257edo

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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 2^1/257, 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.