665edo: Difference between revisions
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{{Infobox ET | |||
| Prime factorization = 3 × 5<sup>2</sup> × 7 | |||
| Step size = 1.80451¢ | |||
| Fifth = 389\665 (701.95¢) | |||
| Semitones = 63:50 (113.68¢ : 90.23¢) | |||
| Consistency = 9 | |||
}} | |||
{{EDO intro|665}} | |||
== Theory == | == Theory == | ||
665edo is best known for its extremely accurate fifth, only 0.00011 cents compressed. 665edo is the denominator of a convergent to log<sub>2</sub>3, after [[41edo]], [[53edo]] and [[306edo]], and before [[15601edo]]. However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It tempers out the [[satanic comma]], {{monzo| -1054 665 }} in the 3-limit; the [[enneadeca]], {{monzo| -14 -19 19 }}, and the [[monzisma]], {{monzo| 54 -37 2 }} in the 5-limit; the [[ragisma]], 4375/4374, the [[meter]], 703125/702464, and {{monzo| 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 [[support]]s 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 === | |||
{{Harmonics in equal|665}} | {{Harmonics in equal|665}} | ||
=== 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. | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
Revision as of 14:52, 16 August 2022
| ← 664edo | 665edo | 666edo → |
(convergent)
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. 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 | -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) | |
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.