631edo
| ← 630edo | 631edo | 632edo → |
631 equal divisions of the octave (abbreviated 631edo or 631ed2), also called 631-tone equal temperament (631tet) or 631 equal temperament (631et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 631 equal parts of about 1.9 ¢ each. Each step represents a frequency ratio of 21/631, or the 631st root of 2.
Theory
631edo is consistent to the 9-odd-limit, with all of the odd harmonics having a flat tendency. Using the patent val, it tempers out 32805/32768 in the 5-limit; 4375/4374 in the 7-limit; 12005/11979 and 41503/41472 in the 11-limit; 1575/1573, 4225/4224, 4459/4455, and 83349/83200 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.212 | -0.260 | -0.839 | -0.423 | +0.188 | +0.043 | -0.472 | -0.360 | -0.841 | +0.851 | -0.699 |
| Relative (%) | -11.1 | -13.7 | -44.1 | -22.3 | +9.9 | +2.3 | -24.8 | -18.9 | -44.2 | +44.8 | -36.8 | |
| Steps (reduced) |
1000 (369) |
1465 (203) |
1771 (509) |
2000 (107) |
2183 (290) |
2335 (442) |
2465 (572) |
2579 (55) |
2680 (156) |
2772 (248) |
2854 (330) | |
Subsets and supersets
631edo is the 115th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-1000 631⟩ | [⟨631 1000]] | +0.0668 | 0.0668 | 3.51 |
| 2.3.5 | 32805/32768, [-50 -71 70⟩ | [⟨631 1000 1465]] | +0.0818 | 0.0585 | 3.08 |
| 2.3.5.7 | 4375/4374, 32805/32768, 678223072849/675000000000 | [⟨631 1000 1465 1771]] | +0.1361 | 0.1067 | 5.61 |
| 2.3.5.7.11 | 4375/4374, 41503/41472, 32805/32768, 12005/11979 | [⟨631 1000 1465 1771 2183]] | +0.0980 | 0.1221 | 6.42 |
| 2.3.5.7.11.13 | 1575/1573, 4375/4374, 4459/4455, 4225/4224, 83349/83200 | [⟨631 1000 1465 1771 2183 2335]] | +0.0797 | 0.1187 | 6.24 |
| 2.3.5.7.11.13.17 | 1225/1224, 1701/1700, 833/832, 1575/1573, 4459/4455, 4225/4224 | [⟨631 1000 1465 1771 2183 2335 2579]] | +0.0809 | 0.1099 | 5.78 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 262\631 | 498.257 | 4/3 | Helmholtz / Pontiac |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct