313edo: Difference between revisions

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-The ''313 equal division'' divides the octave into 313 equal parts of 3.834 cents each. It provides the [[Optimal_patent_val|optimal patent val]] for 11- and 13-limit [[Porwell_temperaments#Hemischis|hemischis temperament]] and the rank three [[Cataharry_family|madagascar temperament]]. It tempers out the schisma, 32805/32768, in the 5-limit; 6144/6125 and 19683/19600 in the 7-limit; 540/539, 5632/5625 and 8019/8000 in the 11-limit; and 351/350 and 676/675 in the 13-limit.
+{{Infobox ET}}
+{{ED intro}}
-=Scales=
+== Theory ==
-[[madagascar19|madagascar19]]      [[Category:313edo]]
+edo 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.
-[[Category:edo]]
-[[Category:todo:add_definition]]
+=== Prime harmonics ===
+{{Harmonics in equal|313}}
+=== Subsets and supersets ===
+edo is the 65th [[prime edo]].
+== 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]]