37th-octave temperaments: Difference between revisions
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{{Infobox fractional-octave|37}} | |||
[[37edo]] has an extremely precise mapping for the 11th harmonic, and it is a strong 2.5.7.13 tuning besides that, therefore various 37th-octave temperaments occur naturally between any two numbers whose greatest common divisor is 37. | [[37edo]] has an extremely precise mapping for the 11th harmonic, and it is a strong 2.5.7.13 tuning besides that, therefore various 37th-octave temperaments occur naturally between any two numbers whose greatest common divisor is 37. | ||
== 37-11-commatic (rank-1) == | |||
{{Main|37-11-comma}} | |||
Subgroup: 2.11 | |||
Comma list: {{monzo|128 -37}} | |||
{{mapping|legend=2| 37 28 }} | |||
: mapping generators: ~35184372088832/34522712143931 = 1\37 | |||
[[Support]]ing [[ET]]s: 37N, N = 1 to 485 | |||
== Rubidium == | == Rubidium == | ||
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[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.3903 | [[Optimal tuning]] ([[POTE]]): ~3/2 = 703.3903 | ||
{{ | {{Optimal ET sequence|legend=1| 37, 74, 111 }} | ||
[[Badness]]: 0.312105 | [[Badness]]: 0.312105 | ||
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Optimal tuning (POTE): ~3/2 = 703.0355 | Optimal tuning (POTE): ~3/2 = 703.0355 | ||
Optimal | {{Optimal ET sequence|legend=1| 37, 74, 111 }} | ||
Badness: 0.101001 | Badness: 0.101001 | ||
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Optimal tuning (POTE): ~3/2 = 703.0520 | Optimal tuning (POTE): ~3/2 = 703.0520 | ||
Optimal | {{Optimal ET sequence|legend=1| 37, 74, 111 }} | ||
Badness: 0.048732 | Badness: 0.048732 | ||
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[[Optimal tuning]] ([[CTE]]): ~24000/16807 = 612.4003 | [[Optimal tuning]] ([[CTE]]): ~24000/16807 = 612.4003 | ||
{{ | {{Optimal ET sequence|legend=1| 37, 222b, 259b, 296, 629 }} | ||
[[Badness]]: 0.784746 | [[Badness]]: 0.784746 | ||
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Optimal tuning (CTE): ~768/359 = 612.4003 | Optimal tuning (CTE): ~768/359 = 612.4003 | ||
Optimal | {{Optimal ET sequence|legend=1| 37, 259b, 296, 629 }} | ||
Badness: 0.167327 | Badness: 0.167327 | ||
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Optimal tuning (CTE): ~462/325 = 612.4206 | Optimal tuning (CTE): ~462/325 = 612.4206 | ||
Optimal | {{Optimal ET sequence|legend=1| 37, 259b, 296, 629f }} | ||
Badness: 0.076183 | Badness: 0.076183 | ||
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Optimal tuning (CTE): ~121/85 = 612.4187 | Optimal tuning (CTE): ~121/85 = 612.4187 | ||
Optimal | {{Optimal ET sequence|legend=1| 37, 259b, 296, 629f }} | ||
Badness: 0.052475 | Badness: 0.052475 | ||
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===13-limit=== | ===13-limit=== | ||
14 periods map to [[13/10]], thus equating a stack of three 11/8 with one 13/10 and making dzelic a [[jacobin]] temperament. | 14 periods map to [[13/10]], thus equating a stack of three 11/8 with one 13/10 and making dzelic a [[6656/6655|jacobin]] temperament. | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Comma list: 4375/4374, 6656/6655, 405769/405504, 34034175/34027136 | Comma list: 4375/4374, 6656/6655, 405769/405504, 34034175/34027136 | ||
Mapping: [{{val|37 0 -23 129 128}}, {{val|0 7 13 -3 0}] | Mapping: [{{val|37 0 -23 129 128 28}}, {{val|0 7 13 -3 0 13}}] | ||
Mapping generators: ~1248/1225 = 1\37, ~117/100 = 271.712 | Mapping generators: ~1248/1225 = 1\37, ~117/100 = 271.712 | ||
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Optimal tuning (CTE): ~11979/10240 = 271.712 | Optimal tuning (CTE): ~11979/10240 = 271.712 | ||
Optimal | {{Optimal ET sequence|legend=1|296, 1369, 1665}}, ... | ||
{{Navbox fractional-octave}} | |||