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. It is the dominant tuning system in the world, and as such is covered by the Xenharmonic Reference for completeness as it is not 'xenharmonic'. Its fifth is at 7 steps, and its major third is at 4 steps.

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)

12edo tempers out the following commas in the 2.3.5.17.19 subgroup:

• 81/80 (equating 9/8 with 10/9)
• 256/255 (equating 16/17 with 16/15)
• 96/95 (equating 6/5 with 19/16)
• 128/125 (causing three 5/4s to reach an octave exactly)

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.

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.

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.

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.

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.