156edo
| ← 155edo | 156edo | 157edo → |
156 equal divisions of the octave (abbreviated 156edo or 156ed2), also called 156-tone equal temperament (156tet) or 156 equal temperament (156et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 156 equal parts of about 7.69 ¢ each. Each step represents a frequency ratio of 21/156, or the 156th root of 2.
The equal temperament tempers out 531441/524288 (Pythagorean comma) and [-27 -2 13⟩ (ditonmic comma) in the 5-limit, as well as [8 14 -13⟩ (parakleisma); 225/224, 250047/250000, and 589824/588245 in the 7-limit. Using the patent val, it tempers out 441/440, 1375/1372, 4375/4356, and 65536/65219 in the 11-limit; 351/350, 364/363, 625/624, 1625/1617, and 13122/13013 in the 13-limit. Using the 156e val, it tempers out 385/384, 540/539, 1331/1323, and 78408/78125 in the 11-limit; 351/350, 625/624, 847/845, and 1001/1000 in the 13-limit. It supports compton and gives a good tuning for the 5- and 7-limit version thereof.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.96 | -1.70 | +0.40 | +2.53 | -2.07 | +2.74 | +2.49 | +2.49 | +1.19 | +1.12 |
| Relative (%) | +0.0 | -25.4 | -22.1 | +5.3 | +32.9 | -26.9 | +35.6 | +32.3 | +32.4 | +15.5 | +14.5 | |
| Steps (reduced) |
156 (0) |
247 (91) |
362 (50) |
438 (126) |
540 (72) |
577 (109) |
638 (14) |
663 (39) |
706 (82) |
758 (134) |
773 (149) | |
Subsets and supersets
Sinece 156 factors into 22 × 3 × 13, 156edo has subset edos 2, 3, 4, 6, 12, 13, 26, 39, 52, and 78. It is the smallest edo to contain both 12edo and 13edo as subsets.