257edo
| ← 256edo | 257edo | 258edo → |
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 4.669 ¢ 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
| 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 (%) | -34 | +26 | -49 | +33 | -7 | -1 | -7 | -48 | +28 | +17 | +44 | |
| 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.