257edo: Difference between revisions

{{EDO intro|257}}

== Theory ==

257edo is in[[consistent]] to the [[5-odd-limit]], with significant errors on [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[9/1|9]].

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

Using the 257c val, {{val| 257 407 '''596''' 721 }} it tempers out 34171875/33554432 ([[ampersand]]) and {{monzo| -3 -23 17}} ([[Maja family|maja comma]]) in the 5-limit; [[225/224]], 1029/1024, and {{monzo| 2 -25 15 1 }} in the 7-limit; [[243/242]], [[385/384]], [[441/440]], and {{monzo| 0 -2 16 -1 -9 }} in the 11-limit, providing for the 11-limit [[Gamelismic clan|miracle temperament]]. Using the 257d val, {{val| 257 407 597 '''722''' }} it tempers out [[1728/1715]], 413343/409600, and [[703125/702464]] in the 7-limit.

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}.

{{Harmonics in equal|257}}

257edo is the 55th [[prime edo]].