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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.  


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 [[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
Temperaments discussed elsewhere include
* ''[[Degrees]]'' → [[Hemimage temperaments #Degrees|Hemimage temperaments]]
* ''[[Degrees]]'' → [[Hemimage temperaments #Degrees|Hemimage temperaments]]


== Soviet Ferris wheel ==
== 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 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 20 [[375/256]]'s at 11 octaves, tempering this interval to 11\20.
 
Subgroup: 2.3.5
 
Comma list: {{monzo|-171 20 60}}
 
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
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.


{{Optimal ET sequence|legend=1|40, 140, 180, 320, 460, 600, 740, 780, 1060, 1240, 1520}}, ...
[[Subgroup]]: 2.3.5.7


=== 7-limit ===
[[Comma list]]: 65625/65536, 1977326743/1968300000


Subgroup: 2.3.5.7
{{Mapping|legend=1| 20 0 57 35 | 0 3 -1 2 }}
: mapping generators: ~16807/16200, ~3456/2401


Comma list: 65625/65536, 1977326743/1968300000
[[Optimal tuning]]s:  
* [[WE]]: ~16807/16200 = 60.0061{{c}}, ~3456/2401 = 634.0252{{c}}
: [[error map]]: {{val| +0.121 +0.121 +0.007 -0.563 }}
* [[CWE]]: ~16807/16200 = 60.0000{{c}}, ~3456/2401 = 633.9770{{c}}
: error map: {{val| 0.000 -0.024 -0.291 -0.872 }}


Mapping: {{val|20 0 57 35}}, {{val|0 3 -1 2}}
{{Optimal ET sequence|legend=1| 140, 320, 460, 600, 1060d }}


Mapping generators: ~16807/16200 = 1\20, ~3456/2401 = 634.023
[[Badness]] (Sintel): 5.25
 
Optimal tuning (CTE): ~3456/2401 = 634.023


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 9801/9800, 65625/65536, 422576/421875
Comma list: 9801/9800, 65625/65536, 422576/421875


Mapping: {{val|20 0 57 35 122}}, {{val|0 3 -1 2 5}}
{{Mapping|legend=0| 20 0 57 35 122 | 0 3 -1 2 5 }}


Mapping generators: ~512/495, ~231/160
Optimal tunings:
* WE: ~512/495 = 60.0085{{c}}, ~231/160 = 633.9990{{c}}
* CWE: ~512/495 = 60.0000{{c}}, ~231/160 = 633.9127{{c}}


Optimal tuning (CTE): ~231/160 = 633.929
{{Optimal ET sequence|legend=0| 140, 320, 460 }}
 
Badness (Sintel): 2.92


=== 13-limit ===
=== 13-limit ===
Line 53: Line 51:
Comma list: 1001/1000, 4225/4224, 4459/4455, 86625/86528
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|legend=0| 20 0 57 35 122 74 | 0 3 -1 2 5 0 }}


Mapping generators: ~3900/3773, ~75/52
Optimal tunings:
* WE: ~512/495 = 60.0082{{c}}, ~75/52 = 633.9971{{c}}
* CWE: ~512/495 = 60.0000{{c}}, ~75/52 = 633.9127{{c}}


Optimal tuning (CTE): ~75/52 = 633.929
{{Optimal ET sequence|legend=0| 140, 320, 460 }}


===17-limit===
Badness (Sintel): 1.48
 
=== 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: 833/832, 1001/1000, 1225/1224, 4225/4224, 4459/4455


Mapping: {{val|20 0 57 35 122 74 124}}, {{val|0 3 -1 2 5 0 -4}}
{{Mapping|legend=0| 20 0 57 35 122 74 124 | 0 3 -1 2 5 0 -4 }}


Mapping generators: ~88/85, ~238/165
Optimal tunings:
* WE: ~88/85 = 60.0081{{c}}, ~75/52 = 633.9982{{c}}
* CWE: ~88/85 = 60.0000{{c}}, ~75/52 = 633.9117{{c}}


Optimal tuning (CTE): ~238/165 = 633.913
{{Optimal ET sequence|legend=0| 140, 320, 460 }}


{{Optimal ET sequence|legend=1|140, 320, 460}}
Badness (Sintel): 1.14
===19-limit===
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 833/832, 1001/1000, 1225/1224, 1729/1728, 2926/2925, 4459/4455
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|legend=0| 20 0 57 35 122 74 124 11 | 0 3 -1 2 5 0 -4 7 }}


Mapping generators: ~88/85, ~238/165
Optimal tunings:
* WE: ~88/85 = 60.0075{{c}}, ~75/52 = 633.9713{{c}}
* CWE: ~88/85 = 60.0000{{c}}, ~75/52 = 633.8948{{c}}


Optimal tuning (CTE): ~238/165 = 633.913
{{Optimal ET sequence|legend=0| 140, 320, 460 }}


{{Optimal ET sequence|legend=1|140, 320, 460}}
Badness (Sintel): 1.16


==Calcium==
== 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.
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 ===
=== 17-limit ===
Both [[5/3]] and [[17/13]] are reached in two generator steps.
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7.11.13.17
Comma list: 9801/9800, 10648/10647, 12376/12375, 37180/37179, 903168/903125


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|legend=0| 20 3 12 -73 -37 8 10 62 145 | 0 10 12 45 37 23 25 8 -19 }}


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


Mapping generators: ~88/85, ~11875/10752
{{Optimal ET sequence|legend=0| 460, 2000, 2460, 4460, 6920 }}


Optimal tuning (CTE): ~11875/10752 = 172.196
Badness (Sintel): 0.839


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


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