1665edo
| ← 1664edo | 1665edo | 1666edo → |
1665 equal divisions of the octave (1665edo), or 1665-tone equal temperament (1665tet), 1665 equal temperament (1665et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 1665 equal parts of about 0.721 ¢ each.
Theory
1665edo is a very strong 5-limit (as well as 2.3.5.11 subgroup) tuning and it is consistent in the 15-odd-limit. In the 5-limit, 1665edo is a tuning for the gross temperament.
1665edo provides the optimal patent val for the rhodium temperament in the 11-limit and also in the 13-limit. In addition, it provides the optimal patent val for dzelic temperament in the 13-limit.
1665cc val is a tuning for the roentgenium temperament, and the patent val tunes the unnamed 111 & 1665 temperament in the 13-limit which has a comma basis {6656/6655, 123201/123200, 250047/250000, 91182091/91125000}.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.000 | +0.027 | -0.007 | -0.177 | +0.033 | -0.167 | +0.270 | +0.145 | +0.194 | +0.333 | +0.190 |
| relative (%) | +0 | +4 | -1 | -25 | +5 | -23 | +37 | +20 | +27 | +46 | +26 | |
| Steps (reduced) |
1665 (0) |
2639 (974) |
3866 (536) |
4674 (1344) |
5760 (765) |
6161 (1166) |
6806 (146) |
7073 (413) |
7532 (872) |
8089 (1429) |
8249 (1589) | |
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 127\1665 | 91.531 | [9 -32 18⟩ | Gross |
| 37 | 377\1665 (17\1665) |
271.711 (12.252) |
117/100 (?) |
Dzelic |
| 45 | 1301\1665 (6\1665) |
937.657 (4.324) |
55/32 (?) |
Rhodium |