639edo

 ← 638edo 639edo 640edo →
Prime factorization 32 × 71
Step size 1.87793¢
Fifth 374\639 (702.347¢)
Semitones (A1:m2) 62:47 (116.4¢ : 88.26¢)
Consistency limit 17
Distinct consistency limit 17

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 of 3 to 17 all tuned sharp. The 639h val gives a reasonable approximation of harmonic 19, where it 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 11-limit ennealimmal and its 13-limit extension ennealimmalis.

Prime harmonics

Approximation of prime harmonics in 639edo
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 = 32 × 71, it 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

Table of rank-2 temperaments by generator
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 / ennealimmalis

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct