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410edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|410}}
{{ED intro}}


== Theory ==
== 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.  
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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=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|410|columns=11}}
{{Harmonics in equal|410}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 410 factors into {{factorization|410}}, 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 ==
== Regular temperament properties ==
{{comma basis begin}}
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 1600000/1594323, 48828125/48771072
| 2401/2400, 1600000/1594323, 48828125/48771072
| {{mapping| 410 650 952 1151 }}
| {{Mapping| 410 650 952 1151 }}
| &minus;0.0753
| −0.0753
| 0.1332
| 0.1332
| 4.55
| 4.55
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| 2.3.5.7.17
| 2.3.5.7.17
| 1225/1224, 2401/2400, 24576/24565, 295936/295245
| 1225/1224, 2401/2400, 24576/24565, 295936/295245
| {{mapping| 410 650 952 1151 1676 }}
| {{Mapping| 410 650 952 1151 1676 }}
| &minus;0.0803
| −0.0803
| 0.1196
| 0.1196
| 4.09
| 4.09
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| 2.3.5.7.17.19
| 2.3.5.7.17.19
| 1216/1215, 1225/1224, 1445/1444, 2401/2400, 24576/24565
| 1216/1215, 1225/1224, 1445/1444, 2401/2400, 24576/24565
| {{mapping| 410 650 952 1151 1676 1742 }}
| {{Mapping| 410 650 952 1151 1676 1742 }}
| &minus;0.1071
| −0.1071
| 0.1245
| 0.1245
| 4.25
| 4.25
{{comma basis end}}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Note: 5-limit temperaments supported by 205et are not shown.  
Note: 5-limit temperaments supported by 205et are not shown.  


{{rank-2 begin}}
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
|-
|-
| 1
| 1
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| 41
| 41
| 61\410<br>(1\410)
| 61\410<br>(1\410)
| 178.54<br>(2.93)
| 178.54<br/>(2.93)
| 567/512<br>(352/351)
| 567/512<br>(352/351)
| [[Hemicountercomp]]
| [[Hemicountercomp]]
{{rank-2 end}}
|}
{{orf}}
<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 ==
== Scales ==
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== Music ==
== Music ==
; [[Mercury Amalgam]] (2023)
; [[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]]
[[Category:Canou]]
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