410edo: Difference between revisions

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== Theory ==

410edo is [[enfactored]] in the [[5-limit]], with the same tuning as [[205edo]] characterized by [[tempering out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 38 -2 -15 }} (luna/hemithirds comma), as well as {{monzo| -29 -11 20 }} (gammic comma) and {{monzo| 47 -15 -10 }} (quintosec comma), but the approximations to [[harmonic]]s [[7/1|7]] and [[13/1|13]] are much improved. It tempers out 2401/2400 ([[breedsma]]), 4802000/4782969 ([[~~canousma~~]]), and 48828125/48771072 (neptunisma) in the [[7-limit]]; [[5632/5625]], [[9801/9800]], [[14641/14580]], and 117649/117612 in the [[11-limit]]; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], and [[4225/4224]] in the 13-limit.

410edo is [[enfactored]] in the [[5-limit]], with the same tuning as [[205edo]] characterized by [[tempering out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 38 -2 -15 }} (luna/hemithirds comma), as well as {{monzo| -29 -11 20 }} (gammic comma) and {{monzo| 47 -15 -10 }} (quintosec comma), but the approximations to [[harmonic]]s [[7/1|7]] and [[13/1|13]] are much improved. It tempers out 2401/2400 ([[breedsma]]), 4802000/4782969 ([[cathartic comma]]), and 48828125/48771072 (neptunisma) in the [[7-limit]]; [[5632/5625]], [[9801/9800]], [[14641/14580]], and 117649/117612 in the [[11-limit]]; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], and [[4225/4224]] in the 13-limit.

410edo provides the [[optimal patent val]] for the 11- and 13-limit [[semiluna]], [[hemiluna]], and [[floral]] temperaments, the rank-3 [[~~semicanou~~]] temperament, and the rank-4 temperament tempering out 14641/14580.

410edo provides the [[optimal patent val]] for the 11- and 13-limit [[semiluna]], [[hemiluna]], and [[floral]] temperaments, the rank-3 [[semicathart]] temperament, and the rank-4 temperament tempering out 14641/14580.

410edo works much better as a no-11 no-13 [[subgroup]] temperament, with a sharp tendency to harmonics up to 29. For example, it tempers out [[1216/1215]], [[1225/1224]], [[1445/1444]], and [[2500/2499]] in the 2.3.5.7.17.19 subgroup.

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* [https://www.youtube.com/watch?v=-bm5UdmveZU ''All Time Best''] – decoid[40], cover of [[Phlub]]

[[Category:Hemiluna]]

[[Category:Semiluna]]

[[Category:Cathartic]]

[[Category:~~Hemiluna~~]]

[[Category:Semicathart]]

[[Category:Semicathartismic]]

[[Category:Listen]]

~~[[Category:Semicanou]]~~

~~[[Category:Semicathartismic]]~~

~~[[Category:Semiluna]]~~

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is [[enfactored]] in the [[5-limit]], with the same tuning as [[205edo]] characterized by [[tempering out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 38 -2 -15 }} (luna/hemithirds comma), as well as {{monzo| -29 -11 20 }} (gammic comma) and {{monzo| 47 -15 -10 }} (quintosec comma), but the approximations to [[harmonic]]s [[7/1|7]] and [[13/1|13]] are much improved. It tempers out 2401/2400 ([[breedsma]]), 4802000/4782969 ([[canousma]]), and 48828125/48771072 (neptunisma) in the [[7-limit]]; [[5632/5625]], [[9801/9800]], [[14641/14580]], and 117649/117612 in the [[11-limit]]; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], and [[4225/4224]] in the 13-limit.
+edo is [[enfactored]] in the [[5-limit]], with the same tuning as [[205edo]] characterized by [[tempering out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 38 -2 -15 }} (luna/hemithirds comma), as well as {{monzo| -29 -11 20 }} (gammic comma) and {{monzo| 47 -15 -10 }} (quintosec comma), but the approximations to [[harmonic]]s [[7/1|7]] and [[13/1|13]] are much improved. It tempers out 2401/2400 ([[breedsma]]), 4802000/4782969 ([[cathartic comma]]), and 48828125/48771072 (neptunisma) in the [[7-limit]]; [[5632/5625]], [[9801/9800]], [[14641/14580]], and 117649/117612 in the [[11-limit]]; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], and [[4225/4224]] in the 13-limit.
-edo provides the [[optimal patent val]] for the 11- and 13-limit [[semiluna]], [[hemiluna]], and [[floral]] temperaments, the rank-3 [[semicanou]] temperament, and the rank-4 temperament tempering out 14641/14580.
+edo provides the [[optimal patent val]] for the 11- and 13-limit [[semiluna]], [[hemiluna]], and [[floral]] temperaments, the rank-3 [[semicathart]] temperament, and the rank-4 temperament tempering out 14641/14580.
 edo works much better as a no-11 no-13 [[subgroup]] temperament, with a sharp tendency to harmonics up to 29. For example, it tempers out [[1216/1215]], [[1225/1224]], [[1445/1444]], and [[2500/2499]] in the 2.3.5.7.17.19 subgroup.
@@ Line 128: / Line 128: @@
 * [https://www.youtube.com/watch?v=-bm5UdmveZU ''All Time Best''] – decoid[40], cover of [[Phlub]]
+[[Category:Hemiluna]]
+[[Category:Semiluna]]
 [[Category:Cathartic]]
-[[Category:Hemiluna]]
+[[Category:Semicathart]]
+[[Category:Semicathartismic]]
 [[Category:Listen]]
-[[Category:Semicanou]]
-[[Category:Semicathartismic]]
-[[Category:Semiluna]]