Didymus rank three family: Difference between revisions
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The '''didymus rank-3 family''' are rank-3 temperaments tempering out the didymus comma, [[81/80]]. If nothing else is tempered out we have a [[7-limit]] [[planar temperament]], with an [[11-limit]] comma we get an 11-limit temperament, and so forth. | The '''didymus rank-3 family''' are rank-3 temperaments tempering out the didymus comma, [[81/80]]. If nothing else is tempered out we have a [[7-limit]] [[planar temperament]], with an [[11-limit]] comma we get an 11-limit temperament, and so forth. | ||
Most of these temperaments were named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100647.html#100648 Yahoo! Tuning Group | ''Higher-rank meantone extensions (muse temperaments?)'']</ref>. | |||
Temperaments discussed elsewhere include: | Temperaments discussed elsewhere include: | ||
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{{Mapping|legend=1| 1 0 -4 0 | 0 1 4 0 | 0 0 0 1 }} | {{Mapping|legend=1| 1 0 -4 0 | 0 1 4 0 | 0 0 0 1 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 696.2387{{c}}, ~7/4 = 964.9090{{c}} | ||
{{Optimal ET sequence|legend=1| 12, 19, 31, 81 }} | {{Optimal ET sequence|legend=1| 12, 19, 31, 81 }} | ||
[[Badness]]: 0.095 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.095 × 10<sup>-3</sup> | ||
== Euterpe == | == Euterpe == | ||
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{{Mapping|legend=1| 1 0 -4 0 1 | 0 1 4 0 -2 | 0 0 0 1 2 }} | {{Mapping|legend=1| 1 0 -4 0 1 | 0 1 4 0 -2 | 0 0 0 1 2 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 696.1982{{c}}, ~7/4 = 968.4280{{c}} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
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{{Optimal ET sequence|legend=1| 12, 17c, 19e, 26, 31, 88 }} | {{Optimal ET sequence|legend=1| 12, 17c, 19e, 26, 31, 88 }} | ||
[[Badness]]: 0.536 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.536 × 10<sup>-3</sup> | ||
== Calliope == | == Calliope == | ||
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{{Mapping|legend=1| 1 0 -4 0 -6 | 0 1 4 0 6 | 0 0 0 1 0 }} | {{Mapping|legend=1| 1 0 -4 0 -6 | 0 1 4 0 6 | 0 0 0 1 0 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 696.1982{{c}}, ~7/4 = 968.4280{{c}} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[11-odd-limit]] | * [[11-odd-limit]] | ||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 0 1/6 }}, {{monzo| 0 0 0 0 2/3 }}, {{monzo| 1 -1 0 1 1/6 }}, {{monzo| 0 0 0 0 1 }}] | : [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 0 1/6 }}, {{monzo| 0 0 0 0 2/3 }}, {{monzo| 1 -1 0 1 1/6 }}, {{monzo| 0 0 0 0 1 }}] | ||
: [[Eigenmonzo basis| | : [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.7/3.11 | ||
{{Optimal ET sequence|legend=1| 7d, 12, 19, 26, 45 }} | {{Optimal ET sequence|legend=1| 7d, 12, 19, 26, 45 }} | ||
[[Badness]]: 0.530 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.530 × 10<sup>-3</sup> | ||
== Erato == | == Erato == | ||
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{{Mapping|legend=1| 1 0 -4 -13 0 | 0 1 4 10 0 | 0 0 0 0 1 }} | {{Mapping|legend=1| 1 0 -4 -13 0 | 0 1 4 10 0 | 0 0 0 0 1 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 696.4949{{c}}, ~11/8 = 547.0252{{c}} | ||
{{Optimal ET sequence|legend=1| 12, 19, 31, 50, 81 }} | {{Optimal ET sequence|legend=1| 12, 19, 31, 50, 81 }} | ||
[[Badness]]: 0.558 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.558 × 10<sup>-3</sup> | ||
=== 13-limit === | === 13-limit === | ||
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Mapping: {{mapping| 1 0 -4 -13 0 -20 | 0 1 4 10 0 15 | 0 0 0 0 1 0 }} | Mapping: {{mapping| 1 0 -4 -13 0 -20 | 0 1 4 10 0 15 | 0 0 0 0 1 0 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 695.9883{{c}}, ~11/8 = 545.6817{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, 19, 31, 50, 81 }} | ||
== Clio == | == Clio == | ||
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{{Mapping|legend=1| 1 0 -4 0 -12 | 0 1 4 0 8 | 0 0 0 1 1 }} | {{Mapping|legend=1| 1 0 -4 0 -12 | 0 1 4 0 8 | 0 0 0 1 1 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 697.2502{{c}}, ~7/4 = 968.6295{{c}} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
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{{Optimal ET sequence|legend=1| 7, 12, 19e, 24, 31, 105, 129 }} | {{Optimal ET sequence|legend=1| 7, 12, 19e, 24, 31, 105, 129 }} | ||
[[Badness]]: 0.738 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.738 × 10<sup>-3</sup> | ||
== Polyhymnia | == Polyhymnia == | ||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
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{{Mapping|legend=1| 1 0 -4 0 11 | 0 1 4 0 -3 | 0 0 0 1 -1 }} | {{Mapping|legend=1| 1 0 -4 0 11 | 0 1 4 0 -3 | 0 0 0 1 -1 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 696.2305{{c}}, ~7/4 = 964.8695{{c}} | ||
{{Optimal ET sequence|legend=1| 7, 12e, 19, 24, 26, 31 }} | {{Optimal ET sequence|legend=1| 7, 12e, 19, 24, 26, 31 }} | ||
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Scales: [[polyhymnia12]] | Scales: [[polyhymnia12]] | ||
== Thalia | == Thalia == | ||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
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{{Mapping|legend=1| 1 0 -4 0 5 | 0 1 4 0 -1 | 0 0 0 1 0 }} | {{Mapping|legend=1| 1 0 -4 0 5 | 0 1 4 0 -1 | 0 0 0 1 0 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 692.0796{{c}}, ~7/4 = 950.2565{{c}} | ||
{{Optimal ET sequence|legend=1| 5, 7, 12e, 14c, 19e }} | {{Optimal ET sequence|legend=1| 5, 7, 12e, 14c, 19e }} | ||
== Melpomene | == Melpomene == | ||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
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{{Mapping|legend=1| 1 0 -4 0 7 | 0 1 4 0 -4 | 0 0 0 1 1 }} | {{Mapping|legend=1| 1 0 -4 0 7 | 0 1 4 0 -4 | 0 0 0 1 1 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 699.2230{{c}}, ~7/4 = 964.2363{{c}} | ||
{{Optimal ET sequence|legend=1| 7d, 12, 17c, 19, 24, 31e, 36 }} | {{Optimal ET sequence|legend=1| 7d, 12, 17c, 19, 24, 31e, 36 }} | ||
== Terpsichore == | |||
== Terpsichore | |||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
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{{Mapping|legend=1| 1 0 -4 0 -2 | 0 1 4 0 7 | 0 0 0 1 -2 }} | {{Mapping|legend=1| 1 0 -4 0 -2 | 0 1 4 0 7 | 0 0 0 1 -2 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 696.2358{{c}}, ~7/4 = 964.0006{{c}} | ||
{{Optimal ET sequence|legend=1| 14c, 17c, 19, 31, 81, 112b }} | {{Optimal ET sequence|legend=1| 14c, 17c, 19, 31, 81, 112b }} | ||
[[Badness]]: 0.850 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.850 × 10<sup>-3</sup> | ||
[[Complexity spectrum]]: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11 | [[Complexity spectrum]]: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11 | ||
== References == | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Pages with mostly numerical content]] | |||
[[Category:Didymus rank-3 family| ]] <!-- main article --> | [[Category:Didymus rank-3 family| ]] <!-- main article --> | ||
[[Category:Didymus| ]] <!-- key article --> | [[Category:Didymus| ]] <!-- key article --> | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||