31edo
From Xenharmonic Reference
31edo, or 31 equal divisions of the octave, is an equal tuning with a step size of approximately 29 cents. Aside from 12edo, it is a popular tuning of meantone and has accurate approximations of harmonics 5 and 7.
Theory
JI approximation
31edo has a somewhat flat prime 3, a slightly sharp prime 5, a slightly flat prime 7, and a flat prime 11. The flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral interval.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -5.2 | +0.8 | -1.1 | -9.4 | +11.1 | +11.2 | +12.2 | -8.9 | +15.6 | +16.3 |
| Relative (%) | 0.0 | -13.4 | +2.0 | -2.8 | -24.2 | +28.6 | +28.9 | +31.4 | -23.0 | +40.3 | +42.0 | |
| Steps
(reduced) |
31
(0) |
49
(18) |
72
(10) |
87
(25) |
107
(14) |
115
(22) |
127
(3) |
132
(8) |
140
(16) |
151
(27) |
154
(30) | |
Scales
31edo does not also temper out 64/63, meaning that it can be used to tune Diasem while representing some simpler 5-limit intervals.
Regular temperaments
Besides meantone, 31edo also supports variations of neutral temperament, slendric, miracle, and orwell.
