20th-octave temperaments: Difference between revisions
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{{Infobox fractional-octave|20}} | {{Infobox fractional-octave|20}} | ||
[[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. | [[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. For example, in the [[17-limit]], one step of 20edo is extremely close to [[88/85]], which serves as a period in all these temperaments. | ||
It is worth noting that [[degrees]] | It is worth noting that [[hemimage temperaments #Degrees|degrees]] 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. | ||
Temperaments discussed elsewhere include | Temperaments discussed elsewhere include | ||
* ''[[Degrees]]'' → [[Hemimage temperaments #Degrees|Hemimage temperaments]] | * ''[[Degrees]]'' → [[Hemimage temperaments #Degrees|Hemimage temperaments]] | ||
* ''[[Soviet ferris wheel]]'' → [[Horwell temperaments #Soviet ferris wheel|Horwell temperaments]] | |||
Considered below is calcium. | |||
== Calcium == | == Calcium == | ||
Latest revision as of 10:17, 28 October 2026
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. For example, in the 17-limit, one step of 20edo is extremely close to 88/85, which serves as a period in all these temperaments.
It is worth noting that degrees 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.
Temperaments discussed elsewhere include
Considered below is calcium.
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