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 | +15 | +33 | -33 | +17 | -22 | -0 | -20 | +30 | +35 | +5 | |
| 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.
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 it is distinct