12edo: Difference between revisions
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'''12edo''' is the equal tuning featuring steps of (1200/12) = 100 cents, by definition, as 12 steps stack to the octave [[Octave|2/1]]. It is the dominant tuning system in the world, and as such is covered by | [[File:Circle of fifths.png|thumb|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 [[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 == | == Theory == | ||
| Line 41: | Line 42: | ||
==== Scales ==== | ==== 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 | 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 | 12edo is small enough that the EDO itself functions as a chromatic scale. | ||
=== Regular temperaments === | === Regular temperaments === | ||
12edo | 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 == | == Notation == | ||
| Line 52: | Line 53: | ||
== Compton temperament == | == 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 | 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. | ||
{{Cat|edos}} | |||
Latest revision as of 23:17, 14 February 2026
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.
| 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) | |
| 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.
