410edo: Difference between revisions

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 The '''410 equal divisions of the octave''' ('''410edo''') is the [[EDO|equal division of the octave]] into 410 parts of 2.9268 [[cent]]s each.
-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]], [[4225/4224]] and 73205/73008 in the 13-limit.
+== 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.
-It 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]] 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.
 === Prime harmonics ===
-{{Primes in edo|410}}
+{{Harmonics in equal|410|columns=11}}
+== 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
+| [{{val| 410 650 952 1151 }}]
+| -0.0753
+| 0.1332
+| 4.55
+|-
+| 2.3.5.7.17
+| 1225/1224, 2401/2400, 24576/24565, 295936/295245
+| [{{val| 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
+| [{{val| 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"
+|+Table of rank-2 temperaments by generator
+! Periods<br>per octave
+! Generator<br>(reduced)
+! Cents<br>(reduced)
+! 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]]
+|}
 [[Category:Equal divisions of the octave]]
+[[Category:Semiluna]]
+[[Category:Hemiluna]]
 [[Category:Canou]]
 [[Category:Semicanou]]
 [[Category:Semicanousmic]]