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 [[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]] ~~temperament~~, 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 [[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.

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.

=== Prime harmonics ===

=== ~~Divisors~~ ===

=== Subsets and supersets ===

Since 410 = 2 × 5 × 41, 410edo has subset edos {{EDOs| 2, 5, 10, 41, 82, and 205 }}. Meanwhile, as every sixth step of [[2460edo]], a step of 410edo is exactly 6 [[mina]]s.

Since 410 factors into 2 × 5 × 41, 410edo has subset edos {{EDOs| 2, 5, 10, 41, 82, and 205 }}. Meanwhile, as every sixth step of [[2460edo]], a step of 410edo is exactly 6 [[mina]]s.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| 2.3.5.7

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

| [{{~~val~~| 410 650 952 1151 }}]

| {{Mapping| 410 650 952 1151 }}

| -0.0753

| −0.0753

| 0.1332

| 4.55

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| 2.3.5.7.17

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

| [{{~~val~~| 410 650 952 1151 1676 }}]

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

| -0.0803

| −0.0803

| 0.1196

| 4.09

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| 2.3.5.7.17.19

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

| [{{~~val~~| 410 650 952 1151 1676 1742 }}]

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

| -0.1071

| −0.1071

| 0.1245

| 4.25

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{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperaments

|-

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| 41

| 61\410 (1\410)

| 178.54 (2.93)

| 178.54 (2.93)

| 567/512 (352/351)

| [[Hemicountercomp]]

|}

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|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]] by [[Eliora]]

[[Category:Canou]]

~~[[Category:Equal divisions of the octave|###]] ~~

[[Category:~~Semiluna~~]]

[[Category:Hemiluna]]

[[Category:~~Canou~~]]

[[Category:Listen]]

[[Category:Semicanou]]

[[Category:Semicanousmic]]

[[Category:~~Listen~~]]

[[Category:Semiluna]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|410}}
+{{ED intro}}
 == 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 [[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]] temperament, 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 [[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.
+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.
 === Prime harmonics ===
-{{Harmonics in equal|410|columns=11}}
+{{Harmonics in equal|410}}
-=== Divisors ===
+=== Subsets and supersets ===
-Since 410 = 2 × 5 × 41, 410edo has subset edos {{EDOs| 2, 5, 10, 41, 82, and 205 }}. Meanwhile, as every sixth step of [[2460edo]], a step of 410edo is exactly 6 [[mina]]s.
+Since 410 factors into 2 × 5 × 41, 410edo has subset edos {{EDOs| 2, 5, 10, 41, 82, and 205 }}. Meanwhile, as every sixth step of [[2460edo]], a step of 410edo is exactly 6 [[mina]]s.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 28: / Line 29: @@
 | 2.3.5.7
 | 2401/2400, 1600000/1594323, 48828125/48771072
-| [{{val| 410 650 952 1151 }}]
+| {{Mapping| 410 650 952 1151 }}
-| -0.0753
+| −0.0753
 | 0.1332
 | 4.55
@@ Line 35: / Line 36: @@
 | 2.3.5.7.17
 | 1225/1224, 2401/2400, 24576/24565, 295936/295245
-| [{{val| 410 650 952 1151 1676 }}]
+| {{Mapping| 410 650 952 1151 1676 }}
-| -0.0803
+| −0.0803
 | 0.1196
 | 4.09
@@ Line 42: / Line 43: @@
 | 2.3.5.7.17.19
 | 1216/1215, 1225/1224, 1445/1444, 2401/2400, 24576/24565
-| [{{val| 410 650 952 1151 1676 1742 }}]
+| {{Mapping| 410 650 952 1151 1676 1742 }}
-| -0.1071
+| −0.1071
 | 0.1245
 | 4.25
@@ Line 52: / Line 53: @@
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 109: / Line 111: @@
 | 41
 | 61\410<br>(1\410)
-| 178.54<br>(2.93)
+| 178.54<br/>(2.93)
 | 567/512<br>(352/351)
 | [[Hemicountercomp]]
 |}
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Scales ==
@@ Line 122: / Line 125: @@
 == 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]] by [[Eliora]]
+[[Category:Canou]]
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
-[[Category:Semiluna]]
 [[Category:Hemiluna]]
-[[Category:Canou]]
+[[Category:Listen]]
 [[Category:Semicanou]]
 [[Category:Semicanousmic]]
-[[Category:Listen]]
+[[Category:Semiluna]]