# 22nd-octave temperaments

(Redirected from Major arcana)

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 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

## 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

## 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

## 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

### 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