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== Theory ==
== Theory ==
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 [[cent]]s 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.
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 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 [[cent]]s 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 13.7 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.


The seventh partial ([[7/4]]) is "represented" 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 [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. 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 [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].
The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by some 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 [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].


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.
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.


12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).


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 offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.


=== Prime harmonics ===
===Prime harmonics===
{{Harmonics in equal}}
{{Harmonics in equal}}


== Intervals ==
==Intervals==
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ Intervals of 12edo
|+Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" |[[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! rowspan="2" |[[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! colspan="4" |Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
! rowspan="2" |Audio
|-
|-
! [[3-limit]]
![[3-limit]]
! [[5-limit]]
![[5-limit]]
! [[7-limit]]
![[7-limit]]
! Other
!Other
|-
|-
| 0
|0
| 0
|0
| 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
|  
|
| [[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)
|[[28/27]] (+37.039)<br>[[21/20]] (+15.533)<br>[[15/14]] (-19.443)
| [[18/17]] (+1.045)<br>[[17/16]] (-4.955)
|[[18/17]] (+1.045)<br>[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|[[File:piano_1_12edo.mp3]]
|-
|-
| 2
|2
| 200
| 200
| Major second
|Major second
| [[9/8]] (-3.910)
|[[9/8]] (-3.910)
| [[10/9]] (+17.596)
|[[10/9]] (+17.596)
| [[28/25]] (+3.802)<br>[[8/7]] (-31.174)
|[[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)
|[[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]]
|-
|-
| 3
|3
| 300
| 300
| Minor third
|Minor third
| [[32/27]] (+5.865)
|[[32/27]] (+5.865)
| [[6/5]] (-15.641)
|[[6/5]] (-15.641)
| [[7/6]] (+33.129)<br>[[25/21]] (-1.847)
|[[7/6]] (+33.129)<br>[[25/21]] (-1.847)
| [[19/16]] (+2.487)<br>[[44/37]] (+0.026)
|[[19/16]] (+2.487)<br>[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|[[File:piano_1_4edo.mp3]]
|-
|-
| 4
|4
| 400
| 400
| Major third
|Major third
| [[81/64]] (-7.820)
|[[81/64]] (-7.820)
| [[5/4]] (+13.686)
|[[5/4]] (+13.686)
| [[63/50]] (-0.108)<br>[[9/7]] (-35.084)
|[[63/50]] (-0.108)<br>[[9/7]] (-35.084)
| [[34/27]] (+0.910)<br>[[24/19]] (-4.442)
|[[34/27]] (+0.910)<br>[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|[[File:piano_1_3edo.mp3]]
|-
|-
| 5
|5
| 500
|500
| Fourth
|Fourth
| [[4/3]] (+1.955)
|[[4/3]] (+1.955)
|  
|
|  
|
|  
|
| [[File:piano_5_12edo.mp3]]
|[[File:piano_5_12edo.mp3]]
|-
|-
| 6
|6
| 600
|600
| [[Tritone]]
|[[Tritone]]
|  
|
|  
|
| [[7/5]] (+17.488)<br>[[10/7]] (-17.488)
|[[7/5]] (+17.488)<br>[[10/7]] (-17.488)
| [[24/17]] (+3.000)<br>[[99/70]] (-0.088)<br>[[17/12]] (-3.000)
|[[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
|7
| 700
|700
| Fifth
|Fifth
| [[3/2]] (-1.955)
|[[3/2]] (-1.955)
|  
|
|  
|
|  
|
| [[File:piano_7_12edo.mp3]]
|[[File:piano_7_12edo.mp3]]
|-
|-
| 8
|8
| 800
| 800
| Minor sixth
|Minor sixth
| [[128/81]] (+7.820)
|[[128/81]] (+7.820)
| [[8/5]] (-13.686)
|[[8/5]] (-13.686)
| [[14/9]] (+35.084)<br>[[100/63]] (+0.108)
|[[14/9]] (+35.084)<br>[[100/63]] (+0.108)
| [[19/12]] (+4.442)<br>[[27/17]] (-0.910)
|[[19/12]] (+4.442)<br>[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|[[File:piano_2_3edo.mp3]]
|-
|-
| 9
|9
| 900
| 900
| Major sixth
|Major sixth
| [[27/16]] (-5.865)
|[[27/16]] (-5.865)
| [[5/3]] (+15.641)
|[[5/3]] (+15.641)
| [[42/25]] (+1.847)<br>[[12/7]] (-33.129)
|[[42/25]] (+1.847)<br>[[12/7]] (-33.129)
| [[37/22]] (-0.026)<br>[[32/19]] (-2.487)
|[[37/22]] (-0.026)<br>[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|[[File:piano_3_4edo.mp3]]
|-
|-
| 10
| 10
| 1000
|1000
| Minor seventh
|Minor seventh
| [[16/9]] (+3.910)
|[[16/9]] (+3.910)
| [[9/5]] (-17.596)
|[[9/5]] (-17.596)
| [[7/4]] (+31.174)<br>[[25/14]] (-3.802)
|[[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)
|[[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]]
|-
|-
| 11
| 11
| 1100
|1100
| Major seventh
|Major seventh
|  
|  
| [[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)
|[[28/15]] (+19.443)<br>[[40/21]] (-15.533)<br>[[27/14]] (-37.039)
| [[32/17]] (+4.955)<br>[[17/9]] (-1.045)
|[[32/17]] (+4.955)<br>[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|[[File:piano_11_12edo.mp3]]
|-
|-
| 12
| 12
| 1200
|1200
| Octave
|Octave
| [[2/1]] (just)
|[[2/1]] (just)
|  
|
|  
|
|  
|
| [[File:piano_1_1edo.mp3]]
|[[File:piano_1_1edo.mp3]]
|}
|}


<nowiki>*</nowiki> based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.
<nowiki>*</nowiki> based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.


== 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.
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.


Line 163: Line 163:


{| class="wikitable center-all"
{| class="wikitable center-all"
|+ Notation of 12edo
|+Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" |[[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" |[[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
|-
! [[5L 2s|Diatonic]] interval names
![[5L 2s|Diatonic]] interval names
! Note names (on D)
!Note names (on D)
|-
|-
| 0
|0
| 0
|0
| '''Perfect unison (P1)'''
|'''Perfect unison (P1)'''
| '''D'''
|'''D'''
|-
|-
| 1
|1
| 100
|100
| Augmented unison (A1)<br>Minor second (m2)
|Augmented unison (A1)<br>Minor second (m2)
| D#<br>Eb
|D#<br>Eb
|-
|-
| 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
| 300
|300
| Augmented second (A2)<br>'''Minor third (m3)'''
|Augmented second (A2)<br>'''Minor third (m3)'''
| E#<br>'''F'''
|E#<br>'''F'''
|-
|-
| 4
|4
| 400
| 400
| Major third (M3)<br>Diminished fourth (d4)
| Major third (M3)<br>Diminished fourth (d4)
| F#<br>Gb
|F#<br>Gb
|-
|-
| 5
|5
| 500
|500
| '''Perfect fourth (P4)'''
|'''Perfect fourth (P4)'''
| '''G'''
|'''G'''
|-
|-
| 6
|6
| 600
|600
| Augmented fourth (A4)<br>Diminished fifth (d5)
|Augmented fourth (A4)<br>Diminished fifth (d5)
| G#<br>Ab
|G#<br>Ab
|-
|-
| 7
|7
| 700
|700
| '''Perfect fifth (P5)'''
|'''Perfect fifth (P5)'''
| A
|A
|-
|-
| 8
|8
| 800
|800
| Augmented fifth (A5)<br>Minor sixth (m6)
|Augmented fifth (A5)<br>Minor sixth (m6)
| A#<br>Bb
|A#<br>Bb
|-
|-
| 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
| 1000
|1000
| Augmented sixth (A6)<br>'''Minor seventh (m7)'''
|Augmented sixth (A6)<br>'''Minor seventh (m7)'''
| B#<br>'''C'''
|B#<br>'''C'''
|-
|-
| 11
| 11
| 1100
|1100
| Major seventh (M7)<br>Diminished octave (d8)
|Major seventh (M7)<br>Diminished octave (d8)
| C#<br>Db
|C#<br>Db
|-
|-
| 12
| 12
| 1200
|1200
| '''Perfect octave (P8)'''
|'''Perfect octave (P8)'''
| '''D'''
|'''D'''
|}
|}


In 12edo:
In 12edo:
* [[ups and downs notation]] is identical to standard notation;
*[[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) 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 (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.


== Solfege ==
==Solfege==
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ Solfege of 12edo
|+Solfege of 12edo
! [[Degree]]
![[Degree]]
! [[Cents]]
![[Cents]]
! Standard [[solfege]]<br>(movable do)
!Standard [[solfege]]<br>(movable do)
! [[Uniform solfege]]<br>(2-3 vowels)
![[Uniform solfege]]<br>(2-3 vowels)
|-
|-
| 0
|0
| 0
|0
| Do
|Do
| Da
| Da
|-
|-
| 1
|1
| 100
|100
| Di (A1)<br>Ra (m2)
|Di (A1)<br>Ra (m2)
| Du (A1)<br>Fra (m2)
|Du (A1)<br>Fra (m2)
|-
|-
| 2
|2
| 200
|200
| Re
|Re
| Ra
| Ra
|-
|-
| 3
|3
| 300
|300
| Ri (A2)<br>Me (m3)
|Ri (A2)<br>Me (m3)
| Ru (A2)<br>Na (m3)
|Ru (A2)<br>Na (m3)
|-
|-
| 4
|4
| 400
|400
| Mi
|Mi
| Ma (M3)<br>Fo (d4)
|Ma (M3)<br>Fo (d4)
|-
|-
| 5
|5
| 500
|500
| Fa
|Fa
| Mu (A3)<br>Fa (P4)
|Mu (A3)<br>Fa (P4)
|-
|-
| 6
|6
| 600
|600
| Fi (A4)<br>Se (d5)
|Fi (A4)<br>Se (d5)
| Pa (A4)<br>Sha (d5)
|Pa (A4)<br>Sha (d5)
|-
|-
| 7
|7
| 700
|700
| So
|So
| Sa
| Sa
|-
|-
| 8
|8
| 800
|800
| Si (A5)<br>Le (m6)
|Si (A5)<br>Le (m6)
| Su (A5)<br>Fla (m6)
|Su (A5)<br>Fla (m6)
|-
|-
| 9
|9
| 900
|900
| La
|La
| La (M6)<br>Tho (d7)
|La (M6)<br>Tho (d7)
|-
|-
| 10
| 10
| 1000
|1000
| Li (A6)<br>Te (m7)
|Li (A6)<br>Te (m7)
| Lu (A6)<br>Tha (m7)
|Lu (A6)<br>Tha (m7)
|-
|-
| 11
| 11
| 1100
| 1100
| Ti
|Ti
| Ta (M7)<br>Do (d8)
|Ta (M7)<br>Do (d8)
|-
|-
| 12
| 12
| 1200
| 1200
| Do
|Do
| Da
|Da
|}
|}


== JI approximation ==
==JI approximation==
=== 15-odd-limit interval mappings ===
===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''.  
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''.  


{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|+ style="white-space:nowrap" |15-odd-limit intervals by direct approximation (even if inconsistent)
|-
|-
! Interval, complement
!Interval, complement
! Error (abs, [[cent|¢]])
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
! Error (rel, [[relative cent|%]])
|-
|-
| '''[[4/3]], [[3/2]]'''
|'''[[4/3]], [[3/2]]'''
| '''1.955'''
|'''1.955'''
| '''2.0'''
|'''2.0'''
|-
|-
| [[9/8]], [[16/9]]
|[[9/8]], [[16/9]]
| 3.910
|3.910
| 3.9
|3.9
|-
|-
| ''[[13/11]], [[22/13]]''
|''[[13/11]], [[22/13]]''
| ''10.790''
|''10.790''
| ''10.8''
|''10.8''
|-
|-
| [[16/15]], [[15/8]]
|[[16/15]], [[15/8]]
| 11.731
|11.731
| 11.7
|11.7
|-
|-
| '''[[5/4]], [[8/5]]'''
|'''[[5/4]], [[8/5]]'''
| '''13.686'''
|'''13.686'''
| '''13.7'''
|'''13.7'''
|-
|-
| [[6/5]], [[5/3]]
|[[6/5]], [[5/3]]
| 15.641
|15.641
| 15.6
|15.6
|-
|-
| [[7/5]], [[10/7]]
|[[7/5]], [[10/7]]
| 17.488
|17.488
| 17.5
|17.5
|-
|-
| [[14/11]], [[11/7]]
|[[14/11]], [[11/7]]
| 17.508
|17.508
| 17.5
|17.5
|-
|-
| [[10/9]], [[9/5]]
|[[10/9]], [[9/5]]
| 17.596
|17.596
| 17.6
|17.6
|-
|-
| [[15/14]], [[28/15]]
|[[15/14]], [[28/15]]
| 19.443
|19.443
| 19.4
|19.4
|-
|-
| ''[[14/13]], [[13/7]]''
|''[[14/13]], [[13/7]]''
| ''28.298''
|''28.298''
| ''28.3''
|''28.3''
|-
|-
| '''[[8/7]], [[7/4]]'''
|'''[[8/7]], [[7/4]]'''
| '''31.174'''
|'''31.174'''
| '''31.2'''
|'''31.2'''
|-
|-
| [[7/6]], [[12/7]]
|[[7/6]], [[12/7]]
| 33.129
|33.129
| 33.1
|33.1
|-
|-
| [[11/10]], [[20/11]]
|[[11/10]], [[20/11]]
| 34.996
|34.996
| 35.0
|35.0
|-
|-
| [[9/7]], [[14/9]]
|[[9/7]], [[14/9]]
| 35.084
|35.084
| 35.1
|35.1
|-
|-
| [[18/13]], [[13/9]]
|[[18/13]], [[13/9]]
| 36.618
|36.618
| 36.7
|36.7
|-
|-
| [[15/11]], [[22/15]]
|[[15/11]], [[22/15]]
| 36.951
|36.951
| 37.0
|37.0
|-
|-
| [[13/12]], [[24/13]]
|[[13/12]], [[24/13]]
| 38.573
|38.573
| 38.6
|38.6
|-
|-
| '''[[16/13]], [[13/8]]'''
|'''[[16/13]], [[13/8]]'''
| '''40.528'''
|'''40.528'''
| '''40.5'''
|'''40.5'''
|-
|-
| ''[[13/10]], [[20/13]]''
|''[[13/10]], [[20/13]]''
| ''45.786''
|''45.786''
| ''45.8''
|''45.8''
|-
|-
| ''[[11/9]], [[18/11]]''
|''[[11/9]], [[18/11]]''
| ''47.408''
|''47.408''
| ''47.4''
|''47.4''
|-
|-
| ''[[15/13]], [[26/15]]''
|''[[15/13]], [[26/15]]''
| ''47.741''
|''47.741''
| ''47.7''
|''47.7''
|-
|-
| '''[[11/8]], [[16/11]]'''
|'''[[11/8]], [[16/11]]'''
| '''48.682'''
|'''48.682'''
| '''48.7'''
|'''48.7'''
|-
|-
| ''[[12/11]], [[11/6]]''
|''[[12/11]], [[11/6]]''
| ''49.323''
|''49.323''
| ''49.3''
|''49.3''
|}
|}
{{15-odd-limit}}
{{15-odd-limit}}


=== Selected 19-limit intervals ===
===Selected 19-limit intervals===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]


Line 431: Line 431:
[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]
[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]


== Regular temperament properties ==
==Regular temperament properties==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" |Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
![[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
![[TE simple badness|Relative]] (%)
|-
|-
| 2.3
|2.3
| {{monzo| -19 12 }}
|{{monzo| -19 12 }}
| [{{val| 12 19 }}]
|[{{val| 12 19 }}]
| +0.617
| +0.617
| 0.617
|0.617
| 0.617
|0.617
|-
|-
| 2.3.5
|2.3.5
| 81/80, 128/125
|81/80, 128/125
| [{{val| 12 19 28 }}]
|[{{val| 12 19 28 }}]
| -1.56
| -1.56
| 3.11
|3.11
| 3.11
|3.11
|-
|-
| 2.3.5.7
|2.3.5.7
| 36/35, 50/49, 64/63
|36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
|[{{val| 12 19 28 34 }}]
| -3.95
| -3.95
| 4.92
|4.92
| 4.94
|4.94
|-
|-
| 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
| [{{val| 12 19 28 34 49 }}]
|[{{val| 12 19 28 34 49 }}]
| -2.92
| -2.92
| 4.86
|4.86
| 4.87
|4.87
|-
|-
| 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
| [{{val| 12 19 28 34 49 51 }}]
|[{{val| 12 19 28 34 49 51 }}]
| -2.53
| -2.53
| 4.52
|4.52
| 4.53
|4.53
|}
|}


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 ETs 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 ET that does this better is 72.
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 ETs 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 ET that does this better is 72.


=== Uniform maps ===
===Uniform maps===
{{Uniform map|13|11.5|12.5}}  
{{Uniform map|13|11.5|12.5}}  


=== Commas ===
===Commas===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.


{| 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>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
![[Monzo]]
! [[Cent]]s
![[Cent]]s
! [[Color name|Color Name]]
![[Color name|Color Name]]
! Name
!Name
|-
|-
| 3
|3
| [[531441/524288|(12 digits)]]
|[[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
|{{monzo| -19 12 }}
| 23.46
|23.46
| Lalawa
|Lalawa
| [[Pythagorean comma]]
|[[Pythagorean comma]]
|-
|-
| 5
|5
| [[648/625]]
|[[648/625]]
| {{monzo| 3 4 -4 }}
|{{monzo| 3 4 -4 }}
| 62.57
|62.57
| Quadgu
|Quadgu
| Diminished comma
|Diminished comma
|-
|-
| 5
|5
| [[128/125]]
|[[128/125]]
| {{monzo| 7 0 -3 }}
|{{monzo| 7 0 -3 }}
| 41.06
|41.06
| Trigu
|Trigu
| Augmented comma
|Augmented comma
|-
|-
| 5
|5
| [[81/80]]
|[[81/80]]
| {{monzo| -4 4 -1 }}
|{{monzo| -4 4 -1 }}
| 21.51
|21.51
| Gu
|Gu
| Syntonic comma
|Syntonic comma
|-
|-
| 5
|5
| [[2048/2025]]
|[[2048/2025]]
| {{monzo| 11 -4 -2 }}
|{{monzo| 11 -4 -2 }}
| 19.55
|19.55
| Sagugu
|Sagugu
| Diaschisma
|Diaschisma
|-
|-
| 5
|5
| [[67108864/66430125|(16 digits)]]
|[[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
|{{monzo| 26 -12 -3 }}
| 17.60
|17.60
| Sasa-trigu
|Sasa-trigu
| [[Misty comma]]
|[[Misty comma]]
|-
|-
| 5
|5
| [[32805/32768]]
|[[32805/32768]]
| {{monzo| -15 8 1 }}
|{{monzo| -15 8 1 }}
| 1.95
|1.95
| Layo
|Layo
| Schisma
|Schisma
|-
|-
| 5
|5
| <abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
|<abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
| {{monzo| 161 -84 -12 }}
|{{monzo| 161 -84 -12 }}
| 0.02
|0.02
| Sepbisa-quadtrigu
|Sepbisa-quadtrigu
| [[Atom]]
|[[Kirnberger's atom]]
|-
|-
| 7
|7
| [[36/35]]
|[[36/35]]
| {{monzo| 2 2 -1 -1 }}
|{{monzo| 2 2 -1 -1 }}
| 48.77
|48.77
| Rugu
|Rugu
| Septimal quartertone
|Septimal quartertone
|-
|-
| 7
|7
| [[50/49]]
|[[50/49]]
| {{monzo| 1 0 2 -2 }}
|{{monzo| 1 0 2 -2 }}
| 34.98
|34.98
| Biruyo
|Biruyo
| Jubilisma
|Jubilisma
|-
|-
| 7
|7
| [[64/63]]
|[[64/63]]
| {{monzo| 6 -2 0 -1 }}
|{{monzo| 6 -2 0 -1 }}
| 27.26
|27.26
| Ru
|Ru
| Septimal comma
|Septimal comma
|-
|-
| 7
|7
| [[3125/3087]]
|[[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
|{{monzo| 0 -2 5 -3 }}
| 21.18
|21.18
| Triru-aquinyo
|Triru-aquinyo
| Gariboh
|Gariboh
|-
|-
| 7
|7
| [[126/125]]
|[[126/125]]
| {{monzo| 1 2 -3 1 }}
|{{monzo| 1 2 -3 1 }}
| 13.79
|13.79
| Zotrigu
|Zotrigu
| Starling comma
|Starling comma
|-
|-
| 7
|7
| [[4000/3969]]
|[[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
|{{monzo| 5 -4 3 -2 }}
| 13.47
|13.47
| Rurutriyo
| Rurutriyo
| Octagar
|Octagar comma
|-
|-
| 7
|7
| <abbr title="321489/320000">(12 digits)</abbr>
|<abbr title="321489/320000">(12 digits)</abbr>
| {{monzo| -9 8 -4 2 }}
|{{monzo| -9 8 -4 2 }}
| 8.04
|8.04
| Labizogugu
|Labizogugu
| [[Varunisma]]
|[[Varunisma]]
|-
|-
| 7
|7
| [[225/224]]
|[[225/224]]
| {{monzo| -5 2 2 -1 }}
|{{monzo| -5 2 2 -1 }}
| 7.71
|7.71
| Ruyoyo
|Ruyoyo
| Marvel comma
|Marvel comma
|-
|-
| 7
|7
| [[3136/3125]]
|[[3136/3125]]
| {{monzo| 6 0 -5 2 }}
|{{monzo| 6 0 -5 2 }}
| 6.08
|6.08
| Zozoquingu
|Zozoquingu
| Hemimean
|Hemimean
|-
|-
| 7
|7
| [[5120/5103]]
|[[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
|{{monzo| 10 -6 1 -1 }}
| 5.76
|5.76
| Saruyo
|Saruyo
| Hemifamity
|Hemifamity
|-
|-
| 7
|7
| [[33554432/33480783|(16 digits)]]
|[[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
|{{monzo| 25 -14 0 -1 }}
| 3.80
|3.80
| Sasaru
|Sasaru
| [[Garischisma]]
|[[Garischisma]]
|-
|-
| 7
|7
| [[703125/702464|(12 digits)]]
|[[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
|{{monzo| -11 2 7 -3 }}
| 1.63
|1.63
| Latriru-asepyo
|Latriru-asepyo
| [[Meter]]
|[[Meter]]
|-
|-
| 7
|7
| <abbr title="250047/250000">(12 digits)</abbr>
|<abbr title="250047/250000">(12 digits)</abbr>
| {{monzo| -4 6 -6 3 }}
|{{monzo| -4 6 -6 3 }}
| 0.33
|0.33
| Trizogugu
| Trizogugu
| [[Landscape comma]]
|[[Landscape comma]]
|-
|-
| 11
|11
| [[99/98]]
|[[128/121]]
| {{monzo| -1 2 0 -2 1 }}
|{{monzo| 7 0 0 0 -2 }}
| 17.58
|97.36
| Loruru
|1uu2
| Mothwellsma
|Axirabian limma
|-
|11
|[[99/98]]
|{{monzo| -1 2 0 -2 1 }}
|17.58
|Loruru
|Mothwellsma
|-
|-
| 11
|11
| [[100/99]]
|[[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
|{{monzo| 2 -2 2 0 -1 }}
| 17.40
|17.40
| Luyoyo
|Luyoyo
| Ptolemisma
|Ptolemisma
|-
|-
| 11
|11
| [[176/175]]
|[[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
|{{monzo| 4 0 -2 -1 1 }}
| 9.86
|9.86
| Lorugugu
|Lorugugu
| Valinorsma
|Valinorsma
|-
|-
| 11
|11
| [[896/891]]
|[[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
|{{monzo| 7 -4 0 1 -1 }}
| 9.69
|9.69
| Saluzo
|Saluzo
| Pentacircle
|Pentacircle
|-
|-
| 11
|11
| [[441/440]]
|[[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
|{{monzo| -3 2 -1 2 -1 }}
| 3.93
|3.93
| Luzozogu
|Luzozogu
| Werckisma
|Werckisma
|-
|-
| 11
|11
| [[9801/9800]]
|[[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
|{{monzo| -3 4 -2 -2 2 }}
| 0.18
|0.18
| Bilorugu
|Bilorugu
| Kalisma
|Kalisma
|-
|-
| 13
|13
| [[91/90]]
|[[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
|{{monzo| -1 -2 -1 1 0 1 }}
| 19.13
|19.13
| Thozogu
|Thozogu
| Superleap
|Superleap
|}
|}
<references/>
<references />


=== Rank-2 temperaments ===
===Rank-2 temperaments===
* [[List of 12et rank two temperaments by badness]]
*[[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
*[[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
*[[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]
*[[Schismic-Pythagorean equivalence continuum]]


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! Periods <br> per 8ve
!Periods <br> per 8ve
! Generator
! Generator
! Pergen
!Pergen
! Temperaments
!Temperaments
|-
|-
| 1
|1
| 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]]
|[[Meantone]] / [[dominant]]
|-
|-
| 2
|2
| 1\12
|1\12
| (P8/2, P5)
|(P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|[[Srutal]] / [[pajara]] / [[injera]]
|-
|-
| 3
|3
| 1\12
|1\12
| (P8/3, P5)
|(P8/3, P5)
| [[Augmented]] / [[lithium]]
|[[Augmented]] / [[lithium]]
|-
|-
| 4
|4
| 1\12
|1\12
| (P8/4, P5)
|(P8/4, P5)
| [[Diminished]]
|[[Diminished]]
|-
|-
| 6
|6
| 1\12
|1\12
| (P8/6, P5)
|(P8/6, P5)
| [[Hexe]]
|[[Hexe]]
|}
|}


== Scales ==
==Scales==
{{Main| List of MOS scales in 12edo }}
{{Main| List of MOS scales in 12edo }}


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


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


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


== Well temperaments ==
==Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''
:''For a list of historical well temperaments, see [[Well temperament]].''


* [[Cauldron]]
*[[Cauldron]]
* [[Bifrost]]
*[[Bifrost]]
* [[Grail]]
*[[Grail]]
* [[Secor5 23TX]]
*[[Secor5 23TX]]
* [[Secor wt10]]
*[[Secor wt10]]
* [[Sabat1]]
*[[Sabat1]]
* [[Sabat2]]
*[[Sabat2]]


== Music ==
==Music==
{{Catrel|12edo tracks}}
{{Catrel|12edo tracks}}


== See also ==
==See also==
* [[Lumatone mapping for 12edo]]
*[[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]
*[[purdal:12-EDD]]


[[Category:Historical]]
[[Category:Historical]]
[[Category:Meantone]]
[[Category:Meantone]]
Retrieved from "https://en.xen.wiki/w/12edo"