768edo
| ← 767edo | 768edo | 769edo → |
768 equal divisions of the octave (abbreviated 768edo or 768ed2), also called 768-tone equal temperament (768tet) or 768 equal temperament (768et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 768 equal parts of about 1.56 ¢ each. Each step represents a frequency ratio of 21/768, or the 768th root of 2.
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
A step of 768edo is known as a 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