313edo
| ← 312edo | 313edo | 314edo → |
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
| 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
| 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
- Hemischis14
- Madagascar19
- Madagascar[9] (or Barbados[9]):
| 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