764edo
| ← 763edo | 764edo | 765edo → |
764 equal divisions of the octave (abbreviated 764edo or 764ed2), also called 764-tone equal temperament (764tet) or 764 equal temperament (764et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 764 equal parts of about 1.57 ¢ each. Each step represents a frequency ratio of 21/764, or the 764th root of 2.
Theory
764edo is a very strong 17-limit system, consistent to the 17-odd-limit or the no-19 no-29 41-odd-limit. It is the fourteenth zeta integral edo. In the 5-limit it tempers out the hemithirds comma, [38 -2 -15⟩; in the 7-limit 4375/4374; in the 11-limit 3025/3024 and 9801/9800; in the 13-limit 1716/1715, 2080/2079, 4096/4095, 4225/4224, 6656/6655 and 10648/10647; and in the 17-limit 2431/2430, 2500/2499, 4914/4913 and 5832/5831. It provides the optimal patent val for the abigail temperament in the 11-limit.
In higher limits, it is a strong no-19 and no-29 37-limit tuning, and an exceptional 2.11.23.31.37 subgroup system, with errors less than 2%.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.139 | +0.074 | +0.284 | -0.009 | -0.214 | +0.280 | -0.654 | -0.002 | -0.781 | -0.009 | -0.035 | -0.267 | +0.524 | +0.462 |
| Relative (%) | +0.0 | +8.9 | +4.7 | +18.1 | -0.6 | -13.6 | +17.8 | -41.7 | -0.1 | -49.7 | -0.6 | -2.2 | -17.0 | +33.4 | +29.4 | |
| Steps (reduced) |
764 (0) |
1211 (447) |
1774 (246) |
2145 (617) |
2643 (351) |
2827 (535) |
3123 (67) |
3245 (189) |
3456 (400) |
3711 (655) |
3785 (729) |
3980 (160) |
4093 (273) |
4146 (326) |
4244 (424) | |
Subsets and supersets
Since 764 factors into 22 × 191, 764edo has subset edos 2, 4, 191, and 382. In addition, one step of 764edo is exactly 22 jinns (22\16808).
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1211 -764⟩ | [⟨764 1211]] | −0.0439 | 0.0439 | 2.80 |
| 2.3.5 | [38 -2 -15⟩, [25 -48 22⟩ | [⟨764 1211 1774]] | −0.0399 | 0.0363 | 2.31 |
| 2.3.5.7 | 4375/4374, 52734375/52706752, [31 -6 -2 -6⟩ | [⟨764 1211 1774 2145]] | −0.0552 | 0.0412 | 2.62 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 131072/130977, 35156250/35153041 | [⟨764 1211 1774 2145 2643]] | −0.0436 | 0.0435 | 2.77 |
| 2.3.5.7.11.13 | 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875 | [⟨764 1211 1774 2145 2643 2827]] | −0.0267 | 0.0548 | 3.49 |
| 2.3.5.7.11.13.17 | 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913 | [⟨764 1211 1774 2145 2643 2827 3123]] | −0.0327 | 0.0528 | 3.36 |
| 2.3.5.7.11.13.17.23 | 1716/1715, 2080/2079, 2024/2023, 2431/2430, 2500/2499, 3520/3519, 4096/4095 | [⟨764 1211 1774 2145 2643 2827 3123 3456]] | −0.0286 | 0.0506 | 3.22 |
- 764et has lower absolute errors than any previous equal temperaments in the 13- and 17-limit. In the 13-limit it beats 684 and is only bettered by 935. In the 17-limit it beats 742 and is only bettered by 814.
- It is best at the no-19 23-limit, where it has a lower relative error than any previous equal temperaments, past 494 and before 1578.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 123\764 | 193.19 | 262144/234375 | Lunatic (7-limit) |
| 1 | 277\764 | 435.08 | 9/7 | Supermajor |
| 2 | 133\764 | 208.90 | 44/39 | Abigail |
| 2 | 277\764 (105\764) |
435.08 (164.92) |
9/7 (11/10) |
Semisupermajor |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct