4501edo
| ← 4500edo | 4501edo | 4502edo → |
4501 equal divisions of the octave (abbreviated 4501edo or 4501ed2), also called 4501-tone equal temperament (4501tet) or 4501 equal temperament (4501et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 4501 equal parts of about 0.267 ¢ each. Each step represents a frequency ratio of 21/4501, or the 4501st root of 2.
4501edo is a very strong high-limit system, distinctly consistent through the 39-odd-limit, and has the lowest 31- and 37-limit relative error of any equal temperament until 16808. The 4501m val likewise performs well in the 41- and 43-limit, with the lowest relative error of any equal temperament until 7361.
Some of the simpler commas it tempers out include 10648/10647 and 140625/140608 in the 13-limit; 14400/14399, 31213/31212, and 37180/37179 in the 17-limit; 10830/10829, 14080/14079, and 27456/27455 in the 19-limit; 11662/11661, 12168/12167, 16929/16928, and 19551/19550 in the 23-limit; 11340/11339, 13312/13311, and 13456/13455 in the 29-limit; 7936/7935, 11935/11934, 15625/15624, 19344/19343, 23716/23715, 24025/24024, and 29792/29791 in the 31-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.022 | +0.000 | +0.025 | +0.026 | +0.086 | +0.088 | +0.021 | +0.119 | +0.061 | +0.043 | +0.067 | -0.091 | +0.102 | -0.055 |
| Relative (%) | +0.0 | +8.4 | +0.2 | +9.5 | +9.8 | +32.1 | +33.0 | +7.8 | +44.8 | +22.8 | +16.2 | +25.0 | -34.2 | +38.2 | -20.4 | |
| Steps (reduced) |
4501 (0) |
7134 (2633) |
10451 (1449) |
12636 (3634) |
15571 (2068) |
16656 (3153) |
18398 (394) |
19120 (1116) |
20361 (2357) |
21866 (3862) |
22299 (4295) |
23448 (943) |
24114 (1609) |
24424 (1919) |
25001 (2496) | |
Subsets and supersets
Since 4501 factors into 7 × 643, 4501edo has subset edos 7 and 643.