22nd-octave temperaments
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular 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.
See also:
Vigintiduo
Vigintiduo sets the 7-limit to that of 22edo and harmonic 11 and the rest are independent generators.
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]]
- mapping generators: ~25/24, ~11/8
- WE: ~25/24 = 54.462 ¢, ~11/8 = 556.713 ¢
- error map: ⟨-1.830, 4.225, -8.737, 7.836, -0.093]
- CWE: ~25/24 = 1\22, ~11/8 = 556.295 ¢
- error map: ⟨0.000 -0.068 -0.088 -0.106 -0.375]
Optimal ET sequence: 22, 66de, 88bde, 110bd, 198bcdde
Badness: 1.599
Vigin
Vigin sets the 11-limit to that of 22edo and harmonic 13 and the rest are independent generators. Can be described as 22 & 22f temperament, and the only non-trivial tuning besides 22edo itself is 44edo, which is also the optimal patent val.
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]]
- mapping generators: ~36/35, ~13/8
- WE: ~36/35 = 54.494 ¢, ~13/8 = 843.820 ¢
- error map: ⟨-1.141, 5.321, -7.141, 9.777, -9.805, -0.130]
- CWE: ~36/35 = 1\22, ~13/8 = 843.919 ¢
- error map: ⟨0.000, 7.136, -4.496, 12.992, -5.863, 3.392]
Badness: 1.233
Major arcana
Major arcana is named after a subset of Tarot cards of which there is 22 and it tempers out the [-193 154 -22⟩ comma in the 5-limit.
It is similar to the icosidillic temperament, having the same pergen (P8/22, P5), however while icosidillic is merely an extension of 5-limit hendecatonic, major arcana stands as an independent and properly not enfactored 22nd-octave comma.
An extension is possible to the 7-limit as the 506 & 682 temperament, however due to inconsistency the temperament branches at the 11-limit. Although notably, a 2.3.5.11/7 extension of remarkably low badness, described as 1848 & 1342, is prescribed. 1848edo as a whole is also a strong tuning despite being only in the optimal ET sequence for 2.3.5.11/7 subgroup version.
Subgroup: 2.3.5
Comma list: [-193 154 -22⟩
Mapping: [⟨22 0 -193], ⟨0 1 7]]
- mapping generators: ~[79 -63 9⟩, ~3
- WE: ~[79 -63 9⟩ = 54.546 ¢, ~3/2 = 701.942
- error map: ⟨0.004, -0.008, 0.003]
- CWE: ~[79 -63 9⟩ = 1\22, ~3/2 = 701.941
- error map: ⟨0.000, -0.014, -0.003]
Optimal ET sequence: 22c, 132cc, 154c, 176, 330, 506, 836, 1342, 4532, 5874, 7216, 8558, 15774
Badness: 28.189
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
- WE: ~8388608/8103375 = 54.546 ¢, ~3/2 = 701.942
- error map: ⟨-0.152, -0.165, 0.275, 0.384]
- CWE: ~8388608/8103375 = 1\22, ~3/2 = 701.941
- error map: ⟨0.000, 0.072, 0.604, 0.829]
Optimal ET sequence: 176, 330d, 506, 682
Badness: 28.987
2.3.5.11/7 subgroup
Subgroup: 2.3.5.11/7
Comma list: 9801/9800, 2.3.5.11/7 [-83 59 -2 -9⟩ (full 11-limit [-83 59 -2 9 -9⟩)
Mapping: [⟨22 0 -193 -160], ⟨0 1 7 5]]
- mapping generators: ~3720087/3604480, ~3
- WE: ~3720087/3604480 = 54.546 ¢, ~3/2 = 701.943
- error map: ⟨0.002, -0.010, 0.013, -0.053]
- CWE: ~3720087/3604480 = 1\22, ~3/2 = 701.942
- error map: ⟨0.000, -0.013, 0.010, -0.053]
Optimal ET sequence: 22*, 154c, 176, 330, 506, 1342, 1848, 2354, 4202, 6556, 8910c, 15466c
Badness: 0.999
Titanium
Titanium is named after the 22nd chemical 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, 4296875/3779136
- WE: ~48125/46656 = 54.545 ¢, ~4296875/3779136 = 224.101
- error map: ⟨-0.002, -0.056, 0.151, -0.121, 0.051]
- CWE: ~48125/46656 = 1\22, ~4296875/3779136 = 224.101
- error map: ⟨0.000, -0.054, 0.153, -0.117, 0.056]
Optimal ET sequence: 198, 616, 814, 1012, 1826
Badness: 2.913
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 13]]
- mapping generators: ~65/63, 338/297
- WE: ~65/63 = 54.545 ¢, ~338/297 = 224.101
- error map: ⟨-0.003, -0.062, 0.142, -0.128, 0.047, 0.045]
- CWE: ~65/63 = 1\22, ~338/297 = 224.101
- error map: ⟨-0.000, -0.060, 0.146, -0.122, 0.055, 0.052]
Optimal ET sequence: 198, 616, 814, 1012, 1826, 2838
Badness: 1.373