# 255edo

 ← 254edo 255edo 256edo →
Prime factorization 3 × 5 × 17
Step size 4.70588¢
Fifth 149\255 (701.176¢)
Semitones (A1:m2) 23:20 (108.2¢ : 94.12¢)
Consistency limit 11
Distinct consistency limit 11

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

Approximation of prime harmonics in 255edo
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

Table of rank-2 temperaments by generator
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)
Hemipental / hemipent (255) / hemipentalis (255f)
5 106\255
(4\255)
498.82
(18.82)
4/3
(81/80)
Pental (5-limit)
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 it is distinct