# 253edo

 ← 252edo 253edo 254edo →
Prime factorization 11 × 23
Step size 4.74308¢
Fifth 148\253 (701.976¢)
(semiconvergent)
Semitones (A1:m2) 24:19 (113.8¢ : 90.12¢)
Consistency limit 17
Distinct consistency limit 17

253 equal divisions of the octave (abbreviated 253edo or 253ed2), also called 253-tone equal temperament (253tet) or 253 equal temperament (253et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 253 equal parts of about 4.74 ¢ each. Each step represents a frequency ratio of 21/253, or the 253rd root of 2.

## Theory

253edo is consistent to the 17-odd-limit, approximating the fifth by 148\253 (0.021284 cents sharper than the just 3/2), and the prime harmonics from 5 to 17 are all slightly flat. The equal temperament tempers out 32805/32768 in the 5-limit; 2401/2400 in the 7-limit; 385/384, 1375/1372 and 4000/3993 in the 11-limit; 325/324, 1575/1573 and 2200/2197 in the 13-limit; 375/374 and 595/594 in the 17-limit. It provides the optimal patent val for the tertiaschis temperament, and a good tuning for the sesquiquartififths temperament in the higher limits.

### Prime harmonics

Approximation of prime harmonics in 253edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.02 -2.12 -1.24 -1.12 -1.00 -0.61 +1.30 -2.19 -0.33 -1.95
Relative (%) +0.0 +0.4 -44.8 -26.1 -23.6 -21.1 -12.8 +27.4 -46.1 -6.9 -41.2
Steps
(reduced)
253
(0)
401
(148)
587
(81)
710
(204)
875
(116)
936
(177)
1034
(22)
1075
(63)
1144
(132)
1229
(217)
1253
(241)

### Subsets and supersets

253 = 11 × 23, and has subset edos 11edo and 23edo.

## Regular temperament properties

Subgroup Comma List Mapping Optimal 8ve
Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [401 -253 [253 401]] -0.007 0.007 0.14
2.3.5 32805/32768, [-4 -37 27 [253 401 587]] +0.300 0.435 9.16
2.3.5.7 2401/2400, 32805/32768, 390625/387072 [253 401 587 710]] +0.335 0.381 8.03
2.3.5.7.11 385/384, 1375/1372, 4000/3993, 19712/19683 [253 401 587 710 875]] +0.333 0.341 7.19
2.3.5.7.11.13 325/324, 385/384, 1375/1372, 1575/1573, 2200/2197 [253 401 587 710 875 936]] +0.323 0.312 6.58
2.3.5.7.11.13.17 325/324, 375/374, 385/384, 595/594, 1275/1274, 2200/2197 [253 401 587 710 875 936 1034]] +0.298 0.295 6.22

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 35\253 166.01 11/10 Tertiaschis
1 37\253 175.49 448/405 Sesquiquartififths
1 105\253 498.02 4/3 Helmholtz
1 123\253 583.40 7/5 Cotritone

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct