22nd-octave temperaments: Difference between revisions

VisualWikitext

@@ Line 1: / Line 1: @@
-[[22edo]] is a notable tuning system, and as such it gives rise to some fractional-octave temperaments.
+{{Technical data page}}
-==Icosidillic==
+{{Infobox fractional-octave|22}}
-Main article: [[Porwell temperaments#Icosidillic]]
+[[22edo]] is a notable tuning system, and as such it gives rise to some fractional-octave temperaments. This page accommodates a number of temperaments that are otherwise difficult to catalog because they belong to multiple families.
-==Major Arcana==
-Named after a subset of Tarot cards of which there is 22. Tempers out the {{monzo|-193 154 -22}} comma in the 5-limit. Can be conceptualized as a more precise version of icosidillic in the 5-limit and the 7-limit, however there's no single way to extend it to the 11-limit due to inconsistency.
+See also:
+* ''[[Icosidillic]]'' → [[Porwell temperaments #Icosidillic|Porwell temperaments]]
+== Vigintiduo ==
+''Vigintiduo'' has the same 7-limit mapping as 22et with harmonic 11 mapped to an independent generator.
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 50/49, 64/63, 245/243
+{{Mapping|legend=1| 22 35 51 62 0 | 0 0 0 0 1 }}
+[[Optimal tuning]] ([[POTE]]): ~36/35 = 1\22, ~11/8 = 557.563
+{{Optimal ET sequence|legend=1| 22, 66de, 88bde, 110bd, 198bcdde }}
+[[Badness]]: 0.048372
+== Vigin ==
+''Vigin'' has the same 11-limit mapping as 22et with harmonic 13 mapped to an independent generator.
+[[Subgroup]]: 2.3.5.7.11.13
+[[Comma list]]: 50/49, 55/54, 64/63, 99/98
+{{Mapping|legend=1| 22 35 51 62 76 0 | 0 0 0 0 0 1 }}
+[[Optimal tuning]] ([[POTE]]): ~33/32 = 1\22, ~13/8 = 844.624
+{{Optimal ET sequence|legend=1| 22, 44 }}
+[[Badness]]: 0.029849
+== Major arcana ==
+''Major arcana'' is named after a subset of Tarot cards of which there is 22. Tempers out the {{monzo|-193 154 -22}} comma in the 5-limit. Can be conceptualized as a more precise version of icosidillic in the 5-limit and the 7-limit bearing the same [[pergen]] (P8/22, P5), however there's no single way to extend it to the 11-limit due to inconsistency.
 Subgroup: 2.3.5
@@ Line 11: / Line 45: @@
 Mapping: [{{val|22 22 -39}}, {{val|0 1 7}}]
-Mapping generators: ~{{monzo|idk}}, ~3/2
+: mapping generators: ~{{monzo|79 -63 9}}, ~3/2
 Optimal tuning (CTE): ~3/2 = 701.942
-Vals: {{val list|176, 336, 506, 836, 1342, 1848}}, ...
+[[Support]]ing [[ET]]s: {{EDOs|176, 330, 506, 836, 1342, 1848}}, ...
 === 7-limit ===
+Subgroup: 2.3.5.7
+Comma list: 14348907/14336000, {{monzo|39 11 -11 -11}}
+Mapping: [{{val|22 0 -193 271}}, {{val|0 1 7 -6}}]
+: mapping generators: ~8388608/8103375, ~3/2
+Optimal tuning (CTE): ~3/2 = 702.014
+[[Support]]ing [[ET]]s: {{EDOs|176, 506, 682}}
+== Titanium ==
+''Titanium'' is named after the 22nd element after the previous temperament called "titanium" was found to be the same thing as [[beep]]. It reaches [[11/8]] and [[7/6]] in one generator, and as a corollary tempers [[96/77]] to 7 steps of 22edo.
+Subgroup: 2.3.5.7.11
+Comma list: 2401/2400, 9801/9800, 2166612408926208/2162550048828125
+Mapping: [{{val|22 2 10 33 72}}, {{val|0 8 10 7 1}}]
+Mapping generators: ~48125/46656 = 1\22, 4296875/3779136 = 224.101
+Optimal tuning (CTE): ~4296875/3779136 = 224.101
+[[Support]]ing [[ET]]s: {{EDOs|198, 616, 814, 1012, 1210}}
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 2401/2400, 9801/9800, 67392/67375, 142884/142805
+Mapping: [{{val|22 2 10 33 72 28}}, {{val|0 8 10 7 1 3}}]
+Mapping generators: ~65/63 = 1\22, 338/297 = 224.100
+Optimal tuning (CTE): ~338/297 = 224.100
+[[Support]]ing [[ET]]s: {{EDOs|198, 616, 814, 1012, 1210}}
+{{Navbox fractional-octave}}
-Subgroup: 2.3.5.7
-Comma list: {{monzo|}}}}
+{{Todo| cleanup }}