457edo
Jump to navigation
Jump to search
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
| ← 456edo | 457edo | 458edo → |
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
| 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
| 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