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
125edo defines the optimal patent val for 7- and 11-limit slender temperament. It 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. 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.
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 | -12 | -24 | +8 | -43 | +45 | +7 | +1 | -45 | -25 | -27 | |
| Steps (reduced) |
125 (0) |
198 (73) |
290 (40) |
351 (101) |
432 (57) |
463 (88) |
511 (11) |
531 (31) |
565 (65) |
607 (107) |
619 (119) | |
Miscellaneous properties
125 is 5 cubed. 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.
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 (Reduced) |
Cents (Reduced) |
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) |
Hemipental |
| 5 | 52\125 (2\125) |
499.2 (19.2) |
4/3 (81/80) |
Pental |