157edo

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← 156edo157edo158edo →
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
1 23\157 175.80 72/65 Quadrafifths
1 46\157 351.59 49/40 Hemififths
1 56\157 428.03 2800/2187 Geb / osiris
1 58\157 443.31 162/125 Warrior
1 64\157 489.17 250/189 Catafourth

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