257edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|257}} | {{EDO intro|257}} | ||
==Theory== | == Theory == | ||
257edo is in[[consistent]] to the 5-limit | 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 | 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, {{ | 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 2.11.13 | 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 === | === Odd harmonics === | ||
{{ | {{Harmonics in equal|257}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
257edo is the 55th [[prime | 257edo is the 55th [[prime edo]]. | ||
Revision as of 08:22, 14 March 2024
| ← 256edo | 257edo | 258edo → |
Theory
257edo is inconsistent to the 5-odd-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 the equal temperament tempers out 393216/390625 (würschmidt comma) in the 5-limit and 1029/1024 and 177147/175000 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) and [-3 -23 17⟩ (maja comma) in the 5-limit; 225/224, 1029/1024, and [2 -25 15 1⟩ in the 7-limit; 243/242, 385/384, 441/440, and [0 -2 16 -1 -9⟩ 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 2.27.15.11.13.37.41.49.53.59.67 subgroup. A comma basis for the 2.27.15.11.13 subgroup is {4225/4224, 256000/255879, 225000/224939, 4159375/4153344}.
Odd harmonics
| 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.