584edo: Difference between revisions
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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
Revision as of 02:45, 16 November 2024
| ← 583edo | 584edo | 585edo → |
Theory
584edo is only consistent to the 5-odd-limit and the error of harmonic 3 is quite large. With reasonable approximations to harmonics 5, 7, 9, 11, 13, and 17, it commends itself as a 2.9.5.7.11.13.17 subgroup tuning.
If we use the harmonic 3 instead, we notice the better-tuned 584d val is enfactored, with the same tuning as 292edo. Therefore, we are left with the patent val, which tempers out 48828125/48771072 and 67108864/66976875, supporting hemiluna.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.785 | -0.012 | -1.018 | -0.485 | -0.633 | -0.117 | +0.772 | -0.161 | +0.432 | -0.233 | +0.493 |
| Relative (%) | +38.2 | -0.6 | -49.5 | -23.6 | -30.8 | -5.7 | +37.6 | -7.8 | +21.0 | -11.3 | +24.0 | |
| Steps (reduced) |
926 (342) |
1356 (188) |
1639 (471) |
1851 (99) |
2020 (268) |
2161 (409) |
2282 (530) |
2387 (51) |
2481 (145) |
2565 (229) |
2642 (306) | |
Regular temperament properties
Template:Comma basis begin |- | 2.3 | [463 -292⟩ | [⟨584 926]] | −0.2476 | 0.2475 | 12.05 |- | 2.3.5 | [3 -18 11⟩, [21 -20 4⟩ | [⟨584 926 1356]] | −0.1633 | 0.2346 | 11.42 |- | 2.3.5.7 | 1500625/1492992, 1605632/1594323, 235298/234375 | [⟨584 926 1356 1639]] | −0.0319 | 0.3052 | 14.85 |- | 2.3.5.7.11 | 5632/5625, 160083/160000, 26411/26244, 968000/964467 | [⟨584 926 1356 1639 2020]] | +0.0111 | 0.2862 | 13.93 |- | 2.3.5.7.11.13 | 2080/2079, 1001/1000, 4096/4095, 85750/85293, 983125/979776 | [⟨584 926 1356 1639 2020 2161]] | +0.0145 | 0.2613 | 12.72 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin |- | 1 | 47\584 | 96.58 | 200/189 | Hemiluna (584, 7-limit) Template:Rank-2 end Template:Orf
Scales
Music
- Are You From The Moon? (2023) – hemiluna in 584edo tuning