169edo
| ← 168edo | 169edo | 170edo → |
169 equal divisions of the octave (abbreviated 169edo or 169ed2), also called 169-tone equal temperament (169tet) or 169 equal temperament (169et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 169 equal parts of about 7.1 ¢ each. Each step represents a frequency ratio of 21/169, or the 169th root of 2.
169edo is inconsistent to the 5-odd-limit and higher limits, with two mappings possible for the 5-limit: ⟨169 268 392] (patent val) and ⟨169 268 393] (169c).
Using the patent val, it tempers out the sycamore comma, 48828125/47775744 and the rodan comma, 131072000/129140163 in the 5-limit; 245/243, 1029/1024, and 9765625/9633792 in the 7-limit; 385/384, 441/440, 896/891, and 312500/307461 in the 11-limit; 676/675, 975/968, and 1625/1617 in the 13-limit. Using the 169d val, it tempers out 225/224, 51200/50421, and 1071875/1062882 in the 7-limit; 2200/2187, 2420/2401, 2560/2541, and 6000/5929 in the 11-limit; 169/168, 364/363, 640/637, and 676/675 in the 13-limit.
Using the 169cdf val, it tempers out the valentine comma, 1990656/1953125 and the vulture comma, 10485760000/10460353203 in the 5-limit; 1728/1715, 5120/5103, and 235298/234375 in the 7-limit; 176/175, 540/539, 8019/8000, and 43923/43904 in the 11-limit; 351/350, 352/351, 640/637, and 676/675 in the 13-limit, supporting the buzzard temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +1.00 | -2.88 | -3.15 | +2.53 | -2.66 | +1.55 | +0.71 | -3.42 | +0.01 | -1.84 |
| Relative (%) | +0.0 | +14.1 | -40.6 | -44.3 | +35.6 | -37.4 | +21.9 | +10.0 | -48.2 | +0.1 | -25.9 | |
| Steps (reduced) |
169 (0) |
268 (99) |
392 (54) |
474 (136) |
585 (78) |
625 (118) |
691 (15) |
718 (42) |
764 (88) |
821 (145) |
837 (161) | |
Subsets and supersets
Since 169 factors into 132, 169edo contains 13edo as its only nontrivial subset. 338edo, which doubles it, provides good correction for the approximations of harmonics 5 and 7.