422edo: Difference between revisions

(8 intermediate revisions by 3 users not shown)

Line 1:

~~{{novelty}}{{stub}}~~{{Infobox ET}}

== Theory ==

422edo is a [[~~The Riemann zeta function and tuning #Zeta EDO lists|~~zeta peak edo]], though not zeta integral nor zeta gap. It is ~~distinctly~~ [[consistent]] through the [[27-odd-limit]], with ~~harmonics~~ of 3 through 23 all tuned sharp. ~~In the 5-limit~~ it tempers out the [[vishnuzma]], {{monzo| 23 6 -14 }}; and in the 7-limit [[4375/4374]] and 589824/588245 ~~so that it~~ [[~~support~~]]s the [[~~gamera~~]] ~~temperament~~, and provides ~~its~~ [[optimal patent val]], and ~~also~~ supports ~~the~~ [[vishnu]] ~~temperament~~.

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 ===

=== Subsets and supersets ===

422edo has subset edos [[2edo]] and [[211edo]].

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

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

Line 24:

Line 25:

| 2.3

| {{monzo| 669 -422 }}

| [{{~~val~~| 422 669 }}]

| {{mapping| 422 669 }}

| -0.1308

| −0.1308

| 0.1308

| 4.60

Line 31:

Line 32:

| 2.3.5

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

| [{{~~val~~| 422 669 980 }}]

| {{mapping| 422 669 980 }}

| -0.1469

| −0.1469

| 0.1092

| 3.84

Line 38:

Line 39:

| 2.3.5.7

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

| [{{~~val~~| 422 669 980 1185 }}]

| {{mapping| 422 669 980 1185 }}

| -0.1852

| −0.1852

| 0.1155

| 4.06

Line 45:

Line 46:

| 2.3.5.7.11

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

| [{{~~val~~| 422 669 980 1185 1460 }}]

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

| -0.1679

| −0.1679

| 0.1090

| 3.83

Line 52:

Line 53:

| 2.3.5.7.11.13

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

| [{{~~val~~| 422 669 980 1185 1460 1562 }}]

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

| -0.1930

| −0.1930

| 0.1142

| 4.02

Line 59:

Line 60:

| 2.3.5.7.11.13.17

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

| [{{~~val~~| 422 669 980 1185 1460 1562 1725 }}]

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

| -0.1744

| −0.1744

| 0.1151

| 4.05

Line 66:

Line 67:

| 2.3.5.7.11.13.17.19

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

| [{{~~val~~| 422 669 980 1185 1460 1562 1725 1793 }}]

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

| -0.1839

| −0.1839

| 0.1106

| 3.89

|-

| 2.3.5.7.11.13.17.19.23

| 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

| 0.1142

| 4.02

|}

* 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]].

* 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 ===

{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per 8ve

|-

! Generator~~ (Reduced)~~

! Periods per 8ve

! Cents~~ (Reduced)~~

! Generator*

! Associated ~~Ratio~~

! Cents*

! Associated ratio*

! Temperaments

|-

Line 99:

Line 108:

| 9/7

| [[Supermajor]]

|-

| 1

| 207\422

| 588.63

| 128/91

| [[Countritonic]]

|-

| 2

Line 113:

Line 128:

|-

| 2

| 153\422 (58\422)

| 153\422 (58\422)

| 435.07 (164.93)

| 435.07 (164.93)

| 9/7 (11/10)

| 9/7 (11/10)

| [[Semisupermajor]]

|}

<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: @@
-{{novelty}}{{stub}}{{Infobox ET}}
+{{Infobox ET}}
-{{EDO intro|422}}
+{{ED intro}}
 == Theory ==
-edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is distinctly [[consistent]] through the [[27-odd-limit]], with harmonics of 3 through 23 all tuned sharp. In the 5-limit it tempers out the [[vishnuzma]], {{monzo| 23 6 -14 }}; and in the 7-limit [[4375/4374]] and 589824/588245 so that it [[support]]s the [[gamera]] temperament, and provides its [[optimal patent val]], and also supports the [[vishnu]] temperament.
+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 ===
-{{Harmonics in equal|422|columns=11}}
+{{Harmonics in equal|422}}
 === Subsets and supersets ===
-edo has subset edos [[2edo]] and [[211edo]].
+Since 422 factors into 2 × 211, 422edo has subset edos [[2edo]] and [[211edo]].
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 24: / Line 25: @@
 | 2.3
 | {{monzo| 669 -422 }}
-| [{{val| 422 669 }}]
+| {{mapping| 422 669 }}
-| -0.1308
+| −0.1308
 | 0.1308
 | 4.60
@@ Line 31: / Line 32: @@
 | 2.3.5
 | {{monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }}
-| [{{val| 422 669 980 }}]
+| {{mapping| 422 669 980 }}
-| -0.1469
+| −0.1469
 | 0.1092
 | 3.84
@@ Line 38: / Line 39: @@
 | 2.3.5.7
 | 4375/4374, 589824/588245, 29360128/29296875
-| [{{val| 422 669 980 1185 }}]
+| {{mapping| 422 669 980 1185 }}
-| -0.1852
+| −0.1852
 | 0.1155
 | 4.06
@@ Line 45: / Line 46: @@
 | 2.3.5.7.11
 | 3025/3024, 4375/4374, 5632/5625, 589824/588245
-| [{{val| 422 669 980 1185 1460 }}]
+| {{mapping| 422 669 980 1185 1460 }}
-| -0.1679
+| −0.1679
 | 0.1090
 | 3.83
@@ Line 52: / Line 53: @@
 | 2.3.5.7.11.13
 | 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625
-| [{{val| 422 669 980 1185 1460 1562 }}]
+| {{mapping| 422 669 980 1185 1460 1562 }}
-| -0.1930
+| −0.1930
 | 0.1142
 | 4.02
@@ Line 59: / Line 60: @@
 | 2.3.5.7.11.13.17
 | 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430
-| [{{val| 422 669 980 1185 1460 1562 1725 }}]
+| {{mapping| 422 669 980 1185 1460 1562 1725 }}
-| -0.1744
+| −0.1744
 | 0.1151
 | 4.05
@@ Line 66: / Line 67: @@
 | 2.3.5.7.11.13.17.19
 | 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197
-| [{{val| 422 669 980 1185 1460 1562 1725 1793 }}]
+| {{mapping| 422 669 980 1185 1460 1562 1725 1793 }}
-| -0.1839
+| −0.1839
 | 0.1106
 | 3.89
+|-
+| 2.3.5.7.11.13.17.19.23
+| 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
+| 0.1142
+| 4.02
 |}
-* 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]].
+* 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 ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per 8ve
+|-
-! Generator<br>(Reduced)
+! Periods<br />per 8ve
-! Cents<br>(Reduced)
+! Generator*
-! Associated<br>Ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 99: / Line 108: @@
 | 9/7
 | [[Supermajor]]
+|-
+| 1
+| 207\422
+| 588.63
+| 128/91
+| [[Countritonic]]
 |-
 | 2
@@ Line 113: / Line 128: @@
 |-
 | 2
-| 153\422<br>(58\422)
+| 153\422<br />(58\422)
-| 435.07<br>(164.93)
+| 435.07<br />(164.93)
-| 9/7<br>(11/10)
+| 9/7<br />(11/10)
 | [[Semisupermajor]]
 |}
+<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]]