257edo: Difference between revisions

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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 257d val, {{val| 257 407 597 '''722''' }} it tempers out [[1728/1715]], 413343/409600, and [[703125/702464]] 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.

In the 7-limit, there are 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 257d val, {{val| 257 407 597 '''722''' }} it tempers out [[1728/1715]], 413343/409600, and [[703125/702464]] 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 257c val, {{val| 257 407 '''596''' 721 }} it tempers out 34171875/33554432 ([[ampersand comma]]) 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, supporting 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}.

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 edo 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 257d val, {{val| 257 407 597 '''722''' }} it tempers out [[1728/1715]], 413343/409600, and [[703125/702464]] 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.
+In the 7-limit, there are 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 257d val, {{val| 257 407 597 '''722''' }} it tempers out [[1728/1715]], 413343/409600, and [[703125/702464]] 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 257c val, {{val| 257 407 '''596''' 721 }} it tempers out 34171875/33554432 ([[ampersand comma]]) 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, supporting 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}.