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.
125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding odd limit. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the patent val). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their octave complements. (Therefore, all 331 other intervals are mapped with strictly less than 4.8 ¢ of error.)
* including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.16 | -2.31 | +0.77 | -4.12 | +4.27 | +0.64 | +0.09 | -4.27 | -2.38 | -2.64 | -1.74 |
| Relative (%) | +0.0 | -12.0 | -24.1 | +8.1 | -42.9 | +44.5 | +6.7 | +0.9 | -44.5 | -24.8 | -27.5 | -18.2 | |
| Steps (reduced) |
125 (0) |
198 (73) |
290 (40) |
351 (101) |
432 (57) |
463 (88) |
511 (11) |
531 (31) |
565 (65) |
607 (107) |
619 (119) |
651 (26) | |
| Harmonic | 41 | 43 | 47 | 53 | 59 | 61 | 67 | |
|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.94 | -2.72 | -3.11 | +0.10 | -3.17 | -3.28 | -2.51 |
| Relative (%) | +30.6 | -28.3 | -32.4 | +1.0 | -33.0 | -34.2 | -26.1 | |
| Steps (reduced) |
670 (45) |
678 (53) |
694 (69) |
716 (91) |
735 (110) |
741 (116) |
758 (8) | |
| Harmonic | 71 | 73 | 79 | 83 | 89 | 97 | 101 | 103 | 107 | 109 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.70 | +2.61 | +0.26 | +1.15 | -4.48 | +0.10 | -2.65 | +1.80 | +3.04 | -0.22 |
| Relative (%) | +28.2 | +27.2 | +2.7 | +12.0 | -46.7 | +1.1 | -27.6 | +18.7 | +31.7 | -2.3 | |
| Steps (reduced) |
769 (19) |
774 (24) |
788 (38) |
797 (47) |
809 (59) |
825 (75) |
832 (82) |
836 (86) |
843 (93) |
846 (96) | |
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