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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2 × 211
{{ED intro}}
| Step size = 2.84360¢
| Fifth = 247\422 (702.37¢)
| Semitones = 41:31 (116.59¢ : 88.15¢)
| Consistency = 27
}}
The '''422 equal divisions of the octave''' ('''422edo'''), or the '''422(-tone) equal temperament''' ('''422tet''', '''422et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 422 [[equal]] parts of 2.84 [[cent]]s each.


== Theory ==
== 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 ===
=== Prime harmonics ===
{{Harmonics in equal|422|columns=11}}
{{Harmonics in equal|422}}
 
=== Subsets and supersets ===
Since 422 factors into 2 × 211, 422edo has subset edos [[2edo]] and [[211edo]].


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! 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
|-
|-
Line 27: Line 25:
| 2.3
| 2.3
| {{monzo| 669 -422 }}
| {{monzo| 669 -422 }}
| [{{val| 422 669 }}]
| {{mapping| 422 669 }}
| -0.1308
| −0.1308
| 0.1308
| 0.1308
| 4.60
| 4.60
Line 34: Line 32:
| 2.3.5
| 2.3.5
| {{monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }}
| {{monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }}
| [{{val| 422 669 980 }}]
| {{mapping| 422 669 980 }}
| -0.1469
| −0.1469
| 0.1092
| 0.1092
| 3.84
| 3.84
Line 41: Line 39:
| 2.3.5.7
| 2.3.5.7
| 4375/4374, 589824/588245, 29360128/29296875
| 4375/4374, 589824/588245, 29360128/29296875
| [{{val| 422 669 980 1185 }}]
| {{mapping| 422 669 980 1185 }}
| -0.1852
| −0.1852
| 0.1155
| 0.1155
| 4.06
| 4.06
Line 48: Line 46:
| 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
| [{{val| 422 669 980 1185 1460 }}]
| {{mapping| 422 669 980 1185 1460 }}
| -0.1679
| −0.1679
| 0.1090
| 0.1090
| 3.83
| 3.83
Line 55: Line 53:
| 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
| [{{val| 422 669 980 1185 1460 1562 }}]
| {{mapping| 422 669 980 1185 1460 1562 }}
| -0.1930
| −0.1930
| 0.1142
| 0.1142
| 4.02
| 4.02
Line 62: Line 60:
| 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
| [{{val| 422 669 980 1185 1460 1562 1725 }}]
| {{mapping| 422 669 980 1185 1460 1562 1725 }}
| -0.1744
| −0.1744
| 0.1151
| 0.1151
| 4.05
| 4.05
Line 69: Line 67:
| 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
| [{{val| 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
| 3.89
| 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 ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| 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 octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
Line 101: Line 108:
| 9/7
| 9/7
| [[Supermajor]]
| [[Supermajor]]
|-
| 1
| 207\422
| 588.63
| 128/91
| [[Countritonic]]
|-
|-
| 2
| 2
Line 115: Line 128:
|-
|-
| 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


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Countritonic]]
[[Category:Gamera]]
[[Category:Gamera]]
[[Category:Vishnu]]
[[Category:Vishnu]]
Retrieved from "https://en.xen.wiki/w/422edo"