125edo

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Prime factorization 53
Step size 9.6¢ 
Fifth 73\125 (700.8¢)
Semitones (A1:m2) 11:10 (105.6¢ : 96¢)
Consistency limit 9
Distinct consistency limit 9

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.

Prime harmonics

Approximation of prime harmonics in 125edo
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

Table of rank-2 temperaments by generator
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