22nd-octave temperaments
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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.
See also:
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: [⟨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
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: [⟨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
Major arcana is 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
Supporting ETs: 176, 330, 506, 836, 1342, 1848, ...
7-limit
Subgroup: 2.3.5.7
Comma list: 14348907/14336000, [39 11 -11 -11⟩
Mapping: [⟨22 0 -193 271], ⟨0 1 7 -6]]
- mapping generators: ~8388608/8103375, ~3/2
Optimal tuning (CTE): ~3/2 = 702.014
Supporting ETs: 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: [⟨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