164edo: Difference between revisions

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== 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.

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.

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=== 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 ==

{~~{comma basis begin}}~~

{| 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

| −0.316

| 0.262

| 3.58

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| 676/675, 256000/255879, 393216/390625

| {{mapping| 164 260 381 607 }}

| ~~−0~~.300

| −0.300

| 0.229

| 3.13

~~{{comma basis end}~~}

|}

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

{| 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

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| 5/4<br />(32805/32768)

| [[Countercomp]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

[[Category:Würschmidt]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|164}}
+{{ED intro}}
 == 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 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 [[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.
@@ Line 13: / Line 13: @@
 === 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 ==
-{{comma basis begin}}
+{| 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 }}
-| &minus;0.316
+| −0.316
 | 0.262
 | 3.58
@@ Line 28: / Line 37: @@
 | 676/675, 256000/255879, 393216/390625
 | {{mapping| 164 260 381 607 }}
-| &minus;0.300
+| −0.300
 | 0.229
 | 3.13
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| 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
@@ Line 89: / Line 105: @@
 | 5/4<br />(32805/32768)
 | [[Countercomp]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 [[Category:Würschmidt]]