12edo: Difference between revisions
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[[File:Circle of fifths.png|thumb|The circle of fifths in 12edo. Source: Wikipedia]] | [[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]]. As the dominant tuning system in the world, it is as such covered by | '''12edo''' is the equal tuning featuring steps of (1200/12) = 100 cents, by definition, as 12 steps stack to the octave [[Octave|2/1]]. As the dominant tuning system in the world, it is as such covered by XenReference for completeness as it is definitionally not "xenharmonic", and to demonstrate how xenharmonic theory describes conventional musical terminology. | ||
12edo has its dominant position due to its approximation of the just perfect fifth, a frequency ratio of [[3/2]], at 7 steps. This approximation is less than 2% of a 12edo step (2 cents) flat of just. | 12edo has its dominant position due to its approximation of the just perfect fifth, a frequency ratio of [[3/2]], at 7 steps. This approximation is less than 2% of a 12edo step (2 cents) flat of just. | ||
Latest revision as of 02:02, 22 October 2026
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 XenReference for completeness as it is definitionally not "xenharmonic", and to demonstrate how xenharmonic theory describes conventional musical terminology.
12edo has its dominant position due to its approximation of the just perfect fifth, a frequency ratio of 3/2, at 7 steps. This approximation is less than 2% of a 12edo step (2 cents) flat of just.
The diatonic scale used in conventional music theory is, in xenharmonic theory, considered to be generated by repeatedly stacking this perfect fifth (i.e. by going around the circle of fifths). It is the moment-of-symmetry scale 5L 2s, meaning that it has 5 large steps and 2 small steps, as equally spaced as possible. In xenharmonic theory, it is common to consider diatonic as a whole as a single scale, being agnostic to rotation and to transposition. This is because xenharmonic theory often deals with scales that don't follow the standard pattern of whole tones and semitones. In 12edo, the diatonic scale has a step ratio of 2:1 (whole steps to half steps, or whole tones to semitones).
The major third of 12edo is flat of the Pythagorean major third (the one generated by the just 3/2 fifth, at about 408 cents), due to the fifth itself being slightly flat. Because of this, it is generally also interpreted as the simple just intonation major third 5/4, which is not generated by a fifth in just intonation. This 'double interpretation' is a temperament, and that particular temperament is called meantone. Because of meantone, the major triad itself represents the just intonation triad 4:5:6, one of the most basic just intonation triads. As a result (due to the rules of regular temperament), the minor third is interpreted as 6/5. 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. This means that 12edo can represent any interval built from combinations of 3/2, 6/5, and 5/4, such as 9/8 (the whole tone) or 25/24 (the semitone between 6/5 and 5/4 themselves).
However, as the 12edo thirds are 16¢ and 14¢ out of tune with 6/5 and 5/4 respectively, another important interpretation is the 2.3.17.19 subgroup, which is tuned more accurately than prime 5. A subgroup is a set of intervals that you can multiply and divide to get other intervals in the same subgroup - which makes it a group in mathematical terms - and is generally defined as all the intervals that you can get by multiplying or dividing (moving up or down by) a certain set of frequency ratios. In the 2.3.17.19 subgroup, these ratios In particular are the 19-limit minor third (19/16, about 298c, and approximated by 12edo's minor third), the perfect fifth (as previously discussed), and the 17-limit semitone (17/16, about 105c, and approximated by 12edo's semitone), alongside the octave.
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)
This means that when looking at just intonation interpretations of 12edo music or scales, these intervals show up wherever 12edo has a single edostep.
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.
| 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) | |
12edo has two qualities of thirds when not including augmented and diminished ones, those being the major third and minor third, and stacking them in either order produces a triad bounded by a perfect fifth.
| Quality | Minor | Major |
|---|---|---|
| Cents | 300 | 400 |
| Just interpretation | 6/5 | 5/4 |
Diatonic thirds are bolded. Note that there are only two third types, both of which are found in the diatonic scale.
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.
Derivation
12edo is derived, as in historical theory, by setting the diatonic and chromatic semitone equal to avoid the problem of "wolf fifths".
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. Another example is the augmented scale, or 3L 3s. 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.
| 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.
| 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
120edo
See 40edo#120edo.
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 (or 81/80 tuned to 16 cents). As such, this is well-tuned in 72edo, and is supported by any multiple of 12 up to and including 300edo.
The canonical extension of Compton to the 7-limit tempers out 225/224, so that 7/4 is mapped to 10 steps of 12edo minus 2 comma steps.
