12edo: Difference between revisions
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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. | ||
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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 | 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]]. | ||
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=== 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 | ||
| | | Lalawama / Poma | ||
| [[Pythagorean comma]] | | [[Pythagorean comma]] | ||
|- | |- | ||
| Line 411: | Line 411: | ||
| {{monzo| 3 4 -4 }} | | {{monzo| 3 4 -4 }} | ||
| 62.57 | | 62.57 | ||
| | | 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 | ||
| | | Saquinguma | ||
| [[Passion comma]] | | [[Passion comma]] | ||
|- | |- | ||
| Line 425: | Line 425: | ||
| {{monzo| 7 0 -3 }} | | {{monzo| 7 0 -3 }} | ||
| 41.06 | | 41.06 | ||
| | | 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 | ||
| | | 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 | ||
| | | Saguguma | ||
| Diaschisma | | Diaschisma | ||
|- | |- | ||
| Line 446: | Line 446: | ||
| {{monzo| 26 -12 -3 }} | | {{monzo| 26 -12 -3 }} | ||
| 17.60 | | 17.60 | ||
| Sasa- | | Sasa-triguma | ||
| [[Misty comma]] | | [[Misty comma]] | ||
|- | |- | ||
| Line 453: | Line 453: | ||
| {{monzo| -15 8 1 }} | | {{monzo| -15 8 1 }} | ||
| 1.95 | | 1.95 | ||
| | | Layoma | ||
| Schisma | | Schisma | ||
|- | |- | ||
| Line 460: | Line 460: | ||
| {{monzo| 161 -84 -12 }} | | {{monzo| 161 -84 -12 }} | ||
| 0.02 | | 0.02 | ||
| Sepbisa- | | 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 | ||
| | | Ruruguma | ||
| Bapbo comma | | Bapbo comma | ||
|- | |- | ||
| Line 474: | Line 474: | ||
| {{monzo| -13 10 0 -1 }} | | {{monzo| -13 10 0 -1 }} | ||
| 50.72 | | 50.72 | ||
| | | 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 | ||
| | | 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 | ||
| | | Biruyoma | ||
| Jubilisma | | Jubilisma | ||
|- | |- | ||
| Line 495: | Line 495: | ||
| {{monzo| -9 6 1 -1 }} | | {{monzo| -9 6 1 -1 }} | ||
| 29.22 | | 29.22 | ||
| | | Laruyoma | ||
| Schismean comma | | Schismean comma | ||
|- | |- | ||
| Line 502: | Line 502: | ||
| {{monzo| 6 -2 0 -1 }} | | {{monzo| 6 -2 0 -1 }} | ||
| 27.26 | | 27.26 | ||
| | | Ruma | ||
| Septimal comma | | Septimal comma | ||
|- | |- | ||
| Line 509: | Line 509: | ||
| {{monzo| 0 -2 5 -3 }} | | {{monzo| 0 -2 5 -3 }} | ||
| 21.18 | | 21.18 | ||
| Triru- | | Triru-aquinyoma | ||
| Gariboh comma | | Gariboh comma | ||
|- | |- | ||
| Line 516: | Line 516: | ||
| {{monzo| 1 2 -3 1 }} | | {{monzo| 1 2 -3 1 }} | ||
| 13.79 | | 13.79 | ||
| | | Zotriguma | ||
| Starling comma | | Starling comma | ||
|- | |- | ||
| Line 523: | Line 523: | ||
| {{monzo| 5 -4 3 -2 }} | | {{monzo| 5 -4 3 -2 }} | ||
| 13.47 | | 13.47 | ||
| | | Rurutriyoma | ||
| Octagar comma | | Octagar comma | ||
|- | |- | ||
| Line 530: | Line 530: | ||
| {{monzo| -9 8 -4 2 }} | | {{monzo| -9 8 -4 2 }} | ||
| 8.04 | | 8.04 | ||
| | | Labizoguguma | ||
| [[Varunisma]] | | [[Varunisma]] | ||
|- | |- | ||
| Line 537: | Line 537: | ||
| {{monzo| -5 2 2 -1 }} | | {{monzo| -5 2 2 -1 }} | ||
| 7.71 | | 7.71 | ||
| | | Ruyoyoma | ||
| Marvel comma | | Marvel comma | ||
|- | |- | ||
| Line 544: | Line 544: | ||
| {{monzo| 6 0 -5 2 }} | | {{monzo| 6 0 -5 2 }} | ||
| 6.08 | | 6.08 | ||
| | | Zozoquinguma | ||
| Hemimean comma | | Hemimean comma | ||
|- | |- | ||
| Line 551: | Line 551: | ||
| {{monzo| 10 -6 1 -1 }} | | {{monzo| 10 -6 1 -1 }} | ||
| 5.76 | | 5.76 | ||
| | | Saruyoma | ||
| Hemifamity comma | | Hemifamity comma | ||
|- | |- | ||
| Line 558: | Line 558: | ||
| {{monzo| 25 -14 0 -1 }} | | {{monzo| 25 -14 0 -1 }} | ||
| 3.80 | | 3.80 | ||
| | | Sasaruma | ||
| [[Garischisma]] | | [[Garischisma]] | ||
|- | |- | ||
| Line 565: | Line 565: | ||
| {{monzo| -11 2 7 -3 }} | | {{monzo| -11 2 7 -3 }} | ||
| 1.63 | | 1.63 | ||
| Latriru- | | Latriru-asepyoma | ||
| [[Metric comma]] | | [[Metric comma]] | ||
|- | |- | ||
| Line 572: | Line 572: | ||
| {{monzo| -4 6 -6 3 }} | | {{monzo| -4 6 -6 3 }} | ||
| 0.33 | | 0.33 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | Luluzozoyoma | ||
| Frostma | | Frostma | ||
|- | |- | ||
| Line 607: | Line 607: | ||
| {{monzo| -1 2 0 -2 1 }} | | {{monzo| -1 2 0 -2 1 }} | ||
| 17.58 | | 17.58 | ||
| | | Loruruma | ||
| Mothwellsma | | Mothwellsma | ||
|- | |- | ||
| Line 614: | Line 614: | ||
| {{monzo| 2 -2 2 0 -1 }} | | {{monzo| 2 -2 2 0 -1 }} | ||
| 17.40 | | 17.40 | ||
| | | Luyoyoma | ||
| Ptolemisma | | Ptolemisma | ||
|- | |- | ||
| Line 621: | Line 621: | ||
| {{monzo| 4 0 -2 -1 1 }} | | {{monzo| 4 0 -2 -1 1 }} | ||
| 9.86 | | 9.86 | ||
| | | Loruguguma | ||
| Valinorsma | | Valinorsma | ||
|- | |- | ||
| Line 628: | Line 628: | ||
| {{monzo| 7 -4 0 1 -1 }} | | {{monzo| 7 -4 0 1 -1 }} | ||
| 9.69 | | 9.69 | ||
| | | 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 | ||
| | | Luzozoguma | ||
| Werckisma | | Werckisma | ||
|- | |- | ||
| Line 642: | Line 642: | ||
| {{monzo| -3 4 -2 -2 2 }} | | {{monzo| -3 4 -2 -2 2 }} | ||
| 0.18 | | 0.18 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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 | ||
| | | 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/> | <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]] | 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 == | ||
Revision as of 00:10, 2 June 2026
| ← 11edo | 12edo | 13edo → |
(convergent)
12 equal divisions of the octave (abbreviated 12edo or 12ed2), also called 12-tone equal temperament (12tet) or 12 equal temperament (12et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 12 equal parts of exactly 100 ¢ each. Each step represents a frequency ratio of 21/12, or the 12th root of 2. It is the predominating tuning system in the world today.
Theory
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave (edo) which can seriously claim to represent 5-limit harmony, and it represents a meantone temperament.
It divides the octave into twelve equal parts, each of exactly 100 cents. It has a fifth which is quite accurate at 700 cents, two cents flat of just. It has a major third which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The minor third is even less accurate, being 15.6 cents flat of just.
Before people used 12edo, people used a variety of historical temperaments such as quarter-comma meantone, and later well temperaments. 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 serialism and much of jazz harmony that derive from 12edo's structure as an equal division rather than its underlying temperament properties.[citation needed]
12edo is the basic example of a 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 chords in functional harmony, for which the 5-limit JI version would be 1–5/4–3/2–16/9, and while 12et officially supports septimal meantone for tempering out 126/125 and 225/224 via its patent val of ⟨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, specially not 11, 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.
Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, 312/219, 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 commas 12et 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.
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.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.96 | +13.69 | +31.17 | +48.68 | -40.53 | -4.96 | +2.49 | -28.27 | -29.58 | -45.04 |
| Relative (%) | +0.0 | -2.0 | +13.7 | +31.2 | +48.7 | -40.5 | -5.0 | +2.5 | -28.3 | -29.6 | -45.0 | |
| Steps (reduced) |
12 (0) |
19 (7) |
28 (4) |
34 (10) |
42 (6) |
44 (8) |
49 (1) |
51 (3) |
54 (6) |
58 (10) |
59 (11) | |
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 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 with an impecable 11-limit, and 60-, 84-, and 96edo all see utilities. Notable rank-2 temperaments that augment 12edo with extra generators include compton and catler.
Intervals
| Degree | Cents | Interval region | Approximated 5-limit JI intervals (error in ¢) |
Audio | Higher limit interpretations[note 1] |
|---|---|---|---|---|---|
| 0 | 0 | Unison (prime) | 1/1 (just) | ||
| 1 | 100 | Minor second | 256/243 (+9.775) 25/24 (+29.328) 16/15 (−11.731) |
28/27 (+37.039), 21/20 (+15.533), 15/14 (−19.443) 17/16 (−4.955), 18/17 (+1.045) 19/18 (+6.397), 20/19 (+11.199) | |
| 2 | 200 | Major second | 9/8 (−3.910) 10/9 (+17.596) |
8/7 (−31.174), 28/25 (+3.802) 17/15 (−16.687), 19/17 (+7.442), 55/49 (+0.020), 64/57 (−0.532) | |
| 3 | 300 | Minor third | 32/27 (+5.865) 6/5 (−15.641) 75/64 (+25.418) |
7/6 (+33.129), 25/21 (−1.847) 19/16 (+2.487) | |
| 4 | 400 | Major third | 81/64 (−7.820) 5/4 (+13.686) 32/25 (-27.373) |
63/50 (−0.108), 9/7 (−35.084) 34/27 (+0.910), 24/19 (−4.442) | |
| 5 | 500 | Fourth | 4/3 (+1.955) 27/20 (-19.551) |
21/16 (-29.219) | |
| 6 | 600 | Tritone | 25/18 (+31.283) 36/25 (-31.283) 45/32 (+9.776) 64/45 (−9.776) |
7/5 (+17.488), 10/7 (−17.488) 24/17 (+3.000), 17/12 (−3.000) 99/70 (−0.088), 140/99 (+0.088) | |
| 7 | 700 | Fifth | 3/2 (−1.955) 40/27 (+19.551) |
32/21 (+29.219) | |
| 8 | 800 | Minor sixth | 128/81 (+7.820) 8/5 (−13.686) 25/16 (+27.373) |
14/9 (+35.084), 100/63 (+0.108) 19/12 (+4.442), 27/17 (−0.910) | |
| 9 | 900 | Major sixth | 27/16 (−5.865) 5/3 (+15.641) 128/75 (-25.418) |
12/7 (−33.129), 42/25 (+1.847) 32/19 (−2.487) | |
| 10 | 1000 | Minor seventh | 16/9 (+3.910) 9/5 (−17.596) |
7/4 (+31.174), 25/14 (−3.802) 30/17 (+16.687), 34/19 (−7.442) 98/55 (-0.020), 57/32 (+0.532) | |
| 11 | 1100 | Major seventh | 243/128 (-9.775) 15/8 (+11.731) 48/25 (−29.328) |
28/15 (+19.443), 40/21 (−15.533), 27/14 (−37.039) 32/17 (+4.955), 17/9 (−1.045) 36/19 (-6.397), 19/10 (-11.199) | |
| 12 | 1200 | Octave | 2/1 (just) |
- ↑ Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations
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.
| Semitones | −2 | −1 | 0 | +1 | +2 |
|---|---|---|---|---|---|
| Symbol |  |
 |
 |
 |
 |
The subsets 1edo, 2edo, 3edo, 4edo and 6edo can all be written using 12edo subset notation.
Any 12edo note or interval can be respelled enharmonically by adding a pythagorean comma to it or subtracting one from it.
| Degree | Cents | Standard notation | |
|---|---|---|---|
| Diatonic (5L 2s) interval names | Note names (on D) | ||
| 0 | 0 | Perfect unison (P1) | D |
| 1 | 100 | Augmented unison (A1) Minor second (m2) |
D# Eb |
| 2 | 200 | Major second (M2) Diminished third (d3) |
E Fb |
| 3 | 300 | Augmented second (A2) Minor third (m3) |
E# F |
| 4 | 400 | Major third (M3) Diminished fourth (d4) |
F# Gb |
| 5 | 500 | Perfect fourth (P4) | G |
| 6 | 600 | Augmented fourth (A4) Diminished fifth (d5) |
G# Ab |
| 7 | 700 | Perfect fifth (P5) | A |
| 8 | 800 | Augmented fifth (A5) Minor sixth (m6) |
A# Bb |
| 9 | 900 | Major sixth (M6) Diminished seventh (d7) |
B Cb |
| 10 | 1000 | Augmented sixth (A6) Minor seventh (m7) |
B# C |
| 11 | 1100 | Major seventh (M7) Diminished octave (d8) |
C# Db |
| 12 | 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 (♯) and flats (♭) for sagittal sharp (
) and sagittal flat (
) respectively.
Sagittal notation
This notation uses the same sagittal sequence as EDOs 5, 19, and 26, is a subset of the notations for EDOs 24, 36, 48, 60, 72, and 84, and is a superset of the notation for 6-EDO.
Evo flavor
Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
Revo flavor
Solfege
| Degree | Cents | Standard solfege (movable do) |
Uniform solfege (2-3 vowels) |
|---|---|---|---|
| 0 | 0 | Do | Da |
| 1 | 100 | Di (A1) Ra (m2) |
Du (A1) Fra (m2) |
| 2 | 200 | Re | Ra |
| 3 | 300 | Ri (A2) Me (m3) |
Ru (A2) Na (m3) |
| 4 | 400 | Mi | Ma (M3) Fo (d4) |
| 5 | 500 | Fa | Mu (A3) Fa (P4) |
| 6 | 600 | Fi (A4) Se (d5) |
Pa (A4) Sha (d5) |
| 7 | 700 | So | Sa |
| 8 | 800 | Si (A5) Le (m6) |
Su (A5) Fla (m6) |
| 9 | 900 | La | La (M6) Tho (d7) |
| 10 | 1000 | Li (A6) Te (m7) |
Lu (A6) Tha (m7) |
| 11 | 1100 | Ti | Ta (M7) Do (d8) |
| 12 | 1200 | Do | Da |
Approximation to JI
15-odd-limit interval mappings
The following tables show how 15-odd-limit intervals are represented in 12edo. Prime harmonics are in bold; inconsistent intervals are in italics.
Note that, since the cent was defined in terms of 12edo, the absolute and relative errors for 12edo are identical.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 3/2, 4/3 | 1.955 | 2.0 |
| 9/8, 16/9 | 3.910 | 3.9 |
| 13/11, 22/13 | 10.790 | 10.8 |
| 15/8, 16/15 | 11.731 | 11.7 |
| 5/4, 8/5 | 13.686 | 13.7 |
| 5/3, 6/5 | 15.641 | 15.6 |
| 7/5, 10/7 | 17.488 | 17.5 |
| 11/7, 14/11 | 17.508 | 17.5 |
| 9/5, 10/9 | 17.596 | 17.6 |
| 15/14, 28/15 | 19.443 | 19.4 |
| 13/7, 14/13 | 28.298 | 28.3 |
| 7/4, 8/7 | 31.174 | 31.2 |
| 7/6, 12/7 | 33.129 | 33.1 |
| 11/10, 20/11 | 34.996 | 35.0 |
| 9/7, 14/9 | 35.084 | 35.1 |
| 13/9, 18/13 | 36.618 | 36.6 |
| 15/11, 22/15 | 36.951 | 37.0 |
| 13/12, 24/13 | 38.573 | 38.6 |
| 13/8, 16/13 | 40.528 | 40.5 |
| 13/10, 20/13 | 45.786 | 45.8 |
| 11/9, 18/11 | 47.408 | 47.4 |
| 15/13, 26/15 | 47.741 | 47.7 |
| 11/8, 16/11 | 48.682 | 48.7 |
| 11/6, 12/11 | 49.363 | 49.4 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 3/2, 4/3 | 1.955 | 2.0 |
| 9/8, 16/9 | 3.910 | 3.9 |
| 15/8, 16/15 | 11.731 | 11.7 |
| 5/4, 8/5 | 13.686 | 13.7 |
| 5/3, 6/5 | 15.641 | 15.6 |
| 7/5, 10/7 | 17.488 | 17.5 |
| 11/7, 14/11 | 17.508 | 17.5 |
| 9/5, 10/9 | 17.596 | 17.6 |
| 15/14, 28/15 | 19.443 | 19.4 |
| 7/4, 8/7 | 31.174 | 31.2 |
| 7/6, 12/7 | 33.129 | 33.1 |
| 11/10, 20/11 | 34.996 | 35.0 |
| 9/7, 14/9 | 35.084 | 35.1 |
| 13/9, 18/13 | 36.618 | 36.6 |
| 15/11, 22/15 | 36.951 | 37.0 |
| 13/12, 24/13 | 38.573 | 38.6 |
| 13/8, 16/13 | 40.528 | 40.5 |
| 11/8, 16/11 | 48.682 | 48.7 |
| 11/6, 12/11 | 50.637 | 50.6 |
| 15/13, 26/15 | 52.259 | 52.3 |
| 11/9, 18/11 | 52.592 | 52.6 |
| 13/10, 20/13 | 54.214 | 54.2 |
| 13/7, 14/13 | 71.702 | 71.7 |
| 13/11, 22/13 | 89.210 | 89.2 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 3/2, 4/3 | 1.955 | 2.0 |
| 9/8, 16/9 | 3.910 | 3.9 |
| 13/11, 22/13 | 10.790 | 10.8 |
| 15/8, 16/15 | 11.731 | 11.7 |
| 5/4, 8/5 | 13.686 | 13.7 |
| 5/3, 6/5 | 15.641 | 15.6 |
| 7/5, 10/7 | 17.488 | 17.5 |
| 11/7, 14/11 | 17.508 | 17.5 |
| 9/5, 10/9 | 17.596 | 17.6 |
| 15/14, 28/15 | 19.443 | 19.4 |
| 13/7, 14/13 | 28.298 | 28.3 |
| 7/4, 8/7 | 31.174 | 31.2 |
| 7/6, 12/7 | 33.129 | 33.1 |
| 11/10, 20/11 | 34.996 | 35.0 |
| 9/7, 14/9 | 35.084 | 35.1 |
| 15/11, 22/15 | 36.951 | 37.0 |
| 13/10, 20/13 | 45.786 | 45.8 |
| 15/13, 26/15 | 47.741 | 47.7 |
| 11/8, 16/11 | 48.682 | 48.7 |
| 11/6, 12/11 | 50.637 | 50.6 |
| 11/9, 18/11 | 52.592 | 52.6 |
| 13/8, 16/13 | 59.472 | 59.5 |
| 13/12, 24/13 | 61.427 | 61.4 |
| 13/9, 18/13 | 63.382 | 63.4 |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-19 12⟩ | [⟨12 19]] | +0.62 | 0.62 | 0.62 |
| 2.3.5 | 81/80, 128/125 | [⟨12 19 28]] | −1.56 | 3.11 | 3.11 |
| 2.3.5.7 | 36/35, 50/49, 64/63 | [⟨12 19 28 34]] | −3.95 | 4.92 | 4.94 |
| 2.3.5.7.17 | 36/35, 50/49, 51/49, 64/63 | [⟨12 19 28 34 49]] | −2.92 | 4.86 | 4.87 |
| 2.3.5.7.17.19 | 36/35, 50/49, 51/49, 57/56, 64/63 | [⟨12 19 28 34 49 51]] | −2.53 | 4.52 | 4.53 |
| 2.3.5.17 | 51/50, 81/80, 128/125 | [⟨12 19 28 49]] | −0.87 | 2.95 | 2.95 |
| 2.3.5.17.19 | 51/50, 76/75, 81/80, 128/125 | [⟨12 19 28 49 51]] | −0.81 | 2.64 | 2.64 |
- 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-, 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.
Uniform maps
| Min. size | Max. size | Wart notation | Map |
|---|---|---|---|
| 11.7554 | 11.8436 | 12cde | ⟨12 19 27 33 41 44] |
| 11.8436 | 11.9329 | 12de | ⟨12 19 28 33 41 44] |
| 11.9329 | 11.9962 | 12e | ⟨12 19 28 34 41 44] |
| 11.9962 | 12.0256 | 12 | ⟨12 19 28 34 42 44] |
| 12.0256 | 12.2743 | 12f | ⟨12 19 28 34 42 45] |
Commas
12edo tempers out the following commas. This assumes the val ⟨12 19 28 34 42 44 49 51].
| Prime limit |
Ratio[note 1] | Monzo | Cents | Color name | Name |
|---|---|---|---|---|---|
| 3 | (12 digits) | [-19 12⟩ | 23.46 | Lalawama / Poma | Pythagorean comma |
| 5 | 648/625 | [3 4 -4⟩ | 62.57 | Quadguma | Diminished comma, greater diesis |
| 5 | (12 digits) | [18 -4 -5⟩ | 60.61 | Saquinguma | Passion comma |
| 5 | 128/125 | [7 0 -3⟩ | 41.06 | Triguma | Augmented comma, lesser diesis |
| 5 | 81/80 | [-4 4 -1⟩ | 21.51 | Guma | Syntonic comma, Didymus' comma, meantone comma |
| 5 | 2048/2025 | [11 -4 -2⟩ | 19.55 | Saguguma | Diaschisma |
| 5 | (16 digits) | [26 -12 -3⟩ | 17.60 | Sasa-triguma | Misty comma |
| 5 | 32805/32768 | [-15 8 1⟩ | 1.95 | Layoma | Schisma |
| 5 | (98 digits) | [161 -84 -12⟩ | 0.02 | Sepbisa-quadtriguma | Kirnberger's atom |
| 7 | 256/245 | [8 0 -1 -2⟩ | 76.03 | Ruruguma | Bapbo comma |
| 7 | 59049/57344 | [-13 10 0 -1⟩ | 50.72 | Laruma | Harrison's comma |
| 7 | 36/35 | [2 2 -1 -1⟩ | 48.77 | Ruguma | Mint comma, septimal quarter tone |
| 7 | 50/49 | [1 0 2 -2⟩ | 34.98 | Biruyoma | Jubilisma |
| 7 | 3645/3584 | [-9 6 1 -1⟩ | 29.22 | Laruyoma | Schismean comma |
| 7 | 64/63 | [6 -2 0 -1⟩ | 27.26 | Ruma | Septimal comma |
| 7 | 3125/3087 | [0 -2 5 -3⟩ | 21.18 | Triru-aquinyoma | Gariboh comma |
| 7 | 126/125 | [1 2 -3 1⟩ | 13.79 | Zotriguma | Starling comma |
| 7 | 4000/3969 | [5 -4 3 -2⟩ | 13.47 | Rurutriyoma | Octagar comma |
| 7 | (12 digits) | [-9 8 -4 2⟩ | 8.04 | Labizoguguma | Varunisma |
| 7 | 225/224 | [-5 2 2 -1⟩ | 7.71 | Ruyoyoma | Marvel comma |
| 7 | 3136/3125 | [6 0 -5 2⟩ | 6.08 | Zozoquinguma | Hemimean comma |
| 7 | 5120/5103 | [10 -6 1 -1⟩ | 5.76 | Saruyoma | Hemifamity comma |
| 7 | (16 digits) | [25 -14 0 -1⟩ | 3.80 | Sasaruma | Garischisma |
| 7 | (12 digits) | [-11 2 7 -3⟩ | 1.63 | Latriru-asepyoma | Metric comma |
| 7 | (12 digits) | [-4 6 -6 3⟩ | 0.33 | Trizoguguma | Landscape comma |
| 11 | 128/121 | [7 0 0 0 -2⟩ | 97.36 | Lulubima | Axirabian limma |
| 11 | 45/44 | [-2 2 1 0 -1⟩ | 38.91 | Luyoma | Undecimal fifth tone |
| 11 | 56/55 | [3 0 -1 1 -1⟩ | 31.19 | Luzoguma | Undecimal tritonic comma |
| 11 | 245/242 | [-1 0 1 2 -2⟩ | 21.33 | Luluzozoyoma | Frostma |
| 11 | 99/98 | [-1 2 0 -2 1⟩ | 17.58 | Loruruma | Mothwellsma |
| 11 | 100/99 | [2 -2 2 0 -1⟩ | 17.40 | Luyoyoma | Ptolemisma |
| 11 | 176/175 | [4 0 -2 -1 1⟩ | 9.86 | Loruguguma | Valinorsma |
| 11 | 896/891 | [7 -4 0 1 -1⟩ | 9.69 | Saluzoma | Pentacircle comma |
| 11 | 441/440 | [-3 2 -1 2 -1⟩ | 3.93 | Luzozoguma | Werckisma |
| 11 | 9801/9800 | [-3 4 -2 -2 2⟩ | 0.18 | Biloruguma | Kalisma |
| 13 | 65/64 | [-6 0 1 0 0 1⟩ | 26.84 | Thoyoma | Wilsorma |
| 13 | 91/90 | [-1 -2 -1 1 0 1⟩ | 19.13 | Thozoguma | Superleap comma, biome comma |
| 13 | 144/143 | [4 2 0 0 -1 -1⟩ | 12.06 | Thuluma | Grossma |
| 13 | 1001/1000 | [-3 0 -3 1 1 1⟩ | 1.73 | Tholozotriguma | Fairytale comma, sinbadma |
| 13 | 4096/4095 | [12 -2 -1 -1 0 -1⟩ | 0.42 | Sathuruguma | Minisma |
| 17 | 51/50 | [-1 1 -2 0 0 0 1⟩ | 34.28 | Soguguma | Large septendecimal sixth tone |
| 17 | 52/51 | [2 -1 0 0 0 1 -1⟩ | 33.62 | Suthoma | Small septendecimal sixth tone |
| 17 | 136/135 | [3 -3 -1 0 0 0 1⟩ | 12.78 | Soguma | Diatisma, fiventeen comma |
| 17 | 256/255 | [8 -1 -1 0 0 0 -1⟩ | 6.78 | Suguma | Charisma, septendecimal kleisma |
| 17 | 289/288 | [-5 -2 0 0 0 0 2⟩ | 6.00 | Sosoma | Semitonisma |
| 17 | 2601/2600 | [-3 2 -2 0 0 -1 2⟩ | 0.67 | Sosothuguguma | Sextantonisma |
| 19 | 39/38 | [-1 1 0 0 0 1 0 -1⟩ | 44.97 | Nuthoma | Undevicesimal two-ninth tone |
| 19 | 96/95 | [5 1 -1 0 0 0 0 -1⟩ | 18.13 | Nuguma | 19th-partial chroma |
| 19 | 153/152 | [-3 2 0 0 0 0 1 -1⟩ | 11.35 | Nusoma | Ganassisma |
| 19 | 171/170 | [-1 2 -1 0 0 0 -1 1⟩ | 10.15 | Nosuguma | Malcolmisma |
| 19 | 324/323 | [2 4 0 0 0 0 -1 -1⟩ | 5.35 | Nusuma | Photisma |
| 19 | 361/360 | [-3 -2 -1 0 0 0 0 2⟩ | 4.80 | Nonoguma | Go comma |
| 19 | 513/512 | [9 3 0 0 0 0 0 -1⟩ | 3.37 | Lanoma | Boethius' comma |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.
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
| Periods per 8ve |
Generator* | Pergen | Temperaments |
|---|---|---|---|
| 1 | 1\12 | (P8, P4/5) | Ripple / passion |
| 1 | 5\12 | (P8, P5) | Meantone / dominant |
| 2 | 5\12 (1\12) | (P8/2, P5) | Srutal / pajara / injera |
| 3 | 5\12 (1\12) | (P8/3, P5) | Augmented / lithium |
| 4 | 5\12 (1\12) | (P8/4, P5) | Diminished |
| 6 | 5\12 (1\12) | (P8/6, P5) | Hexe |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Rank-2 temperaments to which 12et can be detempered:
- Schismic / 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)*
- Aberschismic / Pele (41 & 46 & 53)*
- Cassaschismic (41 & 53 & 270)*
*Through the 12e val
**Through the 12f val
Octave stretch or compression
Whether there is intonational improvement from octave stretch and compression for 12edo varies by context. A slight compression such as what is given by 40ed10 and 34zpi 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 7edf, 19edt, or 31ed6, also makes sense.
Scales
The two most common 12edo mos scales are meantone[5] and meantone[7].
- Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
- Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
- Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
Non-mos scales
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.
- Harmonic major – 2212132
- Melodic major – 2212122
- Hungarian minor – 2131131
- Maqam hijaz / double harmonic major – 1312131
- 5-odd-limit tonality diamond – 3112113
Well temperaments
- For a list of historical well temperaments, see Well temperament.
Music
- See also: Category: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
- 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