331edo: Difference between revisions
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| {{monzo| -1049 331 }} | | {{monzo| -1049 331 }} | ||
| {{mapping| 331 1049 }} | | {{mapping| 331 1049 }} | ||
| 0.1402 | | +0.1402 | ||
| 0.1402 | | 0.1402 | ||
| 3.87 | | 3.87 | ||
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| [[Würschmidt]] (331, 5-limit) | | [[Würschmidt]] (331, 5-limit) | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||
| Line 93: | Line 93: | ||
== Music == | == Music == | ||
; [[User:Francium|Francium]] | ; [[User:Francium|Francium]] | ||
* "Silent Silence" from ''Edson EP'' (2023) | * "Silent Silence" from ''Edson EP'' (2023) – [https://open.spotify.com/track/6q3xr4E4QIL9BMaZyf6LXd Spotify] | [https://francium223.bandcamp.com/track/silent-silence Bandcamp] | [https://www.youtube.com/watch?v=g3FF6oqnilk YouTube] – edson in 331edo tuning | ||
[[Category:Listen]] | [[Category:Listen]] | ||
Revision as of 19:24, 15 January 2025
| ← 330edo | 331edo | 332edo → |
Theory
331edo is only consistent to the 5-odd-limit and the errors of both harmonics 3 and 5 are quite large, commending itself as a temperament of the 2.9.15.7.11.13.17.19 subgroup.
Using the patent val nonetheless, the equal temperament tempers out 5120/5103, 1959552/1953125 and 78125000/78121827 in the 7-limit; 3025/3024, 12005/11979, 16384/16335, 42875/42768, 43923/43750, 78408/78125, and 180224/180075 in the 11-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.37 | +1.60 | -0.85 | -0.89 | -0.26 | +0.56 | -0.66 | +0.18 | -0.23 | +0.52 | -1.08 |
| Relative (%) | +37.7 | +44.2 | -23.4 | -24.5 | -7.2 | +15.4 | -18.1 | +5.0 | -6.4 | +14.3 | -29.9 | |
| Steps (reduced) |
525 (194) |
769 (107) |
929 (267) |
1049 (56) |
1145 (152) |
1225 (232) |
1293 (300) |
1353 (29) |
1406 (82) |
1454 (130) |
1497 (173) | |
Subsets and supersets
331edo is the 67th prime edo. 662edo, which doubles it, gives a good correction to the harmonics 3 and 5.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-1049 331⟩ | [⟨331 1049]] | +0.1402 | 0.1402 | 3.87 |
| 2.9.15 | [-7 17 -12⟩, [-81 12 11⟩ | [⟨331 1049 1293]] | +0.1238 | 0.1168 | 3.22 |
| 2.9.15.7 | 65625/65536, 420175/419904, 80387359983/80000000000 | [⟨331 1049 1293 929]] | +0.1685 | 0.1275 | 3.52 |
| 2.9.15.7.11 | 9801/9800, 41503/41472, 137781/137500, 759375/758912 | [⟨331 1049 1293 929 1145]] | +0.1499 | 0.1200 | 3.31 |
| 2.9.15.7.11.13 | 729/728, 1575/1573, 10648/10647, 41503/41472, 43904/43875 | [⟨331 1049 1293 929 1145 1225]] | +0.0997 | 0.1568 | 4.33 |
| 2.9.15.7.11.13.17 | 729/728, 833/832, 1089/1088, 2025/2023, 10648/10647, 18816/18785 | [⟨331 1049 1293 929 1145 1225 1353]] | +0.0791 | 0.1537 | 4.24 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 107\331 | 387.92 | 5/4 | Würschmidt (331, 5-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct