22nd-octave temperaments: Difference between revisions
→Major Arcana: clarify in what way major arcana and icosidillic are the same |
Move vigintiduo and vigin here |
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{{Fractional-octave navigation|22}} | {{Fractional-octave navigation|22}} | ||
[[22edo]] is a notable tuning system, and as such it gives rise to some fractional-octave temperaments. | [[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. Vigintiduo has the same 7-limit mapping as 22et with harmonic 11 mapped to an independent generator. Vigin has the same 11-limit mapping as 22et with harmonic 13 mapped to an independent generator. | ||
See also: | |||
* ''Icosidillic'' | * ''[[Icosidillic]]'' → [[Porwell temperaments #Icosidillic|Porwell temperaments]] | ||
==Major | == Vigintiduo == | ||
[[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 == | |||
[[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 == | |||
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. | 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. | ||
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=== 7-limit === | === 7-limit === | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
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{{Optimal ET sequence|legend=1|506}} | {{Optimal ET sequence|legend=1|506}} | ||
== | == Bibium == | ||
Bibium is named after an "extended provisional name" for the 22nd element [[wikipedia:titanium|titanium]], which is needed to resolve a name conflict as [[Titanium]] is already a temperament. It reaches [[11/8]] and [[7/6]] in one generator, and as a corollary tempers [[96/77]] to 7 steps of 22edo. | Bibium is named after an "extended provisional name" for the 22nd element [[wikipedia:titanium|titanium]], which is needed to resolve a name conflict as [[Titanium]] is already a temperament. It reaches [[11/8]] and [[7/6]] in one generator, and as a corollary tempers [[96/77]] to 7 steps of 22edo. | ||
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[[Support]]ing [[ET]]s: {{EDOs|198, 616, 814, 1012, 1210}} | [[Support]]ing [[ET]]s: {{EDOs|198, 616, 814, 1012, 1210}} | ||
[[Category:Temperament collections]] | |||
[[Category:Fractional-octave temperaments]] | |||
[[Category:22edo]] | [[Category:22edo]] | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
{{Todo| cleanup }} | |||
Revision as of 11:14, 21 September 2023
Template:Fractional-octave navigation 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. Vigintiduo has the same 7-limit mapping as 22et with harmonic 11 mapped to an independent generator. Vigin has the same 11-limit mapping as 22et with harmonic 13 mapped to an independent generator.
See also:
Vigintiduo
Subgroup: 2.3.5.7.11
Comma list: 50/49, 64/63, 245/243
Mapping: [⟨22 35 51 62 0], ⟨0 0 0 0 1]]
Optimal tuning (POTE): ~36/35 = 1\22, ~11/8 = 557.563
Optimal ET sequence: 22, 66de, 88bde, 110bd, 198bcdde
Badness: 0.048372
Vigin
Subgroup: 2.3.5.7.11.13
Comma list: 50/49, 55/54, 64/63, 99/98
Mapping: [⟨22 35 51 62 76 0], ⟨0 0 0 0 0 1]]
Optimal tuning (POTE): ~33/32 = 1\22, ~13/8 = 844.624
Badness: 0.029849
Major arcana
Named after a subset of Tarot cards of which there is 22. Tempers out the [-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
Comma list: [-193 154 -22⟩
Mapping: [⟨22 22 -39], ⟨0 1 7]]
Mapping generators: ~[79 -63 9⟩, ~3/2
Optimal tuning (CTE): ~3/2 = 701.942
Optimal ET sequence: 176, 336, 506, 836, 1342, 1848, ...
7-limit
Subgroup: 2.3.5.7
Comma list: [51 -24 -8 2⟩, [-20 41 -23 3⟩
Mapping: [⟨22 22 -39 -453], ⟨0 1 7 40]]
Mapping generators: ~16128/15625, ~3/2
Optimal tuning (CTE): ~3/2 = 701.948
Bibium
Bibium is named after an "extended provisional name" for the 22nd element titanium, which is needed to resolve a name conflict as Titanium is already a temperament. 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: [⟨22 2 10 33 72], ⟨0 8 10 7 1]]
Mapping generators: ~48125/46656 = 1\22, 4296875/3779136 = 224.101
Optimal tuning (CTE): ~4296875/3779136 = 224.101
Supporting ETs: 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: [⟨22 2 10 33 72 28], ⟨0 8 10 7 1 3]]
Mapping generators: ~65/63 = 1\22, 338/297 = 224.100
Optimal tuning (CTE): ~338/297 = 224.100