257edo: Difference between revisions
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{{EDO intro|257}} | {{EDO intro|257}} | ||
==Theory== | ==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. | |||
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. | |||
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 === | |||
257edo is the 55th [[prime EDO]]. | 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. | 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|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:Prime EDO]] | [[Category:Prime EDO]] | ||
Revision as of 17:20, 3 January 2024
| ← 256edo | 257edo | 258edo → |
Theory
257edo is inconsistent 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 ⟨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, ⟨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.
Using the 257c val, [257 407 596 721⟩ 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, [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 2.11.13.15.27.37.41.49.53.59.67 subgroup. A comma basis for the 2.11.13.15.27 subgroup is {4225/4224, 256000/255879, 225000/224939, 4159375/4153344}.
Odd harmonics
| 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) | |
Subsets and supersets
257edo is the 55th prime EDO.
Trivia
257 is also the number of sides of a polygon that is known for being constructed with compass and straightedge.