Xenharmonic Wiki

422edo: Difference between revisions

← Older edit
VisualWikitext
ArrowHead294 (talk | contribs)
15,213 edits
mNo edit summary
Countritonic -> ragitritonic
 
(7 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|422}}
{{ED intro}}


== 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. 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 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|columns=11|start=12|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 {{factorization|422}}, 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 ==
{{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
| 2.3
| {{monzo| 669 -422 }}
| {{Monzo| 669 -422 }}
| {{mapping| 422 669 }}
| {{Mapping| 422 669 }}
| &minus;0.1308
| −0.1308
| 0.1308
| 0.1308
| 4.60
| 4.60
|-
|-
| 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 }}
| &minus;0.1469
| −0.1469
| 0.1092
| 0.1092
| 3.84
| 3.84
Line 30: Line 41:
| 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 }}
| &minus;0.1852
| −0.1852
| 0.1155
| 0.1155
| 4.06
| 4.06
Line 37: Line 48:
| 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 }}
| &minus;0.1679
| −0.1679
| 0.1090
| 0.1090
| 3.83
| 3.83
Line 44: Line 55:
| 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 }}
| &minus;0.1930
| −0.1930
| 0.1142
| 0.1142
| 4.02
| 4.02
Line 51: Line 62:
| 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 }}
| &minus;0.1744
| −0.1744
| 0.1151
| 0.1151
| 4.05
| 4.05
Line 58: Line 69:
| 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 }}
| &minus;0.1839
| −0.1839
| 0.1106
| 0.1106
| 3.89
| 3.89
Line 65: Line 76:
| 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 }}
| &minus;0.1675
| −0.1675
| 0.1142
| 0.1142
| 4.02
| 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]].  
* 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 ===
{{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
| 1
Line 91: Line 109:
| 435.07
| 435.07
| 9/7
| 9/7
| [[Supermajor]]
| [[Supermajor (temperament)|Supermajor]]
|-
|-
| 1
| 1
Line 97: Line 115:
| 588.63
| 588.63
| 128/91
| 128/91
| [[Countritonic]]
| [[Ragitritonic]]
|-
|-
| 2
| 2
Line 112: Line 130:
|-
|-
| 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]]
{{rank-2 end}}
|}
{{orf}}
<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]]
Retrieved from "https://en.xen.wiki/w/422edo"