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
Special properties	Highly composite Zeta peak Zeta peak integer Zeta integral Zeta gap

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 because it is the smallest number of equal divisions 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 stretching or compression is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 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, 15.6 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 31 cents, which is 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 1/1–5/4–3/2–16/9, and while 12et officially supports septimal meantone 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.

The commas it tempers out include the Pythagorean comma, 3¹²/2¹⁹, the Didymus' comma, 81/80, the lesser 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.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a well tempered nonet⁠ ⁠^{[clarification needed ]}, and it also contains 2edo, 3edo, 4edo and 6edo as subsets. 12edo is the 5th highly composite edo, 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both strict zeta and highly composite.

12edo 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)

Intervals

Intervals of 12edo

Degree	Cents	Interval region	Approximated JI intervals (error in ¢)				Audio
			3-limit	5-limit	7-limit	Other
0	0	Unison (prime)	1/1 (just)
1	100	Minor second		25/24 (+29.328) 16/15 (-11.731)	28/27 (+37.039) 21/20 (+15.533) 15/14 (-19.443)	18/17 (+1.045) 17/16 (-4.955)
2	200	Major second	9/8 (-3.910)	10/9 (+17.596)	28/25 (+3.802) 8/7 (-31.174)	19/17 (+7.442) 55/49 (+0.020) 64/57 (-0.532) 17/15 (-16.687)
3	300	Minor third	32/27 (+5.865)	6/5 (-15.641)	7/6 (+33.129) 25/21 (-1.847)	19/16 (+2.487) 44/37 (+0.026)
4	400	Major third	81/64 (-7.820)	5/4 (+13.686)	63/50 (-0.108) 9/7 (-35.084)	34/27 (+0.910) 24/19 (-4.442)
5	500	Fourth	4/3 (+1.955)
6	600	Tritone			7/5 (+17.488) 10/7 (-17.488)	24/17 (+3.000) 99/70 (-0.088) 17/12 (-3.000)
7	700	Fifth	3/2 (-1.955)
8	800	Minor sixth	128/81 (+7.820)	8/5 (-13.686)	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)	42/25 (+1.847) 12/7 (-33.129)	37/22 (-0.026) 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) 57/32 (+0.532) 98/55 (-0.020) 34/19 (-7.442)
11	1100	Major seventh		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)
12	1200	Octave	2/1 (just)

Notation

12edo intervals and notes 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

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

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.

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

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 (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, 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 11.5 and 12.5

Min. size	Max. size	Wart notation	Map
11.5000	11.5767	12bbcdddeeeff	⟨12 18 27 32 40 43]
11.5767	11.6722	12bbcdeeeff	⟨12 18 27 33 40 43]
11.6722	11.7071	12cdeeeff	⟨12 19 27 33 40 43]
11.7071	11.7554	12cdeff	⟨12 19 27 33 41 43]
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]
12.2743	12.2853	12ccf	⟨12 19 29 34 42 45]
12.2853	12.2891	12cceef	⟨12 19 29 34 43 45]
12.2891	12.2958	12ccddeef	⟨12 19 29 35 43 45]
12.2958	12.3031	12ccddeefff	⟨12 19 29 35 43 46]
12.3031	12.5000	12bccddeefff	⟨12 20 29 35 43 46]

Commas

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

Prime limit	Ratio	Monzo	Cents	Color name	Name
3	(12 digits)	[-19 12⟩	23.46	Lalawa	Pythagorean comma
5	648/625	[3 4 -4⟩	62.57	Quadgu	Diminished comma, greater diesis
5	(12 digits)	[18 -4 -5⟩	60.61	Saquingu	Passion comma
5	128/125	[7 0 -3⟩	41.06	Trigu	Augmented comma, lesser diesis
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	Kirnberger's atom
7	256/245	[8 0 -1 -2⟩	76.03	Rurugu	Bapbo comma
7	59049/57344	[-13 10 0 -1⟩	50.72	Laru	Harrison's comma
7	36/35	[2 2 -1 -1⟩	48.77	Rugu	Mint comma, septimal quarter tone
7	50/49	[1 0 2 -2⟩	34.98	Biruyo	Jubilisma
7	3645/3584	[-9 6 1 -1⟩	29.22	Laruyo	Schismean comma
7	64/63	[6 -2 0 -1⟩	27.26	Ru	Septimal comma
7	3125/3087	[0 -2 5 -3⟩	21.18	Triru-aquinyo	Gariboh comma
7	126/125	[1 2 -3 1⟩	13.79	Zotrigu	Starling comma
7	4000/3969	[5 -4 3 -2⟩	13.47	Rurutriyo	Octagar comma
7	(12 digits)	[-9 8 -4 2⟩	8.04	Labizogugu	Varunisma
7	225/224	[-5 2 2 -1⟩	7.71	Ruyoyo	Marvel comma
7	3136/3125	[6 0 -5 2⟩	6.08	Zozoquingu	Hemimean comma
7	5120/5103	[10 -6 1 -1⟩	5.76	Saruyo	Hemifamity comma
7	(16 digits)	[25 -14 0 -1⟩	3.80	Sasaru	Garischisma
7	(12 digits)	[-11 2 7 -3⟩	1.63	Latriru-asepyo	Meter comma
7	(12 digits)	[-4 6 -6 3⟩	0.33	Trizogugu	Landscape comma
11	128/121	[7 0 0 0 -2⟩	97.36	1uu2	Axirabian limma
11	45/44	[-2 2 1 0 -1⟩	38.91	Luyo	Undecimal fifth tone
11	56/55	[3 0 -1 1 -1⟩	31.19	Luzogu	Undecimal tritonic comma
11	245/242	[-1 0 1 2 -2⟩	21.33	Luluzozoyo	Frostma
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 comma
11	441/440	[-3 2 -1 2 -1⟩	3.93	Luzozogu	Werckisma
11	9801/9800	[-3 4 -2 -2 2⟩	0.18	Bilorugu	Kalisma
13	65/64	[-6 0 1 0 0 1⟩	26.84	Thoyo	Wilsorma
13	91/90	[-1 -2 -1 1 0 1⟩	19.13	Thozogu	Superleap comma, biome comma
13	144/143	[4 2 0 0 -1 -1⟩	12.06	Thulu	Grossma
13	1001/1000	[-3 0 -3 1 1 1⟩	1.73	Tholozotrigu	Fairytale comma, sinbadma
13	4096/4095	[12 -2 -1 -1 0 -1⟩	0.42	Sathurugu	Schismina
17	51/50	[-1 1 -2 0 0 0 1⟩	34.28	Sogugu	Large septendecimal sixth tone
17	52/51	[2 -1 0 0 0 1 -1⟩	33.62	Sutho	Small septendecimal sixth tone
17	136/135	[3 -3 -1 0 0 0 1⟩	12.78	Sogu	Diatisma, fiventeen comma
17	256/255	[8 -1 -1 0 0 0 -1⟩	6.78	Sugu	Charisma, septendecimal kleisma
17	289/288	[-5 -2 0 0 0 0 2⟩	6.00	Soso	Semitonisma
17	2601/2600	[-3 2 -2 0 0 -1 2⟩	0.67	Sosothugugu	Sextantonisma
19	39/38	[-1 1 0 0 0 1 0 -1⟩	44.97	Nutho	Undevicesimal two-ninth tone
19	96/95	[5 1 -1 0 0 0 0 -1⟩	18.13	Nugu	19th-partial chroma
19	153/152	[-3 2 0 0 0 0 1 -1⟩	11.35	Nuso	Ganassisma
19	171/170	[-1 2 -1 0 0 0 -1 1⟩	10.15	Nosugu	Malcolmisma
19	324/323	[2 4 0 0 0 0 -1 -1⟩	5.35	Nusu	Photisma
19	361/360	[-3 -2 -1 0 0 0 0 2⟩	4.80	Nonogu	Go comma

Rank-2 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 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)	Srutal / pajara / injera
3	5\12 (1\12)	(P8/3, P5)	Augmented / lithium
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

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

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