257edo: Difference between revisions

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==Theory==

257edo is in[[consistent]] to the 5-limit and higher 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 {{val|257 407 597 721}} (patent val), where 257edo tempers out 393216/390625 ([[Würschmidt comma]]) in the 5-limit and 1029/1024, 177147/175000, and 393216/390625 in the 7-limit. Using the 257bd val, {{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.

~~257edo is good at 2.~~11.~~13.37.41.53.59.67 subgroup~~.

Using the 257c val, {{monzo|257 407 '''596''' 721}} 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, {{monzo|257 407 597 '''722'''}} it tempers out 1728/1715, 413343/409600, and 703125/702464 in the 7-limit.

~~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~~.

In higher limits, 257edo is a strong 2.11.13.15.27 subgroup tuning, and it is overall good at the 2.11.13.15.27.37.41.49.53.59.67 subgroup. A comma basis for the 2.11.13.15.27 subgroup is {4225/4224, 256000/255879, 225000/224939, 4159375/4153344}.

=== Odd harmonics ===

=== Subsets and supersets ===

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

== Trivia ==

257 is also the [[wikipedia:257-gon|number of sides of a polygon]] that is known for being constructed with compass and straightedge.

[[Category:Equal divisions of the octave|###]]

[[Category:Prime EDO]]

@@ Line 2: / Line 2: @@
 {{EDO intro|257}}
 ==Theory==
-{{harmonics in equal|257}}
+edo is in[[consistent]] to the 5-limit and higher 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 {{val|257 407 597 721}} (patent val), where 257edo tempers out 393216/390625 ([[Würschmidt comma]])  in the 5-limit and 1029/1024, 177147/175000, and 393216/390625 in the 7-limit. Using the 257bd val, {{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.
-edo is good at 2.11.13.37.41.53.59.67 subgroup.
+Using the 257c val, {{monzo|257 407 '''596''' 721}} 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, {{monzo|257 407 597 '''722'''}} it tempers out 1728/1715, 413343/409600, and 703125/702464 in the 7-limit.
-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.
+In higher limits, 257edo is a strong 2.11.13.15.27 subgroup tuning, and it is overall good at the 2.11.13.15.27.37.41.49.53.59.67 subgroup. A comma basis for the 2.11.13.15.27 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]].
+== Trivia ==
 is also the [[wikipedia:257-gon|number of sides of a polygon]] that is known for being constructed with compass and straightedge.
 [[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Prime EDO]]