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'''410edo''' is the [[EDO|equal division of the octave]] into 410 parts of 2.9268 cents each.
{{Infobox ET}}
{{ED intro}}


It is closely related to [[205edo]], but the patent vals differ on the mapping for 7 and 13. It is contorted in the [[5-limit]], tempering out 1600000/1594323 ([[amity comma]]) and 274877906944/274658203125 (luna/hemithirds comma), as well as 95367431640625/95105071448064 (gammic comma) and 140737488355328/140126044921875 (qintosec comma). Using the [[patent val]], it tempers out 2401/2400 ([[breedsma]]), 4802000/4782969 ([[canousma]]), and 48828125/48771072 in the [[7-limit]]; 5632/5625 (vishdel), 9801/9800 (kalisma), 14641/14580 ([[semicanousma]]), and 117649/117612 in the [[11-limit]]; 676/675 (island comma), 1001/1000 (sinbadma), 1716/1715 (lummic comma), 4096/4095 ([[schismina]]), and 73205/73008 in the 13-limit.
== 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.


It provides the [[optimal patent val]] for [[semicanousmic temperaments|semicanousmic]], the rank-4 temperament that tempers out 14641/14580, and for 11-limit [[Canou family #Semicanou|semicanou]], the rank-3 temperament that tempers out 9801/9800 and 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.
 
=== Prime harmonics ===
{{Harmonics in equal|410}}
 
=== Subsets and supersets ===
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 (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2401/2400, 1600000/1594323, 48828125/48771072
| {{Mapping| 410 650 952 1151 }}
| −0.0753
| 0.1332
| 4.55
|-
| 2.3.5.7.17
| 1225/1224, 2401/2400, 24576/24565, 295936/295245
| {{Mapping| 410 650 952 1151 1676 }}
| −0.0803
| 0.1196
| 4.09
|-
| 2.3.5.7.17.19
| 1216/1215, 1225/1224, 1445/1444, 2401/2400, 24576/24565
| {{Mapping| 410 650 952 1151 1676 1742 }}
| −0.1071
| 0.1245
| 4.25
|}
 
=== Rank-2 temperaments ===
Note: 5-limit temperaments supported by 205et are not shown.
 
{| 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
| 29\410
| 84.88
| 21/20
| [[Amicable]] / amical
|-
| 1
| 33\410
| 96.59
| 143/135
| [[Hemiluna]]
|-
| 1
| 118\410
| 348.29
| 57344/46875
| [[Subneutral]]
|-
| 1
| 199\410
| 582.44
| 7/5
| [[Neptune]]
|-
| 2
| 29\410
| 84.88
| 21/20
| [[Floral]]
|-
| 2
| 66\410
| 193.17
| 121/108
| [[Semiluna]]
|-
| 2
| 6\410
| 17.56
| 99/98
| [[Poseidon]]
|-
| 10
| 85\410<br>(3\410)
| 248.78<br>(8.78)
| 15/13<br>(176/175)
| [[Decoid]]
|-
| 41
| 61\410<br>(1\410)
| 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 ==
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]].
 
* Kartvelian Tetratonic: 120 120 85 85 (simplifies to [[82edo]])
* Kartvelian Decatonic: 48 48 48 48 48 34 34 34 34 34 (simplifies to [[205edo]])
* Kartvelian 22-tonic: 20 20 20 20 20 20 20 20 20 20 20 20 17 17 17 17 17 17 17 17 17
 
== Music ==
; [[Mercury Amalgam]] (2023)
* [https://www.youtube.com/watch?v=-bm5UdmveZU ''All Time Best''] – decoid[40], cover of [[Phlub]]


[[Category:Equal divisions of the octave]]
[[Category:Canou]]
[[Category:Canou]]
[[Category:Hemiluna]]
[[Category:Listen]]
[[Category:Semicanou]]
[[Category:Semicanousmic]]
[[Category:Semicanousmic]]
[[Category:Semiluna]]
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