[919edo|919]]
|-
| [[920edo|920]]
| [[921edo|921]]
| [[922edo|922]]
| [[923edo|923]]
| [[924edo|924]]
| [[925edo|925]]
| [[926edo|926]]
| [[927edo|927]]
| [[928edo|928]]
| [[929edo|929]]
|-
| [[930edo|930]]
| [[931edo|931]]
| [[932edo|932]]
| [[933edo|933]]
| [[934edo|934]]
| [[935edo|935]]
| [[936edo|936]]
| [[937edo|937]]
| [[938edo|938]]
| [[939edo|939]]
|-
| [[940edo|940]]
| [[941edo|941]]
| [[942edo|942]]
| [[943edo|943]]
| [[944edo|944]]
| [[945edo|945]]
| [[946edo|946]]
| [[947edo|947]]
| [[948edo|948]]
| [[949edo|949]]
|-
| [[950edo|950]]
| [[951edo|951]]
| [[952edo|952]]
| [[953edo|953]]
| [[954edo|954]]
| [[955edo|955]]
| [[956edo|956]]
| [[957edo|957]]
| [[958edo|958]]
| [[959edo|959]]
|-
| [[960edo|960]]
| [[961edo|961]]
| [[962edo|962]]
| [[963edo|963]]
| [[964edo|964]]
| [[965edo|965]]
| [[966edo|966]]
| [[967edo|967]]
| [[968edo|968]]
| [[969edo|969]]
|-
| [[970edo|970]]
| [[971edo|971]]
| [[972edo|972]]
| [[973edo|973]]
| [[974edo|974]]
| [[975edo|975]]
| [[976edo|976]]
| [[977edo|977]]
| [[978edo|978]]
| [[979edo|979]]
|-
| [[980edo|980]]
| [[981edo|981]]
| [[982edo|982]]
| [[983edo|983]]
| [[984edo|984]]
| [[985edo|985]]
| [[986edo|986]]
| [[987edo|987]]
| [[988edo|988]]
| [[989edo|989]]
|-
| [[990edo|990]]
| [[991edo|991]]
| [[992edo|992]]
| [[993edo|993]]
| [[994edo|994]]
| [[995edo|995]]
| [[996edo|996]]
| [[997edo|997]]
| [[998edo|998]]
| [[999edo|999]]
|}

=== 1000…1999 ===
{{EDOs
| 1000, 1001, 1012, 1015, 1019, 1051, 1053, 1059, 1063, 1065, 1080, 1092, 1106, 1125, 1131, 1147, 1152, 1166, 1171, 1178, 1193, 1200, 1210, 1224, 1230, 1236, 1240, 1244, 1260, 1272, 1277, 1289, 1308, 1312, 1323, 1330, 1337, 1342, 1361, 1376, 1381, 1395, 1407, 1419, 1429, 1440, 1448, 1489, 1506, 1517, 1520, 1525, 1536, 1547, 1553, 1554, 1559, 1578, 1583, 1590, 1600, 1609, 1612, 1619, 1637, 1641, 1643, 1650, 1665, 1672, 1700, 1730, 1759, 1776, 1778, 1783, 1789, 1793, 1794, 1802, 1803, 1817, 1848, 1861, 1879, 1880, 1889, 1911, 1920, 1944, 1955, 1957, 1983, 1984 }}

=== 2000…9999 ===
{{EDOs
| 2000, 2016, 2019, 2022, 2023, 2024, 2025, 2029, 2048, 2053, 2072, 2081, 2100, 2101, 2113, 2118, 2129, 2153, 2190, 2200, 2207, 2242, 2243, 2320, 2444, 2460, 2477, 2513, 2520, 2544, 2549, 2554, 2619, 2684, 2711, 2730, 2777, 2809, 2814, 2819, 2897, 2901, 2912, 2960, 2964, 3041, 3071, 3072, 3079, 3080, 3125, 3178, 3361, 3395, 3422, 3476, 3498, 3558, 3566, 3578, 3600, 3643, 3684, 3696, 3776, 3889, 3920, 4004, 4007, 4079, 4096, 4172, 4190, 4231, 4296, 4320, 4327, 4349, 4380, 4501, 4650, 4973, 5040, 5280, 5544, 5585, 5809, 5902, 5941, 6079, 6349, 6380, 6650, 6664, 6691, 7033, 7315, 7980, 8103, 8192, 8269, 8404, 8539, 8736, 8745, 9539
}}

=== 10000 and up ===
{{EDOs
| 10009, 10459, 10600, 10729, 11664, 12276, 12348, 12500, 13382, 14124, 14348, 14618, 14842, 15601, 15900, 16218, 16384, 16625, 16808, 17461, 18355, 20203, 20567, 25281, 25282, 28000, 28742, 30103, 30631, 31867, 31920, 32436, 33616, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 99693, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 571611, 762148, 1714833, 1905370, 2547047, 2667518, 2901533, 3159811, 4191814, 6000000, 11358058, 402653184, 5407372813
}}

== Non-integer EDO ==
A non-integer EDO can be defined as using a non-integer divisor to divide the octave. Typically, non-integer EDOs are understood as ''not'' containing the exact octave, so that they remain [[equal tuning]]s. If the exact octave is retained and if the generator resets itself at each period, then this results in a [[MOS scale]] with only 1 small step.

All fractional EDOs are integer equal divisions of another integer interval. For example, (25/2)edo is equivalent to 25ed4. In general:

<math>\displaystyle (p/q) \text{edo} = p \text{-ed} 2^q</math>

for integers ''p'' and ''q''. Many irrational EDOs cannot be converted to integer equal divisions of another integer interval, so they are things of their own.

Non-integer EDOs can be written in decimal form, such as 12.1edo. This is often meant to be approximate, used in the context of [[octave stretch]] of an integer EDO, rather than as a fractional EDO.

== Scale tree ==

The scale tree, or Stern-Brocot tree, provides a visual map of the world of EDOs, based on fifth size.

[[File:The_Scale_Tree.png|alt=The Scale Tree.png|800x1023px|The Scale Tree.png]]

The regular EDOs, up to 72edo:

[[File:Scale_Tree_close-up.png|alt=Scale Tree close-up.png|Scale Tree close-up.png]]

== Pergens ==
{{See also| Pergen #Pergens and EDOs }}

[[Pergen]]s provide a JI-agnostic way to name the rank-2 scales of an EDO. This table lists every possible period/generator combination for EDOs 5–24, and for each coprime combination, the simplest pergen that it can represent. Non-coprime combinations such as {{nowrap|P = 6\12}}, {{nowrap|G = 4\12}} are marked as "-".

{| class="wikitable"
|-
! EDO
! Period
! colspan="11" | Generator in EDO steps
|-
!
! in EDO steps
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
|-
! 5
! 5 = P8
| P4/2
| P5
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 6
! 6 = P8
| P4/2
| -
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/2
| P5
|
|
|
|
|
|
|
|
|
|
|-
! 7
! 7 = P8
| P4/3
| P5/2
| P5
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 8
! 8 = P8
| P4/3
| -
| P5
|
|
|
|
|
|
|
|
|-
! 4 = P8/2
| P5
| -
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 9
! 9 = P8
| P4/4
| P4/2
| -
| P5
|
|
|
|
|
|
|
|-
! 3 = P8/3
| P5
|
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 10
! 10 = P8
| P4/4
| -
| P5/2
| -
|
|
|
|
|
|
|
|-
! 5 = P8/2
| P5
| P4/2
|
|
|
|
|
|
|
|
|
|-
! 11
! 11 = P8
| P4/5
| P5/3
| P5/2
| P11/4
| P5
|
|
|
|
|
|
|-
! rowspan="4" | 12
! 12 = P8
| P4/5
| -
| -
| -
| P5
|
|
|
|
|
|
|-
! 6 = P8/2
| P5
| -
| -
|
|
|
|
|
|
|
|
|-
! 4 = P8/3
| P5
| -
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/4
| P5
|
|
|
|
|
|
|
|
|
|
|-
! 13b
! 13 = P8
| P4/6
| P4/3
| P4/2
| P12/5
| P12/4
| P5
|
|
|
|
|
|-
! rowspan="2" | 14
! 14 = P8
| P4/6
| -
| P4/2
| -
| P11/4
| -
|
|
|
|
|
|-
! 7 = P8/2
| P5
| P4/3
| P4/2
|
|
|
|
|
|
|
|
|-
! rowspan="3" | 15
! 15 = P8
| P4/6
| P4/3
| -
| P12/6
| -
| -
| P11/3
|
|
|
|
|-
! 5 = P8/3
| P5
| P4/2
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/5
| P4/3
|
|
|
|
|
|
|
|
|
|
|-
! rowspan="3" | 16
! 16 = P8
| P4/7
| -
| P5/3
| -
| P12/5
| -
| P5
|
|
|
|
|-
! 8 = P8/2
| P5
| -
| P5/3
| -
|
|
|
|
|
|
|
|-
! 4 = P8/4
| P5
| -
|
|
|
|
|
|
|
|
|
|-
! 17
! 17 = P8
| P4/7
| P5/5
| P11/8
| P11/6
| P5/2
| P11/4
| P5
| P11/3
|
|
|
|-
! rowspan="4" | 18b
! 18 = P8
| P4/8
| -
| -
| -
| P5/2
| -
| P12/4
| -
|
|
|
|-
! 9 = P8/2
| P5
| P4/4
| -
| P4/2
|
|
|
|
|
|
|
|-
! 6 = P8/3
| P5/2
| -
| -
|
|
|
|
|
|
|
|
|-
! 3 = P8/6
| P5
|
|
|
|
|
|
|
|
|
|
|-
! 19
! 19 = P8
| P4/8
| P4/4
| P11/9
| P4/2
| P12/6
| P12/5
| ccP5/7
| P5
| P11/3
|
|
|-
! rowspan="4" | 20
! 20 = P8
| P4/8
| -
| P5/4
| -
| -
| -
| P11/4
| -
| c3P5/8
|
|
|-
! 10 = P8/2
| M2/4
| -
| P5/4
| -
| -
|
|
|
|
|
|
|-
! 5 = P8/4
| P4/2
| P5
|
|
|
|
|
|
|
|
|
|-
! 4 = P8/5
| P5/4
| -
|
|
|
|
|
|
|
|
|
|-
! rowspan="3" | 21
! 21 = P8
| P4/9
| P5/6
| -
| P5/3
| P11/6
| -
| -
| c3P4/9
| -
| P11/3
|
|-
! 7 = P8/3
| P5/2
| P5
| P4/3
|
|
|
|
|
|
|
|
|-
! 3 = P8/7
| P5/3
|
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 22
! 22 = P8
| P4/9
| -
| P4/3
| -
| P12/7
| -
| P12/5
| -
| P5
| -
|
|-
! 11 = P8/2
| M2/4
| P5
| P4/3
| P12/5
| P12/7
|
|
|
|
|
|
|-
! 23
! 23 = P8
| P4/10
| P4/5
| P11/11
| P12/9
| P4/2
| P12/6
| ccP4/8
| ccP4/7
| P12/4
| P5
| P11/3
|-
! rowspan="6" | 24
! 24 = P8
| P4/10
| -
| -
| -
| P4/2
| -
| P5/2
| -
| -
| -
| c4P5/10
|-
! 12 = P8/2
| M2/4
| -
| -
| -
| P4/2
|
|
|
|
|
|
|-
! 8 = P8/3
| P5/2
| -
| P4/2
|
|
|
|
|
|
|
|
|-
! 6 = P8/4
| P4/2
| -
| -
|
|
|
|
|
|
|
|
|-
! 4 = P8/6
| P4/2
| -
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/8
| P5
|
|
|
|
|
|
|
|
|
|
|-
!
!
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
|}

== See also ==
Related topics
* [[Equal-step tuning]]
* [[Highly composite equal division]]
* [[List of rank one temperaments by step size]]
* [[Prime equal division]]

Technical data
* [[Absolute errors of small EDOs]]
* [[Consistency limits of small EDOs]]
* [[Distinct EDO Scales]]
* [[Minimal consistent EDOs]]
* [[Monotonicity levels of small EDOs]]
* [[Relative errors of small EDOs]]

Opinions
* [[Collection of EDO impressions]]

Other
* [[:Category:Equal divisions of the octave]]

== External links ==
* [[Ivor Darreg]], [https://www.webcitation.org/5xZz8RtQB Teen Tunes]

== Notes ==
<references />

[[Category:Equal divisions of the octave| ]] 
[[Category:Acronyms]]
[[Category:Lists of scales]]

1312edo

2025-08-02T22:38:54Z

Eliora: /* Subsets and supersets */

{{Infobox ET}}
{{ED intro}}

1312edo is consistent in the [[7-odd-limit]] and is a satisfactory 2.9.13.23 subgroup tuning, but otherwise it represents low harmonics poorly. It also has a very strong approximation to [[399/256]].

Nonetheless, 1312edo provides the [[optimal patent val]] for the [[bezique]] temperament in the 7, 11, and 13-limit, despite being inconsistent.

=== Odd harmonics ===
{{harmonics in equal|1312}}

=== Subsets and supersets ===

1312edo notably contains [[32edo]] and [[41edo]].

[[Category: Bezique]]

1312edo

2025-08-02T22:38:27Z

Eliora: Created page with "{{Infobox ET}} {{ED intro}} 1312edo is consistent in the 7-odd-limit and is a satisfactory 2.9.13.23 subgroup tuning, but otherwise it represents low harmonics poorly. It also has a very strong approximation to 399/256. Nonetheless, 1312edo provides the optimal patent val for the bezique temperament in the 7, 11, and 13-limit, despite being inconsistent. === Odd harmonics === {{harmonics in equal|1312}} === Subsets and supersets === 1312edo notably c..."

User:Eliora/2592edo

2025-08-02T22:24:27Z

Eliora: Created page with "{{Infobox ET}} {{ED intro}} 2592edo is consistent in the 5-odd-limit, though its approximation of simple harmonics is rather poor. Nonetheless, there are strong direct approximations to 15/14, 10/9, 13/9, 13/10, 15/13. Furthermore, in the 7-limit, it provides the optimal patent val for the 32nd-octave windrose temperament, even if inconsisent. 2592edo overall is best considered for its subsets due to many divisors, see below. === Odd harmonics..."

{{Infobox ET}}
{{ED intro}}

2592edo is consistent in the 5-odd-limit, though its approximation of simple harmonics is rather poor. Nonetheless, there are strong direct approximations to [[15/14]], [[10/9]], [[13/9]], [[13/10]], [[15/13]].

Furthermore, in the 7-limit, it provides the optimal patent val for the 32nd-octave [[windrose]] temperament, even if inconsisent. 2592edo overall is best considered for its subsets due to many divisors, see below.

=== Odd harmonics ===
{{harmonics in equal|2592}}

=== Subsets ===
Since 2592 factors as {{Factorization|2592}}, 2592edo has subset edos {{EDOs|1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 72, 81, 96, 108, 144, 162, 216, 288, 324, 432, 648, 864, 1296}}. Its abundancy index is around 1.94.

Ragismic microtemperaments

2025-06-19T19:24:07Z

Eliora: /* Crazy */ "obvious tuning" is subjective + ironically unclear

{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the ragisma, [[4375/4374]] ({{monzo| -1 -7 4 1 }}). The ragisma is the smallest [[7-limit]] [[superparticular ratio]].

Since {{nowrap|(10/9)4 {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)2 }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.

Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
* ''[[Modus]]'' (+64/63) → [[Tetracot family #Modus|Tetracot family]]
* ''[[Flattone]]'' (+81/80) → [[Meantone family #Flattone|Meantone family]]
* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]
* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]]
* [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]
* [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]]
* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]
* ''[[Zarvo]]'' (+33075/32768) → [[Gravity family #Zarvo|Gravity family]]
* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortonic family #Mitonic|Minortonic family]]
* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnuzmic family #Septimal vishnu|Vishnuzmic family]]
* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]
* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]]
* ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]]
* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]]
* ''[[Chlorine]]'' (+{{monzo| -52 -17 34}}) → [[17th-octave temperaments #Chlorine|17th-octave temperaments]]
* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]
* ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments#Dzelic|37th-octave temperaments]]

== Supermajor ==
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (215)/3, 46 give (219)/5, and 75 give (230)/7. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80-note mos is presumably the place to start, and if that is not enough notes for you, there is always the 171-note mos.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 52734375/52706752

{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082

{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}

[[Badness]]: 0.010836

=== Semisupermajor ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 35156250/35153041

Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082

{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}

Badness: 0.012773

== Enneadecal ==
Enneadecal temperament tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out [[703125/702464]], the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)1/3. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.

''For the 5-limit temperament, see [[19th-octave temperaments#(5-limit) enneadecal]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 703125/702464

{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}

: mapping generators: ~28/27, ~3

[[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)

{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}

[[Badness]]: 0.010954

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 16384/16335

Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}

Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)

{{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }}

Badness: 0.043734

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 2205/2197

Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}

Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)

{{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }}

Badness: 0.033545

=== Hemienneadecal ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 234375/234256

Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}

: mapping generators: ~55/54, ~3

Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)

{{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}

Badness: 0.009985

==== Hemienneadecalis ====
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256

Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}

Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)

{{Optimal ET sequence|legend=1| 152f, 342f, 494 }}

Badness: 0.020782

==== Hemienneadec ====
Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213

Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}

Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444)

{{Optimal ET sequence|legend=1| 152, 342, 494, 1330, 1824, 2318d }}

Badness: 0.030391

==== Semihemienneadecal ====
Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078

Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}

: mapping generators: ~55/54 = 1\38, ~55/54, ~429/250

Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)

{{Optimal ET sequence|legend=1| 190, 304d, 494, 684, 1178, 2850, 4028ce }}

Badness: 0.014694

=== Kalium ===
Named after the 19th element, potassium, and after an archaic variant of the element's name to resolve a name conflict. [[19/16]] can be used as a generator. Since it is enfactored in the 17-limit and lower, it makes no sense to name it for the lower subgroups.

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344

Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}

Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244

{{Optimal ET sequence|legend=1| 855, 988, 1843 }}

== Semidimi ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].''

The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 3955078125/3954653486

{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270

{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}

[[Badness]]: 0.015075

== Brahmagupta ==
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.

Early in the design of the [[Sagittal]] notation system, Secor and Keenan found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4 ¢ many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of Brahmagupta temperament that has pure octaves and pure fifths, which can also be described as a 17-limit extension having 1/7th octave period (171.4286 ¢) and 1/21st apotome generator (5.4136 ¢).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 70368744177664/70338939985125

{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}

: mapping generators: ~1157625/1048576, ~27/20

[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716

{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }}

[[Badness]]: 0.029122

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 4000/3993, 4375/4374, 131072/130977

Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}

Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704

{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }}

Badness: 0.052190

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374

Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}

Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.706

{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771eef }}

Badness: 0.023132

== Abigail ==
Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930]: "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things."</ref>

''For the 5-limit temperament, see [[Very high accuracy temperaments#Abigail]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 2147483648/2144153025

{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }}

: mapping generators: ~46305/32768, ~27/20

[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899

{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}

[[Badness]]: 0.037000

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 131072/130977

Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901

{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }}

Badness: 0.012860

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095

Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903

{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }}

Badness: 0.008856

== Gamera ==
''For the 5-limit temperament, see [[High badness temperaments#Gamera]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 589824/588245

{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }}

: mapping generators: ~2, ~8/7

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336

{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}

[[Badness]]: 0.037648

=== Hemigamera ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 589824/588245

Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}

: mapping generators: ~99/70, ~8/7

Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3370

{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646, 1068d }}

Badness: 0.040955

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024

Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373

{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646f, 1068df }}

Badness: 0.020416

=== Semigamera ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 14641/14580, 15488/15435

Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }}

: mapping generators: ~2, ~77/72

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1642

{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }}

Badness: 0.078

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580

Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }}

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1628

{{Optimal ET sequence|legend=1| 73f, 125f, 198, 323, 521 }}

Badness: 0.044

== Crazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''

Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament. [[1106edo]] is an strong tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}

{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}

: mapping generators: ~332150625/234881024, ~1125/1024

[[Optimal tuning]]s:
* [[CTE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7475
* [[error map]]: {{val| 0.0000 +0.0253 -0.0514 -0.0133 }}
* [[CWE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7474
* error map: {{val| 0.0000 +0.0244 -0.0508 -0.0218 }}

{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}

[[Badness]] (Smith): 0.0394

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 2791309312/2790703125

Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}

Optimal tunings:
* CTE: ~99/70 = 162.7485, ~1125/1024 = 162.7485
* CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481

{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}

Badness (Smith): 0.0170

== Orga ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 54975581388800/54936068900769

{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }}

: mapping generators: ~7411887/5242880, ~1310720/1058841

[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104

{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}

[[Badness]]: 0.040236

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 5767168/5764801

Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103

{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836, 1106 }}

Badness: 0.016188

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360

Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103

{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836f, 1106f }}

Badness: 0.021762

== Seniority ==
{{See also| Very high accuracy temperaments #Senior }}

Aside from the ragisma, the seniority temperament (26 & 145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ({{monzo| -17 62 -35 }}, quadla-sepquingu) is tempered out.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 201768035/201326592

{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3087/2560 = 322.804

{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}

[[Badness]]: 0.044877

=== Senator ===
The senator temperament (26 & 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It can be extended to the 13- and 17-limit immediately, by adding 364/363 and 595/594 to the comma list in this order.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4374, 65536/65219

Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }}

Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793

{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316e, 487ee }}

Badness: 0.092238

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 2200/2197, 4375/4374

Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }}

Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793

{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}

Badness: 0.044662

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197

Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }}

Optimal tuning (POTE): ~77/64 = 322.793

{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}

Badness: 0.026562

== Monzismic ==
: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Monzismic]].

The monzismic temperament (53 & 612) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}

{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207

{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}

[[Badness]]: 0.046569

=== Monzism ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 41503/41472, 184549376/184528125

Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}

Optimal tuning (POTE): ~231/200 = 249.0193

{{Optimal ET sequence|legend=1| 53, 559, 612 }}

Badness: 0.057083

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625

Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}

Optimal tuning (POTE): ~231/200 = 249.0199

{{Optimal ET sequence|legend=1| 53, 559, 612 }}

Badness: 0.053780

== Semidimfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].''

The semidimfourth temperament is featured by a semi-diminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, 235298/234375.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 235298/234375

[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456

{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}

[[Badness]]: 0.055249

=== Neusec ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 235298/234375

Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547

{{Optimal ET sequence|legend=1| 8d, 190, 388 }}

Badness: 0.059127

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374

Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545

{{Optimal ET sequence|legend=1| 8d, 190, 198, 388 }}

Badness: 0.030941

== Acrokleismic ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 2202927104/2197265625

{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }}

: mapping generators: ~2, ~6/5

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557

{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}

[[Badness]]: 0.056184

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 41503/41472, 172032/171875

Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558

{{Optimal ET sequence|legend=1| 19, 251, 270, 829, 1099, 1369, 1639 }}

Badness: 0.036878

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976

Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557

{{Optimal ET sequence|legend=1| 19, 251, 270 }}

Badness: 0.026818

=== Counteracro ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 117649/117612

Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.553

{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}

Badness: 0.042572

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374

Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.554

{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}

Badness: 0.026028

== Quasithird ==
The quasithird temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}.

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| 55 -64 20 }}

{{Mapping|legend=1| 4 0 -11 | 0 5 16 }}

: mapping generators: ~51200000/43046721, ~1594323/1280000

[[Optimal tuning]] ([[POTE]]): ~51200000/43046721, ~1594323/1280000 = 380.395

{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }}

[[Badness]]: 0.099519

=== 7-limit ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}

{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}

[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 1\4, ~5103/4096 = 380.388

{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}

[[Badness]]: 0.061813

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 4296700485/4294967296

Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}

Optimal tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387)

{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448 }}

Badness: 0.021125

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374

Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}

Optimal tuning (POTE): ~81/65 = 380.385 (or ~22/21 = 80.385)

{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}

Badness: 0.029501

== Deca ==
: ''For 5-limit version of this temperament, see [[10th-octave temperaments #Neon]].''

Deca temperament has a period of 1/10 octave and tempers out the [[linus comma]], {{monzo| 11 -10 -10 10 }}, neon comma {{monzo| 21 60 -50 }} and {{monzo| 12 -3 -14 9 }} = 165288374272/164794921875 (satritrizo-asepbigu).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 165288374272/164794921875

{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}

: mapping generators: ~15/14, ~6/5

[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577

{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}

[[Badness]]: 0.080637

Badness (Sintel): 2.041

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 391314/390625

Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}

Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.582

{{Optimal ET sequence|legend=1| 80, 190, 270, 1000, 1270, 1540e, 1810e }}

Badness: 0.024329

Badness (Sintel): 0.804

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374

Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}

Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.602 (~40/39 = 44.398)

{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}

Badness: 0.016810

Badness (Sintel): 0.695

=== no-17's 19-limit ===
Subgroup: 2.3.5.7.11.13.19

Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374, 1521/1520

Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}

Optimal tuning (CTE): ~15/14 = 1\10, ~6/5 = 315.581 (~39/38 = 44.419)

{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}

Badness (Sintel): 0.556

== Keenanose ==
Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}

{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}

: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 21 3 1 -10 }} = 4.4465

{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}

[[Badness]]: 0.0858

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 117649/117612, 67110351/67108864

Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}

Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4465

{{Optimal ET sequence|legend=1| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}

Badness: 0.0308

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612

Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}

Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4466

{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 4048 }}

Badness: 0.0213

== Aluminium ==
Aluminium is named after the 13th element, and tempers out the {{monzo| 92 -39 -13 }} comma which sets [[135/128]] interval to be equal to 1/13th of the octave.

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| 92 -39 -13 }}

[[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}

: mapping generators: ~135/128, ~3

[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897

{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}

[[Badness]]: 0.123

=== 7-limit ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}

[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}

[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 702.0024

{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}

[[Badness]]: 0.126

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 234375/234256, 2097152/2096325

Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}

Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042

{{Optimal ET sequence|legend=1| 494, 1053, 1547, 3588e, 5135e }}

Badness: 0.0421

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078

Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}

Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099

{{Optimal ET sequence|legend=1| 494, 1547, 2041, 4576def }}

Badness: 0.0286

== Countritonic ==
: ''For the 5-limit version of this temperament, see [[Schismic–Mercator equivalence continuum #Countritonic]].''

Countritonic (''co-un-tritonic'') can be described as the 53 & 422 temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 68719476736/68356598625

{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }}

: mapping generators: ~2, ~45927/32768

[[Optimal tuning]] (CTE): ~2 = 1\1, ~45927/32768 = 588.6216

{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}

[[Badness]]: 0.133

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 2621440/2614689

Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 }}

Optimal tuning (CTE): ~2 = 1\1, ~539/384 = 588.6258

{{Optimal ET sequence|legend=1| 53, 316e, 369, 422, 791e, 1213cde }}

Badness: 0.0707

=== 13-limit ===
Subgroup: 2.3.5.7.11

Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625

Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 -74 }}

Optimal tuning (CTE): ~2 = 1\1, ~128/91 = 588.6277

{{Optimal ET sequence|legend=1| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}

Badness: 0.0366

== Quatracot ==
{{See also| Stratosphere }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}

{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }}

: mapping generators: ~2278125/1605632, ~448/405

[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 1\2, ~448/405 = 176.805

{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}

[[Badness]]: 0.175982

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 1265625/1261568

Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 176.806

{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c }}

Badness: 0.041043

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1575/1573, 2200/2197

Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~195/176 = 176.804

{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1690bcc, 2328bccde }}

Badness: 0.022643

== Moulin ==
Moulin has a generator of 22/13, and it is named after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". It can be described as the 494 & 1619 temperament.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}

{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }}

: mapping generators: ~2, ~6422528/3796875

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6422528/3796875 = 910.9323

{{Optimal ET sequence|legend=1| 494, 1125, 1619 }}

[[Badness]]: 0.234

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 759375/758912, 100663296/100656875

Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }}

Optimal tuning (CTE): ~2 = 1\1, ~1024/605 = 910.9323

{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}

Badness: 0.0678

=== 13-limit ===
Since 11/8 is within 23 generators, the 25 tone MOS (4L 21s) of this temperament contains the 8:11:13 triad.

Subgroup: 2.3.5.7.11.13

Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078

Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }}

Optimal tuning (CTE): ~2 = 1\1, ~22/13 = 910.9323

{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}

Badness: 0.0271

== Palladium ==
: ''For the 5-limit version of this temperament, see [[46th-octave temperaments]]''.

The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minortones (10/9) fall short of seven octaves. This temperament can be described as 46 & 414 temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}

{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}

: mapping generators: ~83349/81920, ~3

[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 1\46, ~3/2 = 701.6074

{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }}

[[Badness]]: 0.308505

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 134775333/134217728

Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}

Optimal tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951

{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de }}

Badness: 0.073783

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364

Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}

Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6419

{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334de }}

Badness: 0.040751

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224

Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}

Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425

{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334deg }}

Badness: 0.022441

== Oviminor ==
{{See also| Syntonic–kleismic equivalence continuum }}

Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}

{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }}

: mapping generators: ~2, ~6/5

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 315.7501

{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}

[[Badness]]: 0.582

== Octoid ==
''For the 5-limit temperament, see [[8th-octave temperaments#Octoid (5-limit)]].''

The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 16875/16807

{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}

: mapping generators: ~49/45, ~7/5

[[Optimal tuning]] ([[POTE]]): ~49/45 = 1\8, ~7/5 = 583.940

[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}

[[Badness]]: 0.042670

Scales: [[octoid72]], [[octoid80]]

=== 11-limit ===
The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimaxing the damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, if one wants to use 80edo as the tuning, one must use octopus — not octoid — as 80edo doesn't temper 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 4000/3993

Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}

Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.962

Tuning ranges:
* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

{{Optimal ET sequence|legend=1| 72, 152, 224 }}

Badness: 0.014097

Scales: [[octoid72]], [[octoid80]]

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 1375/1372

Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}

Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.905

{{Optimal ET sequence|legend=1| 72, 152f, 224 }}

Badness: 0.015274

Scales: [[octoid72]], [[octoid80]]

; Music
* ''Dreyfus'' (archived 2010) by [[Gene Ward Smith]] – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – octoid[72] in 224edo tuning

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 715/714, 729/728

Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}

Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.842

{{Optimal ET sequence|legend=1| 72, 152fg, 224, 296, 520g }}

Badness: 0.014304

Scales: [[octoid72]], [[octoid80]]

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714

Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}

Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.932

{{Optimal ET sequence|legend=1| 72, 152fg, 224 }}

Badness: 0.016036

Scales: [[octoid72]], [[octoid80]]

==== Octopus ====
A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4 = 300{{cent}}.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 540/539

Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}

Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.892

{{Optimal ET sequence|legend=1| 72, 152, 224f }}

Badness: 0.021679

Scales: [[octoid72]], [[octoid80]]

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 289/288, 325/324, 540/539

Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}

Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.811

{{Optimal ET sequence|legend=1| 72, 152, 224fg, 296ffg }}

Badness: 0.015614

Scales: [[Octoid72]], [[Octoid80]]

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399

Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}

Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 584.064

{{Optimal ET sequence|legend=1| 72, 152, 224fg, 376ffgh }}

Badness: 0.016321

Scales: [[Octoid72]], [[Octoid80]]

==== Hexadecoid ====
{{ See also | 16th-octave temperaments }}

Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224

Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}

: mapping generators: ~448/429, ~7/5

Optimal tuning (POTE): ~448/429 = 1\16, ~13/8 = 841.015

{{Optimal ET sequence|legend=1| 80, 144, 224 }}

Badness: 0.030818

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224

Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}

Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932

{{Optimal ET sequence|legend=1| 80, 144, 224, 528dg }}

Badness: 0.028611

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444

Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }}

Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896

{{Optimal ET sequence|legend=1| 80, 144, 224, 304dh, 528dghh }}

Badness: 0.023731

== Parakleismic ==
{{Main| Parakleismic }}

In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, and 11-limit adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.

[[Subgroup]]: 2.3.5

[[Comma list]]: 1224440064/1220703125

{{Mapping|legend=1| 1 5 6 | 0 -13 -14 }}

: mapping generators: ~2, ~6/5

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.240

{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}

[[Badness]]: 0.043279

=== 7-limit ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 4375/4374

{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.181

{{Optimal ET sequence|legend=1| 19, 80, 99, 217, 316, 415 }}

[[Badness]]: 0.027431

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 3136/3125, 4375/4374

Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.251

{{Optimal ET sequence|legend=1| 19, 99, 118 }}

Badness: 0.049711

=== Paralytic ===
The ''paralytic'' temperament (118&217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 4375/4374

Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.220

{{Optimal ET sequence|legend=1| 19e, 99e, 118, 217, 335, 552d, 887dd }}

Badness: 0.036027

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374

Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.214

{{Optimal ET sequence|legend=1| 99e, 118, 217, 552d, 769de }}

Badness: 0.044710

==== Paraklein ====
The ''paraklein'' temperament (19e & 118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 625/624, 729/728

Mapping: {{mapping| 1 5 6 12 25 15 | 0 -13 -14 -35 -82 -43 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.225

{{Optimal ET sequence|legend=1| 19e, 99ef, 118, 217ff, 335ff }}

Badness: 0.037618

=== Parkleismic ===
Subgroup: 2.3.5.7.11

Comma list: 176/175, 1375/1372, 2200/2187

Mapping: {{mapping| 1 5 6 12 20 | 0 -13 -14 -35 -63 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.060

{{Optimal ET sequence|legend=1| 19e, 80, 179, 259cd }}

Badness: 0.055884

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 1375/1372

Mapping: {{mapping| 1 5 6 12 20 10 | 0 -13 -14 -35 -63 -24 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.075

{{Optimal ET sequence|legend=1| 19e, 80, 179 }}

Badness: 0.036559

=== Paradigmic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 3136/3125

Mapping: {{mapping| 1 5 6 12 -1 | 0 -13 -14 -35 17 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.096

{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}

Badness: 0.041720

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 540/539, 832/825

Mapping: {{mapping| 1 5 6 12 -1 10 | 0 -13 -14 -35 17 -24 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.080

{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}

Badness: 0.035781

=== Semiparakleismic ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 4375/4374

Mapping: {{mapping| 2 10 12 24 19 | 0 -13 -14 -35 -23 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.181

{{Optimal ET sequence|legend=1| 80, 118, 198, 316, 514c, 830c }}

Badness: 0.034208

==== Semiparamint ====
This extension was named ''semiparakleismic'' in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374

Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.156

{{Optimal ET sequence|legend=1| 80, 118, 198 }}

Badness: 0.033775

==== Semiparawolf ====
This extension was named ''gentsemiparakleismic'' in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 3136/3125

Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }}

Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 315.184

{{Optimal ET sequence|legend=1| 80, 118f, 198f }}

Badness: 0.040467

== Counterkleismic ==
{{See also| High badness temperaments #Counterhanson}}

In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as 19 & 224 temperament (''counterkleismic'', named by analogy to [[catakleismic]] and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 158203125/157351936

{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }}

: mapping generators: ~2, ~5/3

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.060

{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }}

[[Badness]]: 0.090553

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 2097152/2096325

Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.071

{{Optimal ET sequence|legend=1| 19, 205, 224 }}

Badness: 0.070952

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 10985/10976

Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.070

{{Optimal ET sequence|legend=1| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}

Badness: 0.033874

=== Counterlytic ===
Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4375/4374, 496125/495616

Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065

{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}

Badness: 0.065400

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1375/1372, 10985/10976

Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }}

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065

{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}

Badness: 0.029782

== Quincy ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 823543/819200

{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1715 = 16.613

{{Optimal ET sequence|legend=1| 72, 217, 289 }}

[[Badness]]: 0.079657

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 4000/3993, 4375/4374

Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}

Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.613

{{Optimal ET sequence|legend=1| 72, 217, 289 }}

Badness: 0.030875

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675, 4375/4374

Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}

Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602

{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}

Badness: 0.023862

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155

Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}

Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602

{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}

Badness: 0.014741

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675

Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}

Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.594

{{Optimal ET sequence|legend=1| 72, 145, 217 }}

Badness: 0.015197

== Sfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 64827/64000

{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 26.287

{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}

[[Badness]]: 0.123291

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 4375/4374

Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.286

{{Optimal ET sequence|legend=1| 45e, 46, 91e, 137de }}

Badness: 0.054098

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 325/324, 441/440

Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.310

{{Optimal ET sequence|legend=1| 45ef, 46, 91ef, 137def }}

Badness: 0.033067

=== Sfour ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 2401/2376, 4375/4374

Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.246

{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}

Badness: 0.076567

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 364/363, 385/384, 4375/4374

Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.239

{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}

Badness: 0.051893

== Trideci ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tridecatonic]].''

The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from "tridecim" (Latin for "[[wikipedia:13|thirteen]]").

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 83349/81920

{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}

[[Optimal tuning]] ([[POTE]]): ~256/245 = 1\13, ~3/2 = 699.1410

{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }}

[[Badness]]: 0.184585

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 245/242, 385/384, 4375/4374

Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}

Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.6179

{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdde }}

Badness: 0.084590

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 245/242, 325/324, 385/384

Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}

Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.2969

{{Optimal ET sequence|legend=1| 26, 65f, 91f, 156dff }}

Badness: 0.052366

== Counterorson ==
Counterorson tempers out the {{monzo| 147 -103 7 }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]].

Subgroup: 2.3.5.7

Comma list: 4375/4374, {{monzo| 154 -54 -21 -7 }}

Mapping: {{mapping| 1 0 -21 85 | 0 7 103 -363 }}

Optimal tuning (CTE): ~2 = 1\1, ~{{monzo| 66 -23 -9 -3 }} = 271.7113

{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}

Badness: 0.312806

== Notes ==

[[Category:Temperament collections]]
[[Category:Ragismic microtemperaments| ]] 
[[Category:Ragismic| ]] 
[[Category:Rank 2]]
[[Category:Microtemperaments]]
[[Category:Abigail]]
[[Category:Deca]]
[[Category:Enneadecal]]
[[Category:Ennealimmal]]
[[Category:Gamera]]
[[Category:Mitonic]]
[[Category:Octoid]]
[[Category:Parakleismic]]
[[Category:Quincy]]
[[Category:Supermajor]]

EDO

2025-06-14T15:04:51Z

Eliora: /* 1000…1999 */

{{Todo|discuss title}}
{{interwiki
| de = EDO
| en = EDO
| es = EDOs
| ja = オクターブ平均律
| ko = EDO (Korean)
| ro = DEO
}}
An '''equal division of the octave''' ('''EDO''', ''EE-dee-oh''; '''edo''', ''EE-doh'') is a [[Tuning system|tuning]] obtained by dividing the [[octave]] in a certain number of [[Equal-step tuning|equal steps]]. This means that the [[interval]] between any two consecutive pitches is identical.

A tuning with ''n'' equal divisions of the octave is usually called "''n''-edo" ("''n''-EDO"). For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO).

An equal (pitch) division of the octave is the most common type of [[EPD|equal (pitch) division]], which is a kind of [[equal-step tuning]]. Therefore, it is also a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.

== History ==
Tuning theorists first used the term "equal temperament" for edos designed to approximate [[Low-complexity JI|low-complexity just intervals]]. The same term is still used today for all rank-1 [[Regular temperament|temperaments]]. For example, [[15edo]] can be referred to as 15-tone equal temperament (15-TET, 15-tET, 15tet, etc.), or more simply 15 equal temperament (15-ET, 15et, etc.).

The acronym "EDO" was coined by [[Daniel Anthony Stearns]] in 1999, originally standing for "equidistant divisions of the octave"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_65#65 Yahoo! Tuning Group | ''Where F + f = O'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_117#117 Yahoo! Tuning Group | ''f + F and WFS/MOS'']</ref>. More recently, the {{w|anacronym}} "edo", spelled in lowercase and pronounced as a regular word, has also become common.

With the development of [[Edonoi|equal divisions of non-octave intervals (edonoi)]], some people started writing "ed2" ("ED2"), especially when naming a specific tuning.

== Calculating the step size ==
To find the step size of ''n''-edo in terms of [[cent]]s, divide 1200 by ''n''. The size ''s'' of ''k'' steps of ''n''-edo (''k''\''n'') is

<math>\displaystyle s = 1200 \cdot k/n</math>

To find the step size of ''n''-edo in terms of [[frequency ratio]], take the ''n''-th root of 2. For example, the step of 12edo is 21/12 (≈ 1.059). So the ratio ''c'' of the ''k'' steps of ''n''-edo is

<math>\displaystyle c = 2^{k/n}</math>

In particular, when ''k'' is 0, ''c'' is simply 1, because any number to the 0th power is 1. And when {{nowrap|''k'' {{=}} ''n''}}, ''c'' is simply 2, because any number to the 1st power is itself.

== Properties ==
EDO scales are straightforward to work with due to their uniform step size.
Some musicians find the consistency bland, while others appreciate the stable foundation it provides for composition.
The only property shared by all of them is the equality of their step-sizes; otherwise, their individual properties are as different as can be.
Lower-numbered EDOs, especially 5 to 24, possess very strong and unique "characters", which some composers find inspiring.

== Practical advantages ==
=== Fretted instruments ===
If you are a [[guitar]]ist (or a player of some other fretted string instrument, like a bass guitar, Appalachian dulcimer, [[ukulele]], banjo, mandolin, sitar, saz, pipa, or zhong ruan), an EDO will provide you with the simplest possible fretboard layout, as all of the frets will go straight across the fretboard, regardless of how you want to tune the open strings.
Fret crowding can become an issue with smaller divisions, especially high up the neck.
For these cases, [[ed4|equal divisions of the double octave]] or higher multiples offer a compromise solution.

=== Free modulation ===
EDOs allow for modulation to every single key in the tuning, without any alteration in harmonic properties, thus making transposition totally seamless.
This also makes them somewhat easier to learn, as you do not have to memorize the harmonic and melodic variations that appear in various keys (which you would have to learn in JI, an unequal regular temperament, or a well-temperament, especially with smaller numbers of tones).
For those accustomed to the "equality" of 12-TET, the equality of the alternative EDOs can be reassuringly familiar.

== Approaches to exploring EDOs ==
If you are interested in exploring the unique merits and challenges of each EDO, irrespective of any desire to approximate just intonation (or any other a priori musical goal), starting at the bottom and working your way up can be a most illuminating exercise.

If you're a classically trained musician and you'd like to start with some EDOs that have some relationship to common-practice tonal music, starting with reasonably-low EDOs that give a good approximation to the perfect fifth ([[3/2]]) can be rewarding.
These include {{EDOs| 12, 17, 19, 22, 26, 27, 29, 31, 39, 41, 43, 45, 46, 49, 50, and 53 }}.
All of these can be notated with some variant on the [[Circle-of-fifths notation|A–G "circle of fifths" notation]], while other EDOs, including {{EDOs| 24, 34, 36, 38, 44, 48, or 51}} involve multiple such circles.

Some EDOs, such as {{EDOs| 26, 27, 32, 33, or 37 }} have fifths which are reasonably good but quite audibly not just. Other EDOs, such as {{EDOs| 11, 13, 14, 15, 16, 18, 20, 21, 23, or 25 }}, are of interest to the avid seeker of totally unusual sounds that have next-to-no connection with the common practice.

If your interest lies in the nuanced approximation of just intonation through EDOs, then delving into EDOs characterized by a strong [[The Riemann zeta function and tuning#Zeta EDO lists|zeta peak]] could be especially captivating. Such EDOs, including {{EDOs| 12, 19, 22, 27, 31, 34, 41, 46, 53, 58, 60, 65, 68, 72, 77, 80, 84, 87, 94, and 99 }}, offer rich avenues for exploration in the quest for harmonic purity and transparent [[temperament]]s.

EDOs with a less pronounced, yet still noteworthy [[The Riemann zeta function and tuning#Local zeta edos|zeta peak]]—specifically {{EDOs| 10, 14, 15, 16, 17, 21, 24, 26, 29, 32, 36, 37, 38, 39, 43, 45, 48, 49, 50, 56, 62, 63, 82, 89, and 96 }}—present a unique palette for harmony explorers. Although these systems may lack the harmonic precision found in EDOs with more prominent zeta peaks, they strike an intriguing balance between consonance and more distant harmonic textures.

EDOs can be further subdivided and classified according to the size of the fifth, such as with [[Margo Schulter]]'s [[gentle region]] or the distinction between negative, positive, doubly negative and doubly positive of [[R. H. M. Bosanquet]]. [[Kite Giedraitis]] has proposed these six categories, based on the size of the fifth. From narrowest to widest:

* '''Superflat''' EDOs ({{EDOs| 9, 11, 13b, 16, 18b, and 23 }}) have a fifth narrower than four-sevenths of an octave ({{nowrap|4\7 {{=}} 685.714{{c}}}})
* '''Perfect''' EDOs ({{EDOs| 7, 14, 21, 28, and 35 }}) have a fifth equal to {{nowrap|4\7 {{=}} 685.714{{c}}}}
* '''Diatonic''' EDOs ({{EDOs| 12, 17, 19, 22, 24, etc. }}) have a fifth between 685.714{{c}} and 720{{c}}
* '''Pentatonic''' EDOs ({{EDOs| 5, 10, 15, 20, 25, and 30 }}) have a fifth of three-fifths of an octave ({{nowrap|3\5 {{=}} 720{{c}}}})
* '''Supersharp''' EDOs ({{EDOs| 8, 13, and 18 }}) have a fifth wider than 720{{c}}
* '''Trivial''' EDOs ({{EDOs| 1, 2, 3, 4, and 6 }}) have a fifth about 100{{c}} from just, and are contained in 12edo

== Structural properties ==
You will quickly find that the ''factorization'' of the total number of notes in each EDO has consequences for its structure and the way it relates to other EDOs. For example, {{nowrap|6 {{=}} 2 x 3}}, so 6edo contains all of the intervals in both 2edo and 3edo. On the other hand, 7 is a prime number, so no 7edo intervals are redundant with those of smaller EDOs. See [[Prime EDO]] and [[Highly composite EDO]] for more details.

The [[MOS]] paradigm is a fascinating way of thinking about building sub-scales of EDOs and relating them to non-EDO scales, as well as finding common melodic patterns between multiple EDOs.

=== Adding EDOs ===
Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). This really amounts to the consideration of adding the associated [[vals]], which are the mappings to primes larger than 2. An EDO is defined by a certain number of steps equating to 2; if we have more steps equating to 3, we get a 3-limit val, and so forth. So, for example, 12edo can be written {{val| 12 }}, saying that twelve steps maps to 2, but the 3-limit val for 12 is {{val| 12 19 }}, telling us that 19 steps maps to 3, and the 5-limit val is {{val| 12 19 28 }}, telling us that 28 steps maps to 5.

If we add 12 and 19 we get another good division, {{nowrap| 12 + 19 {{=}} 31 }}. We can understand why this works if we look at it as adding vals; {{val| 12 19 28 }} + {{val| 19 30 44 }} = {{val| 31 49 72 }}. The relative error in terms of [[relative cent]]s is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is {{nowrap|[-1.955 13.686]}} (the same as absolute cents) and the error of 19edo is {{nowrap|[-11.429 -11.663]}}, and this sums to {{nowrap|[-13.384 2.023]}}. In relative cents the error of the fifth for 31edo is not much increased from 19edo, and on converting to absolute cents we find it is even better, and the error of the major third is much smaller due to the cancellation. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 31edo) can have a much smaller error due to the cancellation. 24edo's flat fifth and 29edo's sharp fifth can be added to form 53edo.

We may also look at addition of EDOs in terms of MOS; if ''a''\''n'' is a generator for an ''n''-edo MOS, and ''b''\''m'' for an ''m''-edo MOS, where both of these are generators for the same linear temperament, then the mediant, {{nowrap|(''a'' + ''b'')\(''n'' + ''m'')}}, will be a generator for a MOS for the same temperament, this time in {{nowrap|(''n'' + ''m'')}}-edo. A visual way of putting this is that through this addition of ''n'' and ''m'', one becomes the accidentals or black keys, and the other the naturals or white keys. The choice of accidental/natural or black keys/white keys is a question of emphasis on the part of the composer or designer. Furthermore, one may add more than two numbers, hierarchically expanding the possibilities to double flats and sharps and beyond. This can be useful in designing keyboards and systems of notation.

=== Scale size considerations ===
EDOs with fewer than 12 divisions have steps exceeding 100 cents.
Of these, 1, 2, 3, 4, and 6 divide 12 and so are already available.
{{EDOs| 5, 7, and 9 }} have arguably been used in various musical traditions worldwide.

When using EDOs to tune scales or [[regular temperament]]s, the size becomes less conceptually important since not all notes need to be used.
Some of the EDOs which can be used to tune various temperaments are listed on the [[optimal patent val]] page. Tuning a scale in just intonation by one of these EDOs can be regarded as automatically tempering it to the corresponding regular temperament.

To practically tune large edos through software tuning, one may take advantage of [[MIDI]] channels. See [[Tuning per channel]].

All of these tools are also applicable to equal divisions of other ([[nonoctave]]) intervals as well.

== EDOs versus Equal Temperaments ==
See [[EDO vs ET]].

== Individual pages for EDOs ==
=== 0…999 ===
{| class="wikitable center-all mw-collapsible"
|+ style="font-size: 105%; white-space: nowrap;" | 0…99
|-
| [[0edo|0]]
| [[1edo|1]]
| [[2edo|2]]
| [[3edo|3]]
| [[4edo|4]]
| [[5edo|5]]
| [[6edo|6]]
| [[7edo|7]]
| [[8edo|8]]
| [[9edo|9]]
|-
| [[10edo|10]]
| [[11edo|11]]
| [[12edo|12]]
| [[13edo|13]]
| [[14edo|14]]
| [[15edo|15]]
| [[16edo|16]]
| [[17edo|17]]
| [[18edo|18]]
| [[19edo|19]]
|-
| [[20edo|20]]
| [[21edo|21]]
| [[22edo|22]]
| [[23edo|23]]
| [[24edo|24]]
| [[25edo|25]]
| [[26edo|26]]
| [[27edo|27]]
| [[28edo|28]]
| [[29edo|29]]
|-
| [[30edo|30]]
| [[31edo|31]]
| [[32edo|32]]
| [[33edo|33]]
| [[34edo|34]]
| [[35edo|35]]
| [[36edo|36]]
| [[37edo|37]]
| [[38edo|38]]
| [[39edo|39]]
|-
| [[40edo|40]]
| [[41edo|41]]
| [[42edo|42]]
| [[43edo|43]]
| [[44edo|44]]
| [[45edo|45]]
| [[46edo|46]]
| [[47edo|47]]
| [[48edo|48]]
| [[49edo|49]]
|-
| [[50edo|50]]
| [[51edo|51]]
| [[52edo|52]]
| [[53edo|53]]
| [[54edo|54]]
| [[55edo|55]]
| [[56edo|56]]
| [[57edo|57]]
| [[58edo|58]]
| [[59edo|59]]
|-
| [[60edo|60]]
| [[61edo|61]]
| [[62edo|62]]
| [[63edo|63]]
| [[64edo|64]]
| [[65edo|65]]
| [[66edo|66]]
| [[67edo|67]]
| [[68edo|68]]
| [[69edo|69]]
|-
| [[70edo|70]]
| [[71edo|71]]
| [[72edo|72]]
| [[73edo|73]]
| [[74edo|74]]
| [[75edo|75]]
| [[76edo|76]]
| [[77edo|77]]
| [[78edo|78]]
| [[79edo|79]]
|-
| [[80edo|80]]
| [[81edo|81]]
| [[82edo|82]]
| [[83edo|83]]
| [[84edo|84]]
| [[85edo|85]]
| [[86edo|86]]
| [[87edo|87]]
| [[88edo|88]]
| [[89edo|89]]
|-
| [[90edo|90]]
| [[91edo|91]]
| [[92edo|92]]
| [[93edo|93]]
| [[94edo|94]]
| [[95edo|95]]
| [[96edo|96]]
| [[97edo|97]]
| [[98edo|98]]
| [[99edo|99]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 100…199
|-
| [[100edo|100]]
| [[101edo|101]]
| [[102edo|102]]
| [[103edo|103]]
| [[104edo|104]]
| [[105edo|105]]
| [[106edo|106]]
| [[107edo|107]]
| [[108edo|108]]
| [[109edo|109]]
|-
| [[110edo|110]]
| [[111edo|111]]
| [[112edo|112]]
| [[113edo|113]]
| [[114edo|114]]
| [[115edo|115]]
| [[116edo|116]]
| [[117edo|117]]
| [[118edo|118]]
| [[119edo|119]]
|-
| [[120edo|120]]
| [[121edo|121]]
| [[122edo|122]]
| [[123edo|123]]
| [[124edo|124]]
| [[125edo|125]]
| [[126edo|126]]
| [[127edo|127]]
| [[128edo|128]]
| [[129edo|129]]
|-
| [[130edo|130]]
| [[131edo|131]]
| [[132edo|132]]
| [[133edo|133]]
| [[134edo|134]]
| [[135edo|135]]
| [[136edo|136]]
| [[137edo|137]]
| [[138edo|138]]
| [[139edo|139]]
|-
| [[140edo|140]]
| [[141edo|141]]
| [[142edo|142]]
| [[143edo|143]]
| [[144edo|144]]
| [[145edo|145]]
| [[146edo|146]]
| [[147edo|147]]
| [[148edo|148]]
| [[149edo|149]]
|-
| [[150edo|150]]
| [[151edo|151]]
| [[152edo|152]]
| [[153edo|153]]
| [[154edo|154]]
| [[155edo|155]]
| [[156edo|156]]
| [[157edo|157]]
| [[158edo|158]]
| [[159edo|159]]
|-
| [[160edo|160]]
| [[161edo|161]]
| [[162edo|162]]
| [[163edo|163]]
| [[164edo|164]]
| [[165edo|165]]
| [[166edo|166]]
| [[167edo|167]]
| [[168edo|168]]
| [[169edo|169]]
|-
| [[170edo|170]]
| [[171edo|171]]
| [[172edo|172]]
| [[173edo|173]]
| [[174edo|174]]
| [[175edo|175]]
| [[176edo|176]]
| [[177edo|177]]
| [[178edo|178]]
| [[179edo|179]]
|-
| [[180edo|180]]
| [[181edo|181]]
| [[182edo|182]]
| [[183edo|183]]
| [[184edo|184]]
| [[185edo|185]]
| [[186edo|186]]
| [[187edo|187]]
| [[188edo|188]]
| [[189edo|189]]
|-
| [[190edo|190]]
| [[191edo|191]]
| [[192edo|192]]
| [[193edo|193]]
| [[194edo|194]]
| [[195edo|195]]
| [[196edo|196]]
| [[197edo|197]]
| [[198edo|198]]
| [[199edo|199]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 200…299
|-
| [[200edo|200]]
| [[201edo|201]]
| [[202edo|202]]
| [[203edo|203]]
| [[204edo|204]]
| [[205edo|205]]
| [[206edo|206]]
| [[207edo|207]]
| [[208edo|208]]
| [[209edo|209]]
|-
| [[210edo|210]]
| [[211edo|211]]
| [[212edo|212]]
| [[213edo|213]]
| [[214edo|214]]
| [[215edo|215]]
| [[216edo|216]]
| [[217edo|217]]
| [[218edo|218]]
| [[219edo|219]]
|-
| [[220edo|220]]
| [[221edo|221]]
| [[222edo|222]]
| [[223edo|223]]
| [[224edo|224]]
| [[225edo|225]]
| [[226edo|226]]
| [[227edo|227]]
| [[228edo|228]]
| [[229edo|229]]
|-
| [[230edo|230]]
| [[231edo|231]]
| [[232edo|232]]
| [[233edo|233]]
| [[234edo|234]]
| [[235edo|235]]
| [[236edo|236]]
| [[237edo|237]]
| [[238edo|238]]
| [[239edo|239]]
|-
| [[240edo|240]]
| [[241edo|241]]
| [[242edo|242]]
| [[243edo|243]]
| [[244edo|244]]
| [[245edo|245]]
| [[246edo|246]]
| [[247edo|247]]
| [[248edo|248]]
| [[249edo|249]]
|-
| [[250edo|250]]
| [[251edo|251]]
| [[252edo|252]]
| [[253edo|253]]
| [[254edo|254]]
| [[255edo|255]]
| [[256edo|256]]
| [[257edo|257]]
| [[258edo|258]]
| [[259edo|259]]
|-
| [[260edo|260]]
| [[261edo|261]]
| [[262edo|262]]
| [[263edo|263]]
| [[264edo|264]]
| [[265edo|265]]
| [[266edo|266]]
| [[267edo|267]]
| [[268edo|268]]
| [[269edo|269]]
|-
| [[270edo|270]]
| [[271edo|271]]
| [[272edo|272]]
| [[273edo|273]]
| [[274edo|274]]
| [[275edo|275]]
| [[276edo|276]]
| [[277edo|277]]
| [[278edo|278]]
| [[279edo|279]]
|-
| [[280edo|280]]
| [[281edo|281]]
| [[282edo|282]]
| [[283edo|283]]
| [[284edo|284]]
| [[285edo|285]]
| [[286edo|286]]
| [[287edo|287]]
| [[288edo|288]]
| [[289edo|289]]
|-
| [[290edo|290]]
| [[291edo|291]]
| [[292edo|292]]
| [[293edo|293]]
| [[294edo|294]]
| [[295edo|295]]
| [[296edo|296]]
| [[297edo|297]]
| [[298edo|298]]
| [[299edo|299]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 300…399
|-
| [[300edo|300]]
| [[301edo|301]]
| [[302edo|302]]
| [[303edo|303]]
| [[304edo|304]]
| [[305edo|305]]
| [[306edo|306]]
| [[307edo|307]]
| [[308edo|308]]
| [[309edo|309]]
|-
| [[310edo|310]]
| [[311edo|311]]
| [[312edo|312]]
| [[313edo|313]]
| [[314edo|314]]
| [[315edo|315]]
| [[316edo|316]]
| [[317edo|317]]
| [[318edo|318]]
| [[319edo|319]]
|-
| [[320edo|320]]
| [[321edo|321]]
| [[322edo|322]]
| [[323edo|323]]
| [[324edo|324]]
| [[325edo|325]]
| [[326edo|326]]
| [[327edo|327]]
| [[328edo|328]]
| [[329edo|329]]
|-
| [[330edo|330]]
| [[331edo|331]]
| [[332edo|332]]
| [[333edo|333]]
| [[334edo|334]]
| [[335edo|335]]
| [[336edo|336]]
| [[337edo|337]]
| [[338edo|338]]
| [[339edo|339]]
|-
| [[340edo|340]]
| [[341edo|341]]
| [[342edo|342]]
| [[343edo|343]]
| [[344edo|344]]
| [[345edo|345]]
| [[346edo|346]]
| [[347edo|347]]
| [[348edo|348]]
| [[349edo|349]]
|-
| [[350edo|350]]
| [[351edo|351]]
| [[352edo|352]]
| [[353edo|353]]
| [[354edo|354]]
| [[355edo|355]]
| [[356edo|356]]
| [[357edo|357]]
| [[358edo|358]]
| [[359edo|359]]
|-
| [[360edo|360]]
| [[361edo|361]]
| [[362edo|362]]
| [[363edo|363]]
| [[364edo|364]]
| [[365edo|365]]
| [[366edo|366]]
| [[367edo|367]]
| [[368edo|368]]
| [[369edo|369]]
|-
| [[370edo|370]]
| [[371edo|371]]
| [[372edo|372]]
| [[373edo|373]]
| [[374edo|374]]
| [[375edo|375]]
| [[376edo|376]]
| [[377edo|377]]
| [[378edo|378]]
| [[379edo|379]]
|-
| [[380edo|380]]
| [[381edo|381]]
| [[382edo|382]]
| [[383edo|383]]
| [[384edo|384]]
| [[385edo|385]]
| [[386edo|386]]
| [[387edo|387]]
| [[388edo|388]]
| [[389edo|389]]
|-
| [[390edo|390]]
| [[391edo|391]]
| [[392edo|392]]
| [[393edo|393]]
| [[394edo|394]]
| [[395edo|395]]
| [[396edo|396]]
| [[397edo|397]]
| [[398edo|398]]
| [[399edo|399]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 400…499
|-
| [[400edo|400]]
| [[401edo|401]]
| [[402edo|402]]
| [[403edo|403]]
| [[404edo|404]]
| [[405edo|405]]
| [[406edo|406]]
| [[407edo|407]]
| [[408edo|408]]
| [[409edo|409]]
|-
| [[410edo|410]]
| [[411edo|411]]
| [[412edo|412]]
| [[413edo|413]]
| [[414edo|414]]
| [[415edo|415]]
| [[416edo|416]]
| [[417edo|417]]
| [[418edo|418]]
| [[419edo|419]]
|-
| [[420edo|420]]
| [[421edo|421]]
| [[422edo|422]]
| [[423edo|423]]
| [[424edo|424]]
| [[425edo|425]]
| [[426edo|426]]
| [[427edo|427]]
| [[428edo|428]]
| [[429edo|429]]
|-
| [[430edo|430]]
| [[431edo|431]]
| [[432edo|432]]
| [[433edo|433]]
| [[434edo|434]]
| [[435edo|435]]
| [[436edo|436]]
| [[437edo|437]]
| [[438edo|438]]
| [[439edo|439]]
|-
| [[440edo|440]]
| [[441edo|441]]
| [[442edo|442]]
| [[443edo|443]]
| [[444edo|444]]
| [[445edo|445]]
| [[446edo|446]]
| [[447edo|447]]
| [[448edo|448]]
| [[449edo|449]]
|-
| [[450edo|450]]
| [[451edo|451]]
| [[452edo|452]]
| [[453edo|453]]
| [[454edo|454]]
| [[455edo|455]]
| [[456edo|456]]
| [[457edo|457]]
| [[458edo|458]]
| [[459edo|459]]
|-
| [[460edo|460]]
| [[461edo|461]]
| [[462edo|462]]
| [[463edo|463]]
| [[464edo|464]]
| [[465edo|465]]
| [[466edo|466]]
| [[467edo|467]]
| [[468edo|468]]
| [[469edo|469]]
|-
| [[470edo|470]]
| [[471edo|471]]
| [[472edo|472]]
| [[473edo|473]]
| [[474edo|474]]
| [[475edo|475]]
| [[476edo|476]]
| [[477edo|477]]
| [[478edo|478]]
| [[479edo|479]]
|-
| [[480edo|480]]
| [[481edo|481]]
| [[482edo|482]]
| [[483edo|483]]
| [[484edo|484]]
| [[485edo|485]]
| [[486edo|486]]
| [[487edo|487]]
| [[488edo|488]]
| [[489edo|489]]
|-
| [[490edo|490]]
| [[491edo|491]]
| [[492edo|492]]
| [[493edo|493]]
| [[494edo|494]]
| [[495edo|495]]
| [[496edo|496]]
| [[497edo|497]]
| [[498edo|498]]
| [[499edo|499]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 500…599
|-
| [[500edo|500]]
| [[501edo|501]]
| [[502edo|502]]
| [[503edo|503]]
| [[504edo|504]]
| [[505edo|505]]
| [[506edo|506]]
| [[507edo|507]]
| [[508edo|508]]
| [[509edo|509]]
|-
| [[510edo|510]]
| [[511edo|511]]
| [[512edo|512]]
| [[513edo|513]]
| [[514edo|514]]
| [[515edo|515]]
| [[516edo|516]]
| [[517edo|517]]
| [[518edo|518]]
| [[519edo|519]]
|-
| [[520edo|520]]
| [[521edo|521]]
| [[522edo|522]]
| [[523edo|523]]
| [[524edo|524]]
| [[525edo|525]]
| [[526edo|526]]
| [[527edo|527]]
| [[528edo|528]]
| [[529edo|529]]
|-
| [[530edo|530]]
| [[531edo|531]]
| [[532edo|532]]
| [[533edo|533]]
| [[534edo|534]]
| [[535edo|535]]
| [[536edo|536]]
| [[537edo|537]]
| [[538edo|538]]
| [[539edo|539]]
|-
| [[540edo|540]]
| [[541edo|541]]
| [[542edo|542]]
| [[543edo|543]]
| [[544edo|544]]
| [[545edo|545]]
| [[546edo|546]]
| [[547edo|547]]
| [[548edo|548]]
| [[549edo|549]]
|-
| [[550edo|550]]
| [[551edo|551]]
| [[552edo|552]]
| [[553edo|553]]
| [[554edo|554]]
| [[555edo|555]]
| [[556edo|556]]
| [[557edo|557]]
| [[558edo|558]]
| [[559edo|559]]
|-
| [[560edo|560]]
| [[561edo|561]]
| [[562edo|562]]
| [[563edo|563]]
| [[564edo|564]]
| [[565edo|565]]
| [[566edo|566]]
| [[567edo|567]]
| [[568edo|568]]
| [[569edo|569]]
|-
| [[570edo|570]]
| [[571edo|571]]
| [[572edo|572]]
| [[573edo|573]]
| [[574edo|574]]
| [[575edo|575]]
| [[576edo|576]]
| [[577edo|577]]
| [[578edo|578]]
| [[579edo|579]]
|-
| [[580edo|580]]
| [[581edo|581]]
| [[582edo|582]]
| [[583edo|583]]
| [[584edo|584]]
| [[585edo|585]]
| [[586edo|586]]
| [[587edo|587]]
| [[588edo|588]]
| [[589edo|589]]
|-
| [[590edo|590]]
| [[591edo|591]]
| [[592edo|592]]
| [[593edo|593]]
| [[594edo|594]]
| [[595edo|595]]
| [[596edo|596]]
| [[597edo|597]]
| [[598edo|598]]
| [[599edo|599]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 600…699
|-
| [[600edo|600]]
| [[601edo|601]]
| [[602edo|602]]
| [[603edo|603]]
| [[604edo|604]]
| [[605edo|605]]
| [[606edo|606]]
| [[607edo|607]]
| [[608edo|608]]
| [[609edo|609]]
|-
| [[610edo|610]]
| [[611edo|611]]
| [[612edo|612]]
| [[613edo|613]]
| [[614edo|614]]
| [[615edo|615]]
| [[616edo|616]]
| [[617edo|617]]
| [[618edo|618]]
| [[619edo|619]]
|-
| [[620edo|620]]
| [[621edo|621]]
| [[622edo|622]]
| [[623edo|623]]
| [[624edo|624]]
| [[625edo|625]]
| [[626edo|626]]
| [[627edo|627]]
| [[628edo|628]]
| [[629edo|629]]
|-
| [[630edo|630]]
| [[631edo|631]]
| [[632edo|632]]
| [[633edo|633]]
| [[634edo|634]]
| [[635edo|635]]
| [[636edo|636]]
| [[637edo|637]]
| [[638edo|638]]
| [[639edo|639]]
|-
| [[640edo|640]]
| [[641edo|641]]
| [[642edo|642]]
| [[643edo|643]]
| [[644edo|644]]
| [[645edo|645]]
| [[646edo|646]]
| [[647edo|647]]
| [[648edo|648]]
| [[649edo|649]]
|-
| [[650edo|650]]
| [[651edo|651]]
| [[652edo|652]]
| [[653edo|653]]
| [[654edo|654]]
| [[655edo|655]]
| [[656edo|656]]
| [[657edo|657]]
| [[658edo|658]]
| [[659edo|659]]
|-
| [[660edo|660]]
| [[661edo|661]]
| [[662edo|662]]
| [[663edo|663]]
| [[664edo|664]]
| [[665edo|665]]
| [[666edo|666]]
| [[667edo|667]]
| [[668edo|668]]
| [[669edo|669]]
|-
| [[670edo|670]]
| [[671edo|671]]
| [[672edo|672]]
| [[673edo|673]]
| [[674edo|674]]
| [[675edo|675]]
| [[676edo|676]]
| [[677edo|677]]
| [[678edo|678]]
| [[679edo|679]]
|-
| [[680edo|680]]
| [[681edo|681]]
| [[682edo|682]]
| [[683edo|683]]
| [[684edo|684]]
| [[685edo|685]]
| [[686edo|686]]
| [[687edo|687]]
| [[688edo|688]]
| [[689edo|689]]
|-
| [[690edo|690]]
| [[691edo|691]]
| [[692edo|692]]
| [[693edo|693]]
| [[694edo|694]]
| [[695edo|695]]
| [[696edo|696]]
| [[697edo|697]]
| [[698edo|698]]
| [[699edo|699]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 700…799
|-
| [[700edo|700]]
| [[701edo|701]]
| [[702edo|702]]
| [[703edo|703]]
| [[704edo|704]]
| [[705edo|705]]
| [[706edo|706]]
| [[707edo|707]]
| [[708edo|708]]
| [[709edo|709]]
|-
| [[710edo|710]]
| [[711edo|711]]
| [[712edo|712]]
| [[713edo|713]]
| [[714edo|714]]
| [[715edo|715]]
| [[716edo|716]]
| [[717edo|717]]
| [[718edo|718]]
| [[719edo|719]]
|-
| [[720edo|720]]
| [[721edo|721]]
| [[722edo|722]]
| [[723edo|723]]
| [[724edo|724]]
| [[725edo|725]]
| [[726edo|726]]
| [[727edo|727]]
| [[728edo|728]]
| [[729edo|729]]
|-
| [[730edo|730]]
| [[731edo|731]]
| [[732edo|732]]
| [[733edo|733]]
| [[734edo|734]]
| [[735edo|735]]
| [[736edo|736]]
| [[737edo|737]]
| [[738edo|738]]
| [[739edo|739]]
|-
| [[740edo|740]]
| [[741edo|741]]
| [[742edo|742]]
| [[743edo|743]]
| [[744edo|744]]
| [[745edo|745]]
| [[746edo|746]]
| [[747edo|747]]
| [[748edo|748]]
| [[749edo|749]]
|-
| [[750edo|750]]
| [[751edo|751]]
| [[752edo|752]]
| [[753edo|753]]
| [[754edo|754]]
| [[755edo|755]]
| [[756edo|756]]
| [[757edo|757]]
| [[758edo|758]]
| [[759edo|759]]
|-
| [[760edo|760]]
| [[761edo|761]]
| [[762edo|762]]
| [[763edo|763]]
| [[764edo|764]]
| [[765edo|765]]
| [[766edo|766]]
| [[767edo|767]]
| [[768edo|768]]
| [[769edo|769]]
|-
| [[770edo|770]]
| [[771edo|771]]
| [[772edo|772]]
| [[773edo|773]]
| [[774edo|774]]
| [[775edo|775]]
| [[776edo|776]]
| [[777edo|777]]
| [[778edo|778]]
| [[779edo|779]]
|-
| [[780edo|780]]
| [[781edo|781]]
| [[782edo|782]]
| [[783edo|783]]
| [[784edo|784]]
| [[785edo|785]]
| [[786edo|786]]
| [[787edo|787]]
| [[788edo|788]]
| [[789edo|789]]
|-
| [[790edo|790]]
| [[791edo|791]]
| [[792edo|792]]
| [[793edo|793]]
| [[794edo|794]]
| [[795edo|795]]
| [[796edo|796]]
| [[797edo|797]]
| [[798edo|798]]
| [[799edo|799]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 800…899
|-
| [[800edo|800]]
| [[801edo|801]]
| [[802edo|802]]
| [[803edo|803]]
| [[804edo|804]]
| [[805edo|805]]
| [[806edo|806]]
| [[807edo|807]]
| [[808edo|808]]
| [[809edo|809]]
|-
| [[810edo|810]]
| [[811edo|811]]
| [[812edo|812]]
| [[813edo|813]]
| [[814edo|814]]
| [[815edo|815]]
| [[816edo|816]]
| [[817edo|817]]
| [[818edo|818]]
| [[819edo|819]]
|-
| [[820edo|820]]
| [[821edo|821]]
| [[822edo|822]]
| [[823edo|823]]
| [[824edo|824]]
| [[825edo|825]]
| [[826edo|826]]
| [[827edo|827]]
| [[828edo|828]]
| [[829edo|829]]
|-
| [[830edo|830]]
| [[831edo|831]]
| [[832edo|832]]
| [[833edo|833]]
| [[834edo|834]]
| [[835edo|835]]
| [[836edo|836]]
| [[837edo|837]]
| [[838edo|838]]
| [[839edo|839]]
|-
| [[840edo|840]]
| [[841edo|841]]
| [[842edo|842]]
| [[843edo|843]]
| [[844edo|844]]
| [[845edo|845]]
| [[846edo|846]]
| [[847edo|847]]
| [[848edo|848]]
| [[849edo|849]]
|-
| [[850edo|850]]
| [[851edo|851]]
| [[852edo|852]]
| [[853edo|853]]
| [[854edo|854]]
| [[855edo|855]]
| [[856edo|856]]
| [[857edo|857]]
| [[858edo|858]]
| [[859edo|859]]
|-
| [[860edo|860]]
| [[861edo|861]]
| [[862edo|862]]
| [[863edo|863]]
| [[864edo|864]]
| [[865edo|865]]
| [[866edo|866]]
| [[867edo|867]]
| [[868edo|868]]
| [[869edo|869]]
|-
| [[870edo|870]]
| [[871edo|871]]
| [[872edo|872]]
| [[873edo|873]]
| [[874edo|874]]
| [[875edo|875]]
| [[876edo|876]]
| [[877edo|877]]
| [[878edo|878]]
| [[879edo|879]]
|-
| [[880edo|880]]
| [[881edo|881]]
| [[882edo|882]]
| [[883edo|883]]
| [[884edo|884]]
| [[885edo|885]]
| [[886edo|886]]
| [[887edo|887]]
| [[888edo|888]]
| [[889edo|889]]
|-
| [[890edo|890]]
| [[891edo|891]]
| [[892edo|892]]
| [[893edo|893]]
| [[894edo|894]]
| [[895edo|895]]
| [[896edo|896]]
| [[897edo|897]]
| [[898edo|898]]
| [[899edo|899]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 900…999
|-
| [[900edo|900]]
| [[901edo|901]]
| [[902edo|902]]
| [[903edo|903]]
| [[904edo|904]]
| [[905edo|905]]
| [[906edo|906]]
| [[907edo|907]]
| [[908edo|908]]
| [[909edo|909]]
|-
| [[910edo|910]]
| [[911edo|911]]
| [[912edo|912]]
| [[913edo|913]]
| [[914edo|914]]
| [[915edo|915]]
| [[916edo|916]]
| [[917edo|917]]
| [[918edo|918]]
| [[919edo|919]]
|-
| [[920edo|920]]
| [[921edo|921]]
| [[922edo|922]]
| [[923edo|923]]
| [[924edo|924]]
| [[925edo|925]]
| [[926edo|926]]
| [[927edo|927]]
| [[928edo|928]]
| [[929edo|929]]
|-
| [[930edo|930]]
| [[931edo|931]]
| [[932edo|932]]
| [[933edo|933]]
| [[934edo|934]]
| [[935edo|935]]
| [[936edo|936]]
| [[937edo|937]]
| [[938edo|938]]
| [[939edo|939]]
|-
| [[940edo|940]]
| [[941edo|941]]
| [[942edo|942]]
| [[943edo|943]]
| [[944edo|944]]
| [[945edo|945]]
| [[946edo|946]]
| [[947edo|947]]
| [[948edo|948]]
| [[949edo|949]]
|-
| [[950edo|950]]
| [[951edo|951]]
| [[952edo|952]]
| [[953edo|953]]
| [[954edo|954]]
| [[955edo|955]]
| [[956edo|956]]
| [[957edo|957]]
| [[958edo|958]]
| [[959edo|959]]
|-
| [[960edo|960]]
| [[961edo|961]]
| [[962edo|962]]
| [[963edo|963]]
| [[964edo|964]]
| [[965edo|965]]
| [[966edo|966]]
| [[967edo|967]]
| [[968edo|968]]
| [[969edo|969]]
|-
| [[970edo|970]]
| [[971edo|971]]
| [[972edo|972]]
| [[973edo|973]]
| [[974edo|974]]
| [[975edo|975]]
| [[976edo|976]]
| [[977edo|977]]
| [[978edo|978]]
| [[979edo|979]]
|-
| [[980edo|980]]
| [[981edo|981]]
| [[982edo|982]]
| [[983edo|983]]
| [[984edo|984]]
| [[985edo|985]]
| [[986edo|986]]
| [[987edo|987]]
| [[988edo|988]]
| [[989edo|989]]
|-
| [[990edo|990]]
| [[991edo|991]]
| [[992edo|992]]
| [[993edo|993]]
| [[994edo|994]]
| [[995edo|995]]
| [[996edo|996]]
| [[997edo|997]]
| [[998edo|998]]
| [[999edo|999]]
|}

=== 1000…1999 ===
{{EDOs
| 1000, 1001, 1012, 1015, 1019, 1051, 1053, 1059, 1063, 1065, 1080, 1092, 1106, 1125, 1131, 1147, 1152, 1166, 1171, 1178, 1193, 1200, 1210, 1224, 1230, 1236, 1240, 1244, 1260, 1272, 1277, 1289, 1308, 1323, 1330, 1337, 1342, 1361, 1376, 1381, 1395, 1407, 1419, 1429, 1440, 1448, 1489, 1506, 1517, 1520, 1525, 1536, 1547, 1553, 1554, 1559, 1578, 1583, 1590, 1600, 1609, 1612, 1619, 1637, 1641, 1643, 1650, 1665, 1672, 1700, 1730, 1759, 1776, 1778, 1783, 1789, 1793, 1794, 1802, 1803, 1817, 1848, 1861, 1879, 1880, 1889, 1911, 1920, 1944, 1955, 1957, 1983, 1984 }}

=== 2000…9999 ===
{{EDOs
| 2000, 2016, 2019, 2022, 2023, 2024, 2025, 2029, 2048, 2053, 2072, 2081, 2100, 2101, 2113, 2118, 2129, 2153, 2190, 2200, 2207, 2242, 2243, 2320, 2444, 2460, 2477, 2513, 2520, 2544, 2549, 2554, 2619, 2684, 2711, 2730, 2777, 2809, 2814, 2819, 2897, 2901, 2912, 2960, 2964, 3041, 3071, 3072, 3079, 3080, 3125, 3178, 3361, 3395, 3422, 3476, 3498, 3558, 3566, 3578, 3600, 3643, 3684, 3696, 3776, 3889, 3920, 4004, 4007, 4079, 4096, 4172, 4190, 4231, 4296, 4320, 4327, 4349, 4380, 4501, 4650, 4973, 5040, 5280, 5544, 5585, 5809, 5902, 5941, 6079, 6349, 6380, 6650, 6664, 6691, 7033, 7315, 7980, 8103, 8192, 8269, 8404, 8539, 8736, 8745, 9539
}}

=== 10000 and up ===
{{EDOs
| 10009, 10459, 10600, 10729, 11664, 12276, 12348, 12500, 13382, 14124, 14348, 14618, 14842, 15601, 15900, 16218, 16384, 16625, 16808, 17461, 18355, 20203, 20567, 25281, 25282, 28000, 28742, 30103, 30631, 31867, 31920, 32436, 32768, 33616, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 99693, 99694, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 571611, 762148, 1714833, 1905370, 2547047, 2667518, 2901533, 3159811, 4191814, 6000000, 11358058, 402653184, 5407372813
}}

== Non-integer EDO ==
A non-integer EDO can be defined as using a non-integer divisor to divide the octave. Typically, non-integer EDOs are understood as ''not'' containing the exact octave, so that they remain [[equal tuning]]s. If the exact octave is retained and if the generator resets itself at each period, then this results in a [[MOS scale]] with only 1 small step.

All fractional EDOs are integer equal divisions of another integer interval. For example, (25/2)edo is equivalent to 25ed4. In general:

<math>\displaystyle (p/q) \text{edo} = p \text{-ed} 2^q</math>

for integers ''p'' and ''q''. Many irrational EDOs cannot be converted to integer equal divisions of another integer interval, so they are things of their own.

Non-integer EDOs can be written in decimal form, such as 12.1edo. This is often meant to be approximate, used in the context of [[octave stretch]] of an integer EDO, rather than as a fractional EDO.

== Scale tree ==

The scale tree, or Stern-Brocot tree, provides a visual map of the world of EDOs, based on fifth size.

[[File:The_Scale_Tree.png|alt=The Scale Tree.png|800x1023px|The Scale Tree.png]]

The regular EDOs, up to 72edo:

[[File:Scale_Tree_close-up.png|alt=Scale Tree close-up.png|Scale Tree close-up.png]]

== Pergens ==
{{See also| Pergen #Pergens and EDOs }}

[[Pergen]]s provide a JI-agnostic way to name the rank-2 scales of an EDO. This table lists every possible period/generator combination for EDOs 5–24, and for each coprime combination, the simplest pergen that it can represent. Non-coprime combinations such as {{nowrap|P = 6\12}}, {{nowrap|G = 4\12}} are marked as "-".

{| class="wikitable"
|-
! EDO
! Period
! colspan="11" | Generator in EDO steps
|-
!
! in EDO steps
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
|-
! 5
! 5 = P8
| P4/2
| P5
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 6
! 6 = P8
| P4/2
| -
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/2
| P5
|
|
|
|
|
|
|
|
|
|
|-
! 7
! 7 = P8
| P4/3
| P5/2
| P5
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 8
! 8 = P8
| P4/3
| -
| P5
|
|
|
|
|
|
|
|
|-
! 4 = P8/2
| P5
| -
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 9
! 9 = P8
| P4/4
| P4/2
| -
| P5
|
|
|
|
|
|
|
|-
! 3 = P8/3
| P5
|
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 10
! 10 = P8
| P4/4
| -
| P5/2
| -
|
|
|
|
|
|
|
|-
! 5 = P8/2
| P5
| P4/2
|
|
|
|
|
|
|
|
|
|-
! 11
! 11 = P8
| P4/5
| P5/3
| P5/2
| P11/4
| P5
|
|
|
|
|
|
|-
! rowspan="4" | 12
! 12 = P8
| P4/5
| -
| -
| -
| P5
|
|
|
|
|
|
|-
! 6 = P8/2
| P5
| -
| -
|
|
|
|
|
|
|
|
|-
! 4 = P8/3
| P5
| -
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/4
| P5
|
|
|
|
|
|
|
|
|
|
|-
! 13b
! 13 = P8
| P4/6
| P4/3
| P4/2
| P12/5
| P12/4
| P5
|
|
|
|
|
|-
! rowspan="2" | 14
! 14 = P8
| P4/6
| -
| P4/2
| -
| P11/4
| -
|
|
|
|
|
|-
! 7 = P8/2
| P5
| P4/3
| P4/2
|
|
|
|
|
|
|
|
|-
! rowspan="3" | 15
! 15 = P8
| P4/6
| P4/3
| -
| P12/6
| -
| -
| P11/3
|
|
|
|
|-
! 5 = P8/3
| P5
| P4/2
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/5
| P4/3
|
|
|
|
|
|
|
|
|
|
|-
! rowspan="3" | 16
! 16 = P8
| P4/7
| -
| P5/3
| -
| P12/5
| -
| P5
|
|
|
|
|-
! 8 = P8/2
| P5
| -
| P5/3
| -
|
|
|
|
|
|
|
|-
! 4 = P8/4
| P5
| -
|
|
|
|
|
|
|
|
|
|-
! 17
! 17 = P8
| P4/7
| P5/5
| P11/8
| P11/6
| P5/2
| P11/4
| P5
| P11/3
|
|
|
|-
! rowspan="4" | 18b
! 18 = P8
| P4/8
| -
| -
| -
| P5/2
| -
| P12/4
| -
|
|
|
|-
! 9 = P8/2
| P5
| P4/4
| -
| P4/2
|
|
|
|
|
|
|
|-
! 6 = P8/3
| P5/2
| -
| -
|
|
|
|
|
|
|
|
|-
! 3 = P8/6
| P5
|
|
|
|
|
|
|
|
|
|
|-
! 19
! 19 = P8
| P4/8
| P4/4
| P11/9
| P4/2
| P12/6
| P12/5
| ccP5/7
| P5
| P11/3
|
|
|-
! rowspan="4" | 20
! 20 = P8
| P4/8
| -
| P5/4
| -
| -
| -
| P11/4
| -
| c3P5/8
|
|
|-
! 10 = P8/2
| M2/4
| -
| P5/4
| -
| -
|
|
|
|
|
|
|-
! 5 = P8/4
| P4/2
| P5
|
|
|
|
|
|
|
|
|
|-
! 4 = P8/5
| P5/4
| -
|
|
|
|
|
|
|
|
|
|-
! rowspan="3" | 21
! 21 = P8
| P4/9
| P5/6
| -
| P5/3
| P11/6
| -
| -
| c3P4/9
| -
| P11/3
|
|-
! 7 = P8/3
| P5/2
| P5
| P4/3
|
|
|
|
|
|
|
|
|-
! 3 = P8/7
| P5/3
|
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 22
! 22 = P8
| P4/9
| -
| P4/3
| -
| P12/7
| -
| P12/5
| -
| P5
| -
|
|-
! 11 = P8/2
| M2/4
| P5
| P4/3
| P12/5
| P12/7
|
|
|
|
|
|
|-
! 23
! 23 = P8
| P4/10
| P4/5
| P11/11
| P12/9
| P4/2
| P12/6
| ccP4/8
| ccP4/7
| P12/4
| P5
| P11/3
|-
! rowspan="6" | 24
! 24 = P8
| P4/10
| -
| -
| -
| P4/2
| -
| P5/2
| -
| -
| -
| c4P5/10
|-
! 12 = P8/2
| M2/4
| -
| -
| -
| P4/2
|
|
|
|
|
|
|-
! 8 = P8/3
| P5/2
| -
| P4/2
|
|
|
|
|
|
|
|
|-
! 6 = P8/4
| P4/2
| -
| -
|
|
|
|
|
|
|
|
|-
! 4 = P8/6
| P4/2
| -
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/8
| P5
|
|
|
|
|
|
|
|
|
|
|-
!
!
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
|}

== See also ==
Related topics
* [[Equal-step tuning]]
* [[Highly composite equal division]]
* [[List of rank one temperaments by step size]]
* [[Prime equal division]]

Technical data
* [[Absolute errors of small EDOs]]
* [[Consistency limits of small EDOs]]
* [[Distinct EDO Scales]]
* [[Minimal consistent EDOs]]
* [[Monotonicity levels of small EDOs]]
* [[Relative errors of small EDOs]]

Opinions
* [[Collection of EDO impressions]]

Other
* [[:Category:Equal divisions of the octave]]

== External links ==
* [[Ivor Darreg]], [https://www.webcitation.org/5xZz8RtQB Teen Tunes]

== Notes ==
<references />

[[Category:Equal-step tuning]]
[[Category:Equal divisions of the octave| ]] 
[[Category:Acronyms]]
[[Category:Lists of scales]]

1376edo

2025-06-14T15:03:39Z

Eliora: Created page with "{{Infobox ET}} {{ED intro}} 1376edo is consistent in the 15-odd-limit and it is an exceptional 7-limit system. 1376edo supports semidimi. It also supports alphatricot and its 7-limit extension alphatrillium. It suppots 7-limit very high accuracy temperaments {{monzo|0 -11 -7 12}}, {{monzo|1 -15 -18 23}}, {{monzo|-1 4 11 -11}}. It also supports the 32nd-octave temperament germanium, 224 & 1376. In higher limits, a precise extension can be used..."

{{Infobox ET}}
{{ED intro}}

1376edo is consistent in the [[15-odd-limit]] and it is an exceptional 7-limit system.

1376edo supports [[semidimi]]. It also supports [[alphatricot]] and its 7-limit extension [[alphatrillium]]. It suppots 7-limit [[very high accuracy temperaments]] {{monzo|0 -11 -7 12}}, {{monzo|1 -15 -18 23}}, {{monzo|-1 4 11 -11}}. It also supports the 32nd-octave temperament [[germanium]], 224 & 1376.

In higher limits, a precise extension can be used for 2.3.5.7.31, or various satisfactory add-19 extensions.
=== Prime harmonics ===
{{harmonics in equal|1376}}

Since 1376 factors as {{Factorization|1376}}, 1376edo has subset edos {{EDOs|1, 2, 4, 8, 16, 32, 43, 86, 172, 344, 688}}.

EDO

2025-03-15T17:50:34Z

Eliora: /* 2000…9999 */

{{Todo|discuss title}}
{{interwiki
| de = EDO
| en = EDO
| es = EDOs
| ja = オクターブ平均律
| ko = EDO (Korean)
| ro = DEO
}}
An '''equal division of the octave''' ('''EDO''', ''EE-dee-oh''; '''edo''', ''EE-doh'') is a [[Tuning system|tuning]] obtained by dividing the [[octave]] in a certain number of [[Equal-step tuning|equal steps]]. This means that the [[interval]] between any two consecutive pitches is identical.

A tuning with ''n'' equal divisions of the octave is usually called "''n''-edo" ("''n''-EDO"). For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO).

An equal (pitch) division of the octave is the most common type of [[EPD|equal (pitch) division]], which is a kind of [[equal-step tuning]]. Therefore, it is also a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.

== History ==
Tuning theorists first used the term "equal temperament" for edos designed to approximate [[Low-complexity JI|low-complexity just intervals]]. The same term is still used today for all rank-1 [[Regular temperament|temperaments]]. For example, [[15edo]] can be referred to as 15-tone equal temperament (15-TET, 15-tET, 15tet, etc.), or more simply 15 equal temperament (15-ET, 15et, etc.).

The acronym "EDO" was coined by [[Daniel Anthony Stearns]] in 1999, originally standing for "equidistant divisions of the octave"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_65#65 Yahoo! Tuning Group | ''Where F + f = O'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_117#117 Yahoo! Tuning Group | ''f + F and WFS/MOS'']</ref>. More recently, the {{w|anacronym}} "edo", spelled in lowercase and pronounced as a regular word, has also become common.

With the development of [[Edonoi|equal divisions of non-octave intervals (edonoi)]], some people started writing "ed2" ("ED2"), especially when naming a specific tuning.

Several alternate notations have been devised, including "edd" ("EDD"; equal division of the [[octave|ditave]]), "DIV," and "EQ".{{Citation needed|date=July 2021|reason=Who used this term?}}

== Formula ==
To find the step size of ''n''-edo in terms of [[cent]]s, divide 1200 by ''n''. The size ''s'' of ''k'' steps of ''n''-edo (''k''\''n'') is

<math>\displaystyle s = 1200 \cdot k/n</math>

To find the step size of ''n''-edo in terms of [[frequency ratio]], take the ''n''-th root of 2. For example, the step of 12edo is 21/12 (≈ 1.059). So the ratio ''c'' of the ''k'' steps of ''n''-edo is

<math>\displaystyle c = 2^{k/n}</math>

In particular, when ''k'' is 0, ''c'' is simply 1, because any number to the 0th power is 1. And when {{nowrap|''k'' {{=}} ''n''}}, ''c'' is simply 2, because any number to the 1st power is itself.

== EDO FAQ ==
=== What are EDO scales like? ===
Very straightforward to work with, the step size being so even and all. Some find the monotony bland, others find it a safe stable footing for musicmaking. The only property shared by all of them is the equality of their step-sizes; otherwise, their individual properties are as different as can be. The lower-numbered EDOs, especially 5 to 24, possess very strong and unique "characters", which some composers have found to be inspiring in their own right.

=== Why would I want to use an EDO? ===
If you are a [[guitar]]ist (or a player of some other fretted string instrument, like a bass guitar, Appalachian dulcimer, [[ukulele]], banjo, mandolin, sitar, saz, pipa, or zhong ruan), an EDO will provide you with the simplest possible fretboard layout, as all of the frets will go straight across the fretboard, regardless of how you want to tune the open strings. Speaking of string instruments fretted for EDOs, since ascending through the EDOs will crowd a fretboard relatively quickly, especially as one approaches the 30-something EDOs, [[ed4|equal divisions of the double octave]] (or higher multiple of the octave) are a relatively tidy compromise solution to the problem of laying out high-EDO fretboards.

More generally, EDOs allow for modulation to every single key in the tuning, without any alteration in harmonic properties, thus making transposition totally seamless. This also makes them somewhat easier to learn, as you do not have to memorize the harmonic and melodic variations that appear in various keys (which you would have to learn in JI, an unequal regular temperament, or a well-temperament, especially with smaller numbers of tones). For those accustomed to the "equality" of 12-TET, the equality of the alternative EDOs can be reassuringly familiar.

=== How do I explore so many? ===
It depends entirely on your desires as a musician!

If you are interested in exploring the unique merits and challenges of each EDO, irrespective of any desire to approximate Just intonation (or any other a priori musical goal), starting at the bottom and working your way up can be a most illuminating exercise.

If you're a classically-trained musician and you'd like to start with some EDOs that have some relationship to common-practice tonal music, starting with reasonably-low EDOs that give a good approximation to [[3/2]] (the perfect fifth) can be rewarding. These include {{EDOs| 12, 17, 19, 22, 26, 27, 29, 31, 39, 41, 43, 45, 46, 49, 50, and 53 }}. All of these can be notated with some variant on the [[Circle-of-fifths notation|A–G "circle of fifths" notation]], while other EDOs, including {{EDOs| 24, 34, 36, 38, 44, 48, or 51 }} involve multiple such circles.

Some EDOs, such as {{EDOs| 26, 27, 32, 33, or 37 }} have fifths which are reasonably good but quite audibly not just. Other EDOs, such as {{EDOs| 11, 13, 14, 15, 16, 18, 20, 21, 23, or 25 }}, are of interest to the avid seeker of totally unusual sounds that have next-to-no connection with the common practice.

If your interest lies in the nuanced approximation of just intonation through EDOs, then delving into EDOs characterized by a strong [[The Riemann zeta function and tuning#Zeta EDO lists|zeta peak]] could be especially captivating. Such EDOs, including {{EDOs| 12, 19, 22, 27, 31, 34, 41, 46, 53, 58, 60, 65, 68, 72, 77, 80, 84, 87, 94, and 99 }}, offer rich avenues for exploration in the quest for harmonic purity and transparent [[temperament]]s.

EDOs with a less pronounced, yet still noteworthy [[The Riemann zeta function and tuning#Local zeta edos|zeta peak]]—specifically {{EDOs| 10, 14, 15, 16, 17, 21, 24, 26, 29, 32, 36, 37, 38, 39, 43, 45, 48, 49, 50, 56, 62, 63, 82, 89, and 96 }}—present a unique palette for harmony explorers. Although these systems may lack the harmonic precision found in EDOs with more prominent zeta peaks, they strike an intriguing balance between consonance and more distant harmonic textures.

EDOs can be further subdivided and classified according to the size of the fifth, such as with [[Margo Schulter]]'s [[gentle region]] or the distinction between negative, positive, doubly negative and doubly positive of [[R. H. M. Bosanquet]]. [[Kite Giedraitis]] has proposed these six categories, based on the size of the fifth. From narrowest to widest:

* '''Superflat''' EDOs ({{EDOs| 9, 11, 13b, 16, 18b, and 23 }}) have a fifth narrower than four-sevenths of an octave ({{nowrap|4\7 {{=}} 685.714{{c}}}})
* '''Perfect''' EDOs ({{EDOs| 7, 14, 21, 28, and 35 }}) have a fifth equal to {{nowrap|4\7 {{=}} 685.714{{c}}}}
* '''Diatonic''' EDOs ({{EDOs| 12, 17, 19, 22, 24, etc. }}) have a fifth between 685.714{{c}} and 720{{c}}
* '''Pentatonic''' EDOs ({{EDOs| 5, 10, 15, 20, 25, and 30 }}) have a fifth of three-fifths of an octave ({{nowrap|3\5 {{=}} 720{{c}}}})
* '''Supersharp''' EDOs ({{EDOs| 8, 13, and 18 }}) have a fifth wider than 720{{c}}
* '''Trivial''' EDOs ({{EDOs| 1, 2, 3, 4, and 6 }}) have a fifth about 100{{c}} from just, and are contained in 12edo

=== Non-tuning properties ===
You will quickly find that the ''factorization'' of the total number of notes in each EDO has consequences for its structure and the way it relates to other EDOs. For example, {{nowrap|6 {{=}} 2 x 3}}, so 6edo contains all of the intervals in both 2edo and 3edo. On the other hand, 7 is a prime number, so no 7edo intervals are redundant with those of smaller EDOs. See [[Prime EDO]] and [[Highly composite EDO]] for more details.

The [[MOS]] paradigm is a fascinating way of thinking about building sub-scales of EDOs and relating them to non-EDO scales, as well as finding common melodic patterns between multiple EDOs.

=== Adding EDOs ===
Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). This really amounts to the consideration of adding the associated [[vals]], which are the mappings to primes larger than 2. An EDO is defined by a certain number of steps equating to 2; if we have more steps equating to 3, we get a 3-limit val, and so forth. So, for example, 12edo can be written {{val| 12 }}, saying that twelve steps maps to 2, but the 3-limit val for 12 is {{val| 12 19 }}, telling us that 19 steps maps to 3, and the 5-limit val is {{val| 12 19 28 }}, telling us that 28 steps maps to 5.

If we add 12 and 19 we get another good division, {{nowrap| 12 + 19 {{=}} 31 }}. We can understand why this works if we look at it as adding vals; {{val| 12 19 28 }} + {{val| 19 30 44 }} = {{val| 31 49 72 }}. The relative error in terms of [[relative cent]]s is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is {{nowrap|[-1.955 13.686]}} (the same as absolute cents) and the error of 19edo is {{nowrap|[-11.429 -11.663]}}, and this sums to {{nowrap|[-13.384 2.023]}}. In relative cents the error of the fifth for 31edo is not much increased from 19edo, and on converting to absolute cents we find it is even better, and the error of the major third is much smaller due to the cancellation. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 31edo) can have a much smaller error due to the cancellation. 24edo's flat fifth and 29edo's sharp fifth can be added to form 53edo.

We may also look at addition of EDOs in terms of MOS; if ''a''\''n'' is a generator for an ''n''-edo MOS, and ''b''\''m'' for an ''m''-edo MOS, where both of these are generators for the same linear temperament, then the mediant, {{nowrap|(''a'' + ''b'')\(''n'' + ''m'')}}, will be a generator for a MOS for the same temperament, this time in {{nowrap|(''n'' + ''m'')}}-edo. A visual way of putting this is that through this addition of ''n'' and ''m'', one becomes the accidentals or black keys, and the other the naturals or white keys. The choice of accidental/natural or black keys/white keys is a question of emphasis on the part of the composer or designer. Furthermore, one may add more than two numbers, hierarchically expanding the possibilities to double flats and sharps and beyond. This can be useful in designing keyboards and systems of notation.

=== Size of an EDO ===
When an edo divides the octave into fewer than 12 divisions (so that each step exceeds 100 cents), you might call it a [[macrotonal EDO]]. Of these, 1, 2, 3, 4, and 6 divide 12 and so are already available to anyone wishing to explore them. {{EDOs| 5, 7, and 9 }} have arguably been used in various kinds of musical traditions in different parts of the world. [https://soundcloud.com/scottthompson-3/the-13-edos-of-xmas ''The 13 EDOs of Xmas''] by [[Scott Thompson]] is a humorous demonstration of EDOs 1–13.

On the other hand, if you use the edo to tune a scale or [[regular temperament]], the size of the edo does not matter so much (at least conceptually), as you don't need to use all of it. Some of the EDOs which can be used to tune various temperaments are listed on the [[optimal patent val]] page. Tuning a scale in just intonation by one of these EDOs can be regarded as automatically tempering it to the corresponding regular temperament.

To practically tune large edos through software tuning, one may take advantage of [[MIDI]] channels. See [[Tuning per channel]].

All of these tools are also applicable to equal divisions of other ([[nonoctave]]) intervals as well.

=== What's the difference between EDOs and Equal Temperaments? ===
See [[EDO vs ET]].

== Individual pages for EDOs ==
=== 0…999 ===
{| class="wikitable center-all mw-collapsible"
|+ style="font-size: 105%; white-space: nowrap;" | 0…99
|-
| [[0edo|0]]
| [[1edo|1]]
| [[2edo|2]]
| [[3edo|3]]
| [[4edo|4]]
| [[5edo|5]]
| [[6edo|6]]
| [[7edo|7]]
| [[8edo|8]]
| [[9edo|9]]
|-
| [[10edo|10]]
| [[11edo|11]]
| [[12edo|12]]
| [[13edo|13]]
| [[14edo|14]]
| [[15edo|15]]
| [[16edo|16]]
| [[17edo|17]]
| [[18edo|18]]
| [[19edo|19]]
|-
| [[20edo|20]]
| [[21edo|21]]
| [[22edo|22]]
| [[23edo|23]]
| [[24edo|24]]
| [[25edo|25]]
| [[26edo|26]]
| [[27edo|27]]
| [[28edo|28]]
| [[29edo|29]]
|-
| [[30edo|30]]
| [[31edo|31]]
| [[32edo|32]]
| [[33edo|33]]
| [[34edo|34]]
| [[35edo|35]]
| [[36edo|36]]
| [[37edo|37]]
| [[38edo|38]]
| [[39edo|39]]
|-
| [[40edo|40]]
| [[41edo|41]]
| [[42edo|42]]
| [[43edo|43]]
| [[44edo|44]]
| [[45edo|45]]
| [[46edo|46]]
| [[47edo|47]]
| [[48edo|48]]
| [[49edo|49]]
|-
| [[50edo|50]]
| [[51edo|51]]
| [[52edo|52]]
| [[53edo|53]]
| [[54edo|54]]
| [[55edo|55]]
| [[56edo|56]]
| [[57edo|57]]
| [[58edo|58]]
| [[59edo|59]]
|-
| [[60edo|60]]
| [[61edo|61]]
| [[62edo|62]]
| [[63edo|63]]
| [[64edo|64]]
| [[65edo|65]]
| [[66edo|66]]
| [[67edo|67]]
| [[68edo|68]]
| [[69edo|69]]
|-
| [[70edo|70]]
| [[71edo|71]]
| [[72edo|72]]
| [[73edo|73]]
| [[74edo|74]]
| [[75edo|75]]
| [[76edo|76]]
| [[77edo|77]]
| [[78edo|78]]
| [[79edo|79]]
|-
| [[80edo|80]]
| [[81edo|81]]
| [[82edo|82]]
| [[83edo|83]]
| [[84edo|84]]
| [[85edo|85]]
| [[86edo|86]]
| [[87edo|87]]
| [[88edo|88]]
| [[89edo|89]]
|-
| [[90edo|90]]
| [[91edo|91]]
| [[92edo|92]]
| [[93edo|93]]
| [[94edo|94]]
| [[95edo|95]]
| [[96edo|96]]
| [[97edo|97]]
| [[98edo|98]]
| [[99edo|99]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 100…199
|-
| [[100edo|100]]
| [[101edo|101]]
| [[102edo|102]]
| [[103edo|103]]
| [[104edo|104]]
| [[105edo|105]]
| [[106edo|106]]
| [[107edo|107]]
| [[108edo|108]]
| [[109edo|109]]
|-
| [[110edo|110]]
| [[111edo|111]]
| [[112edo|112]]
| [[113edo|113]]
| [[114edo|114]]
| [[115edo|115]]
| [[116edo|116]]
| [[117edo|117]]
| [[118edo|118]]
| [[119edo|119]]
|-
| [[120edo|120]]
| [[121edo|121]]
| [[122edo|122]]
| [[123edo|123]]
| [[124edo|124]]
| [[125edo|125]]
| [[126edo|126]]
| [[127edo|127]]
| [[128edo|128]]
| [[129edo|129]]
|-
| [[130edo|130]]
| [[131edo|131]]
| [[132edo|132]]
| [[133edo|133]]
| [[134edo|134]]
| [[135edo|135]]
| [[136edo|136]]
| [[137edo|137]]
| [[138edo|138]]
| [[139edo|139]]
|-
| [[140edo|140]]
| [[141edo|141]]
| [[142edo|142]]
| [[143edo|143]]
| [[144edo|144]]
| [[145edo|145]]
| [[146edo|146]]
| [[147edo|147]]
| [[148edo|148]]
| [[149edo|149]]
|-
| [[150edo|150]]
| [[151edo|151]]
| [[152edo|152]]
| [[153edo|153]]
| [[154edo|154]]
| [[155edo|155]]
| [[156edo|156]]
| [[157edo|157]]
| [[158edo|158]]
| [[159edo|159]]
|-
| [[160edo|160]]
| [[161edo|161]]
| [[162edo|162]]
| [[163edo|163]]
| [[164edo|164]]
| [[165edo|165]]
| [[166edo|166]]
| [[167edo|167]]
| [[168edo|168]]
| [[169edo|169]]
|-
| [[170edo|170]]
| [[171edo|171]]
| [[172edo|172]]
| [[173edo|173]]
| [[174edo|174]]
| [[175edo|175]]
| [[176edo|176]]
| [[177edo|177]]
| [[178edo|178]]
| [[179edo|179]]
|-
| [[180edo|180]]
| [[181edo|181]]
| [[182edo|182]]
| [[183edo|183]]
| [[184edo|184]]
| [[185edo|185]]
| [[186edo|186]]
| [[187edo|187]]
| [[188edo|188]]
| [[189edo|189]]
|-
| [[190edo|190]]
| [[191edo|191]]
| [[192edo|192]]
| [[193edo|193]]
| [[194edo|194]]
| [[195edo|195]]
| [[196edo|196]]
| [[197edo|197]]
| [[198edo|198]]
| [[199edo|199]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 200…299
|-
| [[200edo|200]]
| [[201edo|201]]
| [[202edo|202]]
| [[203edo|203]]
| [[204edo|204]]
| [[205edo|205]]
| [[206edo|206]]
| [[207edo|207]]
| [[208edo|208]]
| [[209edo|209]]
|-
| [[210edo|210]]
| [[211edo|211]]
| [[212edo|212]]
| [[213edo|213]]
| [[214edo|214]]
| [[215edo|215]]
| [[216edo|216]]
| [[217edo|217]]
| [[218edo|218]]
| [[219edo|219]]
|-
| [[220edo|220]]
| [[221edo|221]]
| [[222edo|222]]
| [[223edo|223]]
| [[224edo|224]]
| [[225edo|225]]
| [[226edo|226]]
| [[227edo|227]]
| [[228edo|228]]
| [[229edo|229]]
|-
| [[230edo|230]]
| [[231edo|231]]
| [[232edo|232]]
| [[233edo|233]]
| [[234edo|234]]
| [[235edo|235]]
| [[236edo|236]]
| [[237edo|237]]
| [[238edo|238]]
| [[239edo|239]]
|-
| [[240edo|240]]
| [[241edo|241]]
| [[242edo|242]]
| [[243edo|243]]
| [[244edo|244]]
| [[245edo|245]]
| [[246edo|246]]
| [[247edo|247]]
| [[248edo|248]]
| [[249edo|249]]
|-
| [[250edo|250]]
| [[251edo|251]]
| [[252edo|252]]
| [[253edo|253]]
| [[254edo|254]]
| [[255edo|255]]
| [[256edo|256]]
| [[257edo|257]]
| [[258edo|258]]
| [[259edo|259]]
|-
| [[260edo|260]]
| [[261edo|261]]
| [[262edo|262]]
| [[263edo|263]]
| [[264edo|264]]
| [[265edo|265]]
| [[266edo|266]]
| [[267edo|267]]
| [[268edo|268]]
| [[269edo|269]]
|-
| [[270edo|270]]
| [[271edo|271]]
| [[272edo|272]]
| [[273edo|273]]
| [[274edo|274]]
| [[275edo|275]]
| [[276edo|276]]
| [[277edo|277]]
| [[278edo|278]]
| [[279edo|279]]
|-
| [[280edo|280]]
| [[281edo|281]]
| [[282edo|282]]
| [[283edo|283]]
| [[284edo|284]]
| [[285edo|285]]
| [[286edo|286]]
| [[287edo|287]]
| [[288edo|288]]
| [[289edo|289]]
|-
| [[290edo|290]]
| [[291edo|291]]
| [[292edo|292]]
| [[293edo|293]]
| [[294edo|294]]
| [[295edo|295]]
| [[296edo|296]]
| [[297edo|297]]
| [[298edo|298]]
| [[299edo|299]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 300…399
|-
| [[300edo|300]]
| [[301edo|301]]
| [[302edo|302]]
| [[303edo|303]]
| [[304edo|304]]
| [[305edo|305]]
| [[306edo|306]]
| [[307edo|307]]
| [[308edo|308]]
| [[309edo|309]]
|-
| [[310edo|310]]
| [[311edo|311]]
| [[312edo|312]]
| [[313edo|313]]
| [[314edo|314]]
| [[315edo|315]]
| [[316edo|316]]
| [[317edo|317]]
| [[318edo|318]]
| [[319edo|319]]
|-
| [[320edo|320]]
| [[321edo|321]]
| [[322edo|322]]
| [[323edo|323]]
| [[324edo|324]]
| [[325edo|325]]
| [[326edo|326]]
| [[327edo|327]]
| [[328edo|328]]
| [[329edo|329]]
|-
| [[330edo|330]]
| [[331edo|331]]
| [[332edo|332]]
| [[333edo|333]]
| [[334edo|334]]
| [[335edo|335]]
| [[336edo|336]]
| [[337edo|337]]
| [[338edo|338]]
| [[339edo|339]]
|-
| [[340edo|340]]
| [[341edo|341]]
| [[342edo|342]]
| [[343edo|343]]
| [[344edo|344]]
| [[345edo|345]]
| [[346edo|346]]
| [[347edo|347]]
| [[348edo|348]]
| [[349edo|349]]
|-
| [[350edo|350]]
| [[351edo|351]]
| [[352edo|352]]
| [[353edo|353]]
| [[354edo|354]]
| [[355edo|355]]
| [[356edo|356]]
| [[357edo|357]]
| [[358edo|358]]
| [[359edo|359]]
|-
| [[360edo|360]]
| [[361edo|361]]
| [[362edo|362]]
| [[363edo|363]]
| [[364edo|364]]
| [[365edo|365]]
| [[366edo|366]]
| [[367edo|367]]
| [[368edo|368]]
| [[369edo|369]]
|-
| [[370edo|370]]
| [[371edo|371]]
| [[372edo|372]]
| [[373edo|373]]
| [[374edo|374]]
| [[375edo|375]]
| [[376edo|376]]
| [[377edo|377]]
| [[378edo|378]]
| [[379edo|379]]
|-
| [[380edo|380]]
| [[381edo|381]]
| [[382edo|382]]
| [[383edo|383]]
| [[384edo|384]]
| [[385edo|385]]
| [[386edo|386]]
| [[387edo|387]]
| [[388edo|388]]
| [[389edo|389]]
|-
| [[390edo|390]]
| [[391edo|391]]
| [[392edo|392]]
| [[393edo|393]]
| [[394edo|394]]
| [[395edo|395]]
| [[396edo|396]]
| [[397edo|397]]
| [[398edo|398]]
| [[399edo|399]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 400…499
|-
| [[400edo|400]]
| [[401edo|401]]
| [[402edo|402]]
| [[403edo|403]]
| [[404edo|404]]
| [[405edo|405]]
| [[406edo|406]]
| [[407edo|407]]
| [[408edo|408]]
| [[409edo|409]]
|-
| [[410edo|410]]
| [[411edo|411]]
| [[412edo|412]]
| [[413edo|413]]
| [[414edo|414]]
| [[415edo|415]]
| [[416edo|416]]
| [[417edo|417]]
| [[418edo|418]]
| [[419edo|419]]
|-
| [[420edo|420]]
| [[421edo|421]]
| [[422edo|422]]
| [[423edo|423]]
| [[424edo|424]]
| [[425edo|425]]
| [[426edo|426]]
| [[427edo|427]]
| [[428edo|428]]
| [[429edo|429]]
|-
| [[430edo|430]]
| [[431edo|431]]
| [[432edo|432]]
| [[433edo|433]]
| [[434edo|434]]
| [[435edo|435]]
| [[436edo|436]]
| [[437edo|437]]
| [[438edo|438]]
| [[439edo|439]]
|-
| [[440edo|440]]
| [[441edo|441]]
| [[442edo|442]]
| [[443edo|443]]
| [[444edo|444]]
| [[445edo|445]]
| [[446edo|446]]
| [[447edo|447]]
| [[448edo|448]]
| [[449edo|449]]
|-
| [[450edo|450]]
| [[451edo|451]]
| [[452edo|452]]
| [[453edo|453]]
| [[454edo|454]]
| [[455edo|455]]
| [[456edo|456]]
| [[457edo|457]]
| [[458edo|458]]
| [[459edo|459]]
|-
| [[460edo|460]]
| [[461edo|461]]
| [[462edo|462]]
| [[463edo|463]]
| [[464edo|464]]
| [[465edo|465]]
| [[466edo|466]]
| [[467edo|467]]
| [[468edo|468]]
| [[469edo|469]]
|-
| [[470edo|470]]
| [[471edo|471]]
| [[472edo|472]]
| [[473edo|473]]
| [[474edo|474]]
| [[475edo|475]]
| [[476edo|476]]
| [[477edo|477]]
| [[478edo|478]]
| [[479edo|479]]
|-
| [[480edo|480]]
| [[481edo|481]]
| [[482edo|482]]
| [[483edo|483]]
| [[484edo|484]]
| [[485edo|485]]
| [[486edo|486]]
| [[487edo|487]]
| [[488edo|488]]
| [[489edo|489]]
|-
| [[490edo|490]]
| [[491edo|491]]
| [[492edo|492]]
| [[493edo|493]]
| [[494edo|494]]
| [[495edo|495]]
| [[496edo|496]]
| [[497edo|497]]
| [[498edo|498]]
| [[499edo|499]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 500…599
|-
| [[500edo|500]]
| [[501edo|501]]
| [[502edo|502]]
| [[503edo|503]]
| [[504edo|504]]
| [[505edo|505]]
| [[506edo|506]]
| [[507edo|507]]
| [[508edo|508]]
| [[509edo|509]]
|-
| [[510edo|510]]
| [[511edo|511]]
| [[512edo|512]]
| [[513edo|513]]
| [[514edo|514]]
| [[515edo|515]]
| [[516edo|516]]
| [[517edo|517]]
| [[518edo|518]]
| [[519edo|519]]
|-
| [[520edo|520]]
| [[521edo|521]]
| [[522edo|522]]
| [[523edo|523]]
| [[524edo|524]]
| [[525edo|525]]
| [[526edo|526]]
| [[527edo|527]]
| [[528edo|528]]
| [[529edo|529]]
|-
| [[530edo|530]]
| [[531edo|531]]
| [[532edo|532]]
| [[533edo|533]]
| [[534edo|534]]
| [[535edo|535]]
| [[536edo|536]]
| [[537edo|537]]
| [[538edo|538]]
| [[539edo|539]]
|-
| [[540edo|540]]
| [[541edo|541]]
| [[542edo|542]]
| [[543edo|543]]
| [[544edo|544]]
| [[545edo|545]]
| [[546edo|546]]
| [[547edo|547]]
| [[548edo|548]]
| [[549edo|549]]
|-
| [[550edo|550]]
| [[551edo|551]]
| [[552edo|552]]
| [[553edo|553]]
| [[554edo|554]]
| [[555edo|555]]
| [[556edo|556]]
| [[557edo|557]]
| [[558edo|558]]
| [[559edo|559]]
|-
| [[560edo|560]]
| [[561edo|561]]
| [[562edo|562]]
| [[563edo|563]]
| [[564edo|564]]
| [[565edo|565]]
| [[566edo|566]]
| [[567edo|567]]
| [[568edo|568]]
| [[569edo|569]]
|-
| [[570edo|570]]
| [[571edo|571]]
| [[572edo|572]]
| [[573edo|573]]
| [[574edo|574]]
| [[575edo|575]]
| [[576edo|576]]
| [[577edo|577]]
| [[578edo|578]]
| [[579edo|579]]
|-
| [[580edo|580]]
| [[581edo|581]]
| [[582edo|582]]
| [[583edo|583]]
| [[584edo|584]]
| [[585edo|585]]
| [[586edo|586]]
| [[587edo|587]]
| [[588edo|588]]
| [[589edo|589]]
|-
| [[590edo|590]]
| [[591edo|591]]
| [[592edo|592]]
| [[593edo|593]]
| [[594edo|594]]
| [[595edo|595]]
| [[596edo|596]]
| [[597edo|597]]
| [[598edo|598]]
| [[599edo|599]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 600…699
|-
| [[600edo|600]]
| [[601edo|601]]
| [[602edo|602]]
| [[603edo|603]]
| [[604edo|604]]
| [[605edo|605]]
| [[606edo|606]]
| [[607edo|607]]
| [[608edo|608]]
| [[609edo|609]]
|-
| [[610edo|610]]
| [[611edo|611]]
| [[612edo|612]]
| [[613edo|613]]
| [[614edo|614]]
| [[615edo|615]]
| [[616edo|616]]
| [[617edo|617]]
| [[618edo|618]]
| [[619edo|619]]
|-
| [[620edo|620]]
| [[621edo|621]]
| [[622edo|622]]
| [[623edo|623]]
| [[624edo|624]]
| [[625edo|625]]
| [[626edo|626]]
| [[627edo|627]]
| [[628edo|628]]
| [[629edo|629]]
|-
| [[630edo|630]]
| [[631edo|631]]
| [[632edo|632]]
| [[633edo|633]]
| [[634edo|634]]
| [[635edo|635]]
| [[636edo|636]]
| [[637edo|637]]
| [[638edo|638]]
| [[639edo|639]]
|-
| [[640edo|640]]
| [[641edo|641]]
| [[642edo|642]]
| [[643edo|643]]
| [[644edo|644]]
| [[645edo|645]]
| [[646edo|646]]
| [[647edo|647]]
| [[648edo|648]]
| [[649edo|649]]
|-
| [[650edo|650]]
| [[651edo|651]]
| [[652edo|652]]
| [[653edo|653]]
| [[654edo|654]]
| [[655edo|655]]
| [[656edo|656]]
| [[657edo|657]]
| [[658edo|658]]
| [[659edo|659]]
|-
| [[660edo|660]]
| [[661edo|661]]
| [[662edo|662]]
| [[663edo|663]]
| [[664edo|664]]
| [[665edo|665]]
| [[666edo|666]]
| [[667edo|667]]
| [[668edo|668]]
| [[669edo|669]]
|-
| [[670edo|670]]
| [[671edo|671]]
| [[672edo|672]]
| [[673edo|673]]
| [[674edo|674]]
| [[675edo|675]]
| [[676edo|676]]
| [[677edo|677]]
| [[678edo|678]]
| [[679edo|679]]
|-
| [[680edo|680]]
| [[681edo|681]]
| [[682edo|682]]
| [[683edo|683]]
| [[684edo|684]]
| [[685edo|685]]
| [[686edo|686]]
| [[687edo|687]]
| [[688edo|688]]
| [[689edo|689]]
|-
| [[690edo|690]]
| [[691edo|691]]
| [[692edo|692]]
| [[693edo|693]]
| [[694edo|694]]
| [[695edo|695]]
| [[696edo|696]]
| [[697edo|697]]
| [[698edo|698]]
| [[699edo|699]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 700…799
|-
| [[700edo|700]]
| [[701edo|701]]
| [[702edo|702]]
| [[703edo|703]]
| [[704edo|704]]
| [[705edo|705]]
| [[706edo|706]]
| [[707edo|707]]
| [[708edo|708]]
| [[709edo|709]]
|-
| [[710edo|710]]
| [[711edo|711]]
| [[712edo|712]]
| [[713edo|713]]
| [[714edo|714]]
| [[715edo|715]]
| [[716edo|716]]
| [[717edo|717]]
| [[718edo|718]]
| [[719edo|719]]
|-
| [[720edo|720]]
| [[721edo|721]]
| [[722edo|722]]
| [[723edo|723]]
| [[724edo|724]]
| [[725edo|725]]
| [[726edo|726]]
| [[727edo|727]]
| [[728edo|728]]
| [[729edo|729]]
|-
| [[730edo|730]]
| [[731edo|731]]
| [[732edo|732]]
| [[733edo|733]]
| [[734edo|734]]
| [[735edo|735]]
| [[736edo|736]]
| [[737edo|737]]
| [[738edo|738]]
| [[739edo|739]]
|-
| [[740edo|740]]
| [[741edo|741]]
| [[742edo|742]]
| [[743edo|743]]
| [[744edo|744]]
| [[745edo|745]]
| [[746edo|746]]
| [[747edo|747]]
| [[748edo|748]]
| [[749edo|749]]
|-
| [[750edo|750]]
| [[751edo|751]]
| [[752edo|752]]
| [[753edo|753]]
| [[754edo|754]]
| [[755edo|755]]
| [[756edo|756]]
| [[757edo|757]]
| [[758edo|758]]
| [[759edo|759]]
|-
| [[760edo|760]]
| [[761edo|761]]
| [[762edo|762]]
| [[763edo|763]]
| [[764edo|764]]
| [[765edo|765]]
| [[766edo|766]]
| [[767edo|767]]
| [[768edo|768]]
| [[769edo|769]]
|-
| [[770edo|770]]
| [[771edo|771]]
| [[772edo|772]]
| [[773edo|773]]
| [[774edo|774]]
| [[775edo|775]]
| [[776edo|776]]
| [[777edo|777]]
| [[778edo|778]]
| [[779edo|779]]
|-
| [[780edo|780]]
| [[781edo|781]]
| [[782edo|782]]
| [[783edo|783]]
| [[784edo|784]]
| [[785edo|785]]
| [[786edo|786]]
| [[787edo|787]]
| [[788edo|788]]
| [[789edo|789]]
|-
| [[790edo|790]]
| [[791edo|791]]
| [[792edo|792]]
| [[793edo|793]]
| [[794edo|794]]
| [[795edo|795]]
| [[796edo|796]]
| [[797edo|797]]
| [[798edo|798]]
| [[799edo|799]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 800…899
|-
| [[800edo|800]]
| [[801edo|801]]
| [[802edo|802]]
| [[803edo|803]]
| [[804edo|804]]
| [[805edo|805]]
| [[806edo|806]]
| [[807edo|807]]
| [[808edo|808]]
| [[809edo|809]]
|-
| [[810edo|810]]
| [[811edo|811]]
| [[812edo|812]]
| [[813edo|813]]
| [[814edo|814]]
| [[815edo|815]]
| [[816edo|816]]
| [[817edo|817]]
| [[818edo|818]]
| [[819edo|819]]
|-
| [[820edo|820]]
| [[821edo|821]]
| [[822edo|822]]
| [[823edo|823]]
| [[824edo|824]]
| [[825edo|825]]
| [[826edo|826]]
| [[827edo|827]]
| [[828edo|828]]
| [[829edo|829]]
|-
| [[830edo|830]]
| [[831edo|831]]
| [[832edo|832]]
| [[833edo|833]]
| [[834edo|834]]
| [[835edo|835]]
| [[836edo|836]]
| [[837edo|837]]
| [[838edo|838]]
| [[839edo|839]]
|-
| [[840edo|840]]
| [[841edo|841]]
| [[842edo|842]]
| [[843edo|843]]
| [[844edo|844]]
| [[845edo|845]]
| [[846edo|846]]
| [[847edo|847]]
| [[848edo|848]]
| [[849edo|849]]
|-
| [[850edo|850]]
| [[851edo|851]]
| [[852edo|852]]
| [[853edo|853]]
| [[854edo|854]]
| [[855edo|855]]
| [[856edo|856]]
| [[857edo|857]]
| [[858edo|858]]
| [[859edo|859]]
|-
| [[860edo|860]]
| [[861edo|861]]
| [[862edo|862]]
| [[863edo|863]]
| [[864edo|864]]
| [[865edo|865]]
| [[866edo|866]]
| [[867edo|867]]
| [[868edo|868]]
| [[869edo|869]]
|-
| [[870edo|870]]
| [[871edo|871]]
| [[872edo|872]]
| [[873edo|873]]
| [[874edo|874]]
| [[875edo|875]]
| [[876edo|876]]
| [[877edo|877]]
| [[878edo|878]]
| [[879edo|879]]
|-
| [[880edo|880]]
| [[881edo|881]]
| [[882edo|882]]
| [[883edo|883]]
| [[884edo|884]]
| [[885edo|885]]
| [[886edo|886]]
| [[887edo|887]]
| [[888edo|888]]
| [[889edo|889]]
|-
| [[890edo|890]]
| [[891edo|891]]
| [[892edo|892]]
| [[893edo|893]]
| [[894edo|894]]
| [[895edo|895]]
| [[896edo|896]]
| [[897edo|897]]
| [[898edo|898]]
| [[899edo|899]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 900…999
|-
| [[900edo|900]]
| [[901edo|901]]
| [[902edo|902]]
| [[903edo|903]]
| [[904edo|904]]
| [[905edo|905]]
| [[906edo|906]]
| [[907edo|907]]
| [[908edo|908]]
| [[909edo|909]]
|-
| [[910edo|910]]
| [[911edo|911]]
| [[912edo|912]]
| [[913edo|913]]
| [[914edo|914]]
| [[915edo|915]]
| [[916edo|916]]
| [[917edo|917]]
| [[918edo|918]]
| [[919edo|919]]
|-
| [[920edo|920]]
| [[921edo|921]]
| [[922edo|922]]
| [[923edo|923]]
| [[924edo|924]]
| [[925edo|925]]
| [[926edo|926]]
| [[927edo|927]]
| [[928edo|928]]
| [[929edo|929]]
|-
| [[930edo|930]]
| [[931edo|931]]
| [[932edo|932]]
| [[933edo|933]]
| [[934edo|934]]
| [[935edo|935]]
| [[936edo|936]]
| [[937edo|937]]
| [[938edo|938]]
| [[939edo|939]]
|-
| [[940edo|940]]
| [[941edo|941]]
| [[942edo|942]]
| [[943edo|943]]
| [[944edo|944]]
| [[945edo|945]]
| [[946edo|946]]
| [[947edo|947]]
| [[948edo|948]]
| [[949edo|949]]
|-
| [[950edo|950]]
| [[951edo|951]]
| [[952edo|952]]
| [[953edo|953]]
| [[954edo|954]]
| [[955edo|955]]
| [[956edo|956]]
| [[957edo|957]]
| [[958edo|958]]
| [[959edo|959]]
|-
| [[960edo|960]]
| [[961edo|961]]
| [[962edo|962]]
| [[963edo|963]]
| [[964edo|964]]
| [[965edo|965]]
| [[966edo|966]]
| [[967edo|967]]
| [[968edo|968]]
| [[969edo|969]]
|-
| [[970edo|970]]
| [[971edo|971]]
| [[972edo|972]]
| [[973edo|973]]
| [[974edo|974]]
| [[975edo|975]]
| [[976edo|976]]
| [[977edo|977]]
| [[978edo|978]]
| [[979edo|979]]
|-
| [[980edo|980]]
| [[981edo|981]]
| [[982edo|982]]
| [[983edo|983]]
| [[984edo|984]]
| [[985edo|985]]
| [[986edo|986]]
| [[987edo|987]]
| [[988edo|988]]
| [[989edo|989]]
|-
| [[990edo|990]]
| [[991edo|991]]
| [[992edo|992]]
| [[993edo|993]]
| [[994edo|994]]
| [[995edo|995]]
| [[996edo|996]]
| [[997edo|997]]
| [[998edo|998]]
| [[999edo|999]]
|}

=== 1000…1999 ===
{{EDOs
| 1000, 1001, 1012, 1015, 1019, 1051, 1053, 1059, 1063, 1065, 1080, 1092, 1106, 1125, 1131, 1147, 1152, 1166, 1171, 1178, 1193, 1200, 1210, 1224, 1230, 1236, 1240, 1244, 1260, 1272, 1277, 1289, 1308, 1323, 1330, 1337, 1342, 1361, 1381, 1395, 1407, 1419, 1429, 1440, 1448, 1489, 1506, 1517, 1520, 1525, 1536, 1547, 1553, 1554, 1559, 1578, 1583, 1590, 1600, 1609, 1612, 1619, 1637, 1641, 1643, 1650, 1665, 1672, 1700, 1730, 1759, 1776, 1778, 1783, 1789, 1793, 1794, 1802, 1803, 1817, 1848, 1861, 1879, 1880, 1889, 1911, 1920, 1944, 1955, 1957, 1983, 1984 }}

=== 2000…9999 ===
{{EDOs
| 2000, 2016, 2019, 2022, 2023, 2024, 2025, 2029, 2048, 2053, 2072, 2100, 2101, 2113, 2118, 2129, 2153, 2190, 2200, 2207, 2242, 2320, 2444, 2460, 2477, 2513, 2520, 2544, 2549, 2554, 2579, 2619, 2684, 2711, 2730, 2777, 2809, 2814, 2897, 2901, 2912, 2960, 2964, 3071, 3072, 3079, 3080, 3125, 3178, 3361, 3395, 3422, 3476, 3498, 3558, 3566, 3578, 3600, 3643, 3684, 3696, 3776, 3889, 3920, 4004, 4007, 4079, 4096, 4172, 4190, 4231, 4296, 4320, 4327, 4349, 4380, 4501, 4650, 4973, 5040, 5280, 5544, 5585, 5809, 5902, 5941, 6079, 6349, 6380, 6650, 6664, 6691, 7033, 7315, 7980, 8103, 8192, 8269, 8404, 8539, 8736, 8745
}}

=== 10000 and up ===
{{EDOs
| 10009, 10459, 10600, 10729, 11664, 12276, 12348, 12500, 13382, 14124, 14348, 14618, 14842, 15601, 15900, 16218, 16384, 16625, 16808, 17461, 18355, 20203, 20567, 25281, 25282, 28000, 28742, 30103, 30631, 31867, 31920, 32436, 32768, 33616, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 99693, 99694, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 571611, 762148, 1714833, 1905370, 2547047, 2667518, 2901533, 4191814, 6000000, 11358058, 402653184, 5407372813
}}

== Non-integer EDO ==
A non-integer EDO can be defined as using a non-integer divisor to divide the octave. Typically, non-integer EDOs are understood as ''not'' containing the exact octave, so that they remain [[equal tuning]]s. If the exact octave is retained and if the generator resets itself at each period, then this results in a [[MOS scale]] with only 1 small step.

All fractional EDOs are integer equal divisions of another integer interval. For example, (25/2)edo is equivalent to 25ed4. In general:

<math>\displaystyle (p/q) \text{edo} = p \text{-ed} 2^q</math>

for integers ''p'' and ''q''. Many irrational EDOs cannot be converted to integer equal divisions of another integer interval, so they are things of their own.

Non-integer EDOs can be written in decimal form, such as 12.1edo. This is often meant to be approximate, used in the context of [[octave stretch]] of an integer EDO, rather than as a fractional EDO.

== Scale tree ==

The scale tree, or Stern-Brocot tree, provides a visual map of the world of EDOs, based on fifth size.

[[File:The_Scale_Tree.png|alt=The Scale Tree.png|800x1023px|The Scale Tree.png]]

The regular EDOs, up to 72edo:

[[File:Scale_Tree_close-up.png|alt=Scale Tree close-up.png|Scale Tree close-up.png]]

== Pergens ==
{{See also| Pergen #Pergens and EDOs }}

[[Pergen]]s provide a JI-agnostic way to name the rank-2 scales of an EDO. This table lists every possible period/generator combination for EDOs 5–24, and for each coprime combination, the simplest pergen that it can represent. Non-coprime combinations such as {{nowrap|P = 6\12}}, {{nowrap|G = 4\12}} are marked as "-".

{| class="wikitable"
|-
! EDO
! Period
! colspan="11" | Generator in EDO steps
|-
!
! in EDO steps
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
|-
! 5
! 5 = P8
| P4/2
| P5
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 6
! 6 = P8
| P4/2
| -
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/2
| P5
|
|
|
|
|
|
|
|
|
|
|-
! 7
! 7 = P8
| P4/3
| P5/2
| P5
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 8
! 8 = P8
| P4/3
| -
| P5
|
|
|
|
|
|
|
|
|-
! 4 = P8/2
| P5
| -
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 9
! 9 = P8
| P4/4
| P4/2
| -
| P5
|
|
|
|
|
|
|
|-
! 3 = P8/3
| P5
|
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 10
! 10 = P8
| P4/4
| -
| P5/2
| -
|
|
|
|
|
|
|
|-
! 5 = P8/2
| P5
| P4/2
|
|
|
|
|
|
|
|
|
|-
! 11
! 11 = P8
| P4/5
| P5/3
| P5/2
| P11/4
| P5
|
|
|
|
|
|
|-
! rowspan="4" | 12
! 12 = P8
| P4/5
| -
| -
| -
| P5
|
|
|
|
|
|
|-
! 6 = P8/2
| P5
| -
| -
|
|
|
|
|
|
|
|
|-
! 4 = P8/3
| P5
| -
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/4
| P5
|
|
|
|
|
|
|
|
|
|
|-
! 13b
! 13 = P8
| P4/6
| P4/3
| P4/2
| P12/5
| P12/4
| P5
|
|
|
|
|
|-
! rowspan="2" | 14
! 14 = P8
| P4/6
| -
| P4/2
| -
| P11/4
| -
|
|
|
|
|
|-
! 7 = P8/2
| P5
| P4/3
| P4/2
|
|
|
|
|
|
|
|
|-
! rowspan="3" | 15
! 15 = P8
| P4/6
| P4/3
| -
| P12/6
| -
| -
| P11/3
|
|
|
|
|-
! 5 = P8/3
| P5
| P4/2
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/5
| P4/3
|
|
|
|
|
|
|
|
|
|
|-
! rowspan="3" | 16
! 16 = P8
| P4/7
| -
| P5/3
| -
| P12/5
| -
| P5
|
|
|
|
|-
! 8 = P8/2
| P5
| -
| P5/3
| -
|
|
|
|
|
|
|
|-
! 4 = P8/4
| P5
| -
|
|
|
|
|
|
|
|
|
|-
! 17
! 17 = P8
| P4/7
| P5/5
| P11/8
| P11/6
| P5/2
| P11/4
| P5
| P11/3
|
|
|
|-
! rowspan="4" | 18b
! 18 = P8
| P4/8
| -
| -
| -
| P5/2
| -
| P12/4
| -
|
|
|
|-
! 9 = P8/2
| P5
| P4/4
| -
| P4/2
|
|
|
|
|
|
|
|-
! 6 = P8/3
| P5/2
| -
| -
|
|
|
|
|
|
|
|
|-
! 3 = P8/6
| P5
|
|
|
|
|
|
|
|
|
|
|-
! 19
! 19 = P8
| P4/8
| P4/4
| P11/9
| P4/2
| P12/6
| P12/5
| ccP5/7
| P5
| P11/3
|
|
|-
! rowspan="4" | 20
! 20 = P8
| P4/8
| -
| P5/4
| -
| -
| -
| P11/4
| -
| c3P5/8
|
|
|-
! 10 = P8/2
| M2/4
| -
| P5/4
| -
| -
|
|
|
|
|
|
|-
! 5 = P8/4
| P4/2
| P5
|
|
|
|
|
|
|
|
|
|-
! 4 = P8/5
| P5/4
| -
|
|
|
|
|
|
|
|
|
|-
! rowspan="3" | 21
! 21 = P8
| P4/9
| P5/6
| -
| P5/3
| P11/6
| -
| -
| c3P4/9
| -
| P11/3
|
|-
! 7 = P8/3
| P5/2
| P5
| P4/3
|
|
|
|
|
|
|
|
|-
! 3 = P8/7
| P5/3
|
|
|
|
|
|
|
|
|
|
|-
! rowspan="2" | 22
! 22 = P8
| P4/9
| -
| P4/3
| -
| P12/7
| -
| P12/5
| -
| P5
| -
|
|-
! 11 = P8/2
| M2/4
| P5
| P4/3
| P12/5
| P12/7
|
|
|
|
|
|
|-
! 23
! 23 = P8
| P4/10
| P4/5
| P11/11
| P12/9
| P4/2
| P12/6
| ccP4/8
| ccP4/7
| P12/4
| P5
| P11/3
|-
! rowspan="6" | 24
! 24 = P8
| P4/10
| -
| -
| -
| P4/2
| -
| P5/2
| -
| -
| -
| c4P5/10
|-
! 12 = P8/2
| M2/4
| -
| -
| -
| P4/2
|
|
|
|
|
|
|-
! 8 = P8/3
| P5/2
| -
| P4/2
|
|
|
|
|
|
|
|
|-
! 6 = P8/4
| P4/2
| -
| -
|
|
|
|
|
|
|
|
|-
! 4 = P8/6
| P4/2
| -
|
|
|
|
|
|
|
|
|
|-
! 3 = P8/8
| P5
|
|
|
|
|
|
|
|
|
|
|-
!
!
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
|}

== Links and articles ==
* [[Alternative Names for EDOs]]
* [[Chuckles McGee's EDO personalities]]
* [[Collection of EDO impressions]]
* [[Consistency limits of small EDOs]]
* [[Distinct EDO Scales]]
* [[Expression to EDO calculator]]
* [[List of rank one temperaments by step size]]
* [[Macrotonal EDO]]
* [[Minimal consistent EDOs]]
* [[Monotonicity levels of small EDOs]]
* [[Relative errors of small EDOs]]
* [[Runoff|Runoff EDOs]]
* [[Absolute errors of small EDOs]]
* [https://www.webcitation.org/5xZz8RtQB Teen Tunes] by [[Ivor Darreg]]
* [[:Category:Equal divisions of the octave]]

== Notes ==
<references />

[[Category:Equal-step tuning]]
[[Category:Equal divisions of the octave| ]] 
[[Category:Acronyms]]
[[Category:Lists of scales]]

2573edo

2025-03-15T17:46:17Z

Eliora: Created page with "{{Infobox ET}} {{EDO intro|2573}} 2573edo is consistent in the 17-odd-limit, being a mostly flat system. It tunes the 31st-octave temperaments#217 & 1178|217 & 1178..."

{{Infobox ET}}
{{EDO intro|2573}}

2573edo is [[consistent]] in the [[17-odd-limit]], being a mostly flat system. It tunes the [[31st-octave temperaments#217 & 1178|217 & 1178]] temperament, for which it provides the [[optimal patent val]] in the 7, 11, 13, 17, and 19-limits (though it is not consistent to the 19-limit).

=== Prime harmonics ===
{{harmonics in equal|2573}}

=== Subsets and supersets ===

2573edo has [[31edo]] and [[83edo]] as subsets.