5809edo
| ← 5808edo | 5809edo | 5810edo → |
5809 equal divisions of the octave (abbreviated 5809edo or 5809ed2), also called 5809-tone equal temperament (5809tet) or 5809 equal temperament (5809et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 5809 equal parts of about 0.207 ¢ each. Each step represents a frequency ratio of 21/5809, or the 5809th root of 2.
5809edo is a fairly strong 19-limit system, consistent to the 21-odd-limit. Moreover, it is a strong no-23 43-limit system, consistent to the no-23 45-odd-limit. Its full 43-limit interpretation using the patent val is also obvious, as 23/22, 23/13, 37/23 and their octave complements exhaust the inconsistently mapped intervals in the full 45-odd-limit.
We may note that it is an egads and euzenius system, supporting hemiegads. Some simpler commas it tempers out in the higher limits include 123201/123200 in the 13-limit; 14400/14399, 194481/194480, and 336141/336140 in the 17-limit; 10830/10829, 23409/23408, 28900/28899, 43681/43680, and 89376/89375 in the 19-limit; 7866/7865, 8625/8624, 21505/21054, and 25921/25920 in the 23-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | -0.0097 | -0.0166 | +0.0155 | +0.0334 | +0.0301 | -0.0148 | -0.0436 | -0.0767 | -0.0024 | +0.0152 | +0.0590 | -0.0040 | -0.0355 | +0.0813 |
| Relative (%) | +0.0 | -4.7 | -8.0 | +7.5 | +16.2 | +14.6 | -7.2 | -21.1 | -37.1 | -1.2 | +7.4 | +28.5 | -2.0 | -17.2 | +39.3 | |
| Steps (reduced) |
5809 (0) |
9207 (3398) |
13488 (1870) |
16308 (4690) |
20096 (2669) |
21496 (4069) |
23744 (508) |
24676 (1440) |
26277 (3041) |
28220 (4984) |
28779 (5543) |
30262 (1217) |
31122 (2077) |
31521 (2476) |
32267 (3222) | |
Subsets and supersets
Since 5809 factors into primes as 37 × 157, 5809edo contains 37edo and 157edo as subsets.