# 21st-octave temperaments

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This page collectes temperaments with a period of 1/21 of an octave.

Although 21edo itself is not remarkably accurate for low-complexity harmonics, some temperaments which are multiples of 21, such as 441, 1407, and 1848 are. 441 and 1848 are also members of zeta edo list.

## 21-23-commatic

Subgroup: 2.23

Comma list: [95 0 0 0 0 0 0 0 -21⟩

- mapping generator: ~529/512 = 1\21

Supporting ETs: 21N, N = 1 to 96, largest: 2016edo

## Akjayland

Subgroup: 2.3.5.7

Comma list: 250047/250000, [43 -1 -13 -4⟩

Mapping: [⟨21 1 38 102], ⟨0 3 1 -4]]

Mapping generators: ~1323/1280, ~131072/91875

Optimal tuning (CTE): ~131072/91875 = 614.9354

Optimal ET sequence: 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c

Badness: 0.0309

### Vasca

Vasca can be described as the 357 & 525 temperament, extended as high as the 23-limit. It tempers out the [95 0 0 0 0 0 0 0 -21⟩, and sets a stack of 21 23/16's equal with 11 octaves. The name derives from elements vanadium (23) and scandium (21), since this uses the 23rd harmonic, which itself is extremely well represented in 21edo.

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 102487/102400, [39 -4 -11 -5 2⟩

Mapping: [⟨21 4 39 98 58], ⟨0 6 2 -8 3 3]]

Mapping generators: ~1323/1280, ~6615/5632

Optimal tuning (CTE): ~6615/5632 = 278.8998

Optimal ET sequence: 168, 357, 525, 882, 1407, 2289e

Badness: 0.0949

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683

Mapping: [⟨21 4 39 98 58 107], ⟨0 6 2 -8 3 -6]]

Optimal tuning (CTE): ~168/143 = 278.9058

Optimal ET sequence: 168, 357, 525, 882

Badness: 0.0551

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 2601/2600, 3025/3024, 4096/4095, 8624/8619, 14641/14625

Mapping: [⟨21 4 39 98 58 107 120], ⟨0 6 2 -8 3 -6 -7]]

Optimal tuning (CTE): ~168/143 = 278.9036

Optimal ET sequence: 168, 357, 525, 882

Badness: 0.0319

#### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095

Mapping: [⟨21 4 39 98 58 107 120 16], ⟨0 6 2 -8 3 -6 -7 15]]

Optimal tuning (CTE): ~168/143 = 278.9036

Optimal ET sequence: 168h, 357, 525, 882, 1407

Badness: 0.0270

#### 23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211

Mapping: [⟨21 34 49 58 73 77 85 91 95], ⟨0 -6 -2 8 -3 6 7 -15 0]]

Optimal tuning (CTE): ~168/143 = 278.8971

Optimal ET sequence: 168h, 357, 525, 882, 1407

Badness: 0.0199

## Scandium

Described as the 525 & 1911 temperament, and named after the 21st element for splitting the octave into 21 parts. Coincidentally, *Encyclopaedia Britannica* entry for scandium was written in the year 1911 which was used as the reason for the naming. Remarkably, unlike akjayland or many temperaments in the thousands which contain 3edo as a subset, it is *not* a landscape system. 39/32 is mapped into 6\21 and 23/16 is, as usual, mapped into 11\21.

Subgroup: 2.3.5.7

Comma list: [47 -7 -7 -7⟩, [-29 0 27 -12⟩

Mapping: [⟨21 0 59 82], ⟨0 13 -4 -9]]

- mapping generators: ~403368/390625 = 1\21, ~160/147 = 146.305

Optimal tuning (CTE): ~160/147 = 146.305

Supporting ETs: 189b, 525, 861, 1050, 1386, 1911, 2436

### 23-limit

Subgroup: 2.3.5.7.11.13.17.19.21.23

Comma list: 2500/2499, 3025/3024, 3060/3059, 3520/3519, 4096/4095, 6175/6174, 79135/79092

Mapping: [⟨21 0 59 82 24 111 114 38 95], ⟨0 13 -4 -9 19 -13 -11 20 0]]

- mapping generators: ~216/209 = 1\21, ~160/147 = 146.308

Optimal tuning (CTE): ~160/147 = 146.305