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| The '''ptolemismic clan''' of [[Rank-3 temperament|rank-3]] [[temperament]]s [[Tempering out|tempers out]] the ptolemisma, [[100/99]]. | | The '''ptolemismic clan''' of [[Rank-3 temperament|rank-3]] [[regular temperament|temperament]]s [[tempering out|tempers out]] the ptolemisma, [[100/99]]. |
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| Temperaments considered below are apollo, supermagic, thrasher, and ares. Discussed elsewhere include:
| | Strong extensions are apollo, supermagic, thrasher, and ares, considered below. Discussed elsewhere are: |
| * ''[[Calliope]]'' (+45/44 or 81/80) → [[Didymus rank three family #Calliope|Didymus rank-3 family]] | | * ''[[Calliope]]'' (+45/44 or 81/80) → [[Didymus rank three family #Calliope|Didymus rank-3 family]] |
| * ''[[Jubilee]]'' (+50/49 or 99/98) → [[Jubilismic family #Jubilee|Jubilismic family]] | | * ''[[Jubilee]]'' (+50/49 or 99/98) → [[Jubilismic family #Jubilee|Jubilismic family]] |
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| == Apollo == | | == Apollo == |
| {{See also| Marvel family #Apollo }}
| | See [[Marvel family #Apollo]]. |
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| [[Subgroup]]: 2.3.5.7.11
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| [[Comma list]]: 100/99, 225/224
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| {{Mapping|legend=1| 1 0 0 -5 2 | 0 1 0 2 -2 | 0 0 1 2 2 }}
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| [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.4846, ~5/4 = 381.6033
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| [[Minimax tuning]]: [[11-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5.11/9
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| {{Optimal ET sequence|legend=1| 12, 19, 22, 34d, 41, 104, 157ce, 198ce, 220ce, 261ce }}
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| [[Projection pair]]s: 7 225/32 11 100/9
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| [[Associated temperament]]: [[magic]]
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| === 13-limit ===
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 100/99, 225/224, 275/273
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| Mapping: {{mapping| 1 0 0 -5 2 7 | 0 1 0 2 -2 -5 | 0 0 1 2 2 2 }}
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| Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.9984, ~5/4 = 381.5352
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| Minimax tuning: 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.11/9.13/9
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| {{Optimal ET sequence|legend=1| 12f, 19f, 22, 29, 34d, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef }}
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| Projection pairs: 7 225/32 11 100/9 13 3200/243
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| == Supermagic == | | == Supermagic == |
| {{See also| Keemic family #Supermagic }}
| | See [[Keemic family #Supermagic]]. |
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| [[Subgroup]]: 2.3.5.7.11
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| [[Comma list]]: 100/99, 385/384
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| {{Mapping|legend=1| 1 0 0 5 2 | 0 1 0 3 -2 | 0 0 1 -3 2 }}
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| [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.1985, ~5/4 = 381.5394
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| {{Optimal ET sequence|legend=1| 7, 15, 19, 22, 37, 41, 104 }}
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| [[Badness]]: 0.641 × 10<sup>-3</sup>
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| Scales: [[supermagic15]]
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| == Thrasher == | | == Thrasher == |
| {{See also| Starling family #Thrasher }}
| | See [[Starling family #Thrasher]]. |
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| [[Subgroup]]: 2.3.5.7.11
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| [[Comma list]]: 56/55, 100/99
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| {{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}
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| Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]
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| Lattice basis:
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| : 6/5 length = 0.9089, 5/4 length = 1.2007
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| : Angle (6/5, 5/4) = 98.8447
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| [[Minimax tuning]]:
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| * [[11-odd-limit]]
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| : [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 3/4 0 1/4 -3/8 }}, {{monzo| 1 1/2 0 1/2 -1/4 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| 2 -1/2 0 1/2 1/4 }}]
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| : [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7.11/9
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| {{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} <nowiki>*</nowiki>
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| <nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
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| [[Badness]]: 0.480 × 10<sup>-3</sup>
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| === 13-limit ===
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 56/55, 91/90, 100/99
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| Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}
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| {{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e, 69bceef }} <nowiki>*</nowiki>
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| <nowiki>*</nowiki> optimal patent val: [[34edo|34]]
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| Badness: 0.876 × 10<sup>-3</sup>
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| === Mockingbird ===
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 40/39, 56/55, 100/99
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| Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}
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| {{Optimal ET sequence|legend=1| 7d, 8d, 12f, 15, 27eff }}
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| Badness: 0.859 × 10<sup>-3</sup>
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| === Catbird ===
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 78/77, 100/99, 126/125
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| Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}
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| {{Optimal ET sequence|legend=1| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
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| Badness: 0.905 × 10<sup>-3</sup>
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| == Ares == | | == Ares == |
| {{Main| Ares }} | | {{Main| Ares }} |
| {{See also| Archytas family #Ares }}
| | See [[Archytas family #Ares]]. |
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| [[Subgroup]]: 2.3.5.7.11
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| [[Comma list]]: 64/63, 100/99
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| {{Mapping|legend=1| 1 0 0 6 2 | 0 1 0 -2 -2 | 0 0 1 0 2 }}
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| Mapping to lattice: [{{val| 0 1 1 -2 0 }}, {{val| 0 0 -1 0 -2 }}]
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| Lattice basis:
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| : 3/2 length = 0.9878, 6/5 length = 1.4312
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| : Angle (3/2, 6/5) = 75.0475 degrees
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| [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 709.6869, ~5/4 = 391.4507
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| [[Minimax tuning]]:
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| * [[11-odd-limit]]
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| : [{{monzo| 1 0 0 0 0 }}, {{monzo| 3/2 1/2 0 -1/4 0 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 -1 0 1/2 0 }}, {{monzo| 3 -1 0 -1/2 1 }}]
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| : [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.11/9.9/7
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| {{Optimal ET sequence|legend=1| 12, 15, 22, 27e, 34d, 49, 181bbcddeeee }}
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| [[Badness]]: 0.469 × 10<sup>-3</sup>
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| Scales: [[ares12]]
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| [[Category:Temperament clans]] | | [[Category:Temperament clans]] |