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The '''313 equal divisions of the octave''' ('''313edo''') is the [[EDO|equal division of the octave]] into 313 parts of 3.83387 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
313edo provides the [[optimal patent val]] for 11- and 13-limit [[hemischis]] temperament and the 13-limit rank-3 temperaments [[madagascar]] and [[hera]]. It tempers out the [[schisma]], 32805/32768, in the 5-limit; [[6144/6125]] and [[19683/19600]] in the 7-limit; [[540/539]], [[5632/5625]], [[8019/8000]] and [[16384/16335]] in the 11-limit; [[351/350]], [[676/675]], [[729/728]], [[1001/1000]], [[2080/2079]] and [[4096/4095]] in the 13-limit.
313edo provides the [[optimal patent val]] for 11- and 13-limit [[hemischis]] temperament and the 13-limit rank-3 temperaments [[madagascar]] and [[hera]]. It tempers out the [[schisma]], 32805/32768, in the 5-limit; [[6144/6125]] and [[19683/19600]] in the 7-limit; [[540/539]], [[5632/5625]], [[8019/8000]] and [[16384/16335]] in the 11-limit; [[351/350]], [[676/675]], [[729/728]], [[1001/1000]], [[2080/2079]] and [[4096/4095]] in the 13-limit.


313edo is the 65th [[prime EDO]].
=== Prime harmonics ===
{{Harmonics in equal|313}}


=== Prime harmonics ===
=== Subsets and supersets ===
{{Primes in edo|313}}
313edo is the 65th [[prime edo]].


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable"
{| class="wikitable center-4 center-5 center-6"
| rowspan="2" |'''Subgroup'''
|-
| rowspan="2" |[[Comma list|'''Comma list''']]
! rowspan="2" | [[Subgroup]]
| rowspan="2" |'''[[Mapping]]'''
! rowspan="2" | [[Comma list]]
| rowspan="2" |'''Optimal''' 
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -496 313 }}
| {{mapping| 313 496 }}
| +0.113
| 0.113
| 2.94
|-
| 2.3.5
| 32805/32768, {{monzo| -1 49 -33 }}
| {{mapping| 313 496 727 }}
| −0.055
| 0.254
| 6.64
|-
| 2.3.5.7
| 6144/6125, 19683/19600, 40500000/40353607
| {{mapping| 313 496 727 879 }}
| −0.143
| 0.268
| 6.99
|-
| 2.3.5.7.11
| 540/539, 5632/5625, 8019/8000, 43923/43904
| {{mapping| 313 496 727 879 1083 }}
| −0.158
| 0.242
| 6.30
|-
| 2.3.5.7.11.13
| 351/350, 540/539, 676/675, 4096/4095, 43923/43904
| {{mapping| 313 496 727 879 1083 1158 }}
| −0.091
| 0.267
| 6.97
|}


'''8ve  stretch (¢)'''
=== Rank-2 temperaments ===
| colspan="2" |'''Tuning  error'''
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
|[[TE error|'''Absolute''']] '''(¢)'''
! Periods<br />per 8ve
|[[TE simple badness|'''Relative''']] '''(%)'''
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
|-
|2.3
| 1
|{{Monzo|-496, 313}}
| 26\313
|[⟨313 496]]
| 99.68
|0.113
| 18/17
|0.113
| [[Quintaschis]]
|2.94
|-
|-
|2.3.5
| 1
|32805/32768, {{Monzo|-1, 49, -33}}
| 65\313
|[⟨313 496 727]]
| 249.20
| -0.055
| 15/13
|0.254
| [[Hemischis]]
|6.80
|-
|-
|2.3.5.7
| 1
|6144/6125, 19683/19600, [7, 10, -5, -4⟩, [5, 4, 6, -9⟩
| 130\313
|[⟨313 496 727 879]]
| 498.40
|<nowiki>-0.142</nowiki>
| 4/3
|0.267
| [[Helmholtz (temperament)|Helmholtz]]
|6.99
|-
|-
|2.3.5.7.11
| 1
|540/539, 5632/5625, 8019/8000, 43923/43904
| 156\313
|[⟨313 496 727 879 1083]]
| 598.08
| -0.158
| 847/600
|0.242
| [[Vydubychi]]
|6.30
|}
|}
<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 ==
* [[Hemischis14]]
* [[Madagascar19]]
* [[Madagascar19]]
* Madagascar[9] (or Barbados[9]):
* Madagascar[9] (or Barbados[9]):


{| class="wikitable right-2"
{| class="wikitable right-2"
|+Madagascar[9] (or Barbados[9]) scale
|+ style="font-size: 105%;" | Madagascar[9] (Barbados[9]) scale
|-
! Step
! Step
! Cents
! Cents
Line 63: Line 109:
| 53 (53\313)
| 53 (53\313)
| 203.195
| 203.195
| [[9/8]] (-0.715 ¢)
| [[9/8]] (−0.715¢)
|-
|-
| 12 (65\313)
| 12 (65\313)
| 249.201
| 249.201
| [[15/13]] (+1.46 ¢)
| [[15/13]] (+1.46¢)
|-
|-
| 53 (118\313)
| 53 (118\313)
| 452.396
| 452.396
| [[13/10]] (-1,818 ¢)
| [[13/10]] (−1,818¢)
|-
|-
| 12 (130\313)
| 12 (130\313)
| 498.403
| 498.403
| [[4/3]] (+0.358 ¢)
| [[4/3]] (+0.358¢)
|-
|-
| 53 (183\313)
| 53 (183\313)
| 701.597
| 701.597
| [[3/2]] (-0.358 ¢)
| [[3/2]] (−0.358¢)
|-
|-
| 12 (195\313)
| 12 (195\313)
| 747.604
| 747.604
| [[20/13]] (+1.818 ¢)
| [[20/13]] (+1.818¢)
|-
|-
| 53 (248\313)
| 53 (248\313)
| 950.799
| 950.799
| [[26/15]] (-1.46 ¢)
| [[26/15]] (−1.46¢)
|-
|-
| 12 (260\313)
| 12 (260\313)
| 996.805
| 996.805
| [[16/9]] (+0.715 ¢)
| [[16/9]] (+0.715¢)
|-
|-
| 53 (313\313)
| 53 (313\313)
| 1200.000
| 1200.000
| [[2/1]] (±0 ¢)
| [[2/1]] (0¢)
|}
|}
<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


== Music ==
== Music ==
{{See also|:Category:313edo tracks}}
{{Catrel| 313edo tracks }}
 
; [[User:Francium|Francium]]
* "Calling Voices" from ''HemischisMatic EP'' (2023) – [https://open.spotify.com/track/6NdgC4jYz5TIvxV09kdpuk Spotify] | [https://francium223.bandcamp.com/track/calling-voices Bandcamp] | [https://youtu.be/ls51r_AICak?si=iaiiHo-Q-Q-74LN9 YouTube] – in Hemischis, 313edo tuning
* "Fried Rolled Up Trousers With Fish Sauce" from ''Unsuspecting Tyrant Double-Decker Beef Fort'' (2026) – [https://open.spotify.com/track/4Gnz6Dagdkg5C35osK1HXa Spotify] | [https://francium223.bandcamp.com/track/fried-rolled-up-trousers-with-fish-sauce Bandcamp] | [https://www.youtube.com/watch?v=ZSq62yTs3p0 YouTube]
 
; [[Sevish]]
* ''[[Desert Island Rain]]''
* ''[[Disorient]]''
* ''[[Never Coming Home]]''
* ''[[Never Coming Home (Remix)]]''
* ''[[Septillion Reptilians]]''


[[Category:Equal divisions of the octave]]
[[Category:Albus]]
[[Category:Prime EDO]]
[[Category:Hemischis]]
[[Category:Hera]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/313edo"