313edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|313}}
{{ED intro}}


== Theory ==
== Theory ==
Line 13: Line 13:
== 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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 32805/32768, {{monzo| -1 49 -33 }}
| 32805/32768, {{monzo| -1 49 -33 }}
| {{mapping| 313 496 727 }}
| {{mapping| 313 496 727 }}
| -0.055
| −0.055
| 0.254
| 0.254
| 6.64
| 6.64
Line 39: Line 40:
| 6144/6125, 19683/19600, 40500000/40353607
| 6144/6125, 19683/19600, 40500000/40353607
| {{mapping| 313 496 727 879 }}
| {{mapping| 313 496 727 879 }}
| -0.143
| −0.143
| 0.268
| 0.268
| 6.99
| 6.99
Line 46: Line 47:
| 540/539, 5632/5625, 8019/8000, 43923/43904
| 540/539, 5632/5625, 8019/8000, 43923/43904
| {{mapping| 313 496 727 879 1083 }}
| {{mapping| 313 496 727 879 1083 }}
| -0.158
| −0.158
| 0.242
| 0.242
| 6.30
| 6.30
Line 53: Line 54:
| 351/350, 540/539, 676/675, 4096/4095, 43923/43904
| 351/350, 540/539, 676/675, 4096/4095, 43923/43904
| {{mapping| 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 60: Line 61:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods<br>per 8ve
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| 498.40
| 498.40
| 4/3
| 4/3
| [[Helmholtz]]
| [[Helmholtz (temperament)|Helmholtz]]
|-
|-
| 1
| 1
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| [[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 ==
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{| 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
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| 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>*</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


== Music ==
== Music ==
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; [[User:Francium|Francium]]
; [[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] – hemischis in 313edo tuning
* "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:Albus]]
[[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"