12edo
| ← 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 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 cents each unless octave stretching or compression is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.
The seventh partial (7/4) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially supports septimal meantone 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, 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.
The commas it tempers out include the Pythagorean comma, 312/219, the Didymus' comma, 81/80, the lesser diesis, 128/125, the diaschisma, 2048/2025, the Archytas' comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a well tempered nonet [clarification needed ], and it also contains 2edo, 3edo, 4edo and 6edo as subsets. 12edo is the 5th highly composite edo, 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both strict zeta and highly composite.
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
| 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) | |
Intervals
| Degree | Cents | Interval region | Approximated JI intervals[note 1] (error in ¢) | Audio | |||
|---|---|---|---|---|---|---|---|
| 3-limit | 5-limit | 7-limit | Other | ||||
| 0 | 0 | Unison (prime) | 1/1 (just) | ||||
| 1 | 100 | Minor second | 25/24 (+29.328) 16/15 (-11.731) |
28/27 (+37.039) 21/20 (+15.533) 15/14 (-19.443) |
18/17 (+1.045) 17/16 (-4.955) |
||
| 2 | 200 | Major second | 9/8 (-3.910) | 10/9 (+17.596) | 28/25 (+3.802) 8/7 (-31.174) |
19/17 (+7.442) 55/49 (+0.020) 64/57 (-0.532) 17/15 (-16.687) |
|
| 3 | 300 | Minor third | 32/27 (+5.865) | 6/5 (-15.641) | 7/6 (+33.129) 25/21 (-1.847) |
19/16 (+2.487) 44/37 (+0.026) |
|
| 4 | 400 | Major third | 81/64 (-7.820) | 5/4 (+13.686) | 63/50 (-0.108) 9/7 (-35.084) |
34/27 (+0.910) 24/19 (-4.442) |
|
| 5 | 500 | Fourth | 4/3 (+1.955) | ||||
| 6 | 600 | Tritone | 7/5 (+17.488) 10/7 (-17.488) |
24/17 (+3.000) 99/70 (-0.088) 17/12 (-3.000) |
|||
| 7 | 700 | Fifth | 3/2 (-1.955) | ||||
| 8 | 800 | Minor sixth | 128/81 (+7.820) | 8/5 (-13.686) | 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) | 42/25 (+1.847) 12/7 (-33.129) |
37/22 (-0.026) 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) 57/32 (+0.532) 98/55 (-0.020) 34/19 (-7.442) |
|
| 11 | 1100 | Major seventh | 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) |
||
| 12 | 1200 | Octave | 2/1 (just) | ||||
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.
| Semitones | −2 | −1 | 0 | +1 | +2 |
|---|---|---|---|---|---|
| Symbol |  |
 |
 |
 |
 |
1edo, 2edo, 3edo, 4edo and 6edo can all be written using 12edo subset notation.
| 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.
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 |
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 (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.
Uniform maps
| Min. size | Max. size | Wart notation | Map |
|---|---|---|---|
| 11.5000 | 11.5767 | 12bbcdddeeeff | ⟨12 18 27 32 40 43] |
| 11.5767 | 11.6722 | 12bbcdeeeff | ⟨12 18 27 33 40 43] |
| 11.6722 | 11.7071 | 12cdeeeff | ⟨12 19 27 33 40 43] |
| 11.7071 | 11.7554 | 12cdeff | ⟨12 19 27 33 41 43] |
| 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] |
| 12.2743 | 12.2853 | 12ccf | ⟨12 19 29 34 42 45] |
| 12.2853 | 12.2891 | 12cceef | ⟨12 19 29 34 43 45] |
| 12.2891 | 12.2958 | 12ccddeef | ⟨12 19 29 35 43 45] |
| 12.2958 | 12.3031 | 12ccddeefff | ⟨12 19 29 35 43 46] |
| 12.3031 | 12.5000 | 12bccddeefff | ⟨12 20 29 35 43 46] |
Commas
12edo tempers out the following commas. This assumes val ⟨12 19 28 34 42 44].
| Prime limit |
Ratio[note 2] | Monzo | Cents | Color name | Name |
|---|---|---|---|---|---|
| 3 | (12 digits) | [-19 12⟩ | 23.46 | Lalawa | Pythagorean comma |
| 5 | 648/625 | [3 4 -4⟩ | 62.57 | Quadgu | Diminished comma, greater diesis |
| 5 | (12 digits) | [18 -4 -5⟩ | 60.61 | Saquingu | Passion comma |
| 5 | 128/125 | [7 0 -3⟩ | 41.06 | Trigu | Augmented comma, lesser diesis |
| 5 | 81/80 | [-4 4 -1⟩ | 21.51 | Gu | Syntonic comma, Didymus' comma, meantone comma |
| 5 | 2048/2025 | [11 -4 -2⟩ | 19.55 | Sagugu | Diaschisma |
| 5 | (16 digits) | [26 -12 -3⟩ | 17.60 | Sasa-trigu | Misty comma |
| 5 | 32805/32768 | [-15 8 1⟩ | 1.95 | Layo | Schisma |
| 5 | (98 digits) | [161 -84 -12⟩ | 0.02 | Sepbisa-quadtrigu | Kirnberger's atom |
| 7 | 256/245 | [8 0 -1 -2⟩ | 76.03 | Rurugu | Bapbo comma |
| 7 | 59049/57344 | [-13 10 0 -1⟩ | 50.72 | Laru | Harrison's comma |
| 7 | 36/35 | [2 2 -1 -1⟩ | 48.77 | Rugu | Mint comma, septimal quarter tone |
| 7 | 50/49 | [1 0 2 -2⟩ | 34.98 | Biruyo | Jubilisma |
| 7 | 3645/3584 | [-9 6 1 -1⟩ | 29.22 | Laruyo | Schismean comma |
| 7 | 64/63 | [6 -2 0 -1⟩ | 27.26 | Ru | Septimal comma |
| 7 | 3125/3087 | [0 -2 5 -3⟩ | 21.18 | Triru-aquinyo | Gariboh comma |
| 7 | 126/125 | [1 2 -3 1⟩ | 13.79 | Zotrigu | Starling comma |
| 7 | 4000/3969 | [5 -4 3 -2⟩ | 13.47 | Rurutriyo | Octagar comma |
| 7 | (12 digits) | [-9 8 -4 2⟩ | 8.04 | Labizogugu | Varunisma |
| 7 | 225/224 | [-5 2 2 -1⟩ | 7.71 | Ruyoyo | Marvel comma |
| 7 | 3136/3125 | [6 0 -5 2⟩ | 6.08 | Zozoquingu | Hemimean comma |
| 7 | 5120/5103 | [10 -6 1 -1⟩ | 5.76 | Saruyo | Hemifamity comma |
| 7 | (16 digits) | [25 -14 0 -1⟩ | 3.80 | Sasaru | Garischisma |
| 7 | (12 digits) | [-11 2 7 -3⟩ | 1.63 | Latriru-asepyo | Meter comma |
| 7 | (12 digits) | [-4 6 -6 3⟩ | 0.33 | Trizogugu | Landscape comma |
| 11 | 128/121 | [7 0 0 0 -2⟩ | 97.36 | 1uu2 | Axirabian limma |
| 11 | 45/44 | [-2 2 1 0 -1⟩ | 38.91 | Luyo | Undecimal fifth tone |
| 11 | 56/55 | [3 0 -1 1 -1⟩ | 31.19 | Luzogu | Undecimal tritonic comma |
| 11 | 245/242 | [-1 0 1 2 -2⟩ | 21.33 | Luluzozoyo | Frostma |
| 11 | 99/98 | [-1 2 0 -2 1⟩ | 17.58 | Loruru | Mothwellsma |
| 11 | 100/99 | [2 -2 2 0 -1⟩ | 17.40 | Luyoyo | Ptolemisma |
| 11 | 176/175 | [4 0 -2 -1 1⟩ | 9.86 | Lorugugu | Valinorsma |
| 11 | 896/891 | [7 -4 0 1 -1⟩ | 9.69 | Saluzo | Pentacircle comma |
| 11 | 441/440 | [-3 2 -1 2 -1⟩ | 3.93 | Luzozogu | Werckisma |
| 11 | 9801/9800 | [-3 4 -2 -2 2⟩ | 0.18 | Bilorugu | Kalisma |
| 13 | 65/64 | [-6 0 1 0 0 1⟩ | 26.84 | Thoyo | Wilsorma |
| 13 | 91/90 | [-1 -2 -1 1 0 1⟩ | 19.13 | Thozogu | Superleap comma, biome comma |
| 13 | 144/143 | [4 2 0 0 -1 -1⟩ | 12.06 | Thulu | Grossma |
| 13 | 1001/1000 | [-3 0 -3 1 1 1⟩ | 1.73 | Tholozotrigu | Fairytale comma, sinbadma |
| 13 | 4096/4095 | [12 -2 -1 -1 0 -1⟩ | 0.42 | Sathurugu | Schismina |
| 17 | 51/50 | [-1 1 -2 0 0 0 1⟩ | 34.28 | Sogugu | Large septendecimal sixth tone |
| 17 | 52/51 | [2 -1 0 0 0 1 -1⟩ | 33.62 | Sutho | Small septendecimal sixth tone |
| 17 | 136/135 | [3 -3 -1 0 0 0 1⟩ | 12.78 | Sogu | Diatisma, fiventeen comma |
| 17 | 256/255 | [8 -1 -1 0 0 0 -1⟩ | 6.78 | Sugu | Charisma, septendecimal kleisma |
| 17 | 289/288 | [-5 -2 0 0 0 0 2⟩ | 6.00 | Soso | Semitonisma |
| 17 | 2601/2600 | [-3 2 -2 0 0 -1 2⟩ | 0.67 | Sosothugugu | Sextantonisma |
| 19 | 39/38 | [-1 1 0 0 0 1 0 -1⟩ | 44.97 | Nutho | Undevicesimal two-ninth tone |
| 19 | 96/95 | [5 1 -1 0 0 0 0 -1⟩ | 18.13 | Nugu | 19th-partial chroma |
| 19 | 153/152 | [-3 2 0 0 0 0 1 -1⟩ | 11.35 | Nuso | Ganassisma |
| 19 | 171/170 | [-1 2 -1 0 0 0 -1 1⟩ | 10.15 | Nosugu | Malcolmisma |
| 19 | 324/323 | [2 4 0 0 0 0 -1 -1⟩ | 5.35 | Nusu | Photisma |
| 19 | 361/360 | [-3 -2 -1 0 0 0 0 2⟩ | 4.80 | Nonogu | Go comma |
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
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
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