410edo: Difference between revisions

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== Theory ==

410edo is [[~~enfactoring|~~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. ~~The equal temperament~~ 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 ([[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]] temperaments, the rank-3 [[semicanou]] temperament, and the rank-4 temperament tempering out 14641/14580.

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| 2401/2400, 1600000/1594323, 48828125/48771072

| {{mapping| 410 650 952 1151 }}

| ~~−0~~.0753

| −0.0753

| 0.1332

| 4.55

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| 1225/1224, 2401/2400, 24576/24565, 295936/295245

| {{mapping| 410 650 952 1151 1676 }}

| ~~−0~~.0803

| −0.0803

| 0.1196

| 4.09

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| 1216/1215, 1225/1224, 1445/1444, 2401/2400, 24576/24565

| {{mapping| 410 650 952 1151 1676 1742 }}

| ~~−0~~.1071

| −0.1071

| 0.1245

| 4.25

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| [[Hemicountercomp]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if ~~it is~~ distinct

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==

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== Music ==

; [[Mercury Amalgam]] (2023)

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

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

[[Category:Canou]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is [[enfactoring|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. The equal temperament 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 ([[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]] temperaments, the rank-3 [[semicanou]] temperament, and the rank-4 temperament tempering out 14641/14580.
@@ Line 30: / Line 30: @@
 | 2401/2400, 1600000/1594323, 48828125/48771072
 | {{mapping| 410 650 952 1151 }}
-| &minus;0.0753
+| −0.0753
 | 0.1332
 | 4.55
@@ Line 37: / Line 37: @@
 | 1225/1224, 2401/2400, 24576/24565, 295936/295245
 | {{mapping| 410 650 952 1151 1676 }}
-| &minus;0.0803
+| −0.0803
 | 0.1196
 | 4.09
@@ Line 44: / Line 44: @@
 | 1216/1215, 1225/1224, 1445/1444, 2401/2400, 24576/24565
 | {{mapping| 410 650 952 1151 1676 1742 }}
-| &minus;0.1071
+| −0.1071
 | 0.1245
 | 4.25
@@ Line 115: / Line 115: @@
 | [[Hemicountercomp]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Scales ==
@@ Line 126: / Line 126: @@
 == Music ==
 ; [[Mercury Amalgam]] (2023)
-* [https://www.youtube.com/watch?v=-bm5UdmveZU ''All Time Best''] &ndash; decoid[40], cover of [[Phlub]]
+* [https://www.youtube.com/watch?v=-bm5UdmveZU ''All Time Best''] – decoid[40], cover of [[Phlub]]
 [[Category:Canou]]