313edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
Line 32: Line 24:
| 32805/32768, {{monzo| -1 49 -33 }}
| 32805/32768, {{monzo| -1 49 -33 }}
| {{mapping| 313 496 727 }}
| {{mapping| 313 496 727 }}
| -0.055
| &minus;0.055
| 0.254
| 0.254
| 6.64
| 6.64
Line 39: Line 31:
| 6144/6125, 19683/19600, 40500000/40353607
| 6144/6125, 19683/19600, 40500000/40353607
| {{mapping| 313 496 727 879 }}
| {{mapping| 313 496 727 879 }}
| -0.143
| &minus;0.143
| 0.268
| 0.268
| 6.99
| 6.99
Line 46: Line 38:
| 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
| &minus;0.158
| 0.242
| 0.242
| 6.30
| 6.30
Line 53: Line 45:
| 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
| &minus;0.091
| 0.267
| 0.267
| 6.97
| 6.97
Line 59: Line 51:


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio
! Temperaments
|-
|-
| 1
| 1
Line 89: Line 76:
| 847/600
| 847/600
| [[Vydubychi]]
| [[Vydubychi]]
|}
{{rank-2 end}}
{{orf}}


== Scales ==
== Scales ==
Line 97: Line 85:


{| class="wikitable right-2"
{| class="wikitable right-2"
|+Madagascar[9] (or Barbados[9]) scale
|+ style="font-size: 105%;" | Madagascar[9] (or Barbados[9]) scale
|-
! Step
! Step
! Cents
! Cents
Line 104: Line 93:
| 53 (53\313)
| 53 (53\313)
| 203.195
| 203.195
| [[9/8]] (-0.715 ¢)
| [[9/8]] (&minus;0.715 ¢)
|-
|-
| 12 (65\313)
| 12 (65\313)
Line 112: Line 101:
| 53 (118\313)
| 53 (118\313)
| 452.396
| 452.396
| [[13/10]] (-1,818 ¢)
| [[13/10]] (&minus;1,818 ¢)
|-
|-
| 12 (130\313)
| 12 (130\313)
Line 120: Line 109:
| 53 (183\313)
| 53 (183\313)
| 701.597
| 701.597
| [[3/2]] (-0.358 ¢)
| [[3/2]] (&minus;0.358 ¢)
|-
|-
| 12 (195\313)
| 12 (195\313)
Line 128: Line 117:
| 53 (248\313)
| 53 (248\313)
| 950.799
| 950.799
| [[26/15]] (-1.46 ¢)
| [[26/15]] (&minus;1.46 ¢)
|-
|-
| 12 (260\313)
| 12 (260\313)
Line 136: Line 125:
| 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
{{orf}}


== 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) &ndash; [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] &ndash; hemischis in 313edo tuning


[[Category:Albus]]
[[Category:Hemischis]]
[[Category:Hemischis]]
[[Category:Hera]]
[[Category:Hera]]
[[Category:Albus]]
[[Category:Listen]]
[[Category:Listen]]

Revision as of 04:34, 16 November 2024

← 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

Template:EDO intro

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

Template:Comma basis begin |- | 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

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

Scales

Madagascar[9] (or 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 ¢)

Template:Orf

Music

See also: Category:313edo tracks
Francium
  • "Calling Voices" from HemischisMatic EP (2023) – Spotify | Bandcamp | YouTube – hemischis in 313edo tuning
Retrieved from "https://en.xen.wiki/index.php?title=313edo&oldid=165904"