641edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{monzo| 1016 -641 }} | | {{monzo| 1016 -641 }} | ||
| {{mapping| 641 1016 }} | | {{mapping| 641 1016 }} | ||
| | | −0.0231 | ||
| 0.0231 | | 0.0231 | ||
| 1.23 | | 1.23 | ||
| Line 37: | Line 29: | ||
| 0.1474 | | 0.1474 | ||
| 7.87 | | 7.87 | ||
{{comma basis end}} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 53: | Line 39: | ||
| 320/243 | | 320/243 | ||
| [[Vulture]] | | [[Vulture]] | ||
{{rank-2 end}} | |||
{{orf}} | |||
== Music == | == Music == | ||
; [[Francium]] | ; [[Francium]] | ||
* "Goofy Individual" from ''Take Advantage'' (2024) | * "Goofy Individual" from ''Take Advantage'' (2024) – [https://open.spotify.com/track/3wNm7mtNBOuE4utwHFSoRW Spotify] | [https://francium223.bandcamp.com/track/goofy-individual Bandcamp] | [https://www.youtube.com/watch?v=c3ZGAeYHmrQ YouTube] | ||
[[Category:Listen]] | |||
Revision as of 03:01, 16 November 2024
| ← 640edo | 641edo | 642edo → |
Theory
641edo is only consistent to the 5-odd-limit. Since both harmonics 7 and 11 are about halfway between its steps, and since harmonic 5 is also off by more than a third step, it can be used as a dual-5 dual-7 dual-11 temperament. Alternatively, it can be used as a 2.3.5.13.17.19 subgroup temperament, as it is consistent in the no-7 no-11 19-odd-limit.
To start with, consider the 641d val ⟨641 1016 1488 1799 2217 2372] in the 13-limit, which tempers out 625/624, 2200/2197, 4459/4455, 14641/14625, and 19712/19683. The alternative 641df val, ⟨641 1016 1488 1799 2217 2371], tempers out 676/675, 1001/1000, 19712/19683, 31213/31104, and 983125/979776. The 641ce val, ⟨641 1016 1489 1800 2218 2372], tempers out 676/675, 1001/1000, 6144/6125, 10985/10976, and 85294/85184.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.073 | -0.666 | +0.909 | +0.146 | -0.928 | +0.034 | -0.593 | -0.119 | +0.147 | -0.890 | +0.743 |
| Relative (%) | +3.9 | -35.6 | +48.5 | +7.8 | -49.6 | +1.8 | -31.7 | -6.4 | +7.8 | -47.5 | +39.7 | |
| Steps (reduced) |
1016 (375) |
1488 (206) |
1800 (518) |
2032 (109) |
2217 (294) |
2372 (449) |
2504 (581) |
2620 (56) |
2723 (159) |
2815 (251) |
2900 (336) | |
Subsets and supersets
641edo is the 116th prime edo. 1282edo, which doubles it, gives a good correction to the harmonics 7 and 11.
Regular temperament properties
Template:Comma basis begin |- | 2.3 | [1016 -641⟩ | [⟨641 1016]] | −0.0231 | 0.0231 | 1.23 |- | 2.3.5 | [24 -21 4⟩, [-56 -13 33⟩ | [⟨641 1016 1488]] | +0.0803 | 0.1474 | 7.87 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin |- | 1 | 254\641 | 475.507 | 320/243 | Vulture Template:Rank-2 end Template:Orf