263edo
| ← 262edo | 263edo | 264edo → |
263 equal divisions of the octave (abbreviated 263edo or 263ed2), also called 263-tone equal temperament (263tet) or 263 equal temperament (263et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 263 equal parts of about 4.56 ¢ each. Each step represents a frequency ratio of 21/263, or the 263rd root of 2.
Theory
263et tempers out 393216/390625 (würschmidt comma) and [50 -33 1⟩ in the 5-limit. Using the patent val, it tempers out 4375/4374, 50421/50000, and 458752/455625 in the 7-limit; 441/440, 3388/3375, 16384/16335, and 26411/26244 in the 11-limit; 364/363, 2080/2079, 2197/2187, and 3584/3575 in the 13-limit; 595/594, 833/832, 936/935, and 1156/1155 in the 17-limit.
Using the 263d val, it tempers out 5120/5103, 16875/16807, and 1959552/1953125 in the 7-limit; 540/539, 1375/1372, 16384/16335, and 43923/43750 in the 11-limit; 351/350, 1001/1000, 1573/1568, 2197/2187, and 4225/4224 in the 13-limit.
Using the 263df val, it tempers out 352/351, 640/637, 729/728, and 3584/3575 in the 13-limit.
Finally, it is accurate for the 17th harmonic, as the denominator of a convergent to log217, after 80 and before 343.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.71 | +1.52 | -1.53 | +0.77 | -0.98 | -0.01 | -0.94 | +1.38 | +1.60 | +0.21 |
| Relative (%) | +0.0 | +15.5 | +33.3 | -33.4 | +16.9 | -21.6 | -0.3 | -20.5 | +30.3 | +35.1 | +4.6 | |
| Steps (reduced) |
263 (0) |
417 (154) |
611 (85) |
738 (212) |
910 (121) |
973 (184) |
1075 (23) |
1117 (65) |
1190 (138) |
1278 (226) |
1303 (251) | |
Subsets and supersets
263edo is the 56th prime edo.
Notable supersets include 789edo, which triples it to achieve extreme accuracy in the 2.5.7 subgroup, and 1578edo, which sextuples it to be extremely strong in the 11-limit add-17 and in higher limits.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [417 -263⟩ | ⟨263 417] | −0.2229 | 0.2229 | 4.89 |
| 2.3.5 | 393216/390625, [50 -33 1⟩ | ⟨263 417 611] | −0.3666 | 0.2728 | 5.98 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 40\263 | 182.51 | 10/9 | Minortone |
| 1 | 85\263 | 387.83 | 5/4 | Würschmidt |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct