457edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| Line 55: | Line 64: | ||
| 0.1920 | | 0.1920 | ||
| 7.31 | | 7.31 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 77: | Line 93: | ||
| 864/625 | | 864/625 | ||
| [[Tritriple]] | | [[Tritriple]] | ||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== Music == | == Music == | ||
Revision as of 13:01, 16 November 2024
| ← 456edo | 457edo | 458edo → |
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 (%) | -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
| 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