761edo
| ← 760edo | 761edo | 762edo → |
761 equal divisions of the octave (abbreviated 761edo or 761ed2), also called 761-tone equal temperament (761tet) or 761 equal temperament (761et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 761 equal parts of about 1.58 ¢ each. Each step represents a frequency ratio of 21/761, or the 761st root of 2.
Theory
761edo is consistent to the 9-odd-limit. As an equal temperament, it tempers out 32805/32768 in the 5-limit; 420175/419904 and [3 13 -15 4⟩ in the 7-limit. The equal temperament is strong in the 2.3.5.13.29.31 subgroup, tempering out 32805/32768, 21141/21125, 3627/3625, 140625/140608 and 45349632/45287125.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.247 | +0.020 | -0.626 | -0.493 | +0.587 | -0.055 | -0.227 | +0.695 | +0.516 | +0.704 | -0.679 |
| Relative (%) | -15.6 | +1.3 | -39.7 | -31.3 | +37.3 | -3.5 | -14.4 | +44.1 | +32.7 | +44.6 | -43.1 | |
| Steps (reduced) |
1206 (445) |
1767 (245) |
2136 (614) |
2412 (129) |
2633 (350) |
2816 (533) |
2973 (690) |
3111 (67) |
3233 (189) |
3343 (299) |
3442 (398) | |
Subsets and supersets
761edo is the 135th prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-1206 761⟩ | [⟨761 1206]] | 0.0778 | 0.0778 | 4.93 |
| 2.3.5 | 32805/32768, [69 81 -85⟩ | [⟨761 1206 1767]] | 0.0490 | 0.0755 | 4.79 |
| 2.3.5.7 | 32805/32768, 420175/419904, [3 13 -15 4⟩ | [⟨761 1206 1767 2136]] | 0.0925 | 0.0998 | 6.33 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 316\761 | 498.292 | 4/3 | Helmholtz |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct