# 20th-octave temperaments

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20edo (in isolation) 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. It's worth noting that Degrees (discussed elsewhere) is a no-31's 41-limit temperament which serves as a well-temperament of 80edo in the corresponding subgroup, as a 60L 20s MOS is sufficient for finding all its primes.

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).

The 5-limit comma is an interval which can also be produced by closing 20 375/256's at 11 octaves, tempering this interval to 11\20.

Subgroup: 2.3.5

Comma list: [-171 20 60⟩

Mapping: ⟨20 0 57], ⟨0 3 -1]

Mapping generators: ~[77 -9 -27⟩ = 1\20, ~208568572998046875/144115188075855872 = 633.970

Optimal tuning (CTE): ~208568572998046875/144115188075855872 = 633.970

Optimal ET sequence: 40, 140, 180, 320, 460, 600, 740, 780, 1060, 1240, 1520, ...

### 7-limit

Subgroup: 2.3.5.7

Comma list: 65625/65536, 1977326743/1968300000

Mapping: ⟨20 0 57 35], ⟨0 3 -1 2]

Mapping generators: ~16807/16200 = 1\20, ~3456/2401 = 634.023

Optimal tuning (CTE): ~3456/2401 = 634.023

### 11-limit

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

Optimal ET sequence: 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

Optimal ET sequence: 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

2000edo and 2460edo are adjacent members of a sequence of tunings with progressively less error in 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

Optimal ET sequence: 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