768edo
Theory
768edo is consistent in the 7-odd-limit. The equal temperament tempers out the mutt comma [-44 -3 21⟩ and the 5-limit commatic comma [-37 38 -10⟩ in the 5-limit, and 65625/65536, 250047/250000, 5250987/5242880, [-12 -5 11 -2⟩, [7 18 -2 -11⟩, and [-36 8 4 5⟩ in the 7-limit.
As a tuning standard
Its adjacent step is known as hexamu (sixth MIDI-resolution unit, 6mu, 26 = 64 equal divisions of the 12edo semitone). The internal data structure of the 6mu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or -) showing the direction of the pitch-bend up or down, and all six of the remaining bits used for the tuning data
Odd harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.393 | -0.376 | -0.076 | +0.245 | +0.097 | -0.268 | -0.638 | -0.149 | +0.110 | +0.277 |
| Relative (%) | +0.0 | -25.1 | -24.1 | -4.9 | +15.7 | +6.2 | -17.1 | -40.8 | -9.6 | +7.1 | +17.7 | |
| Steps (reduced) |
768 (0) |
1217 (449) |
1783 (247) |
2156 (620) |
2657 (353) |
2842 (538) |
3139 (67) |
3262 (190) |
3474 (402) |
3731 (659) |
3805 (733) | |
Subsets and supersets
Since 768 factors into 28 × 3, 768edo has subset edos 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, and 384.
See also
- Equal multiplications of MIDI-resolution units