135edo
| ← 134edo | 135edo | 136edo → |
135 equal divisions of the octave (abbreviated 135edo or 135ed2), also called 135-tone equal temperament (135tet) or 135 equal temperament (135et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 135 equal parts of about 8.89 ¢ each. Each step represents a frequency ratio of 21/135, or the 135th root of 2.
Theory
135edo is consistent to the 7-odd-limit, but there is a large relative delta for the 5th and 13th harmonics. As every other step of the full 13-limit monster – 270et, 135et probably makes more sense as a 2.3.7.11 subgroup temperament, where it tempers out the garischisma and the symbiotic comma. If we consider the full 13-limit, the flat-tending ⟨135 214 313 379 467 499] (135f) and the sharp-tending ⟨135 214 314 379 467 500] (135c) are reasonable choices.
Using the 135f val, it tempers out 32805/32768 (schisma) and [-11 -15 15⟩ (pentadecal comma) in the 5-limit; 225/224, 3125/3087, and 28824005/28697814 in the 7-limit, 385/384, 540/539, 2200/2187, 12005/11979 and the quartisma in the 11-limit; 169/168 and 364/363 in the 13-limit.
Using the 135c val, it tempers out 1594323/1562500 (unicorn comma) and 50331648/48828125 (magus comma) in the 5-limit; 126/125, 10976/10935, and 589824/588245 in the 7-limit; 176/175, 441/440, 14641/14580 and 16384/16335 in the 11-limit; 196/195, 351/350, 352/351, 676/675, and 6656/6655 in the 13-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.27 | -4.09 | +0.06 | -0.21 | +3.92 | +1.71 | -4.18 | +2.84 | +1.53 | +1.63 |
| Relative (%) | +0.0 | +3.0 | -46.0 | +0.7 | -2.3 | +44.1 | +19.3 | -47.0 | +31.9 | +17.3 | +18.3 | |
| Steps (reduced) |
135 (0) |
214 (79) |
313 (43) |
379 (109) |
467 (62) |
500 (95) |
552 (12) |
573 (33) |
611 (71) |
656 (116) |
669 (129) | |
Subsets and supersets
Since 135 factors into 33 × 5, 135edo has subset edos 3, 5, 9, 15, and 45. 270edo, which doubles it, provides good correction for the approximation to harmonics 5, 13, and 19.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [214 -135⟩ | [⟨135 214]] | -0.0843 | 0.0843 | 0.95 |
| 2.3.7 | 33554432/33480783, 40353607/40310784 | [⟨135 214 379]] | -0.0637 | 0.0747 | 0.84 |
| 2.3.7.11 | 19712/19683, 41503/41472, 43923/43904 | [⟨135 214 379 467]] | -0.0328 | 0.0840 | 0.94 |