164edo: Difference between revisions

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== Theory ==

In the 5-limit, ~~164edo~~ tempers out the [[würschmidt comma]], 393216/390625, and the [[vulture comma]], {{monzo| 24 -21 4 }}. It supplies the [[optimal patent val]] for the [[würschmidt]] temperament.

164 = 4 × 41, and 164edo shares its [[perfect fifth|fifth]] with [[41edo]]. In the 5-limit, 164et tempers out the [[würschmidt comma]], 393216/390625, and the [[vulture comma]], {{monzo| 24 -21 4 }}. It supplies the [[optimal patent val]] for the [[würschmidt]] temperament.

In the [[patent val]] {{val| 164 260 381 '''460''' '''567''' 607 }}, it tempers out [[196/195]], [[352/351]], [[385/384]], [[441/440]], [[676/675]], and supplies the optimal patent val for the 7-limit, 1/41 octave period 41&123 temperament, and the 13-limit [[Gamelismic family #Portent|momentous]] temperament, the rank-3 temperament tempering out 196/195, 352/351, 385/384 and 441/440.

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=== Subsets and supersets ===

Since 164 = ~~4 × 41~~, 164edo has subset edos {{EDOs| 2, 4, 41, 82 }}.

Since 164 = {{factorization|164}}, 164edo has subset edos {{EDOs| 2, 4, 41, 82 }}. [[328edo]], which doubles it, provides good correction for the approximation to harmonics 7 and 11, and is [[consistent]] in the [[13-odd-limit]].

== Regular temperament properties ==

@@ Line 3: / Line 3: @@
 == Theory ==
-In the 5-limit, 164edo tempers out the [[würschmidt comma]], 393216/390625, and the [[vulture comma]], {{monzo| 24 -21 4 }}. It supplies the [[optimal patent val]] for the [[würschmidt]] temperament.
+= 4 × 41, and 164edo shares its [[perfect fifth|fifth]] with [[41edo]]. In the 5-limit, 164et tempers out the [[würschmidt comma]], 393216/390625, and the [[vulture comma]], {{monzo| 24 -21 4 }}. It supplies the [[optimal patent val]] for the [[würschmidt]] temperament.
 In the [[patent val]] {{val| 164 260 381 '''460''' '''567''' 607 }}, it tempers out [[196/195]], [[352/351]], [[385/384]], [[441/440]], [[676/675]], and supplies the optimal patent val for the 7-limit, 1/41 octave period 41&amp;123 temperament, and the 13-limit [[Gamelismic family #Portent|momentous]] temperament, the rank-3 temperament tempering out 196/195, 352/351, 385/384 and 441/440.
@@ Line 13: / Line 13: @@
 === Subsets and supersets ===
-Since 164 = 4 × 41, 164edo has subset edos {{EDOs| 2, 4, 41, 82 }}.
+Since 164 = {{factorization|164}}, 164edo has subset edos {{EDOs| 2, 4, 41, 82 }}. [[328edo]], which doubles it, provides good correction for the approximation to harmonics 7 and 11, and is [[consistent]] in the [[13-odd-limit]].
 == Regular temperament properties ==