457edo

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← 456edo457edo458edo →
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 457-tone equal temperament (457tet), 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 of 457edo 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 (%) -33 -12 +4 +34 +4 -10 -45 +3 -30 -29 -27
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