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


== 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, tempering out [[81/80]], equating four [[3/2|perfect fifths]] with the [[5/1|5th harmonic]].


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 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.


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. The [[duodene]] is an unequal 12-note scale in 5-limit just intonation which observes these commas, and can be considered a [[detempering|detemper]] of 12edo. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], and the schisma, [[32805/32768]], and in the 7-limit, the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.
 
12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented, as well as the [[16:19:24]] otonal minor triad.


=== Prime harmonics ===
=== 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]] 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 [[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, provides a great correction for the approximate harmonics 11 and 13. [[36edo]], which triples it, provides a great correction for the approximate harmonic 7. [[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]].


=== Miscellany ===
[[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]].
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.


== Intervals ==
== Intervals ==
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ Intervals of 12edo
|+ style="font-size: 105%;" | Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals<ref group="note">{{sg|limit=2.3.5.7.17.19 subgroup}}</ref> ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
|-
! [[3-limit]]
! [[Degree]]
! [[5-limit]]
! [[Cent]]s
! [[7-limit]]
! [[Interval region]]
! Other
! style="width: 165px;" | Approximated 5-limit<br>JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations<ref group="note">Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations</ref>
|-
|-
| 0
| 0
Line 51: 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>[[16/15]] (−11.731)<br>[[25/24]] (+29.328)
| [[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)
|
| [[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 159: 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"
|+Notation of 12edo
|+ style="font-size: 105%;" | Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
|-
! Diatonic ([[5L 2s]]) interval names
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&nbsp;2s]]) interval names
! Note names (on D)
! Note names (on D)
|-
|-
| 0
| 0
| 0
| 0
|'''Perfect unison (P1)'''
| '''Perfect unison (P1)'''
|'''D'''
| '''D'''
|-
|-
| 1
| 1
Line 195: Line 168:
| 2
| 2
| 200
| 200
|'''Major second (M2)'''<br>Diminished third (d3)
| '''Major second (M2)'''<br>Diminished third (d3)
|'''E'''<br>Fb
| '''E'''<br>Fb
|-
|-
| 3
| 3
Line 210: Line 183:
| 5
| 5
| 500
| 500
|'''Perfect fourth (P4)'''
| '''Perfect fourth (P4)'''
|'''G'''
| '''G'''
|-
|-
| 6
| 6
Line 220: Line 193:
| 7
| 7
| 700
| 700
|'''Perfect fifth (P5)'''
| '''Perfect fifth (P5)'''
| A
| A
|-
|-
Line 230: Line 203:
| 9
| 9
| 900
| 900
|'''Major sixth (M6)'''<br>Diminished seventh (d7)
| '''Major sixth (M6)'''<br>Diminished seventh (d7)
|'''B'''<br>Cb
| '''B'''<br>Cb
|-
|-
| 10
| 10
Line 245: Line 218:
| 12
| 12
| 1200
| 1200
|'''Perfect octave (P8)'''
| '''Perfect octave (P8)'''
|'''D'''
| '''D'''
|}
|}


Line 253: Line 226:
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&#x266F;) and flats (&#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&#x266F;) and flats (&#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.


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


<imagemap>
==== Evo flavor ====
File:12-EDO_Evo_Sagittal.svg
{{Sagittal chart|Evo}}
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====


<imagemap>
==== Revo flavor ====
File:12-EDO_Revo_Sagittal.svg
{{Sagittal chart}}
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 ==
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ Solfege of 12edo
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Degree]]
! [[Cents]]
! [[Cents]]
Line 355: Line 317:
=== 15-odd-limit interval mappings ===
=== 15-odd-limit interval mappings ===
{{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}}
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[34zpi]]
| 12.0231830072926
| 99.8071807833375
| 5.193290
| 1.269599
| 15.899282
| 12edo
| 1197.68616940005
| 10
| 6
|}


== Regular temperament properties ==
== Regular temperament properties ==
Line 401: 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 409: 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 416: 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 423: 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 430: 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 437: 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 444: 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 (12f val) 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 ===
{{Uniform map|13|11.5|12.5}}  
{{Uniform map|edo=12}}


=== 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 470: Line 403:
| {{monzo| -19 12 }}
| {{monzo| -19 12 }}
| 23.46
| 23.46
| Lalawa
| Lalawama / Poma
| [[Pythagorean comma]]
| [[Pythagorean comma]]
|-
|-
Line 477: 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 484: Line 417:
| {{monzo| 18 -4 -5 }}
| {{monzo| 18 -4 -5 }}
| 60.61
| 60.61
| Saquingu
| Saquinguma
| [[Passion comma]]
| [[Passion comma]]
|-
|-
Line 491: 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 498: 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 505: Line 438:
| {{monzo| 11 -4 -2 }}
| {{monzo| 11 -4 -2 }}
| 19.55
| 19.55
| Sagugu
| Saguguma
| Diaschisma
| Diaschisma
|-
|-
Line 512: Line 445:
| {{monzo| 26 -12 -3 }}
| {{monzo| 26 -12 -3 }}
| 17.60
| 17.60
| Sasa-trigu
| Sasa-triguma
| [[Misty comma]]
| [[Misty comma]]
|-
|-
Line 519: Line 452:
| {{monzo| -15 8 1 }}
| {{monzo| -15 8 1 }}
| 1.95
| 1.95
| Layo
| Layoma
| Schisma
| Schisma
|-
|-
Line 526: 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 533: Line 466:
| {{monzo| 8 0 -1 -2 }}
| {{monzo| 8 0 -1 -2 }}
| 76.03
| 76.03
| Rurugu
| Ruruguma
| Bapbo comma
| Bapbo comma
|-
|-
Line 540: 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 547: 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 554: Line 487:
| {{monzo| 1 0 2 -2 }}
| {{monzo| 1 0 2 -2 }}
| 34.98
| 34.98
| Biruyo
| Biruyoma
| Jubilisma
| Jubilisma
|-
|-
Line 561: Line 494:
| {{monzo| -9 6 1 -1 }}
| {{monzo| -9 6 1 -1 }}
| 29.22
| 29.22
| Laruyo
| Laruyoma
| Schismean comma
| Schismean comma
|-
|-
Line 568: Line 501:
| {{monzo| 6 -2 0 -1 }}
| {{monzo| 6 -2 0 -1 }}
| 27.26
| 27.26
| Ru
| Ruma
| Septimal comma
| Septimal comma
|-
|-
Line 575: 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 582: Line 515:
| {{monzo| 1 2 -3 1 }}
| {{monzo| 1 2 -3 1 }}
| 13.79
| 13.79
| Zotrigu
| Zotriguma
| Starling comma
| Starling comma
|-
|-
Line 589: Line 522:
| {{monzo| 5 -4 3 -2 }}
| {{monzo| 5 -4 3 -2 }}
| 13.47
| 13.47
| Rurutriyo
| Rurutriyoma
| Octagar comma
| Octagar comma
|-
|-
Line 596: Line 529:
| {{monzo| -9 8 -4 2 }}
| {{monzo| -9 8 -4 2 }}
| 8.04
| 8.04
| Labizogugu
| Labizoguguma
| [[Varunisma]]
| [[Varunisma]]
|-
|-
Line 603: Line 536:
| {{monzo| -5 2 2 -1 }}
| {{monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Ruyoyo
| Ruyoyoma
| Marvel comma
| Marvel comma
|-
|-
Line 610: Line 543:
| {{monzo| 6 0 -5 2 }}
| {{monzo| 6 0 -5 2 }}
| 6.08
| 6.08
| Zozoquingu
| Zozoquinguma
| Hemimean comma
| Hemimean comma
|-
|-
Line 617: Line 550:
| {{monzo| 10 -6 1 -1 }}
| {{monzo| 10 -6 1 -1 }}
| 5.76
| 5.76
| Saruyo
| Saruyoma
| Hemifamity comma
| Hemifamity comma
|-
|-
Line 624: Line 557:
| {{monzo| 25 -14 0 -1 }}
| {{monzo| 25 -14 0 -1 }}
| 3.80
| 3.80
| Sasaru
| Sasaruma
| [[Garischisma]]
| [[Garischisma]]
|-
|-
Line 631: 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 638: Line 571:
| {{monzo| -4 6 -6 3 }}
| {{monzo| -4 6 -6 3 }}
| 0.33
| 0.33
| Trizogugu
| Trizoguguma
| [[Landscape comma]]
| [[Landscape comma]]
|-
|-
Line 645: 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 652: 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 659: 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 666: Line 599:
| {{monzo| -1 0 1 2 -2 }}
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| 21.33
| Luluzozoyo
| Luluzozoyoma
| Frostma
| Frostma
|-
|-
Line 673: Line 606:
| {{monzo| -1 2 0 -2 1 }}
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| 17.58
| Loruru
| Loruruma
| Mothwellsma
| Mothwellsma
|-
|-
Line 680: Line 613:
| {{monzo| 2 -2 2 0 -1 }}
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| 17.40
| Luyoyo
| Luyoyoma
| Ptolemisma
| Ptolemisma
|-
|-
Line 687: Line 620:
| {{monzo| 4 0 -2 -1 1 }}
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Loruguguma
| Valinorsma
| Valinorsma
|-
|-
Line 694: 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 701: Line 634:
| {{monzo| -3 2 -1 2 -1 }}
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| 3.93
| Luzozogu
| Luzozoguma
| Werckisma
| Werckisma
|-
|-
Line 708: Line 641:
| {{monzo| -3 4 -2 -2 2 }}
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Bilorugu
| Biloruguma
| Kalisma
| Kalisma
|-
|-
Line 715: 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 722: 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 729: 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 736: 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 743: 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 750: 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 757: 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 764: 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 771: 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 778: 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 785: 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 792: 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 799: 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 806: 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 813: 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 820: 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 827: 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 847: 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 874: 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 ==
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 907: 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 ==
* [[Lumatone mapping for 12edo]]
* [[Lumatone mapping for 12edo]]
* [[: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]]
Retrieved from "https://en.xen.wiki/w/12edo"