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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.
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 is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma 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 [[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 flat of just by even more, 15.6 cents.
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.


Historically, 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.


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]].
Line 40: 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 55: 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 163: 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"
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==== 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 376: 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 384: 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 391: 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 398: 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 405: 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 412: 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 419: 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 430: 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 445: Line 403:
| {{monzo| -19 12 }}
| {{monzo| -19 12 }}
| 23.46
| 23.46
| Lalawa
| Lalawama / Poma
| [[Pythagorean comma]]
| [[Pythagorean comma]]
|-
|-
Line 452: 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 459: Line 417:
| {{monzo| 18 -4 -5 }}
| {{monzo| 18 -4 -5 }}
| 60.61
| 60.61
| Saquingu
| Saquinguma
| [[Passion comma]]
| [[Passion comma]]
|-
|-
Line 466: 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 473: 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 480: Line 438:
| {{monzo| 11 -4 -2 }}
| {{monzo| 11 -4 -2 }}
| 19.55
| 19.55
| Sagugu
| Saguguma
| Diaschisma
| Diaschisma
|-
|-
Line 487: Line 445:
| {{monzo| 26 -12 -3 }}
| {{monzo| 26 -12 -3 }}
| 17.60
| 17.60
| Sasa-trigu
| Sasa-triguma
| [[Misty comma]]
| [[Misty comma]]
|-
|-
Line 494: Line 452:
| {{monzo| -15 8 1 }}
| {{monzo| -15 8 1 }}
| 1.95
| 1.95
| Layo
| Layoma
| Schisma
| Schisma
|-
|-
Line 501: 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 508: Line 466:
| {{monzo| 8 0 -1 -2 }}
| {{monzo| 8 0 -1 -2 }}
| 76.03
| 76.03
| Rurugu
| Ruruguma
| Bapbo comma
| Bapbo comma
|-
|-
Line 515: 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 522: 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 529: Line 487:
| {{monzo| 1 0 2 -2 }}
| {{monzo| 1 0 2 -2 }}
| 34.98
| 34.98
| Biruyo
| Biruyoma
| Jubilisma
| Jubilisma
|-
|-
Line 536: Line 494:
| {{monzo| -9 6 1 -1 }}
| {{monzo| -9 6 1 -1 }}
| 29.22
| 29.22
| Laruyo
| Laruyoma
| Schismean comma
| Schismean comma
|-
|-
Line 543: Line 501:
| {{monzo| 6 -2 0 -1 }}
| {{monzo| 6 -2 0 -1 }}
| 27.26
| 27.26
| Ru
| Ruma
| Septimal comma
| Septimal comma
|-
|-
Line 550: 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 557: Line 515:
| {{monzo| 1 2 -3 1 }}
| {{monzo| 1 2 -3 1 }}
| 13.79
| 13.79
| Zotrigu
| Zotriguma
| Starling comma
| Starling comma
|-
|-
Line 564: Line 522:
| {{monzo| 5 -4 3 -2 }}
| {{monzo| 5 -4 3 -2 }}
| 13.47
| 13.47
| Rurutriyo
| Rurutriyoma
| Octagar comma
| Octagar comma
|-
|-
Line 571: Line 529:
| {{monzo| -9 8 -4 2 }}
| {{monzo| -9 8 -4 2 }}
| 8.04
| 8.04
| Labizogugu
| Labizoguguma
| [[Varunisma]]
| [[Varunisma]]
|-
|-
Line 578: Line 536:
| {{monzo| -5 2 2 -1 }}
| {{monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Ruyoyo
| Ruyoyoma
| Marvel comma
| Marvel comma
|-
|-
Line 585: Line 543:
| {{monzo| 6 0 -5 2 }}
| {{monzo| 6 0 -5 2 }}
| 6.08
| 6.08
| Zozoquingu
| Zozoquinguma
| Hemimean comma
| Hemimean comma
|-
|-
Line 592: Line 550:
| {{monzo| 10 -6 1 -1 }}
| {{monzo| 10 -6 1 -1 }}
| 5.76
| 5.76
| Saruyo
| Saruyoma
| Hemifamity comma
| Hemifamity comma
|-
|-
Line 599: Line 557:
| {{monzo| 25 -14 0 -1 }}
| {{monzo| 25 -14 0 -1 }}
| 3.80
| 3.80
| Sasaru
| Sasaruma
| [[Garischisma]]
| [[Garischisma]]
|-
|-
Line 606: 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 613: Line 571:
| {{monzo| -4 6 -6 3 }}
| {{monzo| -4 6 -6 3 }}
| 0.33
| 0.33
| Trizogugu
| Trizoguguma
| [[Landscape comma]]
| [[Landscape comma]]
|-
|-
Line 620: 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 627: 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 634: 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 641: Line 599:
| {{monzo| -1 0 1 2 -2 }}
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| 21.33
| Luluzozoyo
| Luluzozoyoma
| Frostma
| Frostma
|-
|-
Line 648: Line 606:
| {{monzo| -1 2 0 -2 1 }}
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| 17.58
| Loruru
| Loruruma
| Mothwellsma
| Mothwellsma
|-
|-
Line 655: Line 613:
| {{monzo| 2 -2 2 0 -1 }}
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| 17.40
| Luyoyo
| Luyoyoma
| Ptolemisma
| Ptolemisma
|-
|-
Line 662: Line 620:
| {{monzo| 4 0 -2 -1 1 }}
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Loruguguma
| Valinorsma
| Valinorsma
|-
|-
Line 669: 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 676: Line 634:
| {{monzo| -3 2 -1 2 -1 }}
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| 3.93
| Luzozogu
| Luzozoguma
| Werckisma
| Werckisma
|-
|-
Line 683: Line 641:
| {{monzo| -3 4 -2 -2 2 }}
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Bilorugu
| Biloruguma
| Kalisma
| Kalisma
|-
|-
Line 690: 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 697: 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 704: 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 711: 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 718: 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 725: 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 732: 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 739: 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 746: 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 753: 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 760: 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 767: 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 774: 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 781: 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 788: 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 795: 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 802: 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 822: 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 849: 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), [[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 ==
What follows is a comparison of stretched- and compressed-octave 12edo tunings.
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.  
 
; [[WE|12et, 7-limit WE tuning]]  
* Step size: 99.664{{c}}, octave size: 1196.0{{c}}
Compressing the octave of 12edo by 4{{c}} results in much improved primes 5, 7 and 11, but a much worse prime 3. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. [[40ed10]] does this as well. An argument could be made that such tunings [[7-limit|harmonies involving the 7th harmonic]] to regular old 12edo without even needing to add any new notes to the octave. This adds in brand new harmonic possibilities without breaking any common 12-tone music theory.
{{Harmonics in cet|99.664|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 7-limit WE tuning}}
{{Harmonics in cet|99.664|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 7-limit WE tuning (continued)}}
 
; [[zpi|34zpi]]  
* Step size: 99.807{{c}}, octave size: 1197.7{{c}}
Compressing the octave of 12edo by around 2{{c}} results in improved primes 5 and 7, but a worse prime 3. The tuning 34zpi does this. It might be a good tuning for 5-limit [[meantone]], for composers seeking more pure thirds and sixths than regular 12edo. It would be well suited for playing classic pieces written for [[historical temperaments]], as well as being well suited to playing simultaneously with other instruments or voices that use [[just intonation]].
{{Harmonics in cet|99.807|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 34zpi}}
{{Harmonics in cet|99.807|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 34zpi (continued)}}
 
; [[WE|12et, 5-limit WE tuning]]  
* Step size: 99.868{{c}}, octave size: 1198.4{{c}}
Compressing the octave of 12edo by around 1{{c}} results in slightly improved primes 5 and 7, but a slightly prime 3. Its 5-limit WE tuning and 5-limit [[TE]] tuning both do this. This has the same benefits and drawbacks as 34zpi, but both are less intense here compared to 34zpi.
{{Harmonics in cet|99.868|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 5-limit WE tuning}}
{{Harmonics in cet|99.868|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 5-limit WE tuning (continued)}}
 
; 12edo
* Step size: 100.000{{c}}, octave size: 1200.0{{c}}
Pure-octaves 12edo performs well on harmonics 2, 3 and 5 but poorly on harmonics 7, 11 and 13 compared to other edos with a similar number of notes per octave.
{{Harmonics in equal|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12edo}}
{{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12edo (continued)}}
 
; [[31ed6]]  
* Step size: 100.063{{c}}, octave size: 1200.8{{c}}
Stretching the octave of 12edo by a little less than 1{{c}} results in an improved prime 3, but worse prime 5. This loosely resembles the stretched-octave tunings commonly used on pianos. It may better match the [[timbre|slightly inharmonic partials]] of some string instruments. The tuning 31ed6 does this.
{{Harmonics in equal|31|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31ed6}}
{{Harmonics in equal|31|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31ed6 (continued)}}
 
; [[19edt]]  
* Step size: 101.103{{c}}, octave size: 1201.2{{c}}
Stretching the octave of 12edo by a little more than 1{{c}} results in an improved prime 3, but worse prime 5. It may better match the [[timbre|slightly inharmonic partials]] of some string instruments. The tuning 19edt does this.
{{Harmonics in equal|19|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edt}}
{{Harmonics in equal|19|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edt (continued)}}
 
; [[7edf]]  
* Step size: 100.3{{c}}, octave size: 1203.35{{c}}
Stretching the octave of 12edo by around 3{{c}} results in improved primes 3 and 13, but much worse primes 5 and 7. This has similar benefits and drawbacks to [[Pythagorean]] tuning. Most modern music probably won't sound very good here because of the off 5th harmonic. The tuning 7edf does this.
{{Harmonics in equal|7|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edf}}
{{Harmonics in equal|7|3|2|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edf (continued)}}


== Scales ==
== Scales ==
Line 927: 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 932: 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 ==
Retrieved from "https://en.xen.wiki/w/12edo"