Subgroup temperaments: Difference between revisions
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Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]]. | Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]]. | ||
= | = Composite subgroup temperaments = | ||
== 2.3.35 subgroup == | == 2.3.35 subgroup == | ||
=== Shaka === | === Shaka === | ||
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==== 2.9.5.7.13 ==== | ==== 2.9.5.7.13 ==== | ||
{{See also|Chromatic pairs #Baldy}} | |||
Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 & 35 temperament. | |||
[[Subgroup]]: 2.9.5.7.13 | |||
[[Comma list]]: [[225/224]], [[325/324]], [[640/637]] | |||
{{Optimal ET sequence|legend=1| 6, | {{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }} | ||
{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }} | |||
: [[gencom]]: [2 9/8; 225/224 325/324 640/637] | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090 | |||
{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents | |||
Related temperament: [[Schismatic family #Garibaldi|Cassandra]] | Related temperament: [[Schismatic family #Garibaldi|Cassandra]] | ||
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{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }} | {{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }} | ||
== 2.9.5.11 subgroup == | |||
=== Glacial === | |||
{{See also| Chromatic pairs #Glacial }} | |||
[[Subgroup]]: 2.9.5.11.13 | |||
[[Comma list]]: 45/44, 65/64, 81/80 | |||
{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }} | |||
{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }} | |||
: [[gencom]]: [2 9/8; 45/44 65/64 81/80] | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151 | |||
{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents | |||
== 2.9.7 subgroup == | == 2.9.7 subgroup == | ||
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== 2.9.7.11 subgroup == | == 2.9.7.11 subgroup == | ||
=== Apparatus === | |||
[[Subgroup]]: 2.9.7.11 | |||
[[Comma list]]: 41503/41472, 322102/321489 | |||
{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }} | |||
: mapping generators: ~2, ~77/72 | |||
{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }} | |||
: [[gencom]]: [2 77/72; 41503/41472 322102/321489] | |||
[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685 | |||
{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }} | |||
[[Badness]]: 0.00263 | |||
=== Joan === | |||
{{See also| Chromatic pairs #Joan }} | |||
Joan is related to [[casablanca]] as well as to [[orwell]]. | |||
[[Subgroup]]: 2.9.7.11 | |||
[[Comma list]]: 99/98, 9317/9216 | |||
{{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }} | |||
{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }} | |||
: [[gencom]]: [2 11/8; 99/98 9317/9216] | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents | |||
{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents | |||
=== Machine === | === Machine === | ||
Machine is every other step of [[supra]], most interesting for its scale patterns. | Machine is every other step of [[supra]], most interesting for its scale patterns. | ||
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[[Badness]]: 0.00439 | [[Badness]]: 0.00439 | ||
== 2.9.11 subgroup == | == 2.9.11 subgroup == | ||