12edo

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12edo, perhaps better known as 12et since it really is a temperament, 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.

Intervals

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

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

Just approximation

Selected just intervals by error

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13
Error (¢) 0 -1.96 +13.7 +31.2 +48.7 -40.5
Fifthspan 0 +1 +4 -2 +6 -4

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.

Interval, complement Error (abs, ¢)
4/3, 3/2 1.955
9/8, 16/9 3.910
13/11, 22/13 10.790
16/15, 15/8 11.731
5/4, 8/5 13.686
6/5, 5/3 15.641
7/5, 10/7 17.488
14/11, 11/7 17.508
10/9, 9/5 17.596
15/14, 28/15 19.443
14/13, 13/7 28.298
8/7, 7/4 31.174
7/6, 12/7 33.129
11/10, 20/11 34.996
9/7, 14/9 35.084
18/13, 13/9 36.618
15/11, 22/15 36.951
13/12, 24/13 38.573
16/13, 13/8 40.528
13/10, 20/13 45.786
11/9, 18/11 47.408
15/13, 26/15 47.741
11/8, 16/11 48.682
12/11, 11/6 49.323

Selected 19-limit intervals

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

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Temperament measures

Shown below are TE temperament measures (RMS normalized by the rank) of 12et.

3-limit 5-limit 7-limit 2.3.5.7.17.19
Octave stretch (¢) +0.617 -1.56 -3.95 -2.53
Error absolute (¢) 0.617 3.11 4.92 4.52
relative (%) 0.617 3.11 4.94 4.53
  • 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively.
  • 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

Rank two temperaments

Periods
per octave
Generator Temperaments
1 1\12 Ripple
1 5\12 Meantone/dominant
2 1\12 Srutal/pajara/injera
3 1\12 Augmented
4 1\12 Diminished
6 1\12 Hexe

Scales

Main article: MOS in 12edo

Commas

12 EDO tempers out the following commas. (Note: This assumes val 12 19 28 34 42 44].)

Prime
limit
Rational Monzo Cents Color Name Name 1 Name 2 Name 3
3 531441/524288 [-19 12 23.46 Lalawa Pythagorean comma
5 648/625 [3 4 -4 62.57 Quadgu Major diesis Diminished comma
" 128/125 [7 0 -3 41.06 Trigu Diesis Augmented comma
" 81/80 [-4 4 -1 21.51 Gu Syntonic comma Didymos comma Meantone comma
" 2048/2025 [11 -4 -2 19.55 Sagugu Diaschisma
" 5201701/5149091 [26 -12 -3 17.60 Sasa-trigu Misty comma
" 32805/32768 [-15 8 1 1.95 Layo Schisma
" [161 -84 -12 0.02 Sepbisa-quadtrigu Atom
7 36/35 [2 2 -1 -1 48.77 Rugu Septimal quartertone
" 50/49 [1 0 2 -2 34.98 Biruyo Tritonic diesis Jubilisma
" 64/63 [6 -2 0 -1 27.26 Ru Septimal comma Archytas' comma Leipziger Komma
" 3125/3087 [0 -2 5 -3 21.18 Triru-aquinyo Gariboh
" 126/125 [1 2 -3 1 13.79 Zotrigu Septimal semicomma Starling comma
" 4000/3969 [5 -4 3 -2 13.47 Rurutriyo Octagar
" 321489/320000 [-9 8 -4 2 8.04 Labizogugu Varunisma
" 225/224 [-5 2 2 -1 7.71 Ruyoyo Septimal kleisma Marvel comma
" 3136/3125 [6 0 -5 2 6.08 Zozoquingu Hemimean
" 5120/5103 [10 -6 1 -1 5.76 Saruyo Hemifamity
" 33554432/33480783 [25 -14 0 -1 3.80 Sasaru Garischisma
" 703125/702464 [-11 2 7 -3 1.63 Latriru-asepyo Meter
" 250047/250000 [-4 6 -6 3 0.33 Trizogugu Landscape comma
11 99/98 [-1 2 0 -2 1 17.58 Loruru Mothwellsma
" 100/99 [2 -2 2 0 -1 17.40 Luyoyo Ptolemisma
" 176/175 [4 0 -2 -1 1 9.86 Lorugugu Valinorsma
" 896/891 [7 -4 0 1 -1 9.69 Saluzo Pentacircle
" 441/440 [-3 2 -1 2 -1 3.93 Luzozogu Werckisma
" 9801/9800 [-3 4 -2 -2 2 0.18 Bilorugu Kalisma Gauss' comma
13 91/90 [-1 -2 -1 1 0 1 19.13 Thozogu Superleap