# 285edo

 ← 284edo 285edo 286edo →
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

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

Approximation of prime harmonics in 285edo
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

Table of rank-2 temperaments by generator
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)