255edo
| ← 254edo | 255edo | 256edo → |
255 equal divisions of the octave (abbreviated 255edo or 255ed2), also called 255-tone equal temperament (255tet) or 255 equal temperament (255et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 255 equal parts of about 4.71 ¢ each. Each step represents a frequency ratio of 21/255, or the 255th root of 2.
Theory
The equal temperament tempers out the parakleisma, [8 14 -13⟩, and the septendecima, [-52 -17 34⟩, in the 5-limit. In the 7-limit it tempers out cataharry, 19683/19600, mirkwai, 16875/16807 and horwell, 65625/65536, so that it supports the mirkat temperament, and in fact provides the optimal patent val. It also gives the optimal patent val for mirkat in the 11-limit, tempering out 540/539, 1375/1372, 3025/3024 and 8019/8000. In the 13-limit it tempers out 847/845, 625/624, 1575/1573 and 1716/1715.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.78 | -0.43 | +0.59 | -0.73 | +1.83 | -1.43 | -1.04 | +2.31 | +1.01 | -1.51 |
| Relative (%) | +0.0 | -16.5 | -9.2 | +12.4 | -15.5 | +38.8 | -30.3 | -22.2 | +49.2 | +21.5 | -32.0 | |
| Steps (reduced) |
255 (0) |
404 (149) |
592 (82) |
716 (206) |
882 (117) |
944 (179) |
1042 (22) |
1083 (63) |
1154 (134) |
1239 (219) |
1263 (243) | |
Subsets and supersets
Since 255 factors into 3 × 5 × 17, 255edo has subset edos 3, 5, 15, 17, 51, and 85.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-404 255⟩ | [⟨255 404]] | +0.246 | 0.246 | 5.22 |
| 2.3.5 | [8 14 -13⟩, [-36 11 8⟩ | [⟨255 404 592]] | +0.226 | 0.203 | 4.30 |
| 2.3.5.7 | 16875/16807, 19683/19600, 65625/65536 | [⟨255 404 592 716]] | +0.117 | 0.257 | 5.46 |
| 2.3.5.7.11 | 540/539, 1375/1372, 8019/8000, 65625/65536 | [⟨255 404 592 716 882]] | +0.136 | 0.233 | 4.95 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 39\255 | 183.53 | 10/9 | Mirkat (255f) |
| 1 | 52\255 | 244.71 | 15/13 | Subsemifourth (255) |
| 1 | 67\255 | 315.29 | 6/5 | Parakleismic (5-limit) |
| 1 | 74\255 | 348.24 | 11/9 | Eris (255) |
| 3 | 82\255 (3\255) |
385.88 (14.12) |
5/4 (126/125) |
Mutt (7-limit) |
| 5 | 53\255 (2\255) |
249.41 (9.41) |
81/70 (176/175) |
Hemiquintile / hemiquint (255) / hemiquintilis (255f) |
| 5 | 106\255 (4\255) |
498.82 (18.82) |
4/3 (81/80) |
Quintile |
| 17 | 53\255 (7\255) |
249.41 (32.94) |
[-25 -9 17⟩ (1990656/1953125) |
Chlorine (5-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct