665edo: Difference between revisions

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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 [[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.

However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out|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.

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=== Subsets and supersets ===

Since 665 factors into ~~{{factorization|665}}~~, 665edo has subset edos {{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 ([[31920edo|48\31920]]).

Since 665 factors into 5 × 7 × 19, 665edo has subset edos {{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 ([[31920edo|48\31920]]).

[[1330edo]], which doubles 665edo, provides a good correction of the harmonic 11.

=== ~~Miscellaneous properties~~ ===

=== Miscellany ===

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.

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 {{nowrap| 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 ==

@@ Line 5: / Line 5: @@
 edo 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 [[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.
+However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out|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.
 edo 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.
@@ Line 13: / Line 13: @@
 === Subsets and supersets ===
-Since 665 factors into {{factorization|665}}, 665edo has subset edos {{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 ([[31920edo|48\31920]]).
+Since 665 factors into 5 × 7 × 19, 665edo has subset edos {{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 ([[31920edo|48\31920]]).
 [[1330edo]], which doubles 665edo, provides a good correction of the harmonic 11.
-=== Miscellaneous properties ===
+=== Miscellany ===
-A [[maximal evenness]] scale deriving from the {{nowrap|118 &amp; 665}} temperament, known as [[vavoom]], can also theoretically serve as a calendar leap week cycle corresponding to a year length of 365d&nbsp;5h&nbsp;48m&nbsp;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.
+A [[maximal evenness]] scale deriving from the {{nowrap|118 &amp; 665}} temperament, known as [[vavoom]], can also theoretically serve as a calendar leap week cycle corresponding to a year length of {{nowrap| 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 ==