12edo

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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

12edo, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today.

Theory

	prime 2	prime 3	prime 5	prime 7	prime 11	prime 13	prime 17	prime 19
Error (¢)	0	-2.0	+13.7	+31.2	+48.7	-40.5	-5.0	+2.5
nearest edomapping	12	7	4	10	6	8	1	3
Fifthspan	0	+1	+4	-2	+6	-4	-5	-3

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.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a wtn.

Intervals

Steps	Cents	Interval			Approximate JI Ratios*
0	0	unison	P1 P0	D	1/1
1	100	aug 1sn, minor 2nd aug 1sn, minor 1st	A1, m2 A0, m1	D#, Eb	15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
2	200	major 2nd major 1st	M2 M1	E	8/7, 9/8, 10/9, 17/15, 19/17
3	300	minor 3rd minor 2nd	m3 m2	F	7/6, 6/5, 19/16
4	400	major 3rd major 2nd	M3 M2	F#	5/4, 9/7
5	500	perfect 4th perfect 3rd	P4 P3	G	4/3
6	600	aug 4th, dim 5th aug 3rd, dim 4th	A4, d5 A3, d4	G#, Ab	7/5, 10/7, 17/12, 24/17
7	700	perfect 5th perfect 4th	P5 P4	A	3/2
8	800	minor 6th minor 5th	m6 m5	Bb	8/5, 14/9
9	900	major 6th major 5th	M6 M5	B	5/3, 12/7, 32/19
10	1000	minor 7th minor 6th	m7 m6	C	7/4, 9/5, 16/9
11	1100	major 7th major 6th	M7 M6	C#	15/8, 17/9, 28/15, 40/21, 48/25, 27/14
12	1200	perfect 8ve perfect 7th	P8 P7	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

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

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

Temperament measures

Shown below are TE temperament measures (RMS normalized) 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 (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, 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

• List of 12et rank two temperaments by badness
• List of 12et rank two temperaments by complexity
• List of edo-distinct 12f rank two temperaments
• Schismic-Pythagorean equivalence continuum

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

The two most common 12-edo 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)

Pathological Modes

2 1 1 1 1 2 1 1 1 1 2L 8s MOS

3 1 1 1 1 1 1 1 1 1 1L 9s MOS

2 1 1 1 1 1 1 1 1 1 1 1L 10s MOS

Commas

12 EDO tempers out the following commas. This assumes val ⟨12 19 28 34 42 44].

Prime limit	Ratio	Monzo	Cents	Color name	Name(s)
3	(12 digits)	[-19 12⟩	23.46	Lalawa	Pythagorean comma
5	648/625	[3 4 -4⟩	62.57	Quadgu	Major diesis, diminished comma
5	128/125	[7 0 -3⟩	41.06	Trigu	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	Atom
7	36/35	[2 2 -1 -1⟩	48.77	Rugu	Septimal quartertone
7	50/49	[1 0 2 -2⟩	34.98	Biruyo	Tritonic diesis, jubilisma
7	64/63	[6 -2 0 -1⟩	27.26	Ru	Septimal comma, Archytas' comma, Leipziger Komma
7	3125/3087	[0 -2 5 -3⟩	21.18	Triru-aquinyo	Gariboh
7	126/125	[1 2 -3 1⟩	13.79	Zotrigu	Septimal semicomma, 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	Septimal kleisma, 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, Gauss' comma
13	91/90	[-1 -2 -1 1 0 1⟩	19.13	Thozogu	Superleap

See also

• purdal:12-EDD