407edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
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| 2.3 | | 2.3 | ||
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| 0.0638 | | 0.0638 | ||
| 2.16 | | 2.16 | ||
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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {{rank-2 begin}} | ||
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| 1 | | 1 | ||
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| 4/3 | | 4/3 | ||
| [[Helmholtz]] | | [[Helmholtz]] | ||
{{rank-2 end}} | |||
{{orf}} | {{orf}} | ||
[[Category:Subsemifourth]] | [[Category:Subsemifourth]] | ||
Revision as of 01:51, 16 November 2024
| ← 406edo | 407edo | 408edo → |
Theory
407edo is a strong 5-limit system and 2.3.5.11.13.19.23 subgroup system. The equal temperament tempers out 32805/32768 in the 5-limit; using the patent val, 16875/16807, 4096000/4084101, and 26873856/26796875 in the 7-limit. It supports and provides the optimal patent val for the subsemifourth temperament in the 7- and 11-limit. Essentially tempered chords available in 407et include pinkanberry chords.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.24 | -0.07 | +1.20 | +0.03 | -0.23 | +1.19 | +0.28 | -0.26 | -0.58 | -1.06 |
| Relative (%) | +0.0 | -8.0 | -2.5 | +40.7 | +1.1 | -7.9 | +40.3 | +9.4 | -9.0 | -19.8 | -35.8 | |
| Steps (reduced) |
407 (0) |
645 (238) |
945 (131) |
1143 (329) |
1408 (187) |
1506 (285) |
1664 (36) |
1729 (101) |
1841 (213) |
1977 (349) |
2016 (388) | |
Subsets and supersets
407 factors into 11 × 37, with 11edo and 37edo as its subset edos. 814edo, which doubles it, gives a good correction to harmonics 7 and 17, and is a notable full 23-limit temperament.
Regular temperament properties
Template:Comma basis begin |- | 2.3 | [-645 407⟩ | [⟨407 645]] | +0.0742 | 0.0742 | 2.52 |- | 2.3.5 | 32805/32768, [30 47 -45⟩ | [⟨407 645 945]] | +0.0599 | 0.0638 | 2.16 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin |- | 1 | 63\407 | 185.75 | [24 4 -13⟩ | Pirate |- | 1 | 83\407 | 244.72 | 15/13 | Subsemifourth (407f) |- | 1 | 169\407 | 498.28 | 4/3 | Helmholtz Template:Rank-2 end Template:Orf