12edo

From Xenharmonic Wiki
Jump to: navigation, search

12edo, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that 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. Its 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 being 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, 3^12/2^19, 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.

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

Commas

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

Rational Monzo Size (Cents) Name 1 Name 2 Name 3
531441/524288 | -19 12 > 23.46 Pythagorean Comma
648/625 | 3 4 -4 > 62.57 Major Diesis Diminished Comma
128/125 | 7 0 -3 > 41.06 Diesis Augmented Comma
81/80 | -4 4 -1 > 21.51 Syntonic Comma Didymos Comma Meantone Comma
2048/2025 | 11 -4 -2 > 19.55 Diaschisma
5201701/5149091 | 26 -12 -3 > 17.60 Misty Comma
32805/32768 | -15 8 1 > 1.95 Schisma
| 161 -84 -12 > 0.02 Atom
36/35 | 2 2 -1 -1 > 48.77 Septimal Quarter Tone
50/49 | 1 0 2 -2 > 34.98 Tritonic Diesis Jubilisma
64/63 | 6 -2 0 -1 > 27.26 Septimal Comma Archytas' Comma Leipziger Komma
3125/3087 | 0 -2 5 -3 > 21.18 Gariboh
126/125 | 1 2 -3 1 > 13.79 Septimal Semicomma Starling Comma
4000/3969 | 5 -4 3 -2 > 13.47 Octagar
321489/320000 | -9 8 -4 2 > 8.04 Varunisma
225/224 | -5 2 2 -1 > 7.71 Septimal Kleisma Marvel Comma
3136/3125 | 6 0 -5 2 > 6.08 Hemimean
5120/5103 | 10 -6 1 -1 > 5.76 Hemifamity
33554432/33480783 | 25 -14 0 -1 > 3.80 Garischisma
703125/702464 | -11 2 7 -3 > 1.63 Meter
250047/250000 | -4 6 -6 3 > 0.33 Landscape Comma
99/98 | -1 2 0 -2 1 > 17.58 Mothwellsma
100/99 | 2 -2 2 0 -1 > 17.40 Ptolemisma
176/175 | 4 0 -2 -1 1 > 9.86 Valinorsma
896/891 | 7 -4 0 1 -1 > 9.69 Pentacircle
441/440 | -3 2 -1 2 -1 > 3.93 Werckisma
9801/9800 | -3 4 -2 -2 2 > 0.18 Kalisma Gauss' Comma
91/90 | -1 -2 -1 1 0 1 > 19.13 Superleap

Scales

Intervals

alt : Your browser has no SVG support.

12ed2-11-001.svg

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

alt : Your browser has no SVG support.

12ed2-19-001e.svg

Selected just intervals by error

The following table shows how some prominent just intervals are represented in 12edo (ordered by absolute error).

Interval, complement Error (abs., in cents)
4/3, 3/2 1.955
9/8, 16/9 3.910
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
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
11/8, 16/11 48.682
12/11, 11/6 50.637
15/13, 26/15 52.259
11/9, 18/11 52.592
13/10, 20/13 54.214
14/13, 13/7 71.702
13/11, 22/13 89.210