285edo
| ← 284edo | 285edo | 286edo → |
285 equal divisions of the octave (abbreviated 285edo or 285ed2), also called 285-tone equal temperament (285tet) or 285 equal temperament (285et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 285 equal parts of about 4.21 ¢ each. Each step represents a frequency ratio of 21/285, or the 285th root of 2.
Theory
285edo has a sharp tendency. The equal temperament tempers out the misty comma and the enneadeca in the 5-limit; 3136/3125 and 5120/5103 in the 7-limit; 3025/3024 and 3388/3375 in the 11-limit; 352/351, 676/675, 847/845, 1001/1000, and 2080/2079 in the 13-limit. It supports the 13-limit hemimist temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +1.20 | +1.05 | -0.40 | +0.26 | +1.58 | +0.31 | +1.43 | -0.91 | +2.00 | +0.23 |
| Relative (%) | +0.0 | +28.6 | +25.0 | -9.6 | +6.2 | +37.5 | +7.3 | +34.1 | -21.5 | +47.5 | +5.4 | |
| Steps (reduced) |
285 (0) |
452 (167) |
662 (92) |
800 (230) |
986 (131) |
1055 (200) |
1165 (25) |
1211 (71) |
1289 (149) |
1385 (245) |
1412 (272) | |
Subsets and supersets
Since 285 factors into 3 × 5 × 19, 285edo has subset edos 3, 5, 15, 19, 57, and 95.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [452 -285⟩ | [⟨285 452]] | −0.3795 | 0.3794 | 9.01 |
| 2.3.5 | 67108864/66430125, [-14 -19 19⟩ | [⟨285 452 662]] | −0.4043 | 0.3117 | 7.41 |
| 2.3.5.7 | 3136/3125, 5120/5103, 40353607/39858075 | [⟨285 452 662 800]] | −0.2673 | 0.3596 | 8.54 |
| 2.3.5.7.11 | 3025/3024, 3136/3125, 5120/5103, 12005/11979 | [⟨285 452 662 800 986]] | −0.2289 | 0.3307 | 7.85 |
| 2.3.5.7.11.13 | 352/351, 676/675, 847/845, 3025/3024, 12005/11979 | [⟨285 452 662 800 986 1055]] | −0.2618 | 0.3107 | 7.38 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 109\285 | 458.95 | 125/96 | Majvam |
| 3 | 59\285 (36\285) |
248.42 (151.58) |
15/13 (12/11) |
Hemimist |
| 3 | 59\285 (23\285) |
496.84 (96.84) |
4/3 (256/243) |
Misty |
| 19 | 118\285 (2\285) |
496.84 (8.42) |
4/3 (15625/15552) |
Enneadecal (5-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct