20th-octave temperaments: Difference between revisions
creating page, not finished yet |
okay I'll fill in the lower limit versions of calcium later once I have time since I have school, the remarkable fact is how high it goes |
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Comma list: 9801/9800, 65625/65536, 422576/421875 | Comma list: 9801/9800, 65625/65536, 422576/421875 | ||
Mapping: 20 0 57 35 122, 0 3 -1 2 5 | Mapping: {{val|20 0 57 35 122}}, {{val|0 3 -1 2 5}} | ||
Mapping generators: ~512/495, ~231/160 | Mapping generators: ~512/495, ~231/160 | ||
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Comma list: 1001/1000, 4225/4224, 4459/4455, 86625/86528 | Comma list: 1001/1000, 4225/4224, 4459/4455, 86625/86528 | ||
Mapping: 20 0 57 35 122 74, 0 3 -1 2 5 0 | Mapping: {{val|20 0 57 35 122 74}}, {{val|0 3 -1 2 5 0}} | ||
Mapping generators: ~3900/3773, ~75/52 | Mapping generators: ~3900/3773, ~75/52 | ||
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Comma list: 833/832, 1001/1000, 1225/1224, 4225/4224, 4459/4455 | Comma list: 833/832, 1001/1000, 1225/1224, 4225/4224, 4459/4455 | ||
Mapping: 20 0 57 35 122 74 124, 0 3 -1 2 5 0 -4 | Mapping: {{val|20 0 57 35 122 74 124}}, {{val|0 3 -1 2 5 0 -4}} | ||
Mapping generators: ~88/85, ~238/165 | Mapping generators: ~88/85, ~238/165 | ||
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Comma list: 833/832, 1001/1000, 1225/1224, 1729/1728, 2926/2925, 4459/4455 | Comma list: 833/832, 1001/1000, 1225/1224, 1729/1728, 2926/2925, 4459/4455 | ||
Mapping: 20 0 57 35 122 74 124 11, 0 3 -1 2 5 0 -4 7 | Mapping: [{{val|20 0 57 35 122 74 124 11}}, {{val|0 3 -1 2 5 0 -4 7}}] | ||
Mapping generators: ~88/85, ~238/165 | Mapping generators: ~88/85, ~238/165 | ||
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Optimal tuning (CTE): ~238/165 = 633.913 | Optimal tuning (CTE): ~238/165 = 633.913 | ||
Vals: 140, 320, 460 | Vals: {{EDOs|140, 320, 460}} | ||
==Calcium== | ==Calcium== | ||
A highly precise and high-limit 2000 & 2460 temperament, named after the 20th element following the convention of naming some fractional-octave temperaments after chemical elements. | A highly precise and high-limit 2000 & 2460 temperament, named after the 20th element following the convention of naming some fractional-octave temperaments after chemical elements. | ||
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Comma list: 9801/9800, 10626/10625, 12376/12375, 21505/21504, 21736/21735, 89376/89375, 123201/123200 | Comma list: 9801/9800, 10626/10625, 12376/12375, 21505/21504, 21736/21735, 89376/89375, 123201/123200 | ||
Mapping: [{{val|20 3 12 -73 -37 8 10 62 145}}, {{val|0 10 12 45 37 23 25 8 -19}}] | |||
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Mapping generators: ~88/85, ~11875/10752 | Mapping generators: ~88/85, ~11875/10752 | ||
Optimal tuning (CTE): ~11875/10752 = 172.196 | Optimal tuning (CTE): ~11875/10752 = 172.196 | ||
Vals: 460, 2000, 2460 | Vals: {{EDOs|460, 2000, 2460}} | ||
[[Category:20edo]] | |||
[[Category:Temperament collections]] | |||
Revision as of 23:31, 3 November 2022
20edo is not a particularly harmonically interesting EDO, but some of its multiples have high consistency limits (17-limit and higher), and therefore are worthy of being considered in terms of rank-2 temperaments.
In the 17-limit, one step of 20edo is extremely close to 88/85, which serves as a period in two of these temperaments - Soviet Ferris wheel and calcium.
Degrees
Main article: Hemimage temperaments#Degrees
In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.
Subgroup: 2.3.5.7
Comma list: 10976/10935, 390625/388962
Mapping: [⟨20 0 -17 -39], ⟨0 1 2 3]]
Wedgie: ⟨⟨ 20 40 60 17 39 27 ]]
POTE generator: ~3/2 = 703.015
Badness: 0.106471
11-limit
Subgroup: 2.3.5.7.11
Comma list: 1331/1323, 1375/1372, 2200/2187
Mapping: [⟨20 0 -17 -39 -26], ⟨0 1 2 3 3]]
POTE generator: ~3/2 = 703.231
Optimal GPV sequence: Template:Val list
Badness: 0.046770
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 352/351, 1001/1000, 1331/1323
Mapping: [⟨20 0 -17 -39 -26 74], ⟨0 1 2 3 3 0]]
POTE generator: ~3/2 = 703.080
Optimal GPV sequence: Template:Val list
Badness: 0.032718
Soviet Ferris wheel
Defined as the 320 & 460 temperament, and named because it's a period-20 temperament, and there are 20 cabins on a standard ferris wheel found throughout most of Eastern Europe and Central Asia (as in abandoned Pripyat wheel, for example).
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 65625/65536, 422576/421875
Mapping: ⟨20 0 57 35 122], ⟨0 3 -1 2 5]
Mapping generators: ~512/495, ~231/160
Optimal tuning (CTE): ~231/160 = 633.929
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1001/1000, 4225/4224, 4459/4455, 86625/86528
Mapping: ⟨20 0 57 35 122 74], ⟨0 3 -1 2 5 0]
Mapping generators: ~3900/3773, ~75/52
Optimal tuning (CTE): ~75/52 = 633.929
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 833/832, 1001/1000, 1225/1224, 4225/4224, 4459/4455
Mapping: ⟨20 0 57 35 122 74 124], ⟨0 3 -1 2 5 0 -4]
Mapping generators: ~88/85, ~238/165
Optimal tuning (CTE): ~238/165 = 633.913
Vals: 140, 320, 460
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 833/832, 1001/1000, 1225/1224, 1729/1728, 2926/2925, 4459/4455
Mapping: [⟨20 0 57 35 122 74 124 11], ⟨0 3 -1 2 5 0 -4 7]]
Mapping generators: ~88/85, ~238/165
Optimal tuning (CTE): ~238/165 = 633.913
Calcium
A highly precise and high-limit 2000 & 2460 temperament, named after the 20th element following the convention of naming some fractional-octave temperaments after chemical elements.
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 9801/9800, 10626/10625, 12376/12375, 21505/21504, 21736/21735, 89376/89375, 123201/123200
Mapping: [⟨20 3 12 -73 -37 8 10 62 145], ⟨0 10 12 45 37 23 25 8 -19]] Mapping generators: ~88/85, ~11875/10752
Optimal tuning (CTE): ~11875/10752 = 172.196