21st-octave temperaments: Difference between revisions
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Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
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Optimal tuning (CTE): ~6615/5632 = 278.8998 | Optimal tuning (CTE): ~6615/5632 = 278.8998 | ||
Optimal | {{Optimal ET sequence|legend=1| 168, 357, 525, 882, 1407, 2289e }} | ||
Badness: 0.0949 | Badness: 0.0949 | ||
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Optimal tuning (CTE): ~168/143 = 278.9058 | Optimal tuning (CTE): ~168/143 = 278.9058 | ||
Optimal | {{Optimal ET sequence|legend=1| 168, 357, 525, 882 }} | ||
Badness: 0.0551 | Badness: 0.0551 | ||
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Optimal tuning (CTE): ~168/143 = 278.9036 | Optimal tuning (CTE): ~168/143 = 278.9036 | ||
Optimal | {{Optimal ET sequence|legend=1| 168, 357, 525, 882 }} | ||
Badness: 0.0319 | Badness: 0.0319 | ||
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Optimal tuning (CTE): ~168/143 = 278.9036 | Optimal tuning (CTE): ~168/143 = 278.9036 | ||
Optimal | {{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }} | ||
Badness: 0.0270 | Badness: 0.0270 | ||
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Optimal tuning (CTE): ~168/143 = 278.8971 | Optimal tuning (CTE): ~168/143 = 278.8971 | ||
Optimal | {{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }} | ||
Badness: 0.0199 | Badness: 0.0199 | ||
Revision as of 16:08, 7 May 2023
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
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: Template:Val list
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