410edo: Difference between revisions

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== Theory ==

410edo is closely related to [[205edo]], but the [[patent val]] differs on the mappings for 7 and 13. It is [[contorted]] in the [[5-limit]], 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 }} (qintosec comma). 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 closely related to [[205edo]], but the [[patent val]] differs on the mappings for 7 and 13. It is [[contorted]] in the [[5-limit]], 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 }} (qintosec comma). 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 provides the [[optimal patent val]] for the 11- and 13-limit [[semiluna]], [[hemiluna]], and [[floral]] temperament, the rank-3 [[semicanou]] 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.

~~=== Possible usage in Georgian music ===~~

410edo's fifth is on its 240th step, which is a highly composite number. As such, it supports EDFs which are divisors of 240. In addition, it's perfect fourth is on the 170th step, which while is not highly composite, is still notable to carry a few ED4/3 scales. This can be used to play [[Kartvelian scales]].

=== Prime harmonics ===

=== Miscellaneous properties ===

Since 410 = 2 × 5 × 41, 410edo has subset edos {{EDOs| 2, 5, 10, 41, 82, and 205 }}.

410edo's fifth is on its 240th step, which is a highly composite number. As such, it supports edfs which are divisors of 240. In addition, its perfect fourth is on the 170th step, which while is not highly composite, is still notable to carry a few ed4/3 scales. This can be used to play [[Kartvelian scales]].

== Regular temperament properties ==

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 == Theory ==
-edo is closely related to [[205edo]], but the [[patent val]] differs on the mappings for 7 and 13. It is [[contorted]] in the [[5-limit]], 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 }} (qintosec comma). 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 closely related to [[205edo]], but the [[patent val]] differs on the mappings for 7 and 13. It is [[contorted]] in the [[5-limit]], 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 }} (qintosec comma). 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 provides the [[optimal patent val]] for the 11- and 13-limit [[semiluna]], [[hemiluna]], and [[floral]] temperament, the rank-3 [[semicanou]] 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.
-=== Possible usage in Georgian music ===
-edo's fifth is on its 240th step, which is a highly composite number. As such, it supports EDFs which are divisors of 240. In addition, it's perfect fourth is on the 170th step, which while is not highly composite, is still notable to carry a few ED4/3 scales. This can be used to play [[Kartvelian scales]].
 === Prime harmonics ===
 {{Harmonics in equal|410|columns=11}}
+=== Miscellaneous properties ===
+Since 410 = 2 × 5 × 41, 410edo has subset edos {{EDOs| 2, 5, 10, 41, 82, and 205 }}.
+edo's fifth is on its 240th step, which is a highly composite number. As such, it supports edfs which are divisors of 240. In addition, its perfect fourth is on the 170th step, which while is not highly composite, is still notable to carry a few ed4/3 scales. This can be used to play [[Kartvelian scales]].
 == Regular temperament properties ==