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12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.
12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.


The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].
The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, specially not 11, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].


Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3<sup>12</sup>/2<sup>19</sup>]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.
Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3<sup>12</sup>/2<sup>19</sup>]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.
Line 33: Line 33:
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.  
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, 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]].
[[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 with an impecable 11-limit, 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 ==
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The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].
The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].


Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.
Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 382: Line 382:
| 2.64
| 2.64
|}
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of [[13/8]] instead of 8 steps) is lower in relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], [[11-limit|11-]], [[13-limit|13-]], and [[19-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], 19/19e, and 19egh, respectively.
* 12et (using the 12f val, where 9 steps - a major sixth - is used as the approximation of [[13/8]] instead of 8 steps) is lower in relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], [[11-limit|11-]], [[13-limit|13-]], and [[19-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], 19/19e, and 19egh, respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].


Line 389: Line 389:


=== 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 404: Line 404:
| {{monzo| -19 12 }}
| {{monzo| -19 12 }}
| 23.46
| 23.46
| Lalawa
| Lalawama / Poma
| [[Pythagorean comma]]
| [[Pythagorean comma]]
|-
|-
Line 411: Line 411:
| {{monzo| 3 4 -4 }}
| {{monzo| 3 4 -4 }}
| 62.57
| 62.57
| Quadgu
| Quadguma
| Diminished comma, greater diesis
| Diminished comma, greater diesis
|-
|-
Line 418: Line 418:
| {{monzo| 18 -4 -5 }}
| {{monzo| 18 -4 -5 }}
| 60.61
| 60.61
| Saquingu
| Saquinguma
| [[Passion comma]]
| [[Passion comma]]
|-
|-
Line 425: Line 425:
| {{monzo| 7 0 -3 }}
| {{monzo| 7 0 -3 }}
| 41.06
| 41.06
| Trigu
| Triguma
| Augmented comma, lesser diesis
| Augmented comma, lesser diesis
|-
|-
Line 432: Line 432:
| {{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 439: Line 439:
| {{monzo| 11 -4 -2 }}
| {{monzo| 11 -4 -2 }}
| 19.55
| 19.55
| Sagugu
| Saguguma
| Diaschisma
| Diaschisma
|-
|-
Line 446: Line 446:
| {{monzo| 26 -12 -3 }}
| {{monzo| 26 -12 -3 }}
| 17.60
| 17.60
| Sasa-trigu
| Sasa-triguma
| [[Misty comma]]
| [[Misty comma]]
|-
|-
Line 453: Line 453:
| {{monzo| -15 8 1 }}
| {{monzo| -15 8 1 }}
| 1.95
| 1.95
| Layo
| Layoma
| Schisma
| Schisma
|-
|-
Line 460: Line 460:
| {{monzo| 161 -84 -12 }}
| {{monzo| 161 -84 -12 }}
| 0.02
| 0.02
| Sepbisa-quadtrigu
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
| [[Kirnberger's atom]]
|-
|-
Line 467: Line 467:
| {{monzo| 8 0 -1 -2 }}
| {{monzo| 8 0 -1 -2 }}
| 76.03
| 76.03
| Rurugu
| Ruruguma
| Bapbo comma
| Bapbo comma
|-
|-
Line 474: Line 474:
| {{monzo| -13 10 0 -1 }}
| {{monzo| -13 10 0 -1 }}
| 50.72
| 50.72
| Laru
| Laruma
| Harrison's comma
| Harrison's comma
|-
|-
Line 481: Line 481:
| {{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 488: Line 488:
| {{monzo| 1 0 2 -2 }}
| {{monzo| 1 0 2 -2 }}
| 34.98
| 34.98
| Biruyo
| Biruyoma
| Jubilisma
| Jubilisma
|-
|-
Line 495: Line 495:
| {{monzo| -9 6 1 -1 }}
| {{monzo| -9 6 1 -1 }}
| 29.22
| 29.22
| Laruyo
| Laruyoma
| Schismean comma
| Schismean comma
|-
|-
Line 502: Line 502:
| {{monzo| 6 -2 0 -1 }}
| {{monzo| 6 -2 0 -1 }}
| 27.26
| 27.26
| Ru
| Ruma
| Septimal comma
| Septimal comma
|-
|-
Line 509: Line 509:
| {{monzo| 0 -2 5 -3 }}
| {{monzo| 0 -2 5 -3 }}
| 21.18
| 21.18
| Triru-aquinyo
| Triru-aquinyoma
| Gariboh comma
| Gariboh comma
|-
|-
Line 516: Line 516:
| {{monzo| 1 2 -3 1 }}
| {{monzo| 1 2 -3 1 }}
| 13.79
| 13.79
| Zotrigu
| Zotriguma
| Starling comma
| Starling comma
|-
|-
Line 523: Line 523:
| {{monzo| 5 -4 3 -2 }}
| {{monzo| 5 -4 3 -2 }}
| 13.47
| 13.47
| Rurutriyo
| Rurutriyoma
| Octagar comma
| Octagar comma
|-
|-
Line 530: Line 530:
| {{monzo| -9 8 -4 2 }}
| {{monzo| -9 8 -4 2 }}
| 8.04
| 8.04
| Labizogugu
| Labizoguguma
| [[Varunisma]]
| [[Varunisma]]
|-
|-
Line 537: Line 537:
| {{monzo| -5 2 2 -1 }}
| {{monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Ruyoyo
| Ruyoyoma
| Marvel comma
| Marvel comma
|-
|-
Line 544: Line 544:
| {{monzo| 6 0 -5 2 }}
| {{monzo| 6 0 -5 2 }}
| 6.08
| 6.08
| Zozoquingu
| Zozoquinguma
| Hemimean comma
| Hemimean comma
|-
|-
Line 551: Line 551:
| {{monzo| 10 -6 1 -1 }}
| {{monzo| 10 -6 1 -1 }}
| 5.76
| 5.76
| Saruyo
| Saruyoma
| Hemifamity comma
| Hemifamity comma
|-
|-
Line 558: Line 558:
| {{monzo| 25 -14 0 -1 }}
| {{monzo| 25 -14 0 -1 }}
| 3.80
| 3.80
| Sasaru
| Sasaruma
| [[Garischisma]]
| [[Garischisma]]
|-
|-
Line 565: Line 565:
| {{monzo| -11 2 7 -3 }}
| {{monzo| -11 2 7 -3 }}
| 1.63
| 1.63
| Latriru-asepyo
| Latriru-asepyoma
| [[Metric comma]]
| [[Metric comma]]
|-
|-
Line 572: Line 572:
| {{monzo| -4 6 -6 3 }}
| {{monzo| -4 6 -6 3 }}
| 0.33
| 0.33
| Trizogugu
| Trizoguguma
| [[Landscape comma]]
| [[Landscape comma]]
|-
|-
Line 579: Line 579:
| {{monzo| 7 0 0 0 -2 }}
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 97.36
| 1uu2
| Lulubima
| Axirabian limma
| Axirabian limma
|-
|-
Line 586: Line 586:
| {{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 593: Line 593:
| {{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 600: Line 600:
| {{monzo| -1 0 1 2 -2 }}
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| 21.33
| Luluzozoyo
| Luluzozoyoma
| Frostma
| Frostma
|-
|-
Line 607: Line 607:
| {{monzo| -1 2 0 -2 1 }}
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| 17.58
| Loruru
| Loruruma
| Mothwellsma
| Mothwellsma
|-
|-
Line 614: Line 614:
| {{monzo| 2 -2 2 0 -1 }}
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| 17.40
| Luyoyo
| Luyoyoma
| Ptolemisma
| Ptolemisma
|-
|-
Line 621: Line 621:
| {{monzo| 4 0 -2 -1 1 }}
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Loruguguma
| Valinorsma
| Valinorsma
|-
|-
Line 628: Line 628:
| {{monzo| 7 -4 0 1 -1 }}
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| 9.69
| Saluzo
| Saluzoma
| Pentacircle comma
| Pentacircle comma
|-
|-
Line 635: Line 635:
| {{monzo| -3 2 -1 2 -1 }}
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| 3.93
| Luzozogu
| Luzozoguma
| Werckisma
| Werckisma
|-
|-
Line 642: Line 642:
| {{monzo| -3 4 -2 -2 2 }}
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Bilorugu
| Biloruguma
| Kalisma
| Kalisma
|-
|-
Line 649: Line 649:
| {{monzo| -6 0 1 0 0 1 }}
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| 26.84
| Thoyo
| Thoyoma
| Wilsorma
| Wilsorma
|-
|-
Line 656: Line 656:
| {{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 663: Line 663:
| {{monzo| 4 2 0 0 -1 -1 }}
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| 12.06
| Thulu
| Thuluma
| Grossma
| Grossma
|-
|-
Line 670: Line 670:
| {{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 677: Line 677:
| {{monzo| 12 -2 -1 -1 0 -1 }}
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| 0.42
| Sathurugu
| Sathuruguma
| Minisma
| Minisma
|-
|-
Line 684: Line 684:
| {{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 691: Line 691:
| {{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 698: Line 698:
| {{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 705: Line 705:
| {{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 712: Line 712:
| {{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 719: Line 719:
| {{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 726: Line 726:
| {{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 733: Line 733:
| {{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 740: Line 740:
| {{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 747: Line 747:
| {{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 754: Line 754:
| {{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 761: Line 761:
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| 4.80
| Nonogu
| Nonoguma
| Go comma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
|}
<references group="note" />
<references group="note" />
Line 809: Line 816:
| [[Hexe]]
| [[Hexe]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki>*</nowiki> [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
Rank-2 temperaments to which 12et can be [[Detempering|detempered]]:
 
* [[Schismic]] / [[Schismatic family#Schismic, schismatic, a.k.a. helmholtz|helmholtz]] (12 & 53)
* [[Compton]] (12 & 72)
* [[Atomic]] (12 & 612)
* [[Garibaldi]] / [[cassandra]] (41 & 53)*
* [[Heptacot]] (311 & 323)*
* [[Misty]] / murky** (87 & 99ef)
* [[Septimal diaschismic]] (46 & 58)**
 
Rank-3 temperaments to which 12et can be detempered:
 
* [[Odin]] (12 & 84 & 270)
* [[Marvel]] (31 & 41 & 53)*
* [[Hemifamity|Aberschismic]] / [[Pele]] (41 & 46 & 53)*
* [[Cassaschismic]] (41 & 53 & 270)*
 
<nowiki>*</nowiki>Through the 12e val
 
<nowiki>**</nowiki>Through the 12f val


== Octave stretch or compression ==
== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.  


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