20th-octave temperaments

Fractional-octave temperaments

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← 19th-octave • 20th-octave • 21st-octave →

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 is 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

• Degrees → Hemimage temperaments

Soviet ferris wheel

The soviet-ferris-wheel temperament may be described as 320 & 460. It was named by Eliora in 2022 because it has a period of 1/20 octave, 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 twenty 375/256's at 11 octaves, tempering this interval to 11\20.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 1977326743/1968300000

Mapping: [⟨20 0 57 35], ⟨0 3 -1 2]]

mapping generators: ~16807/16200, ~3456/2401

Optimal tunings:

• WE: ~16807/16200 = 60.0061 ¢, ~3456/2401 = 634.0252 ¢

error map: ⟨+0.121 +0.121 +0.007 -0.563]

• CWE: ~16807/16200 = 60.0000 ¢, ~3456/2401 = 633.9770 ¢

error map: ⟨0.000 -0.024 -0.291 -0.872]

Optimal ET sequence: 140, 320, 460, 600, 1060d

Badness (Sintel): 5.25

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]]

Optimal tunings:

• WE: ~512/495 = 60.0085 ¢, ~231/160 = 633.9990 ¢
• CWE: ~512/495 = 60.0000 ¢, ~231/160 = 633.9127 ¢

Optimal ET sequence: 140, 320, 460

Badness (Sintel): 2.92

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]]

Optimal tunings:

• WE: ~512/495 = 60.0082 ¢, ~75/52 = 633.9971 ¢
• CWE: ~512/495 = 60.0000 ¢, ~75/52 = 633.9127 ¢

Optimal ET sequence: 140, 320, 460

Badness (Sintel): 1.48

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]]

Optimal tunings:

• WE: ~88/85 = 60.0081 ¢, ~75/52 = 633.9982 ¢
• CWE: ~88/85 = 60.0000 ¢, ~75/52 = 633.9117 ¢

Optimal ET sequence: 140, 320, 460

Badness (Sintel): 1.14

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]]

Optimal tunings:

• WE: ~88/85 = 60.0075 ¢, ~75/52 = 633.9713 ¢
• CWE: ~88/85 = 60.0000 ¢, ~75/52 = 633.8948 ¢

Optimal ET sequence: 140, 320, 460

Badness (Sintel): 1.16

Calcium

Calcium is a highly precise and high-limit temperament, described as 2000 & 2460. 2000edo and 2460edo are adjacent members of a sequence of tunings with progressively less error in 19-limit. In this temperament, some of the simpler intervals are 5/3 and 17/13, reached in two generator steps, as well as 11/7 and 19/16, reached in eight generator steps.

Calcium was named by Eliora in 2022 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, 10648/10647, 12376/12375, 37180/37179, 903168/903125

Mapping: [⟨20 3 12 -73 -37 8 10], ⟨0 10 12 45 37 23 25]]

mapping generators: ~88/85, ~243/220

Optimal tunings:

• WE: ~88/85 = 59.9999 ¢, ~243/220 = 172.1964 ¢
• CWE: ~88/85 = 60.0000 ¢, ~243/220 = 172.1966 ¢

Optimal ET sequence: 460, 1540d, 2000, 2460, 4460, 6920

Badness (Sintel): 0.909

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 9801/9800, 10648/10647, 12376/12375, 89376/89375, 104976/104975, 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 tunings:

• WE: ~88/85 = 59.9997 ¢, ~169/153 = 172.1962 ¢
• CWE: ~88/85 = 60.0000 ¢, ~169/153 = 172.1969 ¢

Optimal ET sequence: 460, 1540d, 2000, 2460, 4460, 11380h

Badness (Sintel): 0.765

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 9801/9800, 10626/10625, 10648/10647, 12376/12375, 21505/21504, 21736/21735, 89376/89375

Mapping: [⟨20 3 12 -73 -37 8 10 62 145], ⟨0 10 12 45 37 23 25 8 -19]]

Optimal tunings:

• WE: ~88/85 = 59.9999 ¢, ~169/153 = 172.1963 ¢
• CWE: ~88/85 = 60.0000 ¢, ~169/153 = 172.1966 ¢

Optimal ET sequence: 460, 2000, 2460, 4460, 6920

Badness (Sintel): 0.839