22nd-octave temperaments
Template:Fractional-octave navigation 22edo is a notable tuning system, and as such it gives rise to some fractional-octave temperaments.
Temperaments discussed elsewhere include:
- Icosidillic, → Porwell temperaments
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, 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