405edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| Line 42: | Line 34: | ||
| 0.2207 | | 0.2207 | ||
| 7.45 | | 7.45 | ||
{{comma basis end}} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 66: | Line 52: | ||
|- | |- | ||
| 9 | | 9 | ||
| 107\405<br>(17\405) | | 107\405<br />(17\405) | ||
| 317.04<br>(50.37) | | 317.04<br />(50.37) | ||
| 6/5<br>(36/35) | | 6/5<br />(36/35) | ||
| [[Novemkleismic]] | | [[Novemkleismic]] | ||
{{rank-2 end}} | |||
{{orf}} | |||
Revision as of 02:27, 16 November 2024
| ← 404edo | 405edo | 406edo → |
Theory
405edo is enfactored in the 3-limit, with the same tuning as 135edo. Like 135edo, it is consistent to the 7-odd-limit with a poor approximation to the harmonic 5. Using the patent val, the equal temperament tempers out 15625/15552 in the 5-limit; 2100875/2097152, and 2460375/2458624 in the 7-limit; 1375/1372, 4000/3993, 19712/19683, and 41503/41472 in the 11-limit. It supports marthirds, novemkleismic and kleirtismic. It provides the optimal patent val for 7- and 11-limit novemkleismic.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.27 | -1.13 | +0.06 | -0.21 | +0.95 | -1.25 | -1.22 | -0.13 | -1.43 | -1.33 |
| Relative (%) | +0.0 | +9.0 | -38.1 | +2.1 | -7.0 | +32.2 | -42.2 | -41.1 | -4.3 | -48.2 | -45.0 | |
| Steps (reduced) |
405 (0) |
642 (237) |
940 (130) |
1137 (327) |
1401 (186) |
1499 (284) |
1655 (35) |
1720 (100) |
1832 (212) |
1967 (347) |
2006 (386) | |
Subsets and supersets
Since 405 factors into 34 × 5, 405edo has subset edos 3, 5, 9, 15, 27, 45, 81, and 135.
Regular temperament properties
Template:Comma basis begin |- | 2.3.5 | 15625/15552, [110 -65 -3⟩ | [⟨405 642 940]] | +0.1058 | 0.2776 | 9.37 |- | 2.3.5.7 | 15625/15552, 2100875/2097152, 2460375/2458624 | [⟨405 642 940 1137]] | +0.0737 | 0.2467 | 8.33 |- | 2.3.5.7.11 | 1375/1372, 4000/3993, 19712/19683, 41503/41472 | [⟨405 642 940 1137 1401]] | +0.0709 | 0.2207 | 7.45 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin
|-
| 1
| 56\405
| 165.93
| 11/10
| Satin
|-
| 1
| 107\405
| 317.04
| 6/5
| Hanson
|-
| 9
| 107\405
(17\405)
| 317.04
(50.37)
| 6/5
(36/35)
| Novemkleismic
Template:Rank-2 end
Template:Orf