12edo

← 11edo

12edo

13edo →

Prime factorization	22 × 3 (highly composite)
Step size	100 ¢ (by definition)
Fifth	7\12 (700 ¢) (convergent)
Semitones (A1:m2)	1:1 (100 ¢ : 100 ¢)
Consistency limit	9
Distinct consistency limit	5

English Wikipedia has an article on:

12 equal temperament

12 equal divisions of the octave (abbreviated 12edo or 12ed2), also called 12-tone equal temperament (12tet) or 12 equal temperament (12et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 12 equal parts of exactly 100 ¢ each. Each step represents a frequency ratio of 2^1/12, or the 12th root of 2. It is the predominating tuning system in the world today.

Theory

12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave (edo) which can seriously claim to represent 5-limit harmony, and it represents a meantone temperament.

It divides the octave into twelve equal parts, each of exactly 100 cents. It has a fifth which is quite accurate at 700 cents, two cents flat of just. It has a major third which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The minor third is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of historical temperaments such as quarter-comma meantone, and later well temperaments. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as serialism and much of jazz harmony that derive from 12edo's structure as an equal division rather than its underlying temperament properties.^{[citation needed]}

12edo is the basic example of a dodecatonic scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic (7/4) is represented by the diatonic minor seventh, which is sharp by 31 cents, and as such 12edo tempers out 64/63. The deviation explains why minor sevenths tend to stand 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 1–5/4–3/2–16/9, and while 12et officially supports septimal meantone for tempering out 126/125 and 225/224 via its patent val of ⟨12 19 28 34], its approximations of 7-limit intervals are not very accurate. 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.

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, 3¹²/2¹⁹, is tempered out. Three major thirds equal an octave, so the lesser diesis, 128/125, is tempered out. Four minor thirds also equal an octave, so the greater diesis, 648/625, is tempered out. These features have been widely utilized in contemporary music. Other commas 12et tempers out include the diaschisma, 2048/2025, the septimal quartertone, 36/35, and the jubilisma, 50/49. Each 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 also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

Prime harmonics

Approximation of prime harmonics in 12edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-1.96	+13.69	+31.17	+48.68	-40.53	-4.96	+2.49	-28.27	-29.58	-45.04
	Relative (%)	+0.0	-2.0	+13.7	+31.2	+48.7	-40.5	-5.0	+2.5	-28.3	-29.6	-45.0
Steps (reduced)		12 (0)	19 (7)	28 (4)	34 (10)	42 (6)	44 (8)	49 (1)	51 (3)	54 (6)	58 (10)	59 (11)

Subsets and supersets

12edo contains 2edo, 3edo, 4edo, and 6edo as subsets. It is the 5th highly composite edo, 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both strict zeta and highly composite.

24edo, which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. 36edo, which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. 72edo is a notable zeta-record edo, and 60-, 84-, and 96edo all see utilities. Notable rank-2 temperaments that augment 12edo with extra generators include compton and catler.

Intervals

Intervals of 12edo

Degree	Cents	Interval region	Approximated 5-limit JI intervals (error in ¢)	Audio	Higher limit interpretations
0	0	Unison (prime)	1/1 (just)
1	100	Minor second	256/243 (+9.775) 25/24 (+29.328) 16/15 (−11.731)		28/27 (+37.039), 21/20 (+15.533), 15/14 (−19.443) 17/16 (−4.955), 18/17 (+1.045) 19/18 (+6.397), 20/19 (+11.199)
2	200	Major second	9/8 (−3.910) 10/9 (+17.596)		8/7 (−31.174), 28/25 (+3.802) 17/15 (−16.687), 19/17 (+7.442), 55/49 (+0.020), 64/57 (−0.532)
3	300	Minor third	32/27 (+5.865) 6/5 (−15.641) 75/64 (+25.418)		7/6 (+33.129), 25/21 (−1.847) 19/16 (+2.487)
4	400	Major third	81/64 (−7.820) 5/4 (+13.686) 32/25 (-27.373)		63/50 (−0.108), 9/7 (−35.084) 34/27 (+0.910), 24/19 (−4.442)
5	500	Fourth	4/3 (+1.955) 27/20 (-19.551)		21/16 (-29.219)
6	600	Tritone	25/18 (+31.283) 36/25 (-31.283) 45/32 (+9.776) 64/45 (−9.776)		7/5 (+17.488), 10/7 (−17.488) 24/17 (+3.000), 17/12 (−3.000) 99/70 (−0.088), 140/99 (+0.088)
7	700	Fifth	3/2 (−1.955) 40/27 (+19.551)		32/21 (+29.219)
8	800	Minor sixth	128/81 (+7.820) 8/5 (−13.686) 25/16 (+27.373)		14/9 (+35.084), 100/63 (+0.108) 19/12 (+4.442), 27/17 (−0.910)
9	900	Major sixth	27/16 (−5.865) 5/3 (+15.641) 128/75 (-25.418)		12/7 (−33.129), 42/25 (+1.847) 32/19 (−2.487)
10	1000	Minor seventh	16/9 (+3.910) 9/5 (−17.596)		7/4 (+31.174), 25/14 (−3.802) 30/17 (+16.687), 34/19 (−7.442) 98/55 (-0.020), 57/32 (+0.532)
11	1100	Major seventh	243/128 (-9.775) 15/8 (+11.731) 48/25 (−29.328)		28/15 (+19.443), 40/21 (−15.533), 27/14 (−37.039) 32/17 (+4.955), 17/9 (−1.045) 36/19 (-6.397), 19/10 (-11.199)
12	1200	Octave	2/1 (just)

Notation

The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the chain-of-fifths notation or extended Pythagorean notation.

Semitones	−2	−1	0	+1	+2
Symbol

The subsets 1edo, 2edo, 3edo, 4edo and 6edo can all be written using 12edo subset notation.

Any 12edo note or interval can be respelled enharmonically by adding a pythagorean comma to it or subtracting one from it.

Notation of 12edo

Degree	Cents	Standard notation
		Diatonic (5L 2s) interval names	Note names (on D)
0	0	Perfect unison (P1)	D
1	100	Augmented unison (A1) Minor second (m2)	D# Eb
2	200	Major second (M2) Diminished third (d3)	E Fb
3	300	Augmented second (A2) Minor third (m3)	E# F
4	400	Major third (M3) Diminished fourth (d4)	F# Gb
5	500	Perfect fourth (P4)	G
6	600	Augmented fourth (A4) Diminished fifth (d5)	G# Ab
7	700	Perfect fifth (P5)	A
8	800	Augmented fifth (A5) Minor sixth (m6)	A# Bb
9	900	Major sixth (M6) Diminished seventh (d7)	B Cb
10	1000	Augmented sixth (A6) Minor seventh (m7)	B# C
11	1100	Major seventh (M7) Diminished octave (d8)	C# Db
12	1200	Perfect octave (P8)	D

In 12edo:

• Ups and downs notation is identical to standard notation;
• Mixed sagittal notation is identical to standard notation, but pure sagittal notation exchanges sharps (♯) and flats (♭) for sagittal sharp () and sagittal flat () respectively.

Sagittal notation

This notation uses the same sagittal sequence as EDOs 5, 19, and 26, is a subset of the notations for EDOs 24, 36, 48, 60, 72, and 84, and is a superset of the notation for 6-EDO.

Evo flavor

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

Revo flavor

Solfege

Solfege of 12edo

Degree	Cents	Standard solfege (movable do)	Uniform solfege (2-3 vowels)
0	0	Do	Da
1	100	Di (A1) Ra (m2)	Du (A1) Fra (m2)
2	200	Re	Ra
3	300	Ri (A2) Me (m3)	Ru (A2) Na (m3)
4	400	Mi	Ma (M3) Fo (d4)
5	500	Fa	Mu (A3) Fa (P4)
6	600	Fi (A4) Se (d5)	Pa (A4) Sha (d5)
7	700	So	Sa
8	800	Si (A5) Le (m6)	Su (A5) Fla (m6)
9	900	La	La (M6) Tho (d7)
10	1000	Li (A6) Te (m7)	Lu (A6) Tha (m7)
11	1100	Ti	Ta (M7) Do (d8)
12	1200	Do	Da

Approximation to JI

Selected 5-limit intervals approximated in 12edo

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 12edo. Prime harmonics are in bold; inconsistent intervals are in italics.

Note that, since the cent was defined in terms of 12edo, the absolute and relative errors for 12edo are identical.

15-odd-limit intervals in 12edo (direct approximation, even if inconsistent)

Interval and 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
13/11, 22/13	10.790	10.8
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
13/7, 14/13	28.298	28.3
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
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
11/6, 12/11	49.363	49.4

15-odd-limit intervals in 12edo (patent val mapping)

Interval and 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

15-odd-limit intervals by 12f val mapping

Interval and 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
13/11, 22/13	10.790	10.8
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
13/7, 14/13	28.298	28.3
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
15/11, 22/15	36.951	37.0
13/10, 20/13	45.786	45.8
15/13, 26/15	47.741	47.7
11/8, 16/11	48.682	48.7
11/6, 12/11	50.637	50.6
11/9, 18/11	52.592	52.6
13/8, 16/13	59.472	59.5
13/12, 24/13	61.427	61.4
13/9, 18/13	63.382	63.4

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-19 12⟩	[⟨12 19]]	+0.62	0.62	0.62
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
2.3.5.17	51/50, 81/80, 128/125	[⟨12 19 28 49]]	−0.87	2.95	2.95
2.3.5.17.19	51/50, 76/75, 81/80, 128/125	[⟨12 19 28 49 51]]	−0.81	2.64	2.64

• 12et is monotonic to the 11-odd-limit. It is the first equal temperament to achieve this.
• 12et has a lower relative error than any previous equal temperaments in the 3-, 5-, 7-, and 11-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, respectively.
• 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

Uniform maps

13-limit uniform maps between 11.8 and 12.2

Min. size	Max. size	Wart notation	Map
11.7554	11.8436	12cde	⟨12 19 27 33 41 44]
11.8436	11.9329	12de	⟨12 19 28 33 41 44]
11.9329	11.9962	12e	⟨12 19 28 34 41 44]
11.9962	12.0256	12	⟨12 19 28 34 42 44]
12.0256	12.2743	12f	⟨12 19 28 34 42 45]

Commas

12edo tempers out the following commas. This assumes the val ⟨12 19 28 34 42 44 49 51].

Prime limit	Ratio	Monzo	Cents	Color name	Name
3	(12 digits)	[-19 12⟩	23.46	Lalawama / Poma	Pythagorean comma
5	648/625	[3 4 -4⟩	62.57	Quadguma	Diminished comma, greater diesis
5	(12 digits)	[18 -4 -5⟩	60.61	Saquinguma	Passion comma
5	128/125	[7 0 -3⟩	41.06	Triguma	Augmented comma, lesser diesis
5	81/80	[-4 4 -1⟩	21.51	Guma	Syntonic comma, Didymus' comma, meantone comma
5	2048/2025	[11 -4 -2⟩	19.55	Saguguma	Diaschisma
5	(16 digits)	[26 -12 -3⟩	17.60	Sasa-triguma	Misty comma
5	32805/32768	[-15 8 1⟩	1.95	Layoma	Schisma
5	(98 digits)	[161 -84 -12⟩	0.02	Sepbisa-quadtriguma	Kirnberger's atom
7	256/245	[8 0 -1 -2⟩	76.03	Ruruguma	Bapbo comma
7	59049/57344	[-13 10 0 -1⟩	50.72	Laruma	Harrison's comma
7	36/35	[2 2 -1 -1⟩	48.77	Ruguma	Mint comma, septimal quarter tone
7	50/49	[1 0 2 -2⟩	34.98	Biruyoma	Jubilisma
7	3645/3584	[-9 6 1 -1⟩	29.22	Laruyoma	Schismean comma
7	64/63	[6 -2 0 -1⟩	27.26	Ruma	Septimal comma
7	3125/3087	[0 -2 5 -3⟩	21.18	Triru-aquinyoma	Gariboh comma
7	126/125	[1 2 -3 1⟩	13.79	Zotriguma	Starling comma
7	4000/3969	[5 -4 3 -2⟩	13.47	Rurutriyoma	Octagar comma
7	(12 digits)	[-9 8 -4 2⟩	8.04	Labizoguguma	Varunisma
7	225/224	[-5 2 2 -1⟩	7.71	Ruyoyoma	Marvel comma
7	3136/3125	[6 0 -5 2⟩	6.08	Zozoquinguma	Hemimean comma
7	5120/5103	[10 -6 1 -1⟩	5.76	Saruyoma	Hemifamity comma
7	(16 digits)	[25 -14 0 -1⟩	3.80	Sasaruma	Garischisma
7	(12 digits)	[-11 2 7 -3⟩	1.63	Latriru-asepyoma	Metric comma
7	(12 digits)	[-4 6 -6 3⟩	0.33	Trizoguguma	Landscape comma
11	128/121	[7 0 0 0 -2⟩	97.36	Lulubima	Axirabian limma
11	45/44	[-2 2 1 0 -1⟩	38.91	Luyoma	Undecimal fifth tone
11	56/55	[3 0 -1 1 -1⟩	31.19	Luzoguma	Undecimal tritonic comma
11	245/242	[-1 0 1 2 -2⟩	21.33	Luluzozoyoma	Frostma
11	99/98	[-1 2 0 -2 1⟩	17.58	Loruruma	Mothwellsma
11	100/99	[2 -2 2 0 -1⟩	17.40	Luyoyoma	Ptolemisma
11	176/175	[4 0 -2 -1 1⟩	9.86	Loruguguma	Valinorsma
11	896/891	[7 -4 0 1 -1⟩	9.69	Saluzoma	Pentacircle comma
11	441/440	[-3 2 -1 2 -1⟩	3.93	Luzozoguma	Werckisma
11	9801/9800	[-3 4 -2 -2 2⟩	0.18	Biloruguma	Kalisma
13	65/64	[-6 0 1 0 0 1⟩	26.84	Thoyoma	Wilsorma
13	91/90	[-1 -2 -1 1 0 1⟩	19.13	Thozoguma	Superleap comma, biome comma
13	144/143	[4 2 0 0 -1 -1⟩	12.06	Thuluma	Grossma
13	1001/1000	[-3 0 -3 1 1 1⟩	1.73	Tholozotriguma	Fairytale comma, sinbadma
13	4096/4095	[12 -2 -1 -1 0 -1⟩	0.42	Sathuruguma	Minisma
17	51/50	[-1 1 -2 0 0 0 1⟩	34.28	Soguguma	Large septendecimal sixth tone
17	52/51	[2 -1 0 0 0 1 -1⟩	33.62	Suthoma	Small septendecimal sixth tone
17	136/135	[3 -3 -1 0 0 0 1⟩	12.78	Soguma	Diatisma, fiventeen comma
17	256/255	[8 -1 -1 0 0 0 -1⟩	6.78	Suguma	Charisma, septendecimal kleisma
17	289/288	[-5 -2 0 0 0 0 2⟩	6.00	Sosoma	Semitonisma
17	2601/2600	[-3 2 -2 0 0 -1 2⟩	0.67	Sosothuguguma	Sextantonisma
19	39/38	[-1 1 0 0 0 1 0 -1⟩	44.97	Nuthoma	Undevicesimal two-ninth tone
19	96/95	[5 1 -1 0 0 0 0 -1⟩	18.13	Nuguma	19th-partial chroma
19	153/152	[-3 2 0 0 0 0 1 -1⟩	11.35	Nusoma	Ganassisma
19	171/170	[-1 2 -1 0 0 0 -1 1⟩	10.15	Nosuguma	Malcolmisma
19	324/323	[2 4 0 0 0 0 -1 -1⟩	5.35	Nusuma	Photisma
19	361/360	[-3 -2 -1 0 0 0 0 2⟩	4.80	Nonoguma	Go comma
19	513/512	[9 3 0 0 0 0 0 -1⟩	3.37	Lanoma	Boethius' comma

Rank-2 temperaments

Periods per 8ve	Generator*	Pergen	Temperaments
1	1\12	(P8, P4/5)	Ripple, passion
1	5\12	(P8, P5)	Meantone / dominant
2	5\12 (1\12)	(P8/2, P5)	Pajara, injera
3	5\12 (1\12)	(P8/3, P5)	Augmented / august
4	5\12 (1\12)	(P8/4, P5)	Diminished
6	5\12 (1\12)	(P8/6, P5)	Hexe

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Rank-2 temperaments to which 12et can be detempered include compton (12 & 72), schismic/garibaldi (41 & 53), and diaschismic (46 & 58). Rank-3 temperaments to which 12et can be detempered include marvel, starling/thrush, aberschismic, orthoschismic, and cassaschismic. For more comprehensive lists, see:

• 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–commatic equivalence continuum

Octave stretch or compression

Whether there is intonational improvement from octave stretch and compression for 12edo varies by context. A slight compression such as what is given by 40ed10 and 34zpi shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in 7edf, 19edt, or 31ed6, also makes sense.

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

File:12edo modes.pdf

Well temperaments

For a list of historical well temperaments, see Well temperament.

• Cauldron
• Bifrost
• Grail
• Secor5 23TX
• Secor wt10
• Sabat1
• Sabat2

Music

See also: Category:12edo tracks

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

See also

• Lumatone mapping for 12edo
• purdal:12-EDD^{[dead link]}
• Near12 – a just intonation scale where every interval is within 12.5 cents of a 12edo step

External links

• 12-tone equal-temperament on Tonalsoft Encyclopedia