285edo
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Prime factorization
3 × 5 × 19
Step size
4.21053¢
Fifth
167\285 (703.158¢)
Semitones (A1:m2)
29:20 (122.1¢ : 84.21¢)
Consistency limit
7
Distinct consistency limit
7
| ← 284edo | 285edo | 286edo → |
285 equal divisions of the octave (285edo), or 285-tone equal temperament (285tet), 285 equal temperament (285et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 285 equal parts of about 4.21 ¢ each.
Theory
285edo has a sharp tendency. It 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 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +1.20 | +1.05 | -0.40 | -1.80 | +0.26 | +1.58 | -1.95 | +0.31 | +1.43 | +0.80 | -0.91 |
| relative (%) | +29 | +25 | -10 | -43 | +6 | +37 | -46 | +7 | +34 | +19 | -22 | |
| Steps (reduced) |
452 (167) |
662 (92) |
800 (230) |
903 (48) |
986 (131) |
1055 (200) |
1113 (258) |
1165 (25) |
1211 (71) |
1252 (112) |
1289 (149) | |
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 (Reduced) |
Cents (Reduced) |
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) |