# 457edo

 ← 456edo 457edo 458edo →
Prime factorization 457 (prime)
Step size 2.62582¢
Fifth 267\457 (701.094¢)
Semitones (A1:m2) 41:36 (107.7¢ : 94.53¢)
Consistency limit 7
Distinct consistency limit 7

457 equal divisions of the octave (abbreviated 457edo or 457ed2), also called 457-tone equal temperament (457tet) or 457 equal temperament (457et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 457 equal parts of about 2.63 ¢ each. Each step represents a frequency ratio of 21/457, or the 457th root of 2.

## Theory

457edo is consistent to the 7-odd-limit, but the error of harmonic 3 is quite large. The equal temperament tempers out 19683/19600 and 65625/65536 in the 7-limit; 540/539, 8019/8000, and 43923/43904 in the 11-limit.

### Odd harmonics

Approximation of odd harmonics in 457edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.86 -0.32 +0.10 +0.90 +0.10 -0.27 -1.18 +0.08 -0.80 -0.76 -0.70
Relative (%) -32.8 -12.1 +3.9 +34.4 +4.0 -10.1 -44.9 +2.9 -30.3 -28.9 -26.8
Steps
(reduced)
724
(267)
1061
(147)
1283
(369)
1449
(78)
1581
(210)
1691
(320)
1785
(414)
1868
(40)
1941
(113)
2007
(179)
2067
(239)

### Subsets and supersets

457edo is the 88th prime edo.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-724 457 [457 724]] 0.2716 0.2716 10.34
2.3.5 [-36 11 8, [-5 31 -19 [457 724 1061]] 0.2267 0.2307 8.79
2.3.5.7 19683/19600, 65625/65536, 7381125/7340032 [457 724 1061 1283]] 0.1609 0.2300 8.76
2.3.5.7.11 540/539, 8019/8000, 19683/19600, 43923/43904 [457 724 1061 1283 1581]] 0.1227 0.2194 8.36
2.3.5.7.11.13 540/539, 1716/1715, 4225/4224, 41067/40960, 43940/43923 [457 724 1061 1283 1581 1691]] 0.1142 0.2012 7.66
2.3.5.7.11.13.17 936/935, 1089/1088, 1275/1274, 1575/1573, 2601/2600, 4225/4224 [457 724 1061 1283 1581 1691 1868]] 0.0952 0.1920 7.31

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 10\457 26.258 49/48 Sfourth
1 136\457 357.11 49/40 Dodifo
1 213\457 559.30 864/625 Tritriple

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct