# 157edo

 ← 156edo 157edo 158edo →
Prime factorization 157 (prime)
Step size 7.64331¢
Fifth 92\157 (703.185¢)
Semitones (A1:m2) 16:11 (122.3¢ : 84.08¢)
Consistency limit 9
Distinct consistency limit 9

157 equal divisions of the octave (abbreviated 157edo or 157ed2), also called 157-tone equal temperament (157tet) or 157 equal temperament (157et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 157 equal parts of about 7.64 ¢ each. Each step represents a frequency ratio of 21/157, or the 157th root of 2.

## Theory

157et tempers out 78732/78125 (sensipent comma) and [37 -16 -5 (quinticosiennic comma) in the 5-limit; 2401/2400, 5120/5103, and 110592/109375 in the 7-limit (supporting the hemififths and the catafourth temperaments). Using the patent val, it tempers out 176/175, 1331/1323, 3773/3750 and 8019/8000 in the 11-limit; 351/350, 352/351, 847/845, 1573/1568, and 2197/2187 in the 13-limit.

### Odd harmonics

Approximation of odd harmonics in 157edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.23 +3.50 +1.87 +2.46 -1.00 +0.24 -2.92 +2.05 +0.58 +3.10 -1.52
Relative (%) +16.1 +45.7 +24.5 +32.2 -13.1 +3.1 -38.2 +26.8 +7.5 +40.6 -19.9
Steps
(reduced)
249
(92)
365
(51)
441
(127)
498
(27)
543
(72)
581
(110)
613
(142)
642
(14)
667
(39)
690
(62)
710
(82)

### Subsets and supersets

157edo is the 37th prime edo.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [249 -157 [157 249]] -0.388 0.388 5.08
2.3.5 78732/78125, 37 -16 -5] [157 249 365]] -0.760 0.614 8.04
2.3.5.7 2401/2400, 5120/5103, 78732/78125 [157 249 365 441]] -0.737 0.533 6.98
2.3.5.7.11 176/175, 1331/1323, 2401/2400, 5120/5103 [157 249 365 441 543]] -0.532 0.629 8.24
2.3.5.7.11.13 176/175, 351/350, 847/845, 1331/1323, 2197/2187 [157 249 365 441 543 581]] -0.454 0.600 7.86
2.3.5.7.11.13.17 176/175, 256/255, 351/350, 442/441, 715/714, 2197/2187 [157 249 365 441 543 581 642]] -0.461 0.556 7.28
2.3.5.7.11.13.17.19 176/175, 256/255, 286/285, 351/350, 361/360, 442/441, 476/475 [157 249 365 441 543 581 642 667]] -0.420 0.531 6.95

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperament
1 13\157 99.36 18/17 Quinticosiennic