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== Theory == | == Theory == | ||
665edo is best known for its extremely accurate [[3/2|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]], and is the member of this series with the highest 3-2 [[Telicity #k-Strong Telicity|telicity ''k''-strength]] before being finally surpassed in this regard by [[190537edo]]. | 665edo is best known for its extremely accurate [[3/2|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]], and is the member of this series with the highest 3-2 [[Telicity #k-Strong Telicity|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 [[ | 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. | 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 === | === Prime harmonics === | ||
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=== 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 | A [[maximal evenness]] scale deriving from the {{nowrap|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{{frac|17|19}}s, 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. | ||
== Intervals == | == Intervals == | ||
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| 4375/4374, 703125/702464, {{monzo| 36 -5 0 -10 }} | | 4375/4374, 703125/702464, {{monzo| 36 -5 0 -10 }} | ||
| {{mapping| 665 1054 1544 1867 }} | | {{mapping| 665 1054 1544 1867 }} | ||
| | | −0.0015 | ||
| 0.0474 | | 0.0474 | ||
| 2.63 | | 2.63 | ||
| Line 59: | Line 59: | ||
| 4000/3993, 4375/4374, 117649/117612, 131072/130977 | | 4000/3993, 4375/4374, 117649/117612, 131072/130977 | ||
| {{mapping| 665 1054 1544 1867 2301 }} | | {{mapping| 665 1054 1544 1867 2301 }} | ||
| | | −0.0511 | ||
| 0.1078 | | 0.1078 | ||
| 5.97 | | 5.97 | ||
| Line 66: | Line 66: | ||
| 1575/1573, 2080/2079, 4096/4095, 4375/4374, 31250/31213 | | 1575/1573, 2080/2079, 4096/4095, 4375/4374, 31250/31213 | ||
| {{mapping| 665 1054 1544 1867 2301 2461 }} | | {{mapping| 665 1054 1544 1867 2301 2461 }} | ||
| | | −0.0594 | ||
| 0.1002 | | 0.1002 | ||
| 5.55 | | 5.55 | ||
| Line 105: | Line 105: | ||
| [[Enneadecal]] | | [[Enneadecal]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
[[Category:Satanic]] | [[Category:Satanic]] | ||
[[Category:Wizardharry]] | [[Category:Wizardharry]] | ||
[[Category:Monzismic]] | [[Category:Monzismic]] | ||