639edo
| ← 638edo | 639edo | 640edo → |
639 equal divisions of the octave (abbreviated 639edo or 639ed2), also called 639-tone equal temperament (639tet) or 639 equal temperament (639et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 639 equal parts of about 1.88 ¢ each. Each step represents a frequency ratio of 21/639, or the 639th root of 2.
Theory
639edo is distinctly consistent in the 17-odd-limit. It has a sharp tendency, with harmonics 3 to 17 all tuned sharp. The 639h val gives a reasonable approximation of harmonic 19, in which the edo is almost consistent up to the 25-odd-limit, with the exception of 19/16 and 25/16 themselves and their octave complements.
Using this val, the equal temperament tempers out [1 27 -18⟩ (ennealimma) and [55 -1 -23⟩ (counterwürschmidt comma) in the 5-limit; 2401/2400 and 4375/4374 in the 7-limit; 5632/5625 and 19712/19683 in the 11-limit; 2080/2079 and 4459/4455 in the 13-limit; 1156/1155, 2058/2057, and 2601/2600 in the 17-limit; 1216/1215, 1445/1444, 1540/1539, 2376/2375, and 2926/2925 in the 19-limit. It supports ennealimmal and its 13-limit extension enneabiotic.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.392 | +0.541 | +0.188 | +0.795 | +0.787 | +0.209 | -0.799 | +0.834 | -0.469 | +0.504 |
| Relative (%) | +0.0 | +20.9 | +28.8 | +10.0 | +42.3 | +41.9 | +11.1 | -42.6 | +44.4 | -25.0 | +26.9 | |
| Steps (reduced) |
639 (0) |
1013 (374) |
1484 (206) |
1794 (516) |
2211 (294) |
2365 (448) |
2612 (56) |
2714 (158) |
2891 (335) |
3104 (548) |
3166 (610) | |
Subsets and supersets
Since 639 factors into primes as 32 × 71, 639edo has subset edos 3, 9, 71, and 213.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1013 -639⟩ | [⟨639 1013]] | −0.1238 | 0.1238 | 6.59 |
| 2.3.5 | [1 -27 18⟩, [55 -1 -23⟩ | [⟨639 1013 1484]] | −0.1601 | 0.1134 | 6.04 |
| 2.3.5.7 | 2401/2400, 4375/4374, [58 -14 -13 -2⟩ | [⟨639 1013 1484 1794]] | −0.1369 | 0.1062 | 5.65 |
| 2.3.5.7.11 | 2401/2400, 4375/4374, 5632/5625, 161280/161051 | [⟨639 1013 1484 1794 2211]] | −0.1554 | 0.1020 | 5.43 |
| 2.3.5.7.11.13 | 2080/2079, 2401/2400, 4375/4374, 5632/5625, 20480/20449 | [⟨639 1013 1484 1794 2211 2365]] | −0.1650 | 0.0955 | 5.08 |
| 2.3.5.7.11.13.17 | 1156/1155, 2058/2057, 2080/2079, 2401/2400, 4375/4374, 5632/5625 | [⟨639 1013 1484 1794 2211 2365 2612]] | −0.1487 | 0.0970 | 5.16 |
| 2.3.5.7.11.13.17.19 | 1156/1155, 1216/1215, 1445/1444, 2058/2057, 2080/2079, 2376/2375, 2401/2400 | [⟨639 1013 1484 1794 2211 2365 2612, 2715]] (639h) | −0.1618 | 0.0971 | 5.17 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 53\639 | 99.53 | 18/17 | Quindro |
| 1 | 206\639 | 386.85 | 5/4 | Counterwürschmidt |
| 9 | 168\639 (26\639) |
315.49 (48.83) |
6/5 (36/35) |
Ennealimmal / enneabiotic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct