5585edo
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5585 equal divisions of the octave (abbreviated 5585edo or 5585ed2), also called 5585-tone equal temperament (5585tet) or 5585 equal temperament (5585et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 5585 equal parts of about 0.215 ¢ each. Each step represents a frequency ratio of 21/5585, or the 5585th root of 2.
5585edo is a zeta peak edo, which has to do with the fact that it is a strong 13-limit system, with a lower 13-limit relative error than any smaller edo, though 6079, only slightly larger, beats it. A basis for its 13-limit commas is {123201/123200, 151263/151250, 8858304/8857805, 8859375/8859136, 62752536/62748517}.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | -0.0033 | +0.0068 | -0.0166 | +0.0160 | +0.0095 | -0.1031 | +0.0698 | -0.0201 | +0.0378 | -0.0400 |
| Relative (%) | +0.0 | -1.6 | +3.2 | -7.7 | +7.4 | +4.4 | -48.0 | +32.5 | -9.4 | +17.6 | -18.6 | |
| Steps (reduced) |
5585 (0) |
8852 (3267) |
12968 (1798) |
15679 (4509) |
19321 (2566) |
20667 (3912) |
22828 (488) |
23725 (1385) |
25264 (2924) |
27132 (4792) |
27669 (5329) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0436 | +0.0155 | +0.0669 | -0.0814 | -0.0936 | +0.0942 | -0.0361 | -0.0232 | -0.0726 | -0.0366 | +0.0826 |
| Relative (%) | +20.3 | +7.2 | +31.1 | -37.9 | -43.6 | +43.9 | -16.8 | -10.8 | -33.8 | -17.0 | +38.5 | |
| Steps (reduced) |
29095 (1170) |
29922 (1997) |
30306 (2381) |
31022 (3097) |
31990 (4065) |
32855 (4930) |
33123 (5198) |
33879 (369) |
34346 (836) |
34570 (1060) |
35207 (1697) | |