Octagar family: Difference between revisions
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{{Technical data page}} | |||
The '''octagar family''' of [[temperament]]s are [[rank-3 temperament]]s [[tempering out]] the octagar comma, [[4000/3969]]. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7-odd-limit minimax tuning has fifths and 7's 2.245 cents sharp, with just major thirds. | The '''octagar family''' of [[temperament]]s are [[rank-3 temperament]]s [[tempering out]] the octagar comma, [[4000/3969]]. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7-odd-limit minimax tuning has fifths and 7's 2.245 cents sharp, with just major thirds. | ||
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== Octagar == | == Octagar == | ||
This temperament is also known as ''octagari''. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 1 0 1 4 | 0 1 0 -2 | 0 0 2 3 }} | {{Mapping|legend=1| 1 0 1 4 | 0 1 0 -2 | 0 0 2 3 }} | ||
: | : Mapping generators: ~2, ~3, ~63/40 | ||
[[Mapping to lattice]]: [{{val| 0 -1 -2 -1 }}, {{val| 0 -1 0 2 }}] | [[Mapping to lattice]]: [{{val| 0 -1 -2 -1 }}, {{val| 0 -1 0 2 }}] | ||
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: 63/50 length = 0.8966, 21/20 length = 1.0605 | : 63/50 length = 0.8966, 21/20 length = 1.0605 | ||
: Angle (63/50, 21/20) = 97.743 degrees | : Angle (63/50, 21/20) = 97.743 degrees | ||
[[Optimal tuning]] (POTE): ~3/2 = 703.6224, ~21/20 = 89.3227 | |||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit]] | * [[7-odd-limit]] | ||
: {{monzo list| 1 0 0 0 | 5/6 1/3 1/2 -1/3 | 0 0 1 0 | 5/6 -2/3 1/2 2/3 }} | : {{monzo list| 1 0 0 0 | 5/6 1/3 1/2 -1/3 | 0 0 1 0 | 5/6 -2/3 1/2 2/3 }} | ||
: [[Eigenmonzo basis| | : [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3 | ||
* [[9-odd-limit]] | * [[9-odd-limit]] | ||
: {{monzo list| 1 0 0 0 | 5/8 1/2 3/8 -1/4 | 0 0 1 0 | 5/4 -1 3/4 1/2 }} | : {{monzo list| 1 0 0 0 | 5/8 1/2 3/8 -1/4 | 0 0 1 0 | 5/4 -1 3/4 1/2 }} | ||
: [[Eigenmonzo basis| | : [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7 | ||
{{Optimal ET sequence|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }} | {{Optimal ET sequence|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }} | ||
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: 11/7 length = 0.798, 22/21 length = 0.906 | : 11/7 length = 0.798, 22/21 length = 0.906 | ||
: Angle (11/7, 22/21) = 97.747 degrees | : Angle (11/7, 22/21) = 97.747 degrees | ||
[[Optimal tuning]] (POTE): ~3/2 = 703.8837, ~21/20 = 87.8919 | |||
{{Optimal ET sequence|legend=1| 12, 15, 26, 27e, 41, 109e }} | {{Optimal ET sequence|legend=1| 12, 15, 26, 27e, 41, 109e }} | ||
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Mapping: {{mapping| 1 0 1 4 4 2 | 0 1 0 -2 -2 -1 | 0 0 2 3 4 5 }} | Mapping: {{mapping| 1 0 1 4 4 2 | 0 1 0 -2 -2 -1 | 0 0 2 3 4 5 }} | ||
{{Optimal ET sequence | Optimal tuning (POTE): ~3/2 = 701.8881, ~21/20 = 87.4143 | ||
Optimal ET sequence: {{Optimal ET sequence| 12f, 14cf, 15, 26, 29, 41 }} | |||
== Octasand == | |||
[[Subgroup]]: 2.3.5.7.11 | |||
[[Comma list]]: 540/539, 2200/2187 | |||
{{Mapping|legend=1| 1 0 1 4 -5 | 0 1 0 -2 7 | 0 0 2 3 -4 }} | |||
[[Optimal tuning]] (POTE): ~3/2 = 703.5501, ~21/20 = 89.3956 | |||
{{Optimal ET sequence|legend=1| 12e, 14c, 27e, 41, 53, 80, 94, 121 }} | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 325/324, 352/351, 540/539 | |||
Mapping: {{mapping| 1 0 1 4 -5 0 | 0 1 0 -2 7 4 | 0 0 2 3 -4 -4 }} | |||
Optimal tuning (POTE): ~3/2 = 703.5688, ~21/20 = 89.4032 | |||
Optimal ET sequence: {{Optimal ET sequence| 12e, 14c, 27e, 41, 53, 80, 94, 121 }} | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Pages with mostly numerical content]] | |||
[[Category:Octagar family| ]] <!-- main article --> | [[Category:Octagar family| ]] <!-- main article --> | ||
[[Category:Octagar| ]] <!-- key article --> | [[Category:Octagar| ]] <!-- key article --> | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||