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The ''422 equal division'' divides the octave into 422 equal parts of 2.844 cents each.  It is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap, and is distinctly consistent through the 27 limit. In the 5-limit it tempers out the vishnuzma, |23 6 -14>; and in the 7-limit 4375/4374 and 589824/588245 so that it [[support]]s [[Ragismic_microtemperaments#Gamera|gamera temperament]], and provides its [[Optimal_patent_val|optimal patent val]], and also supports the 118&152 temperament. A basis for the 11-limit is 3025/3024, 4375/4374, 5632/5625 and 825000/823543.
{{Infobox ET}}
[[Category:gamera]]
{{ED intro}}
[[Category:vishnuzma]]
 
== 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. 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}}
 
=== Subsets and supersets ===
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]]
! 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
| 4.60
|-
| 2.3.5
| {{monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }}
| {{mapping| 422 669 980 }}
| −0.1469
| 0.1092
| 3.84
|-
| 2.3.5.7
| 4375/4374, 589824/588245, 29360128/29296875
| {{mapping| 422 669 980 1185 }}
| −0.1852
| 0.1155
| 4.06
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5632/5625, 589824/588245
| {{mapping| 422 669 980 1185 1460 }}
| −0.1679
| 0.1090
| 3.83
|-
| 2.3.5.7.11.13
| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625
| {{mapping| 422 669 980 1185 1460 1562 }}
| −0.1930
| 0.1142
| 4.02
|-
| 2.3.5.7.11.13.17
| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430
| {{mapping| 422 669 980 1185 1460 1562 1725 }}
| −0.1744
| 0.1151
| 4.05
|-
| 2.3.5.7.11.13.17.19
| 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
| 0.1106
| 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]].
 
=== 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
| 81\422
| 230.33
| 8/7
| [[Gamera]]
|-
| 1
| 111\422
| 315.64
| 6/5
| [[Egads]]
|-
| 1
| 153\422
| 435.07
| 9/7
| [[Supermajor]]
|-
| 1
| 207\422
| 588.63
| 128/91
| [[Countritonic]]
|-
| 2
| 25\422
| 71.09
| 25/24
| [[Vishnu]] / [[acyuta]]
|-
| 2
| 81\422
| 230.33
| 8/7
| [[Hemigamera]]
|-
| 2
| 153\422<br />(58\422)
| 435.07<br />(164.93)
| 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]]
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