445edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
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| 0.1694 | | 0.1694 | ||
| 6.28 | | 6.28 | ||
{{comma basis end}} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 76: | Line 62: | ||
| 4/3<br>(81/80) | | 4/3<br>(81/80) | ||
| [[Pental (temperament)|Pental]] (5-limit) | | [[Pental (temperament)|Pental]] (5-limit) | ||
{{rank-2 end}} | |||
{{orf}} | |||
Revision as of 02:47, 16 November 2024
| ← 444edo | 445edo | 446edo → |
Theory
445edo is enfactored in the 3-limit with the same tuning as 89edo, but the approximation to some of the higher harmonics are improved. It is consistent to the 7-odd-limit with harmonics 3, 5, 7 all tuned flat, and it allows an extension to the 11-limit. The equal temperament tempers out 2401/2400, 7381125/7340032, 33756345/33554432, 43046721/42875000, and 48828125/48771072 in the 7-limit; 3025/3024, 8019/8000, 24057/24010, 35937/35840, 41503/41472, 137781/137500, 151263/151250, and 234375/234256 in the 11-limit. It notably supports neptune.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.83 | -0.70 | -0.74 | +1.03 | -1.21 | +0.82 | +1.17 | +0.21 | -0.88 | +1.13 | +0.04 |
| Relative (%) | -30.8 | -25.8 | -27.3 | +38.3 | -44.7 | +30.4 | +43.4 | +7.9 | -32.8 | +41.9 | +1.5 | |
| Steps (reduced) |
705 (260) |
1033 (143) |
1249 (359) |
1411 (76) |
1539 (204) |
1647 (312) |
1739 (404) |
1819 (39) |
1890 (110) |
1955 (175) |
2013 (233) | |
Subsets and supersets
Since 445 factors into 5 × 89, 445edo has 5edo and 89edo as its subsets.
Regular temperament properties
Template:Comma basis begin |- | 2.3.5 | [-28 25 -5⟩, [-29 -11 20⟩ | [⟨445 705 1033]] | +0.2748 | 0.2149 | 7.97 |- | 2.3.5.7 | 2401/2400, 7381125/7340032, 43046721/42875000 | [⟨445 705 1033 1249]] | +0.2716 | 0.1862 | 6.90 |- | 2.3.5.7.11 | 2401/2400, 3025/3024, 8019/8000, 234375/234256 | [⟨445 705 1033 1249 1539]] | +0.2870 | 0.1694 | 6.28 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin
|-
| 1
| 13\445
| 35.06
| 1990656/1953125
| Gammic (5-limit)
|-
| 1
| 42\445
| 113.26
| 16/15
| Misneb
|-
| 1
| 216\445
| 582.47
| 7/5
| Neptune (7-limit)
|-
| 5
| 185\445
(7\445)
| 498.88
(18.88)
| 4/3
(81/80)
| Pental (5-limit)
Template:Rank-2 end
Template:Orf