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-odd-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. | ||
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== 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). | 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. | |||
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 | |||
{{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 | Subgroup: 2.3.5.7.11 | ||
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Optimal tuning (CTE): ~238/165 = 633.913 | Optimal tuning (CTE): ~238/165 = 633.913 | ||
{{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 | ||
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Optimal tuning (CTE): ~238/165 = 633.913 | Optimal tuning (CTE): ~238/165 = 633.913 | ||
{{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 | 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 | ||
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=== 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 | ||
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Optimal tuning (CTE): ~169/153 = 172.196 | Optimal tuning (CTE): ~169/153 = 172.196 | ||
{{Optimal ET sequence|legend=1|460, 2000, 2460}} | |||
===23-limit=== | ===23-limit=== | ||
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Optimal tuning (CTE): ~11875/10752 = 172.196 | Optimal tuning (CTE): ~11875/10752 = 172.196 | ||
{{Optimal ET sequence|legend=1|460, 2000, 2460}} | |||
{{Navbox fractional-octave}} | |||
[[Category:20edo]] | [[Category:20edo]] | ||