125edo: Difference between revisions
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The '''125 equal temperament''' divides the octave into 125 equal parts of exactly 9.6 | The '''125 equal divisions of the octave''' ('''125edo'''), or the '''125(-tone) equal temperament''' ('''125tet''', '''125et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 125 [[equal]] parts of exactly 9.6 [[cent]]s each. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. | ||
== Theory == | == Theory == | ||
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[[Category:Theory]] | [[Category:Theory]] | ||
[[Category:Catakleismic]] | [[Category:Catakleismic]] | ||
[[Category:Slender]] | |||
Revision as of 10:31, 29 July 2021
The 125 equal divisions of the octave (125edo), or the 125(-tone) equal temperament (125tet, 125et) when viewed from a regular temperament perspective, divides the octave into 125 equal parts of exactly 9.6 cents each. 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.
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
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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 octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 4\125 | 38.4 | 49/48 | Slender |
| 1 | 19\125 | 182.4 | 10/9 | Minortone / mitonic |
| 1 | 24\125 | 230.4 | 8/7 | Gamera |
| 1 | 33\125 | 316.8 | 6/5 | Hanson / catakleismic |
| 1 | 52\125 | 499.2 | 4/3 | Gracecordial |
| 1 | 61\125 | 585.6 | 7/5 | Merman |
| 5 | 24\125 (1\125) |
230.4 (9.6) |
8/7 (176/175) |
Hemipental |
| 5 | 52\125 (2\125) |
499.2 (19.2) |
4/3 (81/80) |
Pental |