665edo
The 665 equal temperament divides the octave into 665 equal parts of 1.80451 cents each.
Theory
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.000 | -0.148 | +0.197 | +0.863 | +0.375 | -0.294 | +0.231 | -0.304 | +0.799 | +0.829 |
| Relative (%) | +0.0 | -0.0 | -8.2 | +10.9 | +47.8 | +20.8 | -16.3 | +12.8 | -16.9 | +44.3 | +45.9 | |
| Steps (reduced) |
665 (0) |
1054 (389) |
1544 (214) |
1867 (537) |
2301 (306) |
2461 (466) |
2718 (58) |
2825 (165) |
3008 (348) |
3231 (571) |
3295 (635) | |
It is best known for its extremely accurate fifth, only 0.00011 cents compressed. 665edo is the denominator of a convergent to log23, after 41edo, 53edo and 306edo, and before 15601edo. However, it also provides the optimal patent val for the rank four temperament tempering out 4000/3993. It tempers out the 'satanic' comma, |-1054 665> in the 3-limit; the enneadeca, |-14 -19 19> and the monzisma, |54 -37 2> in the 5-limit; the ragisma, 4375/4374, the meter, 703125/702464, and 68719476736/68641485507 in the 7-limit; 4000/3993, 46656/46585, 131072/130977 and 151263/151250 in the 11-limit, providing the optimal patent val for 11-limit Brahmagupta temperament. In the 13-limit it tempers out 1575/1573, 2080/2079, 4096/4095 and 4225/4224; since it tempers out 1575/1573, the nicola, it supports nicolic tempering and hence the nicolic tetrad, for which it provides an excellent tuning. In the 17-limit it tempers out 1156/1155, 1275/1274, 2058/2057, 2500/2499 and 5832/5831; in the 19-limit it tempers out 969/968, 1445/1444, 2432/2431, 3136/3135, 3250/3249 and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184 and 2737/2736.
665edo provides excellent approximations for the 7-limit intervals and harmonics 13, 17, 19 and 23. It is considered as the excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is consistent in the 27-odd-limit (with no elevens). Despite its division number of the octave, 665edo provides poor approximations for the 11-limit intervals, with two mappings possible for the 11/8 fourth: a sharp one from the patent val, and a flat one from the 665e val. Using the 665e val, 41503/41472, 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-limit.
Maximal evenness scale deriving from the 118 & 665 temperament, known as vavoom, can also theoretically serve as a calendar leap week cycle corresponding to a year length of 365d 5h 48m 37+17/19s, about 7 seconds shorter than the average length of the tropical year today. Given the excellence of both 118 and 665 in 5-limit, this is a great point of intersection of solar calendar leap rules and just intonation-based temperaments.