164edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
164 = 4 × 41, and 164edo shares its [[perfect fifth|fifth]] with [[41edo]]. In the 5-limit, 164et tempers out the [[würschmidt comma]], 393216/390625, and the [[vulture comma]], {{monzo| 24 -21 4 }}. It supplies the [[optimal patent val]] for the [[würschmidt]] temperament. | 164 = 4 × 41, and 164edo shares its [[perfect fifth|fifth]] with [[41edo]]. In the 5-limit, 164et tempers out the [[würschmidt comma]], 393216/390625, and the [[vulture comma]], {{monzo| 24 -21 4 }}. It supplies the [[optimal patent val]] for the [[würschmidt]] temperament. | ||
In the [[patent val]] {{val| 164 260 381 '''460''' '''567''' 607 }}, it tempers out [[196/195]], [[352/351]], [[385/384]], [[441/440]], [[676/675]], and supplies the optimal patent val for the 7-limit, 1/41 octave period 41&123 temperament, and the 13-limit [[Gamelismic family #Portent|momentous]] temperament, the rank-3 temperament tempering out 196/195, 352/351, 385/384 and 441/440. | In the [[patent val]] {{val| 164 260 381 '''460''' '''567''' 607 }}, it tempers out [[196/195]], [[352/351]], [[385/384]], [[441/440]], [[676/675]], and supplies the optimal patent val for the 7-limit, 1/41 octave period {{nowrap|41 & 123}} temperament, and the 13-limit [[Gamelismic family #Portent|momentous]] temperament, the rank-3 temperament tempering out 196/195, 352/351, 385/384 and 441/440. | ||
In the alternative val 164de {{val| 164 260 381 '''461''' '''568''' 607 }}, it tempers out [[243/242]], [[351/350]], [[364/363]], [[640/637]], [[676/675]], [[729/728]], and [[1575/1573]]. The 164dg val is a good tuning for 7- to 19-limit [[buzzard]] temperament, although if harmonic 11 is desired it is only easily accessible through the patent mapping. | In the alternative val 164de {{val| 164 260 381 '''461''' '''568''' 607 }}, it tempers out [[243/242]], [[351/350]], [[364/363]], [[640/637]], [[676/675]], [[729/728]], and [[1575/1573]]. The 164dg val is a good tuning for 7- to 19-limit [[buzzard]] temperament, although if harmonic 11 is desired it is only easily accessible through the patent mapping. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 164 = {{factorization|164}}, 164edo has subset edos {{EDOs| 2, 4, 41, 82 }}. [[328edo]], which doubles it, provides good correction for the approximation to harmonics 7 and 11, and is [[consistent]] in the [[13-odd-limit]]. | Since {{nowrap|164 {{=}} {{factorization|164}}}}, 164edo has subset edos {{EDOs| 2, 4, 41, 82 }}. [[328edo]], which doubles it, provides good correction for the approximation to harmonics 7 and 11, and is [[consistent]] in the [[13-odd-limit]]. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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| 393216/390625, {{monzo| 24 -21 4 }} | | 393216/390625, {{monzo| 24 -21 4 }} | ||
| {{mapping| 164 260 381 }} | | {{mapping| 164 260 381 }} | ||
| | | −0.316 | ||
| 0.262 | | 0.262 | ||
| 3.58 | | 3.58 | ||
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| 676/675, 256000/255879, 393216/390625 | | 676/675, 256000/255879, 393216/390625 | ||
| {{mapping| 164 260 381 607 }} | | {{mapping| 164 260 381 607 }} | ||
| | | −0.300 | ||
| 0.229 | | 0.229 | ||
| 3.13 | | 3.13 | ||
| Line 106: | Line 106: | ||
| [[Countercomp]] | | [[Countercomp]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
[[Category:Würschmidt]] | [[Category:Würschmidt]] | ||