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{{interwiki
{{Interwiki
| de = 12edo
| en = 12edo
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| es = 12 EDO
| ja = 12平均律
| ja = 12平均律
| ro = 12DEO
}}
}}
{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2<sup>2</sup> × 3
{{Wikipedia|12 equal temperament}}
| Step size = 100.000¢
{{ED intro}} It is the predominating tuning system in the world today.
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it arose as a temperament, is the predominating tuning system in the world today.


== Theory ==
== Theory ==
{| class="wikitable center-all"
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]].
!
 
! prime 2
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.
! prime 3
 
! prime 5
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}}
! prime 7
 
! prime 11
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.
! prime 13
 
! prime 17
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]].
! prime 19
 
|-
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.
! Error (¢)
 
| 0
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.
| -2.0
 
| +13.7
=== Prime harmonics ===
| +31.2
{{Harmonics in equal|12|prec=2}}
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}


12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 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+2/3 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.
=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite.  


The seventh partial ([[7/4]]) is "approximated" by an interval which is sharp by over 31 cents, and stands 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 supports [[septimal meantone]] (meaning that 12edo tempers out the [[syntonic comma]], and going 10 fifths up, i.e. C-A#, gives 12edo's best approximation to 7/4), its credentials in the 7-limit department are distinctly cheesy. ([[31edo]] is a much better tuning for septimal meantone.) It cannot be said to approximate 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 [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].
[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].


== Intervals ==
== Intervals ==
{| class="wikitable right-1 right-2"
{| class="wikitable center-all"
! Steps
|+ style="font-size: 105%;" | Intervals of 12edo
! Cents
|-
! colspan="3" | Interval
! [[Degree]]
! Approximate JI Ratios*
! [[Cent]]s
! [[Interval region]]
! 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
| 0
| 0
| unison
| Unison (prime)
| P1
| [[1/1]] (just)
| D
| [[File:piano_0_1edo.mp3]]
| 1/1
|
|-
|-
| 1
| 1
| 100
| 100
| aug 1sn, minor 2nd
| Minor second
| A1, m2
| [[256/243]] (+9.775)<br>[[16/15]] (−11.731)<br>[[25/24]] (+29.328)
| D#, Eb
| [[File:piano_1_12edo.mp3]]
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| [[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 2nd
| Major second
| M2
| [[9/8]] (−3.910)<br>[[10/9]] (+17.596)
| E
| [[File:piano_1_6edo.mp3]]
| 8/7, 9/8, 10/9, 17/15, 19/17
| [[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 3rd
| Minor third
| m3
| [[32/27]] (+5.865)<br>[[6/5]] (−15.641)<br>[[75/64]] (+25.418)
| F
| [[File:piano_1_4edo.mp3]]
| 7/6, 6/5, 19/16
| [[7/6]] (+33.129), [[25/21]] (−1.847)<br>[[19/16]] (+2.487)
|-
|-
| 4
| 4
| 400
| 400
| major 3rd
| Major third
| M3
| [[81/64]] (−7.820)<br>[[5/4]] (+13.686)<br> [[32/25]] (-27.373)
| F#
| [[File:piano_1_3edo.mp3]]
| 5/4, 9/7
| [[63/50]] (−0.108), [[9/7]] (−35.084)<br>[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
|-
| 5
| 5
| 500
| 500
| perfect 4th
| Fourth
| P4
| [[4/3]] (+1.955)<br> [[27/20]] (-19.551)
| G
| [[File:piano_5_12edo.mp3]]
| 4/3
| [[21/16]] (-29.219)
|-
|-
| 6
| 6
| 600
| 600
| aug 4th, dim 5th
| [[Tritone]]
| A4, d5
| [[25/18]] (+31.283)<br>[[36/25]] (-31.283)<br>[[45/32]] (+9.776)<br>[[64/45]] (−9.776)
| G#, Ab
| [[File:piano_1_2edo.mp3]]
| 7/5, 10/7, 17/12, 24/17
| [[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
| perfect 5th
| Fifth
| P5
| [[3/2]] (−1.955)<br>[[40/27]] (+19.551)
| A
| [[File:piano_7_12edo.mp3]]
| 3/2
| [[32/21]] (+29.219)
|-
|-
| 8
| 8
| 800
| 800
| minor 6th
| Minor sixth
| m6
| [[128/81]] (+7.820)<br>[[8/5]] (−13.686)<br>[[25/16]] (+27.373)
| Bb
| [[File:piano_2_3edo.mp3]]
| 8/5, 14/9
| [[14/9]] (+35.084), [[100/63]] (+0.108)<br>[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
|-
| 9
| 9
| 900
| 900
| major 6th
| Major sixth
| M6
| [[27/16]] (−5.865)<br>[[5/3]] (+15.641)<br>[[128/75]] (-25.418)
| B
| [[File:piano_3_4edo.mp3]]
| 5/3, 12/7, 32/19
| [[12/7]] (−33.129), [[42/25]] (+1.847)<br>[[32/19]] (−2.487)
|-
|-
| 10
| 10
| 1000
| 1000
| minor 7th
| Minor seventh
| m7
| [[16/9]] (+3.910)<br>[[9/5]] (−17.596)
| C
| [[File:piano_5_6edo.mp3]]
| 7/4, 9/5, 16/9
| [[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 7th
| Major seventh
| M7
| [[243/128]] (-9.775)<br>[[15/8]] (+11.731)<br>[[48/25]] (−29.328)
| C#
| [[File:piano_11_12edo.mp3]]
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| [[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
| 1200
| 1200
| perfect 8ve
| Octave
| P8
| [[2/1]] (just)
| D
| [[File:piano_1_1edo.mp3]]
| 2/1
|
|}
|}
<nowiki>*</nowiki> based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.
<references group="note" />


== Just approximation ==
== Notation ==
=== Selected just intervals by error ===
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.
==== 15-odd-limit interval mappings ====


The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''.  
{{Sharpness-sharp1|12}}
 
The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].
 
Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.


{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
|-
! Interval, complement
! rowspan="2" | [[Degree]]
! Error (abs, [[cent|¢]])
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
|-
| '''[[4/3]], [[3/2]]'''
! Diatonic ([[5L&nbsp;2s]]) interval names
| '''1.955'''
! Note names (on D)
|-
|-
| [[9/8]], [[16/9]]
| 0
| 3.910
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
|-
| ''[[13/11]], [[22/13]]''
| 1
| ''10.790''
| 100
| Augmented unison (A1)<br>Minor second (m2)
| D#<br>Eb
|-
|-
| [[16/15]], [[15/8]]
| 2
| 11.731
| 200
| '''Major second (M2)'''<br>Diminished third (d3)
| '''E'''<br>Fb
|-
|-
| '''[[5/4]], [[8/5]]'''
| 3
| '''13.686'''
| 300
| Augmented second (A2)<br>'''Minor third (m3)'''
| E#<br>'''F'''
|-
| 4
| 400
| Major third (M3)<br>Diminished fourth (d4)
| F#<br>Gb
|-
|-
| [[6/5]], [[5/3]]
| 5
| 15.641
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
|-
| [[7/5]], [[10/7]]
| 6
| 17.488
| 600
| Augmented fourth (A4)<br>Diminished fifth (d5)
| G#<br>Ab
|-
|-
| [[14/11]], [[11/7]]
| 7
| 17.508
| 700
| '''Perfect fifth (P5)'''
| A
|-
|-
| [[10/9]], [[9/5]]
| 8
| 17.596
| 800
| Augmented fifth (A5)<br>Minor sixth (m6)
| A#<br>Bb
|-
|-
| [[15/14]], [[28/15]]
| 9
| 19.443
| 900
| '''Major sixth (M6)'''<br>Diminished seventh (d7)
| '''B'''<br>Cb
|-
|-
| ''[[14/13]], [[13/7]]''
| 10
| ''28.298''
| 1000
| Augmented sixth (A6)<br>'''Minor seventh (m7)'''
| B#<br>'''C'''
|-
|-
| '''[[8/7]], [[7/4]]'''
| 11
| '''31.174'''
| 1100
| Major seventh (M7)<br>Diminished octave (d8)
| C#<br>Db
|-
|-
| [[7/6]], [[12/7]]
| 12
| 33.129
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}
 
In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&#x266F;) and flats (&#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
 
==== Evo flavor ====
{{Sagittal chart|Evo}}
 
Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
 
==== Revo flavor ====
{{Sagittal chart}}
 
== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
|-
| [[11/10]], [[20/11]]
! [[Degree]]
| 34.996
! [[Cents]]
! Standard [[solfege]]<br>(movable do)
! [[Uniform solfege]]<br>(2-3 vowels)
|-
|-
| [[9/7]], [[14/9]]
| 0
| 35.084
| 0
| Do
| Da
|-
|-
| [[18/13]], [[13/9]]
| 1
| 36.618
| 100
| Di (A1)<br>Ra (m2)
| Du (A1)<br>Fra (m2)
|-
|-
| [[15/11]], [[22/15]]
| 2
| 36.951
| 200
| Re
| Ra
|-
|-
| [[13/12]], [[24/13]]
| 3
| 38.573
| 300
| Ri (A2)<br>Me (m3)
| Ru (A2)<br>Na (m3)
|-
|-
| '''[[16/13]], [[13/8]]'''
| 4
| '''40.528'''
| 400
| Mi
| Ma (M3)<br>Fo (d4)
|-
|-
| ''[[13/10]], [[20/13]]''
| 5
| ''45.786''
| 500
| Fa
| Mu (A3)<br>Fa (P4)
|-
|-
| ''[[11/9]], [[18/11]]''
| 6
| ''47.408''
| 600
| Fi (A4)<br>Se (d5)
| Pa (A4)<br>Sha (d5)
|-
|-
| ''[[15/13]], [[26/15]]''
| 7
| ''47.741 ''
| 700
| So
| Sa
|-
|-
| '''[[11/8]], [[16/11]]'''
| 8
| '''48.682'''
| 800
| Si (A5)<br>Le (m6)
| Su (A5)<br>Fla (m6)
|-
|-
| ''[[12/11]], [[11/6]]''
| 9
| ''49.323''
| 900
|}
| La
 
| La (M6)<br>Tho (d7)
==== Selected 19-limit intervals ====
 
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]
 
An expanded version of the above, including some higher-limit intervals:
 
[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]
 
=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et.
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
|-
! colspan="2" |Octave stretch (¢)
| 10
| +0.617
| 1000
| -1.56
| Li (A6)<br>Te (m7)
| -3.95
| Lu (A6)<br>Tha (m7)
| -2.53
|-
|-
! rowspan="2" |Error
| 11
! [[TE error|absolute]] (¢)
| 1100
| 0.617
| Ti
| 3.11
| Ta (M7)<br>Do (d8)
| 4.92
| 4.52
|-
|-
! [[TE simple badness|relative]] (%)
| 12
| 0.617
| 1200
| 3.11
| Do
| 4.94
| Da
| 4.53
|}
|}


* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively.
== Approximation to JI ==
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]


== As a regular temperament ==
=== 15-odd-limit interval mappings ===
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3<sup>12</sup>/2<sup>19</sup>, the Didymus comma, [[81/80]], the 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.
{{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}}


=== Rank two temperaments ===
== Regular temperament properties ==
* [[List of 12et rank two temperaments by badness]]
{| class="wikitable center-4 center-5 center-6"
* [[List of 12et rank two temperaments by complexity]]
|-
* [[List of edo-distinct 12f rank two temperaments]]
! rowspan="2" | [[Subgroup]]
* [[Schismic-Pythagorean equivalence continuum]]
! rowspan="2" | [[Comma list]]
 
! rowspan="2" | [[Mapping]]
{| class="wikitable"
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
! Periods <br> per octave
! [[TE error|Absolute]] (¢)
! Generator
! [[TE simple badness|Relative]] (%)
! Temperaments
|-
|-
| 1
| 2.3
| 1\12
| {{Monzo| -19 12 }}
| [[Ripple]]
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
|-
| 1
| 2.3.5
| 5\12
| 81/80, 128/125
| [[Meantone]]/[[dominant]]
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
|-
| 2
| 2.3.5.7
| 1\12
| 36/35, 50/49, 64/63
| [[Srutal]]/[[pajara]]/[[injera]]
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
|-
| 3
| 2.3.5.7.17
| 1\12
| 36/35, 50/49, 51/49, 64/63
| [[Augmented]]
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
|-
| 4
| 2.3.5.7.17.19
| 1\12
| 36/35, 50/49, 51/49, 57/56, 64/63
| [[Diminished]]
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
|-
| 6
| 2.3.5.17.19
| 1\12
| 51/50, 76/75, 81/80, 128/125
| [[Hexe]]
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this.
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].
=== Uniform maps ===
{{Uniform map|edo=12}}


=== Commas ===
=== Commas ===
12 EDO [[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"
|-
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
! [[Color name]]
! [[Color name]]
! Name(s)
! Name
|-
|-
| 3
| 3
| [[531441/524288|(12 digits)]]
| <abbr title="531441/524288">(12 digits)</abbr>
| {{monzo| -19 12 }}
| {{monzo| -19 12 }}
| 23.46
| 23.46
| Lalawa
| Lalawama / Poma
| [[Pythagorean comma]]
| [[Pythagorean comma]]
|-
|-
Line 359: Line 410:
| {{monzo| 3 4 -4 }}
| {{monzo| 3 4 -4 }}
| 62.57
| 62.57
| Quadgu
| Quadguma
| Major diesis, diminished comma
| Diminished comma, greater diesis
|-
| 5
| <abbr title="262144/253125">(12 digits)</abbr>
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
|-
| 5
| 5
Line 366: Line 424:
| {{monzo| 7 0 -3 }}
| {{monzo| 7 0 -3 }}
| 41.06
| 41.06
| Trigu
| Triguma
| Diesis, augmented comma
| Augmented comma, lesser diesis
|-
|-
| 5
| 5
Line 373: 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
|-
|-
| 5
| 5
Line 380: Line 438:
| {{monzo| 11 -4 -2 }}
| {{monzo| 11 -4 -2 }}
| 19.55
| 19.55
| Sagugu
| Saguguma
| Diaschisma
| Diaschisma
|-
|-
| 5
| 5
| [[67108864/66430125|(16 digits)]]
| <abbr title="67108864/66430125">(16 digits)</abbr>
| {{monzo| 26 -12 -3 }}
| {{monzo| 26 -12 -3 }}
| 17.60
| 17.60
| Sasa-trigu
| Sasa-triguma
| [[Misty comma]]
| [[Misty comma]]
|-
|-
Line 394: Line 452:
| {{monzo| -15 8 1 }}
| {{monzo| -15 8 1 }}
| 1.95
| 1.95
| Layo
| Layoma
| Schisma
| Schisma
|-
|-
Line 401: Line 459:
| {{monzo| 161 -84 -12 }}
| {{monzo| 161 -84 -12 }}
| 0.02
| 0.02
| Sepbisa-quadtrigu
| Sepbisa-quadtriguma
| [[Atom]]
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
|-
| 7
| 7
Line 408: Line 480:
| {{monzo| 2 2 -1 -1 }}
| {{monzo| 2 2 -1 -1 }}
| 48.77
| 48.77
| Rugu
| Ruguma
| Septimal quartertone
| Mint comma, septimal quarter tone
|-
|-
| 7
| 7
Line 415: Line 487:
| {{monzo| 1 0 2 -2 }}
| {{monzo| 1 0 2 -2 }}
| 34.98
| 34.98
| Biruyo
| Biruyoma
| Tritonic diesis, jubilisma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
|-
| 7
| 7
Line 422: Line 501:
| {{monzo| 6 -2 0 -1 }}
| {{monzo| 6 -2 0 -1 }}
| 27.26
| 27.26
| Ru
| Ruma
| Septimal comma, Archytas' comma, Leipziger Komma
| Septimal comma
|-
|-
| 7
| 7
Line 429: Line 508:
| {{monzo| 0 -2 5 -3 }}
| {{monzo| 0 -2 5 -3 }}
| 21.18
| 21.18
| Triru-aquinyo
| Triru-aquinyoma
| Gariboh
| Gariboh comma
|-
|-
| 7
| 7
Line 436: Line 515:
| {{monzo| 1 2 -3 1 }}
| {{monzo| 1 2 -3 1 }}
| 13.79
| 13.79
| Zotrigu
| Zotriguma
| Septimal semicomma, starling comma
| Starling comma
|-
|-
| 7
| 7
Line 443: Line 522:
| {{monzo| 5 -4 3 -2 }}
| {{monzo| 5 -4 3 -2 }}
| 13.47
| 13.47
| Rurutriyo
| Rurutriyoma
| Octagar
| Octagar comma
|-
|-
| 7
| 7
Line 450: Line 529:
| {{monzo| -9 8 -4 2 }}
| {{monzo| -9 8 -4 2 }}
| 8.04
| 8.04
| Labizogugu
| Labizoguguma
| [[Varunisma]]
| [[Varunisma]]
|-
|-
Line 457: Line 536:
| {{monzo| -5 2 2 -1 }}
| {{monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Ruyoyo
| Ruyoyoma
| Septimal kleisma, marvel comma
| Marvel comma
|-
|-
| 7
| 7
Line 464: Line 543:
| {{monzo| 6 0 -5 2 }}
| {{monzo| 6 0 -5 2 }}
| 6.08
| 6.08
| Zozoquingu
| Zozoquinguma
| Hemimean
| Hemimean comma
|-
|-
| 7
| 7
Line 471: Line 550:
| {{monzo| 10 -6 1 -1 }}
| {{monzo| 10 -6 1 -1 }}
| 5.76
| 5.76
| Saruyo
| Saruyoma
| Hemifamity
| Hemifamity comma
|-
|-
| 7
| 7
Line 478: Line 557:
| {{monzo| 25 -14 0 -1 }}
| {{monzo| 25 -14 0 -1 }}
| 3.80
| 3.80
| Sasaru
| Sasaruma
| [[Garischisma]]
| [[Garischisma]]
|-
|-
Line 485: Line 564:
| {{monzo| -11 2 7 -3 }}
| {{monzo| -11 2 7 -3 }}
| 1.63
| 1.63
| Latriru-asepyo
| Latriru-asepyoma
| [[Meter]]
| [[Metric comma]]
|-
|-
| 7
| 7
Line 492: Line 571:
| {{monzo| -4 6 -6 3 }}
| {{monzo| -4 6 -6 3 }}
| 0.33
| 0.33
| Trizogugu
| Trizoguguma
| [[Landscape comma]]
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
|-
| 11
| 11
Line 499: Line 606:
| {{monzo| -1 2 0 -2 1 }}
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| 17.58
| Loruru
| Loruruma
| Mothwellsma
| Mothwellsma
|-
|-
Line 506: Line 613:
| {{monzo| 2 -2 2 0 -1 }}
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| 17.40
| Luyoyo
| Luyoyoma
| Ptolemisma
| Ptolemisma
|-
|-
Line 513: Line 620:
| {{monzo| 4 0 -2 -1 1 }}
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Loruguguma
| Valinorsma
| Valinorsma
|-
|-
Line 520: Line 627:
| {{monzo| 7 -4 0 1 -1 }}
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| 9.69
| Saluzo
| Saluzoma
| Pentacircle
| Pentacircle comma
|-
|-
| 11
| 11
Line 527: Line 634:
| {{monzo| -3 2 -1 2 -1 }}
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| 3.93
| Luzozogu
| Luzozoguma
| Werckisma
| Werckisma
|-
|-
Line 534: Line 641:
| {{monzo| -3 4 -2 -2 2 }}
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Bilorugu
| Biloruguma
| Kalisma, Gauss' comma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
|-
| 13
| 13
Line 541: Line 655:
| {{monzo| -1 -2 -1 1 0 1 }}
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| 19.13
| Thozogu
| Thozoguma
| Superleap
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
|}
<references/>
<references group="note" />
 
=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods<br>per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
<nowiki>*</nowiki> [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 & 72), [[schismic]]/[[garibaldi]] (41 & 53), and [[diaschismic]] (46 & 58). Rank-3 temperaments to which 12et can be detempered include [[marvel]], [[starling]]/[[thrush]], [[aberschismic]], [[orthoschismic]], and [[cassaschismic]]. For more comprehensive lists, see:
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]
 
== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.


== Scales ==
== Scales ==
{{main| MOS in 12edo }}
{{See also| List of MOS scales in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
 
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diatonic: [[5L 2s]] – 2221221 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
* Pentatonic: [[2L 3s]] – 22323 (generator = 7\12)
==Pathological Modes==
 
3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS
The diminished and augmented scales are also MOS scales.
* Diminished: [[4L 4s]] – 12121212 (generator = 1\12, period = 3\12)
* Augmented: [[3L 3s]] – 131313 (generator = 1\12, period = 4\12)
 
Other widely used scales include:
* Melodic minor – 2122221
* Harmonic minor – 2122131
* Harmonic major – 2212131
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
 
== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''


2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS
* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]


2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS
== Music ==
{{Catrel|12edo tracks}}
 
The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.


== See also ==
== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step


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


[[Category:12edo]]
[[Category:3-limit record edos|##]] <!-- 2-digit number -->
[[Category:Equal divisions of the octave]]
[[Category:Historical]]
[[Category:Meantone]]
[[Category:Meantone]]
[[Category:Zeta]]
Retrieved from "https://en.xen.wiki/w/12edo"