665edo
| ← 664edo | 665edo | 666edo → |
(convergent)
665 equal divisions of the octave (abbreviated 665edo or 665ed2), also called 665-tone equal temperament (665tet) or 665 equal temperament (665et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 665 equal parts of about 1.8 ¢ each. Each step represents a frequency ratio of 21/665, or the 665th root of 2.
Theory
665edo 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, and is the member of this series with the highest 3-2 telicity k-strength before being finally surpassed in this regard by 190537edo.
However, it also provides the optimal patent val for the rank-4 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 [36 -5 0 -10⟩ in the 7-limit; 4000/3993, 46656/46585, 131072/130977 and 151263/151250 in the 11-limit, providing the optimal patent val for the 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 chords, 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). 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.
Prime harmonics
| 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 | -8 | +11 | +48 | +21 | -16 | +13 | -17 | +44 | +46 | |
| Steps (reduced) |
665 (0) |
1054 (389) |
1544 (214) |
1867 (537) |
2301 (306) |
2461 (466) |
2718 (58) |
2825 (165) |
3008 (348) |
3231 (571) |
3295 (635) | |
Subsets and supersets
Since 665 factors into 5 × 7 × 19, 665edo has subset edos 5, 7, 19, 35, 95, and 133. One step of 665edo has been proposed as an interval size measure, called a Delfi unit. A Delfi unit is exactly 48 imps (48\31920).
1330edo, which doubles 665edo, provides a good correction of the harmonic 11.
Miscellaneous properties
A 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.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-1054 665⟩ | [⟨665 1054]] | +0.0000 | 0.0000 | 0.00 |
| 2.3.5 | [-14 -19 19⟩, [54 -37 2⟩ | [⟨665 1054 1544]] | +0.0213 | 0.0301 | 1.67 |
| 2.3.5.7 | 4375/4374, 703125/702464, [36 -5 0 -10⟩ | [⟨665 1054 1544 1867]] | -0.0015 | 0.0474 | 2.63 |
| 2.3.5.7.11 | 4000/3993, 4375/4374, 117649/117612, 131072/130977 | [⟨665 1054 1544 1867 2301]] | -0.0511 | 0.1078 | 5.97 |
| 2.3.5.7.11.13 | 1575/1573, 2080/2079, 4096/4095, 4375/4374, 31250/31213 | [⟨665 1054 1544 1867 2301 2461]] | -0.0594 | 0.1002 | 5.55 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 62\665 | 111.88 | 16/15 | Vavoom |
| 1 | 138\665 | 249.02 | [-26 18 -1⟩ | Monzismic |
| 7 | 288\665 (3\665) |
519.70 (5.41) |
27/20 (325/324) |
Brahmagupta |
| 19 | 276\665 (4\665) |
498.05 (7.21) |
4/3 (225//224) |
Enneadecal |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct