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20th-octave temperaments: Difference between revisions

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[[20edo]] is not a particularly harmonically interesting EDO, but some of its multiples have high consistency limits (17-limit and higher), and therefore are worthy of being considered in terms of rank-2 temperaments.
{{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.


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.
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.


== Degrees ==
Temperaments discussed elsewhere include
''Main article: [[Hemimage temperaments#Degrees]]''
* ''[[Degrees]]'' → [[Hemimage temperaments #Degrees|Hemimage temperaments]]


In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.
== 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).


Subgroup: 2.3.5.7
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.


[[Comma list]]: 10976/10935, 390625/388962
Subgroup: 2.3.5


[[Mapping]]: [{{val| 20 0 -17 -39 }}, {{val| 0 1 2 3 }}]
Comma list: {{monzo|-171 20 60}}


{{Multival|legend=1| 20 40 60 17 39 27 }}
Mapping: {{val|20 0 57}}, {{val|0 3 -1}}


[[POTE generator]]: ~3/2 = 703.015
Mapping generators: ~{{monzo|77 -9 -27}} = 1\20, ~208568572998046875/144115188075855872 = 633.970


{{Val list|legend=1| 60, 80, 140, 640b, 780b, 920b }}
Optimal tuning (CTE): ~208568572998046875/144115188075855872 = 633.970


[[Badness]]: 0.106471
{{Optimal ET sequence|legend=1|40, 140, 180, 320, 460, 600, 740, 780, 1060, 1240, 1520}}, ...
===11-limit===
Subgroup: 2.3.5.7.11


Comma list: 1331/1323, 1375/1372, 2200/2187
=== 7-limit ===


Mapping: [{{val| 20 0 -17 -39 -26 }}, {{val| 0 1 2 3 3 }}]
Subgroup: 2.3.5.7


POTE generator: ~3/2 = 703.231
Comma list: 65625/65536, 1977326743/1968300000


Optimal GPV sequence: {{Val list| 60e, 80, 140, 360, 500be, 860bde }}
Mapping: {{val|20 0 57 35}}, {{val|0 3 -1 2}}


Badness: 0.046770
Mapping generators: ~16807/16200 = 1\20, ~3456/2401 = 634.023
===13-limit===
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 352/351, 1001/1000, 1331/1323
Optimal tuning (CTE): ~3456/2401 = 634.023


Mapping: [{{val| 20 0 -17 -39 -26 74 }}, {{val| 0 1 2 3 3 0 }}]
=== 11-limit ===
 
POTE generator: ~3/2 = 703.080
 
Optimal GPV sequence: {{Val list| 60e, 80, 140, 500be, 640be, 780be }}
 
Badness: 0.032718
==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).


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 80: Line 70:
Optimal tuning (CTE): ~238/165 = 633.913
Optimal tuning (CTE): ~238/165 = 633.913


Vals: 140, 320, 460
{{Optimal ET sequence|legend=1|140, 320, 460}}
===19-limit===
===19-limit===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19
Line 92: Line 82:
Optimal tuning (CTE): ~238/165 = 633.913
Optimal tuning (CTE): ~238/165 = 633.913


Vals: {{EDOs|140, 320, 460}}
{{Optimal ET sequence|legend=1|140, 320, 460}}
 
==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.
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 ===
=== 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


Line 108: Line 101:


=== 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


Line 118: Line 113:
Optimal tuning (CTE): ~169/153 = 172.196
Optimal tuning (CTE): ~169/153 = 172.196


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


===23-limit===
===23-limit===
Line 131: Line 126:
Optimal tuning (CTE): ~11875/10752 = 172.196
Optimal tuning (CTE): ~11875/10752 = 172.196


Vals: {{EDOs|460, 2000, 2460}}
{{Optimal ET sequence|legend=1|460, 2000, 2460}}
 
{{Navbox fractional-octave}}


[[Category:20edo]]
[[Category:20edo]]
[[Category:Temperament collections]]
Retrieved from "https://en.xen.wiki/w/20th-octave_temperaments"