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
- 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 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