313edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable" | {| 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 | | 2.3 | ||
|{{ | | {{monzo| -496 313 }} | ||
|[ | | [{{val| 313 496 }}] | ||
|0.113 | | +0.113 | ||
|0.113 | | 0.113 | ||
|2.94 | | 2.94 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|32805/32768, {{ | | 32805/32768, {{monzo| -1 49 -33 }} | ||
|[ | | [{{val| 313 496 727 }}] | ||
| -0.055 | | -0.055 | ||
|0.254 | | 0.254 | ||
|6. | | 6.64 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|6144/6125, 19683/19600, | | 6144/6125, 19683/19600, 40500000/40353607 | ||
|[ | | [{{val| 313 496 727 879 }}] | ||
| | | -0.143 | ||
|0. | | 0.268 | ||
|6.99 | | 6.99 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|540/539, 5632/5625, 8019/8000, 43923/43904 | | 540/539, 5632/5625, 8019/8000, 43923/43904 | ||
|[ | | [{{val| 313 496 727 879 1083 }}] | ||
| -0.158 | | -0.158 | ||
|0.242 | | 0.242 | ||
|6.30 | | 6.30 | ||
|- | |||
| 2.3.5.7.11.13 | |||
| 351/350, 540/539, 676/675, 4096/4095, 43923/43904 | |||
| [{{val| 313 496 727 879 1083 1158 }}] | |||
| -0.091 | |||
| 0.267 | |||
| 6.97 | |||
|} | |} | ||
Revision as of 13:17, 31 March 2022
The 313 equal divisions of the octave (313edo) is the equal division of the octave into 313 parts of 3.83387 cents each.
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.
313edo is the 65th prime EDO.
Prime harmonics
Script error: No such module "primes_in_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 |
Scales
- 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 ¢) |