313edo: Difference between revisions

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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 313 (prime)
{{ED intro}}
| Step size = 3.83387¢
| Fifth = 183\313 (701.60¢)
| Semitones = 29:24 (111.18¢ : 92.01¢)
| Consistency = 11
}}
{{EDO intro|313}}


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


== 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|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
|-
|-
Line 29: Line 25:
| 2.3
| 2.3
| {{monzo| -496 313 }}
| {{monzo| -496 313 }}
| [{{val| 313 496 }}]
| {{mapping| 313 496 }}
| +0.113
| +0.113
| 0.113
| 0.113
Line 36: Line 32:
| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| -1 49 -33 }}
| 32805/32768, {{monzo| -1 49 -33 }}
| [{{val| 313 496 727 }}]
| {{mapping| 313 496 727 }}
| -0.055
| −0.055
| 0.254
| 0.254
| 6.64
| 6.64
Line 43: Line 39:
| 2.3.5.7
| 2.3.5.7
| 6144/6125, 19683/19600, 40500000/40353607
| 6144/6125, 19683/19600, 40500000/40353607
| [{{val| 313 496 727 879 }}]
| {{mapping| 313 496 727 879 }}
| -0.143
| −0.143
| 0.268
| 0.268
| 6.99
| 6.99
Line 50: Line 46:
| 2.3.5.7.11
| 2.3.5.7.11
| 540/539, 5632/5625, 8019/8000, 43923/43904
| 540/539, 5632/5625, 8019/8000, 43923/43904
| [{{val| 313 496 727 879 1083 }}]
| {{mapping| 313 496 727 879 1083 }}
| -0.158
| −0.158
| 0.242
| 0.242
| 6.30
| 6.30
Line 57: Line 53:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 351/350, 540/539, 676/675, 4096/4095, 43923/43904
| 351/350, 540/539, 676/675, 4096/4095, 43923/43904
| [{{val| 313 496 727 879 1083 1158 }}]
| {{mapping| 313 496 727 879 1083 1158 }}
| -0.091
| −0.091
| 0.267
| 0.267
| 6.97
| 6.97
Line 65: Line 61:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods<br>per Octave
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Generator<br>(Reduced)
|-
! Cents<br>(Reduced)
! Periods<br />per 8ve
! Associated<br>Ratio
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
| 1
| 26\313
| 99.68
| 18/17
| [[Quintaschis]]
|-
|-
| 1
| 1
| 65\313
| 65\313
| 249.201
| 249.20
| 15/13
| 15/13
| [[Hemischis]]
| [[Hemischis]]
|-
| 1
| 130\313
| 498.40
| 4/3
| [[Helmholtz (temperament)|Helmholtz]]
|-
|-
| 1
| 1
| 156\313
| 156\313
| 598.083
| 598.08
| 847/600
| 847/600
| [[Vydubychi]]
| [[Vydubychi]]
|}
|}
<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 96: 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:313edo| ]] <!-- main article -->
[[Category:Albus]]
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
[[Category:Hemischis]]
[[Category:Hemischis]]
[[Category:Hera]]
[[Category:Hera]]
[[Category:Albus]]
[[Category:Listen]]

Latest revision as of 13:02, 29 April 2026

← 312edo 313edo 314edo →
Prime factorization 313 (prime)
Step size 3.83387 ¢ 
Fifth 183\313 (701.597 ¢)
Semitones (A1:m2) 29:24 (111.2 ¢ : 92.01 ¢)
Consistency limit 11
Distinct consistency limit 11

313 equal divisions of the octave (abbreviated 313edo or 313ed2), also called 313-tone equal temperament (313tet) or 313 equal temperament (313et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 313 equal parts of about 3.83 ¢ each. Each step represents a frequency ratio of 21/313, or the 313th root of 2.

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.

Prime harmonics

Approximation of prime harmonics in 313edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.36 +0.91 +1.14 +0.76 -0.91 -1.44 +1.53 +0.48 +1.73 +1.29
Relative (%) +0.0 -9.3 +23.7 +29.8 +19.8 -23.8 -37.6 +39.9 +12.5 +45.2 +33.7
Steps
(reduced) 		313
(0) 		496
(183) 		727
(101) 		879
(253) 		1083
(144) 		1158
(219) 		1279
(27) 		1330
(78) 		1416
(164) 		1521
(269) 		1551
(299)

Subsets and supersets

313edo is the 65th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-496 313⟩ [⟨313 496]] +0.113 0.113 2.94
2.3.5 32805/32768, [-1 49 -33⟩ [⟨313 496 727]] −0.055 0.254 6.64
2.3.5.7 6144/6125, 19683/19600, 40500000/40353607 [⟨313 496 727 879]] −0.143 0.268 6.99
2.3.5.7.11 540/539, 5632/5625, 8019/8000, 43923/43904 [⟨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 [⟨313 496 727 879 1083 1158]] −0.091 0.267 6.97

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 26\313 99.68 18/17 Quintaschis
1 65\313 249.20 15/13 Hemischis
1 130\313 498.40 4/3 Helmholtz
1 156\313 598.08 847/600 Vydubychi

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

Madagascar[9] (Barbados[9]) scale
Step Cents JI Interpretation
53 (53\313) 203.195 9/8 (−0.715¢)
12 (65\313) 249.201 15/13 (+1.46¢)
53 (118\313) 452.396 13/10 (−1,818¢)
12 (130\313) 498.403 4/3 (+0.358¢)
53 (183\313) 701.597 3/2 (−0.358¢)
12 (195\313) 747.604 20/13 (+1.818¢)
53 (248\313) 950.799 26/15 (−1.46¢)
12 (260\313) 996.805 16/9 (+0.715¢)
53 (313\313) 1200.000 2/1 (0¢)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

See also: Category:313edo tracks
Francium
  • "Calling Voices" from HemischisMatic EP (2023) – Spotify | Bandcamp | YouTube – in Hemischis, 313edo tuning
  • "Fried Rolled Up Trousers With Fish Sauce" from Unsuspecting Tyrant Double-Decker Beef Fort (2026) – Spotify | Bandcamp | YouTube
Sevish
Retrieved from "https://en.xen.wiki/index.php?title=313edo&oldid=228963"