665edo: Difference between revisions
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{{Infobox ET | {{Infobox ET | ||
| Prime factorization = | | Prime factorization = 5 × 7 × 19 | ||
| Step size = 1.80451¢ | | Step size = 1.80451¢ | ||
| Fifth = 389\665 (701.95¢) | | Fifth = 389\665 (701.95¢) | ||
| Line 19: | Line 19: | ||
=== Miscellaneous properties === | === 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. | 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 == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal 8ve<br>stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| -1054 665 }} | |||
| [{{val| 665 1054 }}] | |||
| +0.0000 | |||
| 0.0000 | |||
| 0.00 | |||
|- | |||
| 2.3.5 | |||
| {{monzo| -14 -19 19 }}, {{monzo| 54 -37 2 }} | |||
| [{{val| 665 1054 1544 }}] | |||
| +0.0213 | |||
| 0.0301 | |||
| 1.67 | |||
|- | |||
| 2.3.5.7 | |||
| 4375/4374, 703125/702464, {{monzo| 36 -5 0 -10 }} | |||
| [{{val| 665 1054 1544 1867 }}] | |||
| -0.0015 | |||
| 0.0474 | |||
| 2.63 | |||
|- | |||
| 2.3.5.7.11 | |||
| 4000/3993, 4375/4374, 117649/117612, 131072/130977 | |||
| [{{val| 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 | |||
| [{{val| 665 1054 1544 1867 2301 2461 }}] | |||
| -0.0594 | |||
| 0.1002 | |||
| 5.55 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per Octave | |||
! Generator<br>(Reduced) | |||
! Cents<br>(Reduced) | |||
! Associated<br>Ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 62\665 | |||
| 111.88 | |||
| 16/15 | |||
| [[Vavoom]] | |||
|- | |||
| 1 | |||
| 138\665 | |||
| 249.02 | |||
| {{monzo| -26 18 -1 }} | |||
| [[Monzismic]] | |||
|- | |||
| 7 | |||
| 288\665<br>(3\665) | |||
| 519.70<br>(5.41) | |||
| 27/20<br>(325/324) | |||
| [[Brahmagupta]] | |||
|- | |||
| 19 | |||
| 276\665<br>(4\665) | |||
| 498.05<br>(7.21) | |||
| 4/3<br>(225//224) | |||
| [[Enneadecal]] | |||
|} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
Revision as of 15:17, 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.
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 Octave |
Generator (Reduced) |
Cents (Reduced) |
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 |