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{{interwiki
{{Interwiki
| en = 12edo
| en = 12edo
| de = 12-EDO
| de = 12-EDO
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== Theory ==
== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.
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.


The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is 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 chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone 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 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].
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. 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.


The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3<sup>12</sup>/2<sup>19</sup>]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these 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.
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 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.
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|3<sup>12</sup>/2<sup>19</sup>]], 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. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], 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.


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}
{{Harmonics in equal|12|prec=2}}
=== Octave stretch ===
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 the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 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.


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


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


== Intervals ==
== Intervals ==
Line 34: Line 37:
|+ style="font-size: 105%;" | Intervals of 12edo
|+ style="font-size: 105%;" | Intervals of 12edo
|-
|-
! rowspan="2" | [[Degree]]
! [[Degree]]
! rowspan="2" | [[Cent]]s
! [[Cent]]s
! rowspan="2" | [[Interval region]]
! [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals<ref group="note">{{sg|limit=2.3.5.7.17.19 subgroup}}</ref> ([[error]] in [[¢]])
! style="width: 165px;" | Approximated 5-limit<br>JI intervals (error in [[¢]])
! rowspan="2" | Audio
! 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>
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
|-
| 0
| 0
Line 49: Line 48:
| Unison (prime)
| Unison (prime)
| [[1/1]] (just)
| [[1/1]] (just)
|
|
|
| [[File:piano_0_1edo.mp3]]
| [[File:piano_0_1edo.mp3]]
|
|-
|-
| 1
| 1
| 100
| 100
| Minor second
| Minor second
|
| [[256/243]] (+9.775)<br>[[25/24]] (+29.328)<br>[[16/15]] (−11.731)
| [[25/24]] (+29.328)<br>[[16/15]] (−11.731)
| [[28/27]] (+37.039)<br>[[21/20]] (+15.533)<br>[[15/14]] (−19.443)
| [[18/17]] (+1.045)<br>[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)<br>[[17/16]] (−4.955), [[18/17]] (+1.045)<br>[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
|-
| 2
| 2
| 200
| 200
| Major second
| Major second
| [[9/8]] (−3.910)
| [[9/8]] (−3.910)<br>[[10/9]] (+17.596)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)<br>[[8/7]] (−31.174)
| [[19/17]] (+7.442)<br>[[55/49]] (+0.020)<br>[[64/57]] (−0.532)<br>[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)<br>[[17/15]] (−16.687), [[19/17]] (+7.442),<br>[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
|-
| 3
| 3
| 300
| 300
| Minor third
| Minor third
| [[32/27]] (+5.865)
| [[32/27]] (+5.865)<br>[[6/5]] (−15.641)<br>[[75/64]] (+25.418)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)<br>[[25/21]] (−1.847)
| [[19/16]] (+2.487)<br>[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)<br>[[19/16]] (+2.487)
|-
|-
| 4
| 4
| 400
| 400
| Major third
| Major third
| [[81/64]] (−7.820)
| [[81/64]] (−7.820)<br>[[5/4]] (+13.686)<br> [[32/25]] (-27.373)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)<br>[[9/7]] (−35.084)
| [[34/27]] (+0.910)<br>[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)<br>[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
|-
| 5
| 5
| 500
| 500
| Fourth
| Fourth
| [[4/3]] (+1.955)
| [[4/3]] (+1.955)<br> [[27/20]] (-19.551)
|
|
|
| [[File:piano_5_12edo.mp3]]
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
|-
| 6
| 6
| 600
| 600
| [[Tritone]]
| [[Tritone]]
|
| [[25/18]] (+31.283)<br>[[36/25]] (-31.283)<br>[[45/32]] (+9.776)<br>[[64/45]] (−9.776)
| [[45/32]] (+9.776)<br>[[64/45]] (−9.776)
| [[7/5]] (+17.488)<br>[[10/7]] (−17.488)
| [[24/17]] (+3.000)<br>[[99/70]] (−0.088)<br>[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)<br>[[24/17]] (+3.000), [[17/12]] (−3.000)<br>[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
|-
| 7
| 7
| 700
| 700
| Fifth
| Fifth
| [[3/2]] (−1.955)
| [[3/2]] (−1.955)<br>[[40/27]] (+19.551)
|
|
|
| [[File:piano_7_12edo.mp3]]
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
|-
| 8
| 8
| 800
| 800
| Minor sixth
| Minor sixth
| [[128/81]] (+7.820)
| [[128/81]] (+7.820)<br>[[8/5]] (−13.686)<br>[[25/16]] (+27.373)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)<br>[[100/63]] (+0.108)
| [[19/12]] (+4.442)<br>[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)<br>[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
|-
| 9
| 9
| 900
| 900
| Major sixth
| Major sixth
| [[27/16]] (−5.865)
| [[27/16]] (−5.865)<br>[[5/3]] (+15.641)<br>[[128/75]] (-25.418)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)<br>[[12/7]] (−33.129)
| [[37/22]] (−0.026)<br>[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)<br>[[32/19]] (−2.487)
|-
|-
| 10
| 10
| 1000
| 1000
| Minor seventh
| Minor seventh
| [[16/9]] (+3.910)
| [[16/9]] (+3.910)<br>[[9/5]] (−17.596)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)<br>[[25/14]] (−3.802)
| [[30/17]] (+16.687)<br>[[57/32]] (+0.532)<br>[[98/55]] (−0.020)<br>[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)<br>[[30/17]] (+16.687), [[34/19]] (−7.442)<br>[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
|-
| 11
| 11
| 1100
| 1100
| Major seventh
| Major seventh
|
| [[243/128]] (-9.775)<br>[[15/8]] (+11.731)<br>[[48/25]] (−29.328)
| [[15/8]] (+11.731)<br>[[48/25]] (−29.328)
| [[28/15]] (+19.443)<br>[[40/21]] (−15.533)<br>[[27/14]] (−37.039)
| [[32/17]] (+4.955)<br>[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)<br>[[32/17]] (+4.955), [[17/9]] (−1.045)<br>[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
|-
| 12
| 12
Line 157: Line 132:
| Octave
| Octave
| [[2/1]] (just)
| [[2/1]] (just)
|
|
|
| [[File:piano_1_1edo.mp3]]
| [[File:piano_1_1edo.mp3]]
|
|}
|}
<references group="note" />


== Notation ==
== Notation ==
12edo intervals and notes 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.
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}}
{{Sharpness-sharp1|12}}


{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].
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 diminished 2nd to it or subtracting one from it.
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"
{| class="wikitable center-all"
Line 256: Line 230:


==== Evo flavor ====
==== Evo flavor ====
<imagemap>
{{Sagittal chart|Evo}}
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
</imagemap>


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


==== Revo flavor ====
==== Revo flavor ====
<imagemap>
{{Sagittal chart}}
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
</imagemap>


== Solfege ==
== Solfege ==
Line 356: Line 318:
{{Q-odd-limit intervals|12}}
{{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}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}
=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}


== Regular temperament properties ==
== Regular temperament properties ==
Line 384: Line 332:
|-
|-
| 2.3
| 2.3
| {{monzo| -19 12 }}
| {{Monzo| -19 12 }}
| {{mapping| 12 19 }}
| {{Mapping| 12 19 }}
| +0.62
| +0.62
| 0.62
| 0.62
Line 392: Line 340:
| 2.3.5
| 2.3.5
| 81/80, 128/125
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| {{Mapping| 12 19 28 }}
| −1.56
| −1.56
| 3.11
| 3.11
Line 399: Line 347:
| 2.3.5.7
| 2.3.5.7
| 36/35, 50/49, 64/63
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| {{Mapping| 12 19 28 34 }}
| −3.95
| −3.95
| 4.92
| 4.92
Line 406: Line 354:
| 2.3.5.7.17
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| −2.92
| 4.86
| 4.86
Line 413: Line 361:
| 2.3.5.7.17.19
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| −2.53
| 4.52
| 4.52
Line 420: Line 368:
| 2.3.5.17
| 2.3.5.17
| 51/50, 81/80, 128/125
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| {{Mapping| 12 19 28 49 }}
| −0.87
| −0.87
| 2.95
| 2.95
Line 427: Line 375:
| 2.3.5.17.19
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| −0.81
| 2.64
| 2.64
| 2.64
| 2.64
|}
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, 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 72.
* 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 maps ===
Line 438: Line 388:


=== Commas ===
=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.
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"
{| class="commatable wikitable center-all left-3 right-4 left-6"
Line 453: Line 403:
| {{monzo| -19 12 }}
| {{monzo| -19 12 }}
| 23.46
| 23.46
| Lalawa
| Lalawama / Poma
| [[Pythagorean comma]]
| [[Pythagorean comma]]
|-
|-
Line 460: Line 410:
| {{monzo| 3 4 -4 }}
| {{monzo| 3 4 -4 }}
| 62.57
| 62.57
| Quadgu
| Quadguma
| Diminished comma, greater diesis
| Diminished comma, greater diesis
|-
|-
Line 467: Line 417:
| {{monzo| 18 -4 -5 }}
| {{monzo| 18 -4 -5 }}
| 60.61
| 60.61
| Saquingu
| Saquinguma
| [[Passion comma]]
| [[Passion comma]]
|-
|-
Line 474: Line 424:
| {{monzo| 7 0 -3 }}
| {{monzo| 7 0 -3 }}
| 41.06
| 41.06
| Trigu
| Triguma
| Augmented comma, lesser diesis
| Augmented comma, lesser diesis
|-
|-
Line 481: Line 431:
| {{monzo| -4 4 -1 }}
| {{monzo| -4 4 -1 }}
| 21.51
| 21.51
| Gu
| Guma
| Syntonic comma, Didymus' comma, meantone comma
| Syntonic comma, Didymus' comma, meantone comma
|-
|-
Line 488: Line 438:
| {{monzo| 11 -4 -2 }}
| {{monzo| 11 -4 -2 }}
| 19.55
| 19.55
| Sagugu
| Saguguma
| Diaschisma
| Diaschisma
|-
|-
Line 495: Line 445:
| {{monzo| 26 -12 -3 }}
| {{monzo| 26 -12 -3 }}
| 17.60
| 17.60
| Sasa-trigu
| Sasa-triguma
| [[Misty comma]]
| [[Misty comma]]
|-
|-
Line 502: Line 452:
| {{monzo| -15 8 1 }}
| {{monzo| -15 8 1 }}
| 1.95
| 1.95
| Layo
| Layoma
| Schisma
| Schisma
|-
|-
Line 509: Line 459:
| {{monzo| 161 -84 -12 }}
| {{monzo| 161 -84 -12 }}
| 0.02
| 0.02
| Sepbisa-quadtrigu
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
| [[Kirnberger's atom]]
|-
|-
Line 516: Line 466:
| {{monzo| 8 0 -1 -2 }}
| {{monzo| 8 0 -1 -2 }}
| 76.03
| 76.03
| Rurugu
| Ruruguma
| Bapbo comma
| Bapbo comma
|-
|-
Line 523: Line 473:
| {{monzo| -13 10 0 -1 }}
| {{monzo| -13 10 0 -1 }}
| 50.72
| 50.72
| Laru
| Laruma
| Harrison's comma
| Harrison's comma
|-
|-
Line 530: Line 480:
| {{monzo| 2 2 -1 -1 }}
| {{monzo| 2 2 -1 -1 }}
| 48.77
| 48.77
| Rugu
| Ruguma
| Mint comma, septimal quarter tone
| Mint comma, septimal quarter tone
|-
|-
Line 537: Line 487:
| {{monzo| 1 0 2 -2 }}
| {{monzo| 1 0 2 -2 }}
| 34.98
| 34.98
| Biruyo
| Biruyoma
| Jubilisma
| Jubilisma
|-
|-
Line 544: Line 494:
| {{monzo| -9 6 1 -1 }}
| {{monzo| -9 6 1 -1 }}
| 29.22
| 29.22
| Laruyo
| Laruyoma
| Schismean comma
| Schismean comma
|-
|-
Line 551: Line 501:
| {{monzo| 6 -2 0 -1 }}
| {{monzo| 6 -2 0 -1 }}
| 27.26
| 27.26
| Ru
| Ruma
| Septimal comma
| Septimal comma
|-
|-
Line 558: Line 508:
| {{monzo| 0 -2 5 -3 }}
| {{monzo| 0 -2 5 -3 }}
| 21.18
| 21.18
| Triru-aquinyo
| Triru-aquinyoma
| Gariboh comma
| Gariboh comma
|-
|-
Line 565: Line 515:
| {{monzo| 1 2 -3 1 }}
| {{monzo| 1 2 -3 1 }}
| 13.79
| 13.79
| Zotrigu
| Zotriguma
| Starling comma
| Starling comma
|-
|-
Line 572: Line 522:
| {{monzo| 5 -4 3 -2 }}
| {{monzo| 5 -4 3 -2 }}
| 13.47
| 13.47
| Rurutriyo
| Rurutriyoma
| Octagar comma
| Octagar comma
|-
|-
Line 579: Line 529:
| {{monzo| -9 8 -4 2 }}
| {{monzo| -9 8 -4 2 }}
| 8.04
| 8.04
| Labizogugu
| Labizoguguma
| [[Varunisma]]
| [[Varunisma]]
|-
|-
Line 586: Line 536:
| {{monzo| -5 2 2 -1 }}
| {{monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Ruyoyo
| Ruyoyoma
| Marvel comma
| Marvel comma
|-
|-
Line 593: Line 543:
| {{monzo| 6 0 -5 2 }}
| {{monzo| 6 0 -5 2 }}
| 6.08
| 6.08
| Zozoquingu
| Zozoquinguma
| Hemimean comma
| Hemimean comma
|-
|-
Line 600: Line 550:
| {{monzo| 10 -6 1 -1 }}
| {{monzo| 10 -6 1 -1 }}
| 5.76
| 5.76
| Saruyo
| Saruyoma
| Hemifamity comma
| Hemifamity comma
|-
|-
Line 607: Line 557:
| {{monzo| 25 -14 0 -1 }}
| {{monzo| 25 -14 0 -1 }}
| 3.80
| 3.80
| Sasaru
| Sasaruma
| [[Garischisma]]
| [[Garischisma]]
|-
|-
Line 614: Line 564:
| {{monzo| -11 2 7 -3 }}
| {{monzo| -11 2 7 -3 }}
| 1.63
| 1.63
| Latriru-asepyo
| Latriru-asepyoma
| [[Metric comma]]
| [[Metric comma]]
|-
|-
Line 621: Line 571:
| {{monzo| -4 6 -6 3 }}
| {{monzo| -4 6 -6 3 }}
| 0.33
| 0.33
| Trizogugu
| Trizoguguma
| [[Landscape comma]]
| [[Landscape comma]]
|-
|-
Line 628: Line 578:
| {{monzo| 7 0 0 0 -2 }}
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 97.36
| 1uu2
| Lulubima
| Axirabian limma
| Axirabian limma
|-
|-
Line 635: Line 585:
| {{monzo| -2 2 1 0 -1 }}
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| 38.91
| Luyo
| Luyoma
| Undecimal fifth tone
| Undecimal fifth tone
|-
|-
Line 642: Line 592:
| {{monzo| 3 0 -1 1 -1 }}
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| 31.19
| Luzogu
| Luzoguma
| Undecimal tritonic comma
| Undecimal tritonic comma
|-
|-
Line 649: Line 599:
| {{monzo| -1 0 1 2 -2 }}
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| 21.33
| Luluzozoyo
| Luluzozoyoma
| Frostma
| Frostma
|-
|-
Line 656: Line 606:
| {{monzo| -1 2 0 -2 1 }}
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| 17.58
| Loruru
| Loruruma
| Mothwellsma
| Mothwellsma
|-
|-
Line 663: Line 613:
| {{monzo| 2 -2 2 0 -1 }}
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| 17.40
| Luyoyo
| Luyoyoma
| Ptolemisma
| Ptolemisma
|-
|-
Line 670: Line 620:
| {{monzo| 4 0 -2 -1 1 }}
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Loruguguma
| Valinorsma
| Valinorsma
|-
|-
Line 677: Line 627:
| {{monzo| 7 -4 0 1 -1 }}
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| 9.69
| Saluzo
| Saluzoma
| Pentacircle comma
| Pentacircle comma
|-
|-
Line 684: Line 634:
| {{monzo| -3 2 -1 2 -1 }}
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| 3.93
| Luzozogu
| Luzozoguma
| Werckisma
| Werckisma
|-
|-
Line 691: Line 641:
| {{monzo| -3 4 -2 -2 2 }}
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Bilorugu
| Biloruguma
| Kalisma
| Kalisma
|-
|-
Line 698: Line 648:
| {{monzo| -6 0 1 0 0 1 }}
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| 26.84
| Thoyo
| Thoyoma
| Wilsorma
| Wilsorma
|-
|-
Line 705: Line 655:
| {{monzo| -1 -2 -1 1 0 1 }}
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| 19.13
| Thozogu
| Thozoguma
| Superleap comma, biome comma
| Superleap comma, biome comma
|-
|-
Line 712: Line 662:
| {{monzo| 4 2 0 0 -1 -1 }}
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| 12.06
| Thulu
| Thuluma
| Grossma
| Grossma
|-
|-
Line 719: Line 669:
| {{monzo| -3 0 -3 1 1 1 }}
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| 1.73
| Tholozotrigu
| Tholozotriguma
| Fairytale comma, sinbadma
| Fairytale comma, sinbadma
|-
|-
Line 726: Line 676:
| {{monzo| 12 -2 -1 -1 0 -1 }}
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| 0.42
| Sathurugu
| Sathuruguma
| Schismina
| Minisma
|-
|-
| 17
| 17
Line 733: Line 683:
| {{monzo| -1 1 -2 0 0 0 1 }}
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| 34.28
| Sogugu
| Soguguma
| Large septendecimal sixth tone
| Large septendecimal sixth tone
|-
|-
Line 740: Line 690:
| {{monzo| 2 -1 0 0 0 1 -1 }}
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| 33.62
| Sutho
| Suthoma
| Small septendecimal sixth tone
| Small septendecimal sixth tone
|-
|-
Line 747: Line 697:
| {{monzo| 3 -3 -1 0 0 0 1 }}
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| 12.78
| Sogu
| Soguma
| Diatisma, fiventeen comma
| Diatisma, fiventeen comma
|-
|-
Line 754: Line 704:
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| 6.78
| Sugu
| Suguma
| Charisma, septendecimal kleisma
| Charisma, septendecimal kleisma
|-
|-
Line 761: Line 711:
| {{monzo| -5 -2 0 0 0 0 2 }}
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| 6.00
| Soso
| Sosoma
| Semitonisma
| Semitonisma
|-
|-
Line 768: Line 718:
| {{monzo| -3 2 -2 0 0 -1 2 }}
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| 0.67
| Sosothugugu
| Sosothuguguma
| Sextantonisma
| Sextantonisma
|-
|-
Line 775: Line 725:
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| 44.97
| Nutho
| Nuthoma
| Undevicesimal two-ninth tone
| Undevicesimal two-ninth tone
|-
|-
Line 782: Line 732:
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| 18.13
| Nugu
| Nuguma
| 19th-partial chroma
| 19th-partial chroma
|-
|-
Line 789: Line 739:
| {{monzo| -3 2 0 0 0 0 1 -1}}
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| 11.35
| Nuso
| Nusoma
| Ganassisma
| Ganassisma
|-
|-
Line 796: Line 746:
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| 10.15
| Nosugu
| Nosuguma
| Malcolmisma
| Malcolmisma
|-
|-
Line 803: Line 753:
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| 5.35
| Nusu
| Nusuma
| Photisma
| Photisma
|-
|-
Line 810: Line 760:
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| 4.80
| Nonogu
| Nonoguma
| Go comma
| 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 ===
=== Rank-2 temperaments ===
* [[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–Pythagorean equivalence continuum]]
{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
Line 830: Line 783:
| 1\12
| 1\12
| (P8, P4/5)
| (P8, P4/5)
| [[Ripple]] / [[passion]]
| [[Ripple]], [[passion]]
|-
|-
| 1
| 1
| 5\12
| 5\12
| (P8, P5)
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
|-
| 2
| 2
| 5\12 (1\12)
| 5\12 (1\12)
| (P8/2, P5)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
| [[Pajara]], [[injera]]
|-
|-
| 3
| 3
| 5\12 (1\12)
| 5\12 (1\12)
| (P8/3, P5)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
|-
| 4
| 4
Line 857: Line 810:
| [[Hexe]]
| [[Hexe]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<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), [[garibaldi]] (41 & 53), and [[diaschismic]] (46 & 58). 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 ==
== Scales ==
Line 890: Line 852:
== Music ==
== Music ==
{{Catrel|12edo tracks}}
{{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 ==
== See also ==
Line 895: Line 859:
* [[:purdal:12-EDD]]{{dead link}}
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step
== Notes ==
<references group="note" />


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


[[Category:3-limit record edos|##]] <!-- 2-digit number -->
[[Category:Historical]]
[[Category:Historical]]
[[Category:Meantone]]
[[Category:Meantone]]

Revision as of 09:16, 3 June 2026

← 11edo 12edo 13edo →
Prime factorization 22 × 3 (highly composite)
Step size 100 ¢ (by definition) 
Fifth 7\12 (700 ¢)
(convergent)
Semitones (A1:m2) 1:1 (100 ¢ : 100 ¢)
Consistency limit 9
Distinct consistency limit 5
English Wikipedia has an article on:

12 equal divisions of the octave (abbreviated 12edo or 12ed2), also called 12-tone equal temperament (12tet) or 12 equal temperament (12et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 12 equal parts of exactly 100 ¢ each. Each step represents a frequency ratio of 21/12, or the 12th root of 2. 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.

It divides the octave into twelve equal parts, each of exactly 100 cents. It has a fifth which is quite accurate at 700 cents, two cents flat of just. It has a major third which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The 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 temperaments. 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 serialism and much of jazz harmony that derive from 12edo's structure as an equal division rather than its underlying temperament properties.[citation needed]

12edo is the basic example of a 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 chords in functional harmony, for which the 5-limit JI version would be 1–5/4–3/2–16/9, and while 12et officially supports septimal meantone for tempering out 126/125 and 225/224 via its patent val of ⟨12 19 28 34], its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 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, 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. Other commas 12et tempers out include the diaschisma, 2048/2025, 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.

Prime harmonics

Approximation of prime harmonics in 12edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.96 +13.69 +31.17 +48.68 -40.53 -4.96 +2.49 -28.27 -29.58 -45.04
Relative (%) +0.0 -2.0 +13.7 +31.2 +48.7 -40.5 -5.0 +2.5 -28.3 -29.6 -45.0
Steps
(reduced) 		12
(0) 		19
(7) 		28
(4) 		34
(10) 		42
(6) 		44
(8) 		49
(1) 		51
(3) 		54
(6) 		58
(10) 		59
(11)

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 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 60-, 84-, and 96edo all see utilities. Notable rank-2 temperaments that augment 12edo with extra generators include compton and catler.

Intervals

Intervals of 12edo
Degree Cents Interval region Approximated 5-limit
JI intervals (error in ¢) 		Audio Higher limit interpretations[note 1]
0 0 Unison (prime) 1/1 (just)
1 100 Minor second 256/243 (+9.775)
25/24 (+29.328)
16/15 (−11.731) 		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) 		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) 		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) 		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) 		21/16 (-29.219)
6 600 Tritone 25/18 (+31.283)
36/25 (-31.283)
45/32 (+9.776)
64/45 (−9.776) 		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) 		32/21 (+29.219)
8 800 Minor sixth 128/81 (+7.820)
8/5 (−13.686)
25/16 (+27.373) 		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) 		12/7 (−33.129), 42/25 (+1.847)
32/19 (−2.487)
10 1000 Minor seventh 16/9 (+3.910)
9/5 (−17.596) 		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) 		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)
  1. ↑ Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations

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.

Semitones −2 −1 0 +1 +2
Symbol  
 		 
 		 
 		 
 		 

The subsets 1edo, 2edo, 3edo, 4edo and 6edo can all be written using 12edo subset notation.

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

Notation of 12edo
Degree Cents 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 () and sagittal flat () respectively.

Sagittal notation

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

Evo flavor

12-EDO_Evo_Sagittal.svg

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

Revo flavor

12-EDO_Revo_Sagittal.svg

Solfege

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

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Selected 5-limit intervals approximated in 12edo

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 12edo. Prime harmonics are in bold; inconsistent intervals are in italics.

Note that, since the cent was defined in terms of 12edo, the absolute and relative errors for 12edo are identical.

15-odd-limit intervals in 12edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 1.955 2.0
9/8, 16/9 3.910 3.9
13/11, 22/13 10.790 10.8
15/8, 16/15 11.731 11.7
5/4, 8/5 13.686 13.7
5/3, 6/5 15.641 15.6
7/5, 10/7 17.488 17.5
11/7, 14/11 17.508 17.5
9/5, 10/9 17.596 17.6
15/14, 28/15 19.443 19.4
13/7, 14/13 28.298 28.3
7/4, 8/7 31.174 31.2
7/6, 12/7 33.129 33.1
11/10, 20/11 34.996 35.0
9/7, 14/9 35.084 35.1
13/9, 18/13 36.618 36.6
15/11, 22/15 36.951 37.0
13/12, 24/13 38.573 38.6
13/8, 16/13 40.528 40.5
13/10, 20/13 45.786 45.8
11/9, 18/11 47.408 47.4
15/13, 26/15 47.741 47.7
11/8, 16/11 48.682 48.7
11/6, 12/11 49.363 49.4
15-odd-limit intervals in 12edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 1.955 2.0
9/8, 16/9 3.910 3.9
15/8, 16/15 11.731 11.7
5/4, 8/5 13.686 13.7
5/3, 6/5 15.641 15.6
7/5, 10/7 17.488 17.5
11/7, 14/11 17.508 17.5
9/5, 10/9 17.596 17.6
15/14, 28/15 19.443 19.4
7/4, 8/7 31.174 31.2
7/6, 12/7 33.129 33.1
11/10, 20/11 34.996 35.0
9/7, 14/9 35.084 35.1
13/9, 18/13 36.618 36.6
15/11, 22/15 36.951 37.0
13/12, 24/13 38.573 38.6
13/8, 16/13 40.528 40.5
11/8, 16/11 48.682 48.7
11/6, 12/11 50.637 50.6
15/13, 26/15 52.259 52.3
11/9, 18/11 52.592 52.6
13/10, 20/13 54.214 54.2
13/7, 14/13 71.702 71.7
13/11, 22/13 89.210 89.2
15-odd-limit intervals by 12f val mapping
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 1.955 2.0
9/8, 16/9 3.910 3.9
13/11, 22/13 10.790 10.8
15/8, 16/15 11.731 11.7
5/4, 8/5 13.686 13.7
5/3, 6/5 15.641 15.6
7/5, 10/7 17.488 17.5
11/7, 14/11 17.508 17.5
9/5, 10/9 17.596 17.6
15/14, 28/15 19.443 19.4
13/7, 14/13 28.298 28.3
7/4, 8/7 31.174 31.2
7/6, 12/7 33.129 33.1
11/10, 20/11 34.996 35.0
9/7, 14/9 35.084 35.1
15/11, 22/15 36.951 37.0
13/10, 20/13 45.786 45.8
15/13, 26/15 47.741 47.7
11/8, 16/11 48.682 48.7
11/6, 12/11 50.637 50.6
11/9, 18/11 52.592 52.6
13/8, 16/13 59.472 59.5
13/12, 24/13 61.427 61.4
13/9, 18/13 63.382 63.4

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-19 12⟩ [⟨12 19]] +0.62 0.62 0.62
2.3.5 81/80, 128/125 [⟨12 19 28]] −1.56 3.11 3.11
2.3.5.7 36/35, 50/49, 64/63 [⟨12 19 28 34]] −3.95 4.92 4.94
2.3.5.7.17 36/35, 50/49, 51/49, 64/63 [⟨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 [⟨12 19 28 34 49 51]] −2.53 4.52 4.53
2.3.5.17 51/50, 81/80, 128/125 [⟨12 19 28 49]] −0.87 2.95 2.95
2.3.5.17.19 51/50, 76/75, 81/80, 128/125 [⟨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-, 5-, 7-, and 11-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 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 72.

Uniform maps

13-limit uniform maps between 11.8 and 12.2
Min. size Max. size Wart notation Map
11.7554 11.8436 12cde ⟨12 19 27 33 41 44]
11.8436 11.9329 12de ⟨12 19 28 33 41 44]
11.9329 11.9962 12e ⟨12 19 28 34 41 44]
11.9962 12.0256 12 ⟨12 19 28 34 42 44]
12.0256 12.2743 12f ⟨12 19 28 34 42 45]

Commas

12edo tempers out the following commas. This assumes the val ⟨12 19 28 34 42 44 49 51].

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

Rank-2 temperaments

Periods
per 8ve 		Generator* Pergen Temperaments
1 1\12 (P8, P4/5) Ripple, passion
1 5\12 (P8, P5) Meantone / dominant
2 5\12 (1\12) (P8/2, P5) Pajara, injera
3 5\12 (1\12) (P8/3, P5) Augmented / august
4 5\12 (1\12) (P8/4, P5) Diminished
6 5\12 (1\12) (P8/6, P5) Hexe

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Rank-2 temperaments to which 12et can be detempered include compton (12 & 72), garibaldi (41 & 53), and diaschismic (46 & 58). For more comprehensive lists, see:

Octave stretch or compression

Whether there is intonational improvement from octave stretch and compression for 12edo varies by context. A slight compression such as what is given by 40ed10 and 34zpi shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 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

Main article: List of MOS scales in 12edo

The two most common 12edo mos scales are meantone[5] and meantone[7].

  • Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
  • Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
  • Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

Non-mos scales

Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

  • Harmonic major – 2212132
  • Melodic major – 2212122
  • Hungarian minor – 2131131
  • Maqam hijaz / double harmonic major – 1312131
  • 5-odd-limit tonality diamond – 3112113

File:12edo modes.pdf

Well temperaments

For a list of historical well temperaments, see Well temperament.

Music

See also: Category: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

External links

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