12edo

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Prime factorization 22 × 3
Step size 100¢by definition
Fifth 7\12 (700¢)
(convergent)
Semitones (A1:m2) 1:1 (100¢ : 100¢)
Consistency limit 9
Distinct consistency limit 5
Special properties
English Wikipedia has an article on:

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 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 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.

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 312/219, 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[clarification needed], 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 full zeta and highly composite, 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.

Prime harmonics

Approximation of prime harmonics in 12edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.0 -2.0 +13.7 +31.2 +48.7 -40.5 -5.0 +2.5 -28.3 -29.6 -45.0
relative (%) +0 -2 +14 +31 +49 -41 -5 +2 -28 -30 -45
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

Intervals of 12edo
Degree Cents Interval region Approximated JI intervals* (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)

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

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
Heji4.svg
Heji11.svg
Heji18.svg
Heji25.svg
Heji32.svg

1edo, 2edo, 3edo, 4edo and 6edo can all be written using 12edo subset notation.

Notation of 12edo
Degree Cents Standard notation
Diatonic 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 (b) for sagittal sharp (Sagittal sharp.png) and sagittal flat (Sagittal flat.png) respectively.

Solfege

Solfege of 12edo
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

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Selected 5-limit intervals approximated in 12edo

15-odd-limit interval mappings

The following table shows how 15-odd-limit intervals are represented in 12edo. Prime harmonics are in bold; inconsistent intervals are in italic.

15-odd-limit intervals by direct approximation (even if inconsistent)
Interval, complement Error (abs, ¢) Error (rel, %)
4/3, 3/2 1.955 2.0
9/8, 16/9 3.910 3.9
13/11, 22/13 10.790 10.8
16/15, 15/8 11.731 11.7
5/4, 8/5 13.686 13.7
6/5, 5/3 15.641 15.6
7/5, 10/7 17.488 17.5
14/11, 11/7 17.508 17.5
10/9, 9/5 17.596 17.6
15/14, 28/15 19.443 19.4
14/13, 13/7 28.298 28.3
8/7, 7/4 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
18/13, 13/9 36.618 36.7
15/11, 22/15 36.951 37.0
13/12, 24/13 38.573 38.6
16/13, 13/8 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
12/11, 11/6 49.323 49.3
15-odd-limit intervals by patent val mapping
Interval, 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
2.3.5.17 51/50, 81/80, 128/125 [12 19 28 49]] −0.87 2.95 2.95
2.3.5.7.17 36/35, 50/49, 51/49, 64/63 [12 19 28 34 49]] −2.92 4.86 4.87
2.3.5.17.19 51/50, 76/75, 81/80, 128/125 [12 19 28 49 51]] −0.81 2.64 2.64
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

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

13-limit uniform maps between 11.5 and 12.5
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[1] Monzo Cents Color Name Name
3 (12 digits) [-19 12 23.46 Lalawa Pythagorean comma
5 648/625 [3 4 -4 62.57 Quadgu Greater diesis, diminished comma
5 (12 digits) [18 -4 -5 60.61 Saquingu Passion comma
5 128/125 [7 0 -3 41.06 Trigu Lesser diesis, augmented comma
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 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
  1. Ratios longer than 10 digits are presented by placeholders with informative hints

Rank-2 temperaments

Periods
per 8ve
Generator Pergen Temperaments
1 1\12 (P8, P4/5) Ripple / passion
1 5\12 (P8, P5) Meantone / dominant
2 1\12 (P8/2, P5) Srutal / pajara / injera
3 1\12 (P8/3, P5) Augmented / lithium
4 1\12 (P8/4, P5) Diminished
6 1\12 (P8/6, P5) Hexe

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

File:12edo modes.pdf

Well temperaments

For a list of historical well temperaments, see Well temperament.

Music

See also: Category:12edo tracks

See also