456edo: Difference between revisions
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[[ | 456edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[152edo]], defined by [[tempering out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 32 -7 -9 }} ([[escapade comma]]), as well as {{monzo| 23 6 -14 }} ([[vishnuzma]]), {{monzo| 41 -20 -4 }} (undim comma), and {{monzo| -14 -19 19 }} ([[enneadeca]]). In the 7-limit, it tempers out [[10976/10935]], 235298/234375, and 134217728/133984375, providing the [[optimal patent val]] for the [[chromat]] temperament. | ||
=== Odd harmonics === | |||
{{Harmonics in equal|456}} | |||
=== Subsets and supersets === | |||
Since 456 factors into {{factorization|456}}, 456edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 19, 24, 38, 57, 76, 114, 152, and 228 }}. | |||
[[Category:Chromat]] | |||
Latest revision as of 22:41, 20 February 2025
| ← 455edo | 456edo | 457edo → |
456 equal divisions of the octave (abbreviated 456edo or 456ed2), also called 456-tone equal temperament (456tet) or 456 equal temperament (456et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 456 equal parts of about 2.63 ¢ each. Each step represents a frequency ratio of 21/456, or the 456th root of 2.
456edo is enfactored in the 5-limit, with the same tuning as 152edo, defined by tempering out 1600000/1594323 (amity comma) and [32 -7 -9⟩ (escapade comma), as well as [23 6 -14⟩ (vishnuzma), [41 -20 -4⟩ (undim comma), and [-14 -19 19⟩ (enneadeca). In the 7-limit, it tempers out 10976/10935, 235298/234375, and 134217728/133984375, providing the optimal patent val for the chromat temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.68 | +0.53 | -0.40 | -1.28 | +1.31 | -1.05 | +1.20 | +0.31 | -0.14 | +0.27 | +0.67 |
| Relative (%) | +25.7 | +20.1 | -15.4 | -48.6 | +49.9 | -40.1 | +45.8 | +11.7 | -5.5 | +10.3 | +25.6 | |
| Steps (reduced) |
723 (267) |
1059 (147) |
1280 (368) |
1445 (77) |
1578 (210) |
1687 (319) |
1782 (414) |
1864 (40) |
1937 (113) |
2003 (179) |
2063 (239) | |
Subsets and supersets
Since 456 factors into 23 × 3 × 19, 456edo has subset edos 2, 3, 4, 6, 8, 12, 19, 24, 38, 57, 76, 114, 152, and 228.