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== Theory ==
== 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]].  
422edo is [[consistency|distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. It is also nearly consistent in the [[31-odd-limit]] with the only inconsistent intervals being [[29/27]], [[29/21]] and their [[octave complement]]s, and picks up only one extra inconsistent interval pair in the [[35-odd-limit]], [[35/29]] and its octave complement. It is a [[zeta peak edo]].
 
As an equal temperament, it [[tempering out|tempers out]] the [[vishnu comma]], {{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 and provides the [[optimal patent val]]s for [[gamera]] in the 7-limit, [[hemigamera]] in the 13-limit, and [[ragitritonic]] in the 11- and 13-limit. Other notable temperaments it supports are [[vishnu]] and [[semisupermajor]].  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|422}}
{{Harmonics in equal|422|columns=11}}
{{Harmonics in equal|422|start=12|collapsed=1|title=Approximation of prime harmonics in 422edo (continued)}}
{{Harmonics in equal|422|columns=11|start=12|collapsed=1|title=Approximation of prime harmonics in 422edo (continued)}}
{{Harmonics in equal|422|start=23|collapsed=1|title=Approximation of prime harmonics in 422edo (continued)}}
{{Harmonics in equal|422|columns=11|start=23|collapsed=1|title=Approximation of prime harmonics in 422edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 422 factors into 2 × 211, 422edo has subset edos [[2edo]] and [[211edo]].
Since 422 factors into primes as {{nowrap| 2 × 211 }}, 422edo has subset edos [[2edo]] and [[211edo]].


== Regular temperament properties ==
== Regular temperament properties ==
Line 19: Line 21:
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 669 -422 }}
| {{Monzo| 669 -422 }}
| {{mapping| 422 669 }}
| {{Mapping| 422 669 }}
| −0.1308
| −0.1308
| 0.1308
| 0.1308
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|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }}
| {{Monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }}
| {{mapping| 422 669 980 }}
| {{Mapping| 422 669 980 }}
| −0.1469
| −0.1469
| 0.1092
| 0.1092
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| 2.3.5.7
| 2.3.5.7
| 4375/4374, 589824/588245, 29360128/29296875
| 4375/4374, 589824/588245, 29360128/29296875
| {{mapping| 422 669 980 1185 }}
| {{Mapping| 422 669 980 1185 }}
| −0.1852
| −0.1852
| 0.1155
| 0.1155
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| 2.3.5.7.11
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5632/5625, 589824/588245
| 3025/3024, 4375/4374, 5632/5625, 589824/588245
| {{mapping| 422 669 980 1185 1460 }}
| {{Mapping| 422 669 980 1185 1460 }}
| −0.1679
| −0.1679
| 0.1090
| 0.1090
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625
| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625
| {{mapping| 422 669 980 1185 1460 1562 }}
| {{Mapping| 422 669 980 1185 1460 1562 }}
| −0.1930
| −0.1930
| 0.1142
| 0.1142
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430
| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430
| {{mapping| 422 669 980 1185 1460 1562 1725 }}
| {{Mapping| 422 669 980 1185 1460 1562 1725 }}
| −0.1744
| −0.1744
| 0.1151
| 0.1151
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| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197
| 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197
| {{mapping| 422 669 980 1185 1460 1562 1725 1793 }}
| {{Mapping| 422 669 980 1185 1460 1562 1725 1793 }}
| −0.1839
| −0.1839
| 0.1106
| 0.1106
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| 2.3.5.7.11.13.17.19.23
| 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
| 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 }}
| {{Mapping| 422 669 980 1185 1460 1562 1725 1793 1909 }}
| −0.1675
| −0.1675
| 0.1142
| 0.1142
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 435.07
| 435.07
| 9/7
| 9/7
| [[Supermajor]]
| [[Supermajor (temperament)|Supermajor]]
|-
|-
| 1
| 1
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| 588.63
| 588.63
| 128/91
| 128/91
| [[Countritonic]]
| [[Ragitritonic]]
|-
|-
| 2
| 2
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|-
|-
| 2
| 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]]
| [[Semisupermajor]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<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:Countritonic]]
[[Category:Ragitritonic]]
[[Category:Gamera]]
[[Category:Gamera]]
[[Category:Vishnu]]
[[Category:Vishnu]]
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