125edo
| ← 124edo | 125edo | 126edo → |
125 equal divisions of the octave (abbreviated 125edo or 125ed2), also called 125-tone equal temperament (125tet) or 125 equal temperament (125et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 125 equal parts of exactly 9.6 ¢ each. Each step represents a frequency ratio of 21/125, or the 125th root of 2.
Theory
The equal temperament tempers out 15625/15552 in the 5-limit; 225/224 and 4375/4374 in the 7-limit; 385/384 and 540/539 in the 11-limit. It defines the optimal patent val for 7- and 11-limit slender temperament. In the 13-limit the 125f val ⟨125 198 290 351 432 462] does a better job, where it tempers out 169/168, 325/324, 351/350, 625/624 and 676/675, providing a good tuning for catakleismic. Among well-known intervals, the approximation of 10/9, as 19 steps, is notable for being a strong convergent, within 0.004 cents.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.16 | -2.31 | +0.77 | -4.12 | +4.27 | +0.64 | +0.09 | -4.27 | -2.38 | -2.64 |
| Relative (%) | +0.0 | -12.0 | -24.1 | +8.1 | -42.9 | +44.5 | +6.7 | +0.9 | -44.5 | -24.8 | -27.5 | |
| Steps (reduced) |
125 (0) |
198 (73) |
290 (40) |
351 (101) |
432 (57) |
463 (88) |
511 (11) |
531 (31) |
565 (65) |
607 (107) |
619 (119) | |
Octave stretch
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly stretching the octave, though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as 198edt and 323ed6 are great demonstrations of this.
Subsets and supersets
Since 125 factors into primes as 53, 125edo contains 5edo and 25edo as subset edos. Being the cube closest to division of the octave by the Germanic long hundred, 125edo has a unit step which is the cubic (fine) relative cent of 1edo. Using every 9th step, or 1ed86.4c still encapsulates many of its best-tuned harmonics.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-198 125⟩ | [⟨125 198]] | +0.364 | 0.364 | 3.80 |
| 2.3.5 | 15625/15552, 17433922005/17179869184 | [⟨125 198 290]] | +0.575 | 0.421 | 4.39 |
| 2.3.5.7 | 225/224, 4375/4374, 589824/588245 | [⟨125 198 290 351]] | +0.362 | 0.519 | 5.40 |
| 2.3.5.7.11 | 225/224, 385/384, 1331/1323, 4375/4374 | [⟨125 198 290 351 432]] | +0.528 | 0.570 | 5.94 |
| 2.3.5.7.11.13 | 169/168, 225/224, 325/324, 385/384, 1331/1323 | [⟨125 198 290 351 432 462]] (125f) | +0.680 | 0.622 | 6.47 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 4\125 | 38.4 | 49/48 | Slender |
| 1 | 12\125 | 115.2 | 77/72 | Semigamera |
| 1 | 19\125 | 182.4 | 10/9 | Mitonic |
| 1 | 24\125 | 230.4 | 8/7 | Gamera |
| 1 | 33\125 | 316.8 | 6/5 | Catakleismic |
| 1 | 52\125 | 499.2 | 4/3 | Gracecordial |
| 1 | 61\125 | 585.6 | 7/5 | Merman |
| 5 | 26\125 (1\125) |
249.6 (9.6) |
81/70 (176/175) |
Hemiquintile |
| 5 | 52\125 (2\125) |
499.2 (19.2) |
4/3 (81/80) |
Quintile |
| 5 | 33\125 (8\125) |
316.8 (76.8) |
6/5 (24/23~23/22~22/21) |
Thunderclysmic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct