129edo
| ← 128edo | 129edo | 130edo → |
129 equal divisions of the octave (abbreviated 129edo or 129ed2), also called 129-tone equal temperament (129tet) or 129 equal temperament (129et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 129 equal parts of about 9.3 ¢ each. Each step represents a frequency ratio of 21/129, or the 129th root of 2.
129edo is inconsistent to the 5-odd-limit and both harmonics 3 and 5 are about halfway between its steps. The patent val is enfactored in the 5-limit, with the same tuning as 43edo. It is the last patent val that tempers out 81/80 so as to support meantone and its higher-limit expansions. It also tempers out 1029/1024 and 1728/1715 in the 7-limit; 176/175 and 540/539 in the 11-limit; 507/500, 676/675 and 847/845 in the 13-limit; 221/220 in the 17-limit; 171/170 and 286/285 in the 19-limit. It provides the optimal patent val for the 11-limit rank-3 clio temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.28 | +4.38 | -1.38 | +0.74 | -2.48 | -3.32 | +0.10 | -2.63 | +0.16 | +3.64 | +4.28 |
| Relative (%) | -46.0 | +47.1 | -14.9 | +8.0 | -26.7 | -35.7 | +1.1 | -28.3 | +1.7 | +39.1 | +46.1 | |
| Steps (reduced) |
204 (75) |
300 (42) |
362 (104) |
409 (22) |
446 (59) |
477 (90) |
504 (117) |
527 (11) |
548 (32) |
567 (51) |
584 (68) | |
Subsets and supersets
Since 129 factors into 3 × 43, 129edo contains 3edo and 43edo as its subsets.