12edo

The circle of fifths in 12edo. Source: Wikipedia

12edo is the equal tuning featuring steps of (1200/12) = 100 cents, by definition, as 12 steps stack to the octave 2/1. As the dominant tuning system in the world, it is as such covered by the Xenharmonic Reference for completeness as it is definitionally not "xenharmonic".

12edo's most important feature is its approximation of 3/2 at 7 steps, which is less than 2¢ flat of just, and which, when stacked against the octave, produces the basic diatonic scale, 2-2-1-2-2-2-1. As the fifth is still slightly flat, this has allowed 12edo to be conventionally interpreted as a tuning of Meantone and hence a 5-limit system, where the minor third of 3 steps and the major third of 4 steps represent 6/5 and 5/4 respectively. Also important to 12edo's structure is the fact that these thirds derive from 4edo and 3edo, allowing for the construction of scales that repeat at fractions of the octave.

However, as the 12edo thirds are 16¢ and 14¢ out of tune with the 5-limit, another important interpretation is the 2.3.17.19 subgroup, which is tuned more accurately than prime 5. In particular, the minor third closely approximates 19/16, and the 100¢ semitone can be thought of as 17/16~18/17.

Theory

Edostep interpretations

12edo's edostep has the following interpretations in the 2.3.5 subgroup:

• 16/15 (the difference between 5/4 and 4/3)
• 25/24 (the difference between 6/5 and 5/4)
• 27/25 (the difference between 10/9 and 6/5)

When primes 17 and 19 are included, it also serves as 17/16, 18/17, 19/18, and 20/19.

12edo tempers out the following important commas in its 5-limit:

• 81/80 (equating 9/8 with 10/9, and four 3/2s to 5/1)
• 128/125 (causing three 5/4s to reach an octave exactly)
• 648/625 (causing four 6/5s to reach an octave exactly)
• 2048/2025 (splitting 9/8 into two 16/15s).

It can be defined in the subgroup 2.3.5.17.19 by equalizing the arithmetic progression 15::20, and therefore tempering out square superparticulars S16 through S19, and combinations thereof.

JI approximation

12edo is conventionally seen as a 2.3.5 edo, though perhaps more salient to conventional Western musical practice is the fact that it contains the basic diatonic scale, 2-2-1-2-2-2-1.

Approximation of prime harmonics in 12edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	-2.0	+13.7	+31.2	+48.7	-40.5	-5.0	+2.5	-28.3	-29.6	-45.0
	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)

Thirds in 12edo

Quality	Minor	Major
Cents	300	400
Just interpretation	6/5	5/4

Diatonic thirds are bolded.

Notation

12edo has a standard notation system, consistent with classical theory. As a result, ups and downs notation, KISS notation for diatonic, Pythagorean notation, and sagittal notation all converge on 12edo.

Compositional theory

Regular temperaments

Notable 5-limit temperaments supported by 12edo are Augmented (12 & 15), Diminished (12 & 16), and Meantone (12 & 19). These temperaments lead to the 3L 3s (or 3L 6s) MOS, the 4L 4s MOS, and the 5L 2s (or 3L 2s) MOS, respectively. Of these, it is a particularly good tuning of Diminished.

Chords

12edo is notable for its tritone of exactly 600c, major third of exactly 400c, and minor third of exactly 300c. This makes available a fully symmetrical diminished seventh chord and also a fully symmetrical augmented triad, and enables tritone substitution of dominant tetrads.

Due to 12edo's accuracy in the 2.3.17.19 subgroup, the minor triad of [0 3 7] can be analyzed as 16:19:24, which some theorists believe to contribute to its stable sound.

Scales

12edo, due to its large number of factors, contains many MOS scales with periods that are some fraction of the octave. One well-known example is 4L 4s, which is known as the octatonic scale. Additionally, it contains 6edo as a subset, which is the whole tone scale.

12edo is small enough that the EDO itself functions as a chromatic scale.

Multiples

24edo

Main article: 24edo

24edo corrects 12edo's approximate 11th and 13th harmonics (which fall about halfway between the steps in 12edo) and it can be used in the 2.3.(5).11.13.17.19 subgroup, however its 5 now accumulates error too quickly and is less usable than 12edo's.

Approximation of prime harmonics in 24edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	-2.0	+13.7	-18.8	-1.3	+9.5	-5.0	+2.5	+21.7	+20.4	+5.0
	Relative (%)	0.0	-3.9	+27.4	-37.7	-2.6	+18.9	-9.9	+5.0	+43.5	+40.8	+9.9
Steps (reduced)		24 (0)	38 (14)	56 (8)	67 (19)	83 (11)	89 (17)	98 (2)	102 (6)	109 (13)	117 (21)	119 (23)

36edo

36edo is another way to extend 12edo while maintaining a manageable amount of notes, correcting the 7th and 13th harmonics (in a way distinct from 24edo). Its 5 is still inherited from 12edo, although it is now about halfway between the steps. It is a good tuning for the 2.3.7.13.17.19 subgroup, and its 8/7 makes it a good tuning for slendric temperament.

Approximation of prime harmonics in 36edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	-2.0	+13.7	-2.2	+15.3	-7.2	-5.0	+2.5	+5.1	+3.8	-11.7
	Relative (%)	0.0	-5.9	+41.1	-6.5	+46.0	-21.6	-14.9	+7.5	+15.2	+11.3	-35.1
Steps (reduced)		36 (0)	57 (21)	84 (12)	101 (29)	125 (17)	133 (25)	147 (3)	153 (9)	163 (19)	175 (31)	178 (34)

72edo

Main article: 72edo

612edo

See 34edo#612edo.

Compton temperament

If 12edo is taken as a temperament of Pythagorean tuning instead of a 2.3.5 temperament, an independent dimension for any other prime (conventionally 5) may be added. This temperament, called Compton, takes advantage of the fact that due to 12edo's extraordinary accuracy in the 3-limit, higher-limit intervals tend to be off by roughly consistent offsets from 12edo ones. Compton temperament tempers out the Pythagorean comma (531441/524288, the ~24c difference between the Pythagorean chroma and minor second), and equates any mosdiatonic interval with its enharmonic counterpart. However, intervals of 5 are distinct, with 5/4 usually being tuned around 384 cents. As such, this is well-tuned in 72edo, and is supported by any multiple of 12 up to and including 300edo.