12edo

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.

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, 18/17, 21/20, 25/24, 28/27
2	200	major 2nd	M2	E	8/7, 9/8, 10/9
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.

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	Cents	Color Name	Name 1	Name 2	Name 3
648/625	[3 4 -4⟩	62.57	Quadgu	Major Diesis	Diminished Comma
36/35	[2 2 -1 -1⟩	48.77	Rugu	Septimal Quarter Tone
128/125	[7 0 -3⟩	41.06	Trigu	Diesis	Augmented Comma
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
531441/524288	[-19 12⟩	23.46	Lalawa	Pythagorean Comma
81/80	[-4 4 -1⟩	21.51	Gu	Syntonic Comma	Didymos Comma	Meantone Comma
3125/3087	[0 -2 5 -3⟩	21.18	Triru-aquinyo	Gariboh
2048/2025	[11 -4 -2⟩	19.55	Sagugu	Diaschisma
91/90	[-1 -2 -1 1 0 1⟩	19.13	Thozogu	Superleap
5201701/5149091	[26 -12 -3⟩	17.60	Sasa-trigu	Misty Comma
99/98	[-1 2 0 -2 1⟩	17.58	Loruru	Mothwellsma
100/99	[2 -2 2 0 -1⟩	17.40	Luyoyo	Ptolemisma
126/125	[1 2 -3 1⟩	13.79	Zotrigu	Septimal Semicomma	Starling Comma
4000/3969	[5 -4 3 -2⟩	13.47	Rurutriyo	Octagar
176/175	[4 0 -2 -1 1⟩	9.86	Lorugugu	Valinorsma
896/891	[7 -4 0 1 -1⟩	9.69	Saluzo	Pentacircle
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
441/440	[-3 2 -1 2 -1⟩	3.93	Luzozogu	Werckisma
33554432/33480783	[25 -14 0 -1⟩	3.80	Sasaru	Garischisma
32805/32768	[-15 8 1⟩	1.95	Layo	Schisma
703125/702464	[-11 2 7 -3⟩	1.63	Latriru-asepyo	Meter
250047/250000	[-4 6 -6 3⟩	0.33	Trizogugu	Landscape Comma
9801/9800	[-3 4 -2 -2 2⟩	0.18	Bilorugu	Kalisma	Gauss' Comma
	[161 -84 -12⟩	0.02	Sepbisa-quadtrigu	Atom

Scales

MOS in 12edo

Intervals

12ed2-11-001.svg

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

12ed2-19-001e.svg

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

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
15/13, 26/15	2.259
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
11/8, 16/11	48.682
12/11, 11/6	49.323