Trivial EDO
A trivial EDO is a small EDO which is not xenharmonic due to being a subset of 12edo. There are five trivial edos: 1edo, 2edo, 3edo, 4edo, and 6edo.
1edo
1edo is equivalent to the 2-limit (the system consisting of only octaves) and the 1-odd-limit, and of the primes only meaningfully represents 2/1. One step of 1edo (that is, the octave itself) serves as the period of most octave-equivalent temperaments.
2edo
2edo may be considered to represent the 2.17/3.7/5.45.23 subgroup, or any other subgroup consisting of only octaves and tritones as generators. Its step is the semioctave, which appears as a critical structural interval in temperaments like pajara (separating 5/4 from 7/4), and it additionally contains the simplest form of functional contrast (between tonic and antitonic).
2edo represents a simplification of the structure of a trine, and counts as a trine itself.
3edo
The step of 3edo serves as 12edo's major third and also as 15edo's. When interpreted as 5/4, it supports augmented temperament, as 3edo is equivalent to an augmented triad in 12edo. It also serves as the period of misty temperament, which splits the deficit between three 5/4s and the octave into three. In general, 3edo provides a basic outline for triadic harmony. 3edo may be considered a 2.5.11/7.19/3 temperament, being the first edo to approximate a low prime other than 2.
3edo also simplifies 3:4:5-based harmony, reflecting the inaccurate temperament Father which equates 5/4 and 4/3.
4edo
The step of 4edo serves as 12edo's minor third, which also appears in 16edo, supporting diminished temperament if interpreted as 6/5. 4edo's step may also be interpreted as 19/16, supporting the temperament wherein four 19/16s are equated to the octave. Additionally, the semioctave retains its interpretations from 2edo, so that 4edo may be considered a 2.5/3.7/3.17/3.27.19.23 temperament. 4edo outlines tetradic harmony, and distinctly (although inaccurately) represents 4:5:6:7.
6edo
6edo is large enough to be usable as a melodic system in its own right, and is more familiar as the 12edo wholetone scale. It provides the basic structure of the 2.5.7 subgroup (serving a similar role for it as 9edo does for the 2.3.5 subgroup), and additionally inherits 2edo's tritone representations and 3edo's mappings, making it a 2.9.5.7.11.17/3.19/3.23 subgroup temperament, although the 11 is one of the less accurate intervals along with the 7 and 23, and 13 is actually better approximated than 11. The basic 4:5:7 chord is mapped to 0-2-5 steps of 6edo.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | +98.0 | +13.7 | +31.2 | +48.7 | -40.5 | +95.0 | -97.5 | -28.3 | -29.6 | +55.0 |
| Relative (%) | 0.0 | +49.0 | +6.8 | +15.6 | +24.3 | -20.3 | +47.5 | -48.8 | -14.1 | -14.8 | +27.5 | |
| Steps
(reduced) |
6
(0) |
10
(4) |
14
(2) |
17
(5) |
21
(3) |
22
(4) |
25
(1) |
25
(1) |
27
(3) |
29
(5) |
30
(6) | |
