164edo: Difference between revisions

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-The ''164 equal division'' divides the octave into 164 equal parts of 7.317 cents each. In the 5-limit it tempers out the würschmidt comma, 393216/390625, and supplies the [[Optimal_patent_val|optimal patent val]] for [[Würschmidt_family|würschmidt temperament]]. In higher limits, also supplies the optimal patent val for the 7-limit, 1/41 octave period 41&amp;123 temperament, and the 13-limit [[Gamelismic_family#Portent|momentous temperament]], the rank-three temperament tempering out 196/195, 352/351, 385/384 and 441/440.
+{{Infobox ET}}
+{{ED intro}}
-= 4 * 41, with divisors 2, 4, 41, 82
+== Theory ==
+= 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 {{nowrap|41 &amp; 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.
+=== Prime harmonics ===
+{{Harmonics in equal|164}}
+=== Subsets and supersets ===
+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 ==
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3.5
+| 393216/390625, {{monzo| 24 -21 4 }}
+| {{mapping| 164 260 381 }}
+| −0.316
+| 0.262
+| 3.58
+|-
+| 2.3.5.13
+| 676/675, 256000/255879, 393216/390625
+| {{mapping| 164 260 381 607 }}
+| −0.300
+| 0.229
+| 3.13
+|}
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
+|-
+| 1
+| 47\164
+| 343.90
+| 8000/6561
+| [[Geb]]
+|-
+| 1
+| 49\164
+| 358.54
+| 16/13
+| [[Restles]] (164)
+|-
+| 1
+| 53\164
+| 387.80
+| 5/4
+| [[Würschmidt]]
+|-
+| 1
+| 53\164
+| 475.61
+| 320/243
+| [[Vulture]]
+|-
+| 1
+| 69\164
+| 504.88
+| 104976/78125
+| [[Countermeantone]]
+|-
+| 2
+| 17\164
+| 124.39
+| 275/256
+| [[Semivulture]] (164)
+|-
+| 2
+| 25\164
+| 182.93
+| 10/9
+| [[Unidecmic]]
+|-
+| 4
+| 68\164<br />(14\164)
+| 497.56<br />(102.44)
+| 4/3<br />(35/33)
+| [[Undim]] (164deff) / [[unlit]] (164f)
+|-
+| 41
+| 53\164<br />(1\164)
+| 387.80<br />(7.32)
+| 5/4<br />(32805/32768)
+| [[Countercomp]]
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+[[Category:Würschmidt]]