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 with large relative error for the 5th and 13th harmonics. As every other step of the full 13-limit monster – 270et, 135et supports an easy but extremely accurate equal tuning of the 2.3.7.11-subgroup temperament gary. As an equal temperament, it is characterized by tempering out the garischisma, the symbiotic comma and the argyria, which indirectly also tempers out the septiennealimma, the olympia and the chrysia. On top of this, it also has fairly good approximations to primes 17, 29, and 31.
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) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.46 | -2.40 | +4.04 | +1.16 | -2.39 | -1.39 | +3.12 | +0.69 | -1.92 | +3.32 | -0.09 |
| Relative (%) | -27.6 | -27.0 | +45.4 | +13.1 | -26.9 | -15.7 | +35.0 | +7.8 | -21.6 | +37.4 | -1.0 | |
| Steps (reduced) |
703 (28) |
723 (48) |
733 (58) |
750 (75) |
773 (98) |
794 (119) |
801 (126) |
819 (9) |
830 (20) |
836 (26) |
851 (41) | |
Subsets and supersets
Since 135 factors into primes as 33 × 5, 135edo has subset edos 3, 5, 9, 15, 27, and 45. 270edo, which doubles it, notably provides extremely good corrections 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 |
| 2.3.7.11.17 | 1089/1088, 2058/2057, 5832/5831, 19712/19683 | [⟨135 214 379 467 552]] | −0.1100 | 0.1716 | 1.93 |