445edo
| ← 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 | 185\445 (7\445) |
498.88 (18.88) |
4/3 (81/80) |
Pental (5-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct