787edo
| ← 786edo | 787edo | 788edo → |
787 equal divisions of the octave (abbreviated 787edo or 787ed2), also called 787-tone equal temperament (787tet) or 787 equal temperament (787et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 787 equal parts of about 1.52 ¢ each. Each step represents a frequency ratio of 21/787, or the 787th root of 2.
Theory
787edo is consistent to the 7-odd-limit and the error of its harmonic 3 is large. The accuracy of the equal temperament is higher in the upper harmonics, making it strong in the 2.17.19.23.31 subgroup, tempering out 151604477/151519232, 573114368/572870539, 68939809/68876792 and 4127463571456/4124314002323. It can also be used in the 2.3.5.7.13.41.43.47 subgroup, tempering out 729/728, 2401/2400, 1025/1024, 8127/8125, 1601613/1600000, 40625/40608 and 546875/545792.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.557 | -0.545 | -0.592 | +0.410 | +0.652 | -0.375 | +0.423 | +0.254 | -0.181 | +0.375 | -0.066 |
| Relative (%) | -36.5 | -35.7 | -38.8 | +26.9 | +42.7 | -24.6 | +27.7 | +16.7 | -11.9 | +24.6 | -4.3 | |
| Steps (reduced) |
1247 (460) |
1827 (253) |
2209 (635) |
2495 (134) |
2723 (362) |
2912 (551) |
3075 (714) |
3217 (69) |
3343 (195) |
3457 (309) |
3560 (412) | |
Subsets and supersets
787edo is the 138th prime edo. 1574edo, which doubles it, gives a good correction to its harmonic 3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [2495 -787⟩ | [⟨787 2495]] | –0.0647 | 0.0647 | 4.24 |