257edo: Difference between revisions
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{{EDO intro|257}} | |||
==Theory== | |||
{{harmonics in equal|257}} | |||
257edo is good at 2.11.13.37.41.53.59.67 subgroup. | |||
It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 7-limit: <257 407 597 721| (patent val), <257 408 597 722| (257bd), <257 407 596 721| (257c), and <257 407 597 722| (257d). Using the patent val, it tempers out 393216/390625 ([[Würschmidt comma]]) and |-36 33 -7> in the 5-limit; 1029/1024, 177147/175000, and 393216/390625 in the 7-limit. Using the 257bd val, it tempers out 15625/15552 (kleisma) and |69 -42 -1> in the 5-limit; 4000/3969, 6144/6125, and 40353607/39858075 in the 7-limit. Using the 257c val, it tempers out 34171875/33554432 (ampersand comma) and 762939453125/753145430616 ([[Maja family|maja comma]]) in the 5-limit; 225/224, 1029/1024, and 854492187500/847288609443 in the 7-limit; 243/242, 385/384, 441/440, and 152587890625/148550704533 in the 11-limit, providing for the 11-limit [[Gamelismic clan|miracle temperament]]. Using the 257d val, it tempers out 1728/1715, 413343/409600, and 703125/702464 in the 7-limit. | |||
257edo is the 55th [[prime EDO]]. | 257edo is the 55th [[prime EDO]]. | ||
Revision as of 11:52, 19 July 2022
Theory
| 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) | |
257edo is good at 2.11.13.37.41.53.59.67 subgroup.
It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 7-limit: <257 407 597 721| (patent val), <257 408 597 722| (257bd), <257 407 596 721| (257c), and <257 407 597 722| (257d). Using the patent val, it tempers out 393216/390625 (Würschmidt comma) and |-36 33 -7> in the 5-limit; 1029/1024, 177147/175000, and 393216/390625 in the 7-limit. Using the 257bd val, it tempers out 15625/15552 (kleisma) and |69 -42 -1> in the 5-limit; 4000/3969, 6144/6125, and 40353607/39858075 in the 7-limit. Using the 257c val, it tempers out 34171875/33554432 (ampersand comma) and 762939453125/753145430616 (maja comma) in the 5-limit; 225/224, 1029/1024, and 854492187500/847288609443 in the 7-limit; 243/242, 385/384, 441/440, and 152587890625/148550704533 in the 11-limit, providing for the 11-limit miracle temperament. Using the 257d val, it tempers out 1728/1715, 413343/409600, and 703125/702464 in the 7-limit.
257edo is the 55th prime EDO.
257 is also the number of sides of a polygon that is known for being constructed with compass and straightedge.