# 257edo

 ← 256edo 257edo 258edo →
Prime factorization 257 (prime)
Step size 4.66926¢
Fifth 150\257 (700.389¢)
Semitones (A1:m2) 22:21 (102.7¢ : 98.05¢)
Dual sharp fifth 151\257 (705.058¢)
Dual flat fifth 150\257 (700.389¢)
Dual major 2nd 44\257 (205.447¢)
Consistency limit 3
Distinct consistency limit 3

257 equal divisions of the octave (abbreviated 257edo or 257ed2), also called 257-tone equal temperament (257tet) or 257 equal temperament (257et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 257 equal parts of about 4.67 ¢ each. Each step represents a frequency ratio of 21/257, or the 257th root of 2.

257edo is inconsistent to the 5-odd-limit, with significant errors on harmonics 3, 5, 7, and 9.

In the 7-limit, there is a number of mappings to be considered. First is the 257 407 597 721] (patent val), where the equal temperament tempers out 393216/390625 (würschmidt comma) in the 5-limit and 1029/1024 and 177147/175000 in the 7-limit. Using the 257d val, 257 407 597 722] it tempers out 1728/1715, 413343/409600, and 703125/702464 in the 7-limit. Using the 257bd val, 257 408 597 722], it tempers out 15625/15552 (kleisma) in the 5-limit; 4000/3969, 6144/6125, and 40353607/39858075 in the 7-limit.

Using the 257c val, 257 407 596 721] it tempers out 34171875/33554432 (ampersand) and [-3 -23 17 (maja comma) in the 5-limit; 225/224, 1029/1024, and [2 -25 15 1 in the 7-limit; 243/242, 385/384, 441/440, and [0 -2 16 -1 -9 in the 11-limit, providing for the 11-limit miracle temperament.

In higher limits, 257edo is a strong 2.11.13.15.27 subgroup tuning, and it is overall good at the 2.27.15.11.13.37.41.49.53.59.67 subgroup. A comma basis for the 2.27.15.11.13 subgroup is {4225/4224, 256000/255879, 225000/224939, 4159375/4153344}.

### Odd harmonics

Approximation of odd harmonics in 257edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.57 +1.23 -2.29 +1.54 -0.35 -0.06 -0.33 -2.23 +1.32 +0.81 +2.08
Relative (%) -33.5 +26.4 -49.0 +32.9 -7.4 -1.3 -7.1 -47.8 +28.3 +17.4 +44.5
Steps
(reduced)
407
(150)
597
(83)
721
(207)
815
(44)
889
(118)
951
(180)
1004
(233)
1050
(22)
1092
(64)
1129
(101)
1163
(135)

### Subsets and supersets

257edo is the 55th prime edo.