445edo: Difference between revisions
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! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
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|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| {{ | | {{Monzo| -28 25 -5 }}, {{monzo| -29 -11 20 }} | ||
| {{ | | {{Mapping| 445 705 1033 }} | ||
| +0.2748 | | +0.2748 | ||
| 0.2149 | | 0.2149 | ||
| Line 32: | Line 32: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 2401/2400, 7381125/7340032, 43046721/42875000 | | 2401/2400, 7381125/7340032, 43046721/42875000 | ||
| {{ | | {{Mapping| 445 705 1033 1249 }} | ||
| +0.2716 | | +0.2716 | ||
| 0.1862 | | 0.1862 | ||
| Line 39: | Line 39: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 2401/2400, 3025/3024, 8019/8000, 234375/234256 | | 2401/2400, 3025/3024, 8019/8000, 234375/234256 | ||
| {{ | | {{Mapping| 445 705 1033 1249 1539 }} | ||
| +0.2870 | | +0.2870 | ||
| 0.1694 | | 0.1694 | ||
| Line 49: | Line 49: | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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|- | |- | ||
| 5 | | 5 | ||
| | | 7\445 | ||
| | | 18.88 | ||
| | | 81/80 | ||
| [[ | | [[Quintile]] (5-limit) | ||
|} | |} | ||
<nowiki />* | <nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | ||
Revision as of 06:13, 4 September 2026
| ← 444edo | 445edo | 446edo → |
445 equal divisions of the octave (abbreviated 445edo or 445ed2), also called 445-tone equal temperament (445tet) or 445 equal temperament (445et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 445 equal parts of about 2.7 ¢ each. Each step represents a frequency ratio of 21/445, or the 445th root of 2.
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
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 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 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 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 | 7\445 | 18.88 | 81/80 | Quintile (5-limit) |