665edo: Difference between revisions

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== 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-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 log23, 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 [[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.

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=== Subsets and supersets ===

Since 665 ~~= 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]]).

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]]).

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

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! Generator*

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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]], 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]].
+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 [[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.
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 === Subsets and supersets ===
-Since 665 = 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]]).
+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]]).
 [[1330edo]], which doubles 665edo, provides a good correction of the harmonic 11.
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 ! Generator*
 ! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-