21st-octave temperaments

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.

Akjayland

See also: Landscape microtemperaments #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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV sequence: 168h, 357, 525, 882, 1407

Badness: 0.0199