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 ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! 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]] (¢)
! [[TE error|Absolute]] (¢)
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| 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 }}
| -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 }}
| -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 }}
| -0.1071
| −0.1071
| 0.1245
| 0.1245
| 4.25
| 4.25
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{| class="wikitable center-all left-5"
{| 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
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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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]]
|}
|}
<nowiki>*</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 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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* [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:Semiluna]]
[[Category:Canou]]
[[Category:Hemiluna]]
[[Category:Hemiluna]]
[[Category:Canou]]
[[Category:Listen]]
[[Category:Semicanou]]
[[Category:Semicanou]]
[[Category:Semicanousmic]]
[[Category:Semicanousmic]]
[[Category:Listen]]
[[Category:Semiluna]]

Latest revision as of 13:38, 13 March 2026

← 409edo 410edo 411edo →
Prime factorization 2 × 5 × 41
Step size 2.92683 ¢ 
Fifth 240\410 (702.439 ¢) (→ 24\41)
Semitones (A1:m2) 40:30 (117.1 ¢ : 87.8 ¢)
Consistency limit 9
Distinct consistency limit 9

410 equal divisions of the octave (abbreviated 410edo or 410ed2), also called 410-tone equal temperament (410tet) or 410 equal temperament (410et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 410 equal parts of about 2.93 ¢ each. Each step represents a frequency ratio of 21/410, or the 410th root of 2.

Theory

410edo is enfactored in the 5-limit, with the same tuning as 205edo characterized by tempering out 1600000/1594323 (amity comma) and [38 -2 -15⟩ (luna/hemithirds comma), as well as [-29 -11 20⟩ (gammic comma) and [47 -15 -10⟩ (quintosec comma), but the approximations to harmonics 7 and 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 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

Approximation of prime harmonics in 410edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.48 +0.03 -0.05 -1.07 -0.53 +0.41 +1.02 +0.99 +0.67 -0.65
Relative (%) +0.0 +16.5 +0.9 -1.6 -36.7 -18.0 +14.0 +35.0 +34.0 +22.8 -22.0
Steps
(reduced) 		410
(0) 		650
(240) 		952
(132) 		1151
(331) 		1418
(188) 		1517
(287) 		1676
(36) 		1742
(102) 		1855
(215) 		1992
(352) 		2031
(391)

Subsets and supersets

Since 410 factors into 2 × 5 × 41, 410edo has subset edos 2, 5, 10, 41, 82, and 205. Meanwhile, as every sixth step of 2460edo, a step of 410edo is exactly 6 minas.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5.7 2401/2400, 1600000/1594323, 48828125/48771072 [⟨410 650 952 1151]] −0.0753 0.1332 4.55
2.3.5.7.17 1225/1224, 2401/2400, 24576/24565, 295936/295245 [⟨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 [⟨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.

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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
(3\410) 		248.78
(8.78) 		15/13
(176/175) 		Decoid
41 61\410
(1\410) 		178.54
(2.93) 		567/512
(352/351) 		Hemicountercomp

* Octave-reduced form, reduced to the first half-octave, and 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)
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