20th-octave temperaments

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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

Vals: 140, 320, 460

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

Both 5/3 and 17/13 are reached in two generator steps.

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

Both 11/7 and 19/16 are reached in eight generator steps.

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

Vals: 460, 2000, 2460