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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 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.
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. 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.
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.
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}}
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.
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.
Line 23: Line 21:
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]].
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.
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. 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.
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 ===
=== Prime harmonics ===
Line 39: 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="2" | Approximated [[JI]] intervals ([[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>
! [[5-limit]]
! style="width: 330px;" | Higher limits<ref group="note">{{sg|limit=2.3.5.7.17.19 subgroup}}</ref>
|-
|-
| 0
| 0
Line 52: 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)
| [[256/243]] (+9.775)<br>[[16/15]] (−11.731)<br>[[25/24]] (+29.328)
| [[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)
| [[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)
| [[File:piano_1_12edo.mp3]]
|-
|-
| 2
| 2
Line 66: Line 62:
| Major second
| Major second
| [[9/8]] (−3.910)<br>[[10/9]] (+17.596)
| [[9/8]] (−3.910)<br>[[10/9]] (+17.596)
| [[8/7]] (−31.174), [[28/25]] (+3.802)<br>[[17/15]] (−16.687), [[19/17]] (+7.442), [[64/57]] (−0.532)
| [[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
Line 73: Line 69:
| Minor third
| Minor third
| [[32/27]] (+5.865)<br>[[6/5]] (−15.641)<br>[[75/64]] (+25.418)
| [[32/27]] (+5.865)<br>[[6/5]] (−15.641)<br>[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)<br>[[19/16]] (+2.487)
| [[7/6]] (+33.129), [[25/21]] (−1.847)<br>[[19/16]] (+2.487)
| [[File:piano_1_4edo.mp3]]
|-
|-
| 4
| 4
Line 80: Line 76:
| Major third
| Major third
| [[81/64]] (−7.820)<br>[[5/4]] (+13.686)<br> [[32/25]] (-27.373)
| [[81/64]] (−7.820)<br>[[5/4]] (+13.686)<br> [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)<br>[[34/27]] (+0.910), [[24/19]] (−4.442)
| [[63/50]] (−0.108), [[9/7]] (−35.084)<br>[[34/27]] (+0.910), [[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
|-
| 5
| 5
Line 87: Line 83:
| Fourth
| Fourth
| [[4/3]] (+1.955)<br> [[27/20]] (-19.551)
| [[4/3]] (+1.955)<br> [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
| [[21/16]] (-29.219)
| [[File:piano_5_12edo.mp3]]
|-
|-
| 6
| 6
Line 94: Line 90:
| [[Tritone]]
| [[Tritone]]
| [[25/18]] (+31.283)<br>[[36/25]] (-31.283)<br>[[45/32]] (+9.776)<br>[[64/45]] (−9.776)
| [[25/18]] (+31.283)<br>[[36/25]] (-31.283)<br>[[45/32]] (+9.776)<br>[[64/45]] (−9.776)
| [[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)
| [[7/5]] (+17.488), [[10/7]] (−17.488)<br>[[24/17]] (+3.000), [[17/12]] (−3.000)<br>[[99/70]] (−0.088), [[140/99]] (+0.088)
| [[File:piano_1_2edo.mp3]]
|-
|-
| 7
| 7
Line 101: Line 97:
| Fifth
| Fifth
| [[3/2]] (−1.955)<br>[[40/27]] (+19.551)
| [[3/2]] (−1.955)<br>[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
| [[32/21]] (+29.219)
| [[File:piano_7_12edo.mp3]]
|-
|-
| 8
| 8
Line 108: Line 104:
| Minor sixth
| Minor sixth
| [[128/81]] (+7.820)<br>[[8/5]] (−13.686)<br>[[25/16]] (+27.373)
| [[128/81]] (+7.820)<br>[[8/5]] (−13.686)<br>[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)<br>[[19/12]] (+4.442), [[27/17]] (−0.910)
| [[14/9]] (+35.084), [[100/63]] (+0.108)<br>[[19/12]] (+4.442), [[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
|-
| 9
| 9
Line 115: Line 111:
| Major sixth
| Major sixth
| [[27/16]] (−5.865)<br>[[5/3]] (+15.641)<br>[[128/75]] (-25.418)
| [[27/16]] (−5.865)<br>[[5/3]] (+15.641)<br>[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)<br>[[32/19]] (−2.487)
| [[12/7]] (−33.129), [[42/25]] (+1.847)<br>[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
|-
| 10
| 10
Line 122: Line 118:
| Minor seventh
| Minor seventh
| [[16/9]] (+3.910)<br>[[9/5]] (−17.596)
| [[16/9]] (+3.910)<br>[[9/5]] (−17.596)
| [[7/4]] (+31.174), [[25/14]] (−3.802)<br>[[30/17]] (+16.687), [[34/19]] (−7.442)<br>[[57/32]] (+0.532)
| [[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
Line 129: Line 125:
| Major seventh
| Major seventh
| [[243/128]] (-9.775)<br>[[15/8]] (+11.731)<br>[[48/25]] (−29.328)
| [[243/128]] (-9.775)<br>[[15/8]] (+11.731)<br>[[48/25]] (−29.328)
| [[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)
| [[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)
| [[File:piano_11_12edo.mp3]]
|-
|-
| 12
| 12
Line 136: Line 132:
| Octave
| Octave
| [[2/1]] (just)
| [[2/1]] (just)
|
| [[File:piano_1_1edo.mp3]]
| [[File:piano_1_1edo.mp3]]
|
|}
|}
<references group="note" />
<references group="note" />
Line 148: Line 144:
The subsets {{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 336: 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 344: 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 351: 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 358: 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 365: 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 372: 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 379: 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-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], [[11-limit|11-]], [[13-limit|13-]], and [[19-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], 19/19e, and 19egh, respectively.
* 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]].
* 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]].


Line 391: 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 406: Line 403:
| {{monzo| -19 12 }}
| {{monzo| -19 12 }}
| 23.46
| 23.46
| Lalawa
| Lalawama / Poma
| [[Pythagorean comma]]
| [[Pythagorean comma]]
|-
|-
Line 413: 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 420: Line 417:
| {{monzo| 18 -4 -5 }}
| {{monzo| 18 -4 -5 }}
| 60.61
| 60.61
| Saquingu
| Saquinguma
| [[Passion comma]]
| [[Passion comma]]
|-
|-
Line 427: 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 434: 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 441: Line 438:
| {{monzo| 11 -4 -2 }}
| {{monzo| 11 -4 -2 }}
| 19.55
| 19.55
| Sagugu
| Saguguma
| Diaschisma
| Diaschisma
|-
|-
Line 448: Line 445:
| {{monzo| 26 -12 -3 }}
| {{monzo| 26 -12 -3 }}
| 17.60
| 17.60
| Sasa-trigu
| Sasa-triguma
| [[Misty comma]]
| [[Misty comma]]
|-
|-
Line 455: Line 452:
| {{monzo| -15 8 1 }}
| {{monzo| -15 8 1 }}
| 1.95
| 1.95
| Layo
| Layoma
| Schisma
| Schisma
|-
|-
Line 462: 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 469: Line 466:
| {{monzo| 8 0 -1 -2 }}
| {{monzo| 8 0 -1 -2 }}
| 76.03
| 76.03
| Rurugu
| Ruruguma
| Bapbo comma
| Bapbo comma
|-
|-
Line 476: 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 483: 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 490: Line 487:
| {{monzo| 1 0 2 -2 }}
| {{monzo| 1 0 2 -2 }}
| 34.98
| 34.98
| Biruyo
| Biruyoma
| Jubilisma
| Jubilisma
|-
|-
Line 497: Line 494:
| {{monzo| -9 6 1 -1 }}
| {{monzo| -9 6 1 -1 }}
| 29.22
| 29.22
| Laruyo
| Laruyoma
| Schismean comma
| Schismean comma
|-
|-
Line 504: Line 501:
| {{monzo| 6 -2 0 -1 }}
| {{monzo| 6 -2 0 -1 }}
| 27.26
| 27.26
| Ru
| Ruma
| Septimal comma
| Septimal comma
|-
|-
Line 511: 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 518: Line 515:
| {{monzo| 1 2 -3 1 }}
| {{monzo| 1 2 -3 1 }}
| 13.79
| 13.79
| Zotrigu
| Zotriguma
| Starling comma
| Starling comma
|-
|-
Line 525: Line 522:
| {{monzo| 5 -4 3 -2 }}
| {{monzo| 5 -4 3 -2 }}
| 13.47
| 13.47
| Rurutriyo
| Rurutriyoma
| Octagar comma
| Octagar comma
|-
|-
Line 532: Line 529:
| {{monzo| -9 8 -4 2 }}
| {{monzo| -9 8 -4 2 }}
| 8.04
| 8.04
| Labizogugu
| Labizoguguma
| [[Varunisma]]
| [[Varunisma]]
|-
|-
Line 539: Line 536:
| {{monzo| -5 2 2 -1 }}
| {{monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Ruyoyo
| Ruyoyoma
| Marvel comma
| Marvel comma
|-
|-
Line 546: Line 543:
| {{monzo| 6 0 -5 2 }}
| {{monzo| 6 0 -5 2 }}
| 6.08
| 6.08
| Zozoquingu
| Zozoquinguma
| Hemimean comma
| Hemimean comma
|-
|-
Line 553: Line 550:
| {{monzo| 10 -6 1 -1 }}
| {{monzo| 10 -6 1 -1 }}
| 5.76
| 5.76
| Saruyo
| Saruyoma
| Hemifamity comma
| Hemifamity comma
|-
|-
Line 560: Line 557:
| {{monzo| 25 -14 0 -1 }}
| {{monzo| 25 -14 0 -1 }}
| 3.80
| 3.80
| Sasaru
| Sasaruma
| [[Garischisma]]
| [[Garischisma]]
|-
|-
Line 567: 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 574: Line 571:
| {{monzo| -4 6 -6 3 }}
| {{monzo| -4 6 -6 3 }}
| 0.33
| 0.33
| Trizogugu
| Trizoguguma
| [[Landscape comma]]
| [[Landscape comma]]
|-
|-
Line 581: 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 588: 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 595: 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 602: Line 599:
| {{monzo| -1 0 1 2 -2 }}
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| 21.33
| Luluzozoyo
| Luluzozoyoma
| Frostma
| Frostma
|-
|-
Line 609: Line 606:
| {{monzo| -1 2 0 -2 1 }}
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| 17.58
| Loruru
| Loruruma
| Mothwellsma
| Mothwellsma
|-
|-
Line 616: Line 613:
| {{monzo| 2 -2 2 0 -1 }}
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| 17.40
| Luyoyo
| Luyoyoma
| Ptolemisma
| Ptolemisma
|-
|-
Line 623: Line 620:
| {{monzo| 4 0 -2 -1 1 }}
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Loruguguma
| Valinorsma
| Valinorsma
|-
|-
Line 630: 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 637: Line 634:
| {{monzo| -3 2 -1 2 -1 }}
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| 3.93
| Luzozogu
| Luzozoguma
| Werckisma
| Werckisma
|-
|-
Line 644: Line 641:
| {{monzo| -3 4 -2 -2 2 }}
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Bilorugu
| Biloruguma
| Kalisma
| Kalisma
|-
|-
Line 651: 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 658: 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 665: 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 672: 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 679: Line 676:
| {{monzo| 12 -2 -1 -1 0 -1 }}
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| 0.42
| Sathurugu
| Sathuruguma
| Minisma
| Minisma
|-
|-
Line 686: 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 693: 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 700: 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 707: 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 714: 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 721: 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 728: 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 735: 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 742: 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 749: 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 756: 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 763: 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" />
<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 784: 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 811: Line 810:
| [[Hexe]]
| [[Hexe]]
|}
|}
<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
<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 ==
== 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.
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 ==
{{Main| List of MOS scales in 12edo }}
{{See also| List of MOS scales in 12edo }}


The two most common 12edo mos scales are meantone[5] and meantone[7].
The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Diatonic: [[5L 2s]] – 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Pentatonic: [[2L 3s]] – 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)


=== Non-mos scales ===
The diminished and augmented scales are also MOS scales.
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.
* Diminished: [[4L 4s]] – 12121212 (generator = 1\12, period = 3\12)
* Augmented: [[3L 3s]] – 131313 (generator = 1\12, period = 4\12)


* Harmonic major – 2212132
Other widely used scales include:
* Melodic major – 2212122
* Melodic minor – 2122221
* Harmonic minor – 2122131
* Harmonic major – 2212131
* Hungarian minor – 2131131
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]


== Well temperaments ==
== Well temperaments ==
Retrieved from "https://en.xen.wiki/w/12edo"