# 12edo

(Redirected from 12 EDO)
 ← 11edo 12edo 13edo →
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
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12 equal divisions of the octave (12edo), or 12-tone equal temperament (12tet), 12 equal temperament (12et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 12 equal parts of exactly 100 ¢ each. It is the predominating tuning system in the world today.

## 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 cents 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.

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 supports septimal meantone via the 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 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 wtn[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 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).

### 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

12edo intervals have standard names from classical music theory.

Steps Cents Approximate JI Ratios* Interval solfege Audio
0 0 1/1 unison P1 D Do
1 100 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27 aug 1sn, minor 2nd A1, m2 D#, Eb Di / Ra
2 200 8/7, 9/8, 10/9, 17/15, 19/17 major 2nd M2 E Re
3 300 7/6, 6/5, 19/16 minor 3rd m3 F Ri / Me
4 400 5/4, 9/7 major 3rd M3 F# Mi
5 500 4/3 perfect 4th P4 G Fa
6 600 7/5, 10/7, 17/12, 24/17 aug 4th, dim 5th A4, d5 G#, Ab Fi / Se
7 700 3/2 perfect 5th P5 A So
8 800 8/5, 14/9 minor 6th m6 Bb Si / Le
9 900 5/3, 12/7, 32/19 major 6th M6 B La
10 1000 7/4, 9/5, 16/9 minor 7th m7 C Li / Te
11 1100 15/8, 17/9, 28/15, 40/21, 48/25, 27/14 major 7th M7 C# Ti
12 1200 2/1 perfect 8ve P8 D Do

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

## JI approximation

### 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

### Selected 19-limit intervals

An expanded version of the above, including some higher-limit intervals:

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-19 12 [12 19]] +0.617 0.617 0.617
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

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.

### 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 Diminished comma
5 128/125 [7 0 -3 41.06 Trigu Augmented comma
5 81/80 [-4 4 -1 21.51 Gu Syntonic 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 Atom
7 36/35 [2 2 -1 -1 48.77 Rugu Septimal quartertone
7 50/49 [1 0 2 -2 34.98 Biruyo Jubilisma
7 64/63 [6 -2 0 -1 27.26 Ru Septimal comma
7 3125/3087 [0 -2 5 -3 21.18 Triru-aquinyo Gariboh
7 126/125 [1 2 -3 1 13.79 Zotrigu Starling comma
7 4000/3969 [5 -4 3 -2 13.47 Rurutriyo Octagar
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
7 5120/5103 [10 -6 1 -1 5.76 Saruyo Hemifamity
7 (16 digits) [25 -14 0 -1 3.80 Sasaru Garischisma
7 (12 digits) [-11 2 7 -3 1.63 Latriru-asepyo Meter
7 (12 digits) [-4 6 -6 3 0.33 Trizogugu Landscape comma
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
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 91/90 [-1 -2 -1 1 0 1 19.13 Thozogu Superleap
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
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

Main article: List of MOS scales in 12edo

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.