257edo: Difference between revisions

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'''257edo''' is the [[EDO|equal division of the octave]] into 257 parts of 4.66926 [[cent]]s each. 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 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]].

[[~~Category:Equal divisions of~~ the ~~octave~~]]

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.

[[~~Category:Prime EDO~~]]

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}.

=== Odd harmonics ===

=== Subsets and supersets ===

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

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-'''257edo''' is the [[EDO|equal division of the octave]] into 257 parts of 4.66926 [[cent]]s each. It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 7-limit: &lt;257 407 597 721| (patent val), &lt;257 408 597 722| (257bd), &lt;257 407 596 721| (257c), and &lt;257 407 597 722| (257d). Using the patent val, it tempers out 393216/390625 ([[Würschmidt comma]]) and |-36 33 -7&gt; 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&gt; 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.
+{{Infobox ET}}
+{{ED intro}}
-edo is the 55th [[prime EDO]].
+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]].
-[[Category:Equal divisions of the octave]]
+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.
-[[Category:Prime EDO]]
+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}.
+=== Odd harmonics ===
+{{Harmonics in equal|257}}
+=== Subsets and supersets ===
+edo is the 55th [[prime edo]].