139edo

 ← 138edo 139edo 140edo →
Prime factorization 139 (prime)
Step size 8.63309¢
Fifth 81\139 (699.281¢)
Semitones (A1:m2) 11:12 (94.96¢ : 103.6¢)
Consistency limit 3
Distinct consistency limit 3

139 equal divisions of the octave (abbreviated 139edo or 139ed2), also called 139-tone equal temperament (139tet) or 139 equal temperament (139et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 139 equal parts of about 8.63 ¢ each. Each step represents a frequency ratio of 21/139, or the 139th root of 2.

139edo is inconsistent to the 5-odd-limit and higher limits, with three mappings possible for the 5-limit: 139 220 323] (patent val), 139 221 323] (139b), and 139 220 322] (139c).

Using the patent val, it tempers out 1990656/1953125 (valentine comma) and 43046721/41943040 (python comma) in the 5-limit; 126/125, 1029/1024, and 4782969/4705960 in the 7-limit, supporting the 7-limit valentine temperament; 540/539, 1944/1925, 2835/2816, and 12005/11979 in the 11-limit; 364/363, 676/675, 1287/1280, 1701/1690, and 1716/1715 in the 13-limit. Using the alternative 139f val, it tempers out 144/143, 196/195, 351/350, 4096/4095, and 4455/4394 in the 13-limit.

Using the 139df val, it tempers out 225/224, 51200/50421, and 157464/153125 in the 7-limit; 99/98, 176/175, and 15309/15125 in the 11-limit, supporting the 11-limit interpental temperament; 144/143, 648/637, 847/845, 1575/1573, and 3159/3125 in the 13-limit.

Using the 139ce val, it tempers out 3125/3072 (magic comma) and [-42 28 -1 in the 5-limit; 1029/1024, 3125/3087, and 19683/19600 in the 7-limit, supporting the 7-limit trismegistus temperament; 540/539, 1331/1323, and 1815/1792 in the 11-limit; 275/273, 325/324, 847/845, 1287/1280, and 1575/1573 in the 13-limit.

Using the 139bdf val, it tempers out 393216/390625 (würschmidt comma), and 409600000/387420489 in the 5-limit; 245/243, 3125/3087, and 131072/127575 in the 7-limit; 176/175, 1232/1215, and 2560/2541 in the 11-limit; 275/273, 512/507, 625/624, 847/845, and 1625/1617 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 139edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -2.67 +2.18 -1.92 +3.28 +1.20 -3.12 -0.50 -1.36 -3.99 +4.04 +1.94
Relative (%) -31.0 +25.2 -22.2 +38.0 +13.9 -36.1 -5.8 -15.7 -46.2 +46.8 +22.5
Steps
(reduced)
220
(81)
323
(45)
390
(112)
441
(24)
481
(64)
514
(97)
543
(126)
568
(12)
590
(34)
611
(55)
629
(73)

Subsets and supersets

139edo is the 34th prime edo, following 137edo and before 149edo.