20th-octave temperaments: Difference between revisions
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[[20edo]] is not a particularly harmonically interesting | [[20edo]] is not a particularly harmonically interesting edo, but some of its multiples have high consistency limits (17-odd-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 | 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. | ||
Temperaments discussed elsewhere include | |||
'' | * ''[[Degrees]]'' → [[Hemimage temperaments #Degrees|Hemimage temperaments]] | ||
== Soviet Ferris wheel == | |||
==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). | 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). | ||
Revision as of 08:41, 18 November 2022
20edo is not a particularly harmonically interesting edo, but some of its multiples have high consistency limits (17-odd-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.
Temperaments discussed elsewhere include
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.
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 9801/9800, 12376/12375, 37180/37179, 123201/123200, 903168/903125
Mapping: [⟨20 3 12 -73 -37 8 10], ⟨0 10 12 45 37 23 25]]
Mapping generators: ~88/85, ~243/220
Optimal tuning (CTE): ~243/220 = 172.196
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 9801/9800, 12376/12375, 89376/89375, 104976/104975, 123201/123200, 1549184/1549125
Mapping: [⟨20 3 12 -73 -37 8 10 62], ⟨0 10 12 45 37 23 25 8]]
Mapping generators: ~88/85, ~169/153
Optimal tuning (CTE): ~169/153 = 172.196
Vals: 460, 2000, 2460
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