125edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| Line 48: | Line 57: | ||
| 0.622 | | 0.622 | ||
| 6.47 | | 6.47 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 106: | Line 122: | ||
| 4/3<br>(81/80) | | 4/3<br>(81/80) | ||
| [[Pental (temperament)|Pental]] | | [[Pental (temperament)|Pental]] | ||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[Category:Catakleismic]] | [[Category:Catakleismic]] | ||
Revision as of 12:53, 16 November 2024
| ← 124edo | 125edo | 126edo → |
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.
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) | |
Subsets and supersets
Since 125 factors into 53, 125edo contains 5edo and 25edo as its subsets. 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* | 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) |
Hemipental |
| 5 | 52\125 (2\125) |
499.2 (19.2) |
4/3 (81/80) |
Pental |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct