313edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-06-23 04:29:29 UTC</tt>.<br>
 
: The original revision id was <tt>238344103</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
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.
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 
<h4>Original Wikitext content:</h4>
=== Prime harmonics ===
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=Scales=  
{{Harmonics in equal|313}}
[[madagascar19]]</pre></div>
 
<h4>Original HTML content:</h4>
=== Subsets and supersets ===
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;313edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Scales&lt;/h1&gt;
313edo is the 65th [[prime edo]].
&lt;a class="wiki_link" href="/madagascar19"&gt;madagascar19&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
 
== 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
| {{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
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />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 (temperament)|Helmholtz]]
|-
| 1
| 156\313
| 598.08
| 847/600
| [[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 ==
* [[Hemischis14]]
* [[Madagascar19]]
* Madagascar[9] (or Barbados[9]):
 
{| class="wikitable right-2"
|+ style="font-size: 105%;" | 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¢)
|}
<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 ==
{{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:Albus]]
[[Category:Hemischis]]
[[Category:Hera]]
[[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"