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.

Fractional-octave temperaments

← 21st-octave • 22nd-octave • 23rd-octave →

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

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

Optimal tunings:

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

Optimal tunings:

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]

Optimal ET sequence: 22, 44

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 a 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

Optimal tunings:

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

Optimal tunings:

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

Optimal tunings:

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

Optimal tunings: