422edo: Difference between revisions

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== Theory ==

422edo is a [[zeta peak edo]], though not zeta integral nor zeta gap. It is [[consistency|distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. ~~The~~ equal temperament [[tempering out|tempers out]] the [[vishnuzma]], {{monzo| 23 6 -14 }} and the countritonic comma, {{monzo| 33 -34 9 }}, in the 5-limit; [[4375/4374]] and [[589824/588245]] in the 7-limit; [[3025/3024]], [[5632/5625]], and [[9801/9800]] in the 11-limit; [[1716/1715]], [[2080/2079]], and [[2200/2197]] in the 13-limit; [[1156/1155]], [[1275/1274]], and [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]], and [[2432/2431]] in the 19-limit; and [[736/735]], [[1496/1495]], and [[1863/1862]] in the 23-limit. It [[support]]s provides the [[optimal patent val]]s for [[gamera]] in the 7-limit and [[hemigamera]] in the 13-limit. Other notable temperaments it supports are [[vishnu]], [[semisupermajor]], and [[countritonic]].

422edo is a [[zeta peak edo]], though not zeta integral nor zeta gap. It is [[consistency|distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. As an equal temperament, it [[tempering out|tempers out]] the [[vishnuzma]], {{monzo| 23 6 -14 }} and the countritonic comma, {{monzo| 33 -34 9 }}, in the 5-limit; [[4375/4374]] and [[589824/588245]] in the 7-limit; [[3025/3024]], [[5632/5625]], and [[9801/9800]] in the 11-limit; [[1716/1715]], [[2080/2079]], and [[2200/2197]] in the 13-limit; [[1156/1155]], [[1275/1274]], and [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]], and [[2432/2431]] in the 19-limit; and [[736/735]], [[1496/1495]], and [[1863/1862]] in the 23-limit. It [[support]]s provides the [[optimal patent val]]s for [[gamera]] in the 7-limit and [[hemigamera]] in the 13-limit. Other notable temperaments it supports are [[vishnu]], [[semisupermajor]], and [[countritonic]].

=== Prime harmonics ===

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=== Subsets and supersets ===

Since 422 factors into ~~{{factorization|422}}~~, 422edo has subset edos [[2edo]] and [[211edo]].

Since 422 factors into 2 × 211, 422edo has subset edos [[2edo]] and [[211edo]].

== 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

| {{monzo| 669 -422 }}

| {{mapping| 422 669 }}

| ~~−0~~.1308

| −0.1308

| 0.1308

| 4.60

Line 24:

Line 33:

| {{monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }}

| {{mapping| 422 669 980 }}

| ~~−0~~.1469

| −0.1469

| 0.1092

| 3.84

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| 4375/4374, 589824/588245, 29360128/29296875

| {{mapping| 422 669 980 1185 }}

| ~~−0~~.1852

| −0.1852

| 0.1155

| 4.06

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| 3025/3024, 4375/4374, 5632/5625, 589824/588245

| {{mapping| 422 669 980 1185 1460 }}

| ~~−0~~.1679

| −0.1679

| 0.1090

| 3.83

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| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625

| {{mapping| 422 669 980 1185 1460 1562 }}

| ~~−0~~.1930

| −0.1930

| 0.1142

| 4.02

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| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430

| {{mapping| 422 669 980 1185 1460 1562 1725 }}

| ~~−0~~.1744

| −0.1744

| 0.1151

| 4.05

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| 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197

| {{mapping| 422 669 980 1185 1460 1562 1725 1793 }}

| ~~−0~~.1839

| −0.1839

| 0.1106

| 3.89

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| 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715

| {{mapping| 422 669 980 1185 1460 1562 1725 1793 1909 }}

| ~~−0~~.1675

| −0.1675

| 0.1142

| 4.02

~~{{comma basis end}~~}

|}

* 422et has lower absolute errors than any previous equal temperaments in the 17-, 19-, and 23-limit. In the 17- and 19-limit it beats [[400edo|400]] and is bettered by [[460edo|460]]. In the 23-limit it beats [[373edo|373g]] and is bettered by [[525edo|525]].

=== 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 116:

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| 9/7<br />(11/10)

| [[Semisupermajor]]

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

|}

~~{{orf}}~~

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

[[Category:Countritonic]]

[[Category:Gamera]]

[[Category:Vishnu]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|422}}
+{{ED intro}}
 == Theory ==
-edo is a [[zeta peak edo]], though not zeta integral nor zeta gap. It is [[consistency|distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. The equal temperament [[tempering out|tempers out]] the [[vishnuzma]], {{monzo| 23 6 -14 }} and the countritonic comma, {{monzo| 33 -34 9 }}, in the 5-limit; [[4375/4374]] and [[589824/588245]] in the 7-limit; [[3025/3024]], [[5632/5625]], and [[9801/9800]] in the 11-limit; [[1716/1715]], [[2080/2079]], and [[2200/2197]] in the 13-limit; [[1156/1155]], [[1275/1274]], and [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]], and [[2432/2431]] in the 19-limit; and [[736/735]], [[1496/1495]], and [[1863/1862]] in the 23-limit. It [[support]]s provides the [[optimal patent val]]s for [[gamera]] in the 7-limit and [[hemigamera]] in the 13-limit. Other notable temperaments it supports are [[vishnu]], [[semisupermajor]], and [[countritonic]].
+edo is a [[zeta peak edo]], though not zeta integral nor zeta gap. It is [[consistency|distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. As an equal temperament, it [[tempering out|tempers out]] the [[vishnuzma]], {{monzo| 23 6 -14 }} and the countritonic comma, {{monzo| 33 -34 9 }}, in the 5-limit; [[4375/4374]] and [[589824/588245]] in the 7-limit; [[3025/3024]], [[5632/5625]], and [[9801/9800]] in the 11-limit; [[1716/1715]], [[2080/2079]], and [[2200/2197]] in the 13-limit; [[1156/1155]], [[1275/1274]], and [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]], and [[2432/2431]] in the 19-limit; and [[736/735]], [[1496/1495]], and [[1863/1862]] in the 23-limit. It [[support]]s provides the [[optimal patent val]]s for [[gamera]] in the 7-limit and [[hemigamera]] in the 13-limit. Other notable temperaments it supports are [[vishnu]], [[semisupermajor]], and [[countritonic]].
 === Prime harmonics ===
@@ Line 9: / Line 9: @@
 === Subsets and supersets ===
-Since 422 factors into {{factorization|422}}, 422edo has subset edos [[2edo]] and [[211edo]].
+Since 422 factors into 2 × 211, 422edo has subset edos [[2edo]] and [[211edo]].
 == 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
 | {{monzo| 669 -422 }}
 | {{mapping| 422 669 }}
-| &minus;0.1308
+| −0.1308
 | 0.1308
 | 4.60
@@ Line 24: / Line 33: @@
 | {{monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }}
 | {{mapping| 422 669 980 }}
-| &minus;0.1469
+| −0.1469
 | 0.1092
 | 3.84
@@ Line 31: / Line 40: @@
 | 4375/4374, 589824/588245, 29360128/29296875
 | {{mapping| 422 669 980 1185 }}
-| &minus;0.1852
+| −0.1852
 | 0.1155
 | 4.06
@@ Line 38: / Line 47: @@
 | 3025/3024, 4375/4374, 5632/5625, 589824/588245
 | {{mapping| 422 669 980 1185 1460 }}
-| &minus;0.1679
+| −0.1679
 | 0.1090
 | 3.83
@@ Line 45: / Line 54: @@
 | 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625
 | {{mapping| 422 669 980 1185 1460 1562 }}
-| &minus;0.1930
+| −0.1930
 | 0.1142
 | 4.02
@@ Line 52: / Line 61: @@
 | 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430
 | {{mapping| 422 669 980 1185 1460 1562 1725 }}
-| &minus;0.1744
+| −0.1744
 | 0.1151
 | 4.05
@@ Line 59: / Line 68: @@
 | 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197
 | {{mapping| 422 669 980 1185 1460 1562 1725 1793 }}
-| &minus;0.1839
+| −0.1839
 | 0.1106
 | 3.89
@@ Line 66: / Line 75: @@
 | 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715
 | {{mapping| 422 669 980 1185 1460 1562 1725 1793 1909 }}
-| &minus;0.1675
+| −0.1675
 | 0.1142
 | 4.02
-{{comma basis end}}
+|}
 * 422et has lower absolute errors than any previous equal temperaments in the 17-, 19-, and 23-limit. In the 17- and 19-limit it beats [[400edo|400]] and is bettered by [[460edo|460]]. In the 23-limit it beats [[373edo|373g]] and is bettered by [[525edo|525]].
 === 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 116: / Line 132: @@
 | 9/7<br />(11/10)
 | [[Semisupermajor]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 [[Category:Countritonic]]
 [[Category:Gamera]]
 [[Category:Vishnu]]