313edo

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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

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

• Hemischis14
• Madagascar19
• Madagascar[9] (or Barbados[9]):

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 ¢)

Music

See also: Category:313edo tracks

Francium

• "Calling Voices" from HemischisMatic EP (2023) – Spotify | Bandcamp | YouTube – hemischis in 313edo tuning