125edo

← 124edo

125edo

126edo →

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

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.

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. 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

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 distinct