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. 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.
[[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.  


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.
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]]


== Soviet Ferris wheel ==
Considered below is calcium.  
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.
== 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.


Subgroup: 2.3.5
Calcium was named by [[Eliora]] in 2022 after the 20th element, following the convention of naming some fractional-octave temperaments after chemical elements.


Comma list: {{monzo|-171 20 60}}
=== 17-limit ===
 
Mapping: {{val|20 0 57}}, {{val|0 3 -1}}
 
Mapping generators: ~{{monzo|77 -9 -27}} = 1\20, ~208568572998046875/144115188075855872 = 633.970
 
Optimal tuning (CTE): ~208568572998046875/144115188075855872 = 633.970
 
{{Optimal ET sequence|legend=1|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: {{val|20 0 57 35}}, {{val|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: {{val|20 0 57 35 122}}, {{val|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: {{val|20 0 57 35 122 74}}, {{val|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
Subgroup: 2.3.5.7.11.13.17


Comma list: 833/832, 1001/1000, 1225/1224, 4225/4224, 4459/4455
Comma list: 9801/9800, 10648/10647, 12376/12375, 37180/37179, 903168/903125
 
Mapping: {{val|20 0 57 35 122 74 124}}, {{val|0 3 -1 2 5 0 -4}}
 
Mapping generators: ~88/85, ~238/165
 
Optimal tuning (CTE): ~238/165 = 633.913
 
{{Optimal ET sequence|legend=1|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: [{{val|20 0 57 35 122 74 124 11}}, {{val|0 3 -1 2 5 0 -4 7}}]
 
Mapping generators: ~88/85, ~238/165
 
Optimal tuning (CTE): ~238/165 = 633.913
 
{{Optimal ET sequence|legend=1|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|legend=0| 20 3 12 -73 -37 8 10 | 0 10 12 45 37 23 25 }}
: mapping generators: ~88/85, ~243/220


Mapping: [{{val|20 3 12 -73 -37 8 10}}, {{val|0 10 12 45 37 23 25}}]
Optimal tunings:  
* WE: ~88/85 = 59.9999{{c}}, ~243/220 = 172.1964{{c}}
* CWE: ~88/85 = 60.0000{{c}}, ~243/220 = 172.1966{{c}}


Mapping generators: ~88/85, ~243/220
{{Optimal ET sequence|legend=0| 460, 1540d, 2000, 2460, 4460, 6920 }}


Optimal tuning (CTE): ~243/220 = 172.196
Badness (Sintel): 0.909


=== 19-limit ===
=== 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
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 9801/9800, 12376/12375, 89376/89375, 104976/104975, 123201/123200, 1549184/1549125
Comma list: 9801/9800, 10648/10647, 12376/12375, 89376/89375, 104976/104975, 1549184/1549125


Mapping: [{{val|20 3 12 -73 -37 8 10 62}}, {{val|0 10 12 45 37 23 25 8}}]
{{Mapping|legend=0| 20 3 12 -73 -37 8 10 62 | 0 10 12 45 37 23 25 8 }}
: mapping generators: ~88/85, ~169/153


Mapping generators: ~88/85, ~169/153
Optimal tunings:
* WE: ~88/85 = 59.9997{{c}}, ~169/153 = 172.1962{{c}}
* CWE: ~88/85 = 60.0000{{c}}, ~169/153 = 172.1969{{c}}


Optimal tuning (CTE): ~169/153 = 172.196
{{Optimal ET sequence|legend=1| 460, 1540d, 2000, 2460, 4460, 11380h }}


{{Optimal ET sequence|legend=1|460, 2000, 2460}}
Badness (Sintel): 0.765


===23-limit===
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
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
Comma list: 9801/9800, 10626/10625, 10648/10647, 12376/12375, 21505/21504, 21736/21735, 89376/89375


Mapping: [{{val|20 3 12 -73 -37 8 10 62 145}}, {{val|0 10 12 45 37 23 25 8 -19}}]
{{Mapping|legend=0| 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 tunings:
* WE: ~88/85 = 59.9999{{c}}, ~169/153 = 172.1963{{c}}
* CWE: ~88/85 = 60.0000{{c}}, ~169/153 = 172.1966{{c}}


Optimal tuning (CTE): ~11875/10752 = 172.196
{{Optimal ET sequence|legend=0| 460, 2000, 2460, 4460, 6920 }}
 
Badness (Sintel): 0.839


{{Optimal ET sequence|legend=1|460, 2000, 2460}}


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}
[[Category:20edo]]
[[Category:20edo]]

Latest revision as of 10:17, 28 October 2026

← 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. 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


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