User:Francium/3631edo
| ← 3630edo | 3631edo | 3632edo → |
(semiconvergent)
3631 equal divisions of the octave (abbreviated 3631edo or 3631ed2), also called 3631-tone equal temperament (3631tet) or 3631 equal temperament (3631et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3631 equal parts of about 0.33 ¢ each. Each step represents a frequency ratio of 21/3631, or the 3631st root of 2.
Theory
3631edo is consistent to the 9-odd-limit, tempering out 43046721/43025920, 78125000/78121827 and [79 -6 -13 -14⟩ in the 7-limit. Its harmonic 3 has an extremely low error of 0.1 percent. It is strong in the 2.3.5.11.23 subgroup, tempering out 64009/64000, 19133125/19131876, 184549376/184528125, 510526328421483/510320967680000. As an equal temperament, 3631edo supports quectismic in the 5-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.000 | +0.026 | +0.163 | -0.065 | -0.098 | +0.140 | -0.074 | -0.018 | -0.109 | +0.104 |
| Relative (%) | +0.0 | +0.1 | +7.9 | +49.4 | -19.6 | -29.7 | +42.2 | -22.5 | -5.3 | -32.9 | +31.3 | |
| Steps (reduced) |
3631 (0) |
5755 (2124) |
8431 (1169) |
10194 (2932) |
12561 (1668) |
13436 (2543) |
14842 (318) |
15424 (900) |
16425 (1901) |
17639 (3115) |
17989 (3465) | |
Subsets and supersets
3631edo is the 508th prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [5755 -3631⟩ | [⟨3631 5755]] | −0.0001 | 0.0001 | 0.03 |
| 2.3.5 | [91 -12- 31⟩, [-33 97 -52⟩ | [⟨3631 5755 8431]] | −0.0038 | 0.0053 | 1.60 |