139edo
| ← 138edo | 139edo | 140edo → |
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
| 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.