|
|
| (12 intermediate revisions by 3 users not shown) |
| Line 1: |
Line 1: |
| The '''ptolemismic clan''' of [[Rank-3 temperament|rank-3]] [[temperament]]s [[Tempering out|tempers out]] the ptolemisma, [[100/99]]. | | {{Technical data page}} |
| | The '''ptolemismic clan''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the ptolemisma, [[100/99]]. |
|
| |
|
| Temperaments considered below are apollo, supermagic, thrasher, and ares. Discussed elsewhere include:
| | Strong extensions are apollo, keemic, 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-3 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]] |
| * ''[[Sonic]]'' (+55/54) → [[Porcupine rank three family #Sonic|Porcupine rank-3 family]] | | * ''[[Sonic]]'' (+55/54) → [[Porcupine rank-3 family #Sonic|Porcupine rank-3 family]] |
| * ''[[Nakika]]'' (+245/242) → [[Octagar family #Nakika|Octagar family]] | | * ''[[Nakika]]'' (+245/242) → [[Octagar family #Nakika|Octagar family]] |
| * ''[[Octarod]]'' (+245/243) → [[Sensamagic family #Octarod|Sensamagic family]] | | * ''[[Octarod]]'' (+245/243) → [[Sensamagic family #Octarod|Sensamagic family]] |
| Line 10: |
Line 11: |
| For the rank-4 temperament, see [[Rank-4 temperament #Ptolemismic (100/99)]]. | | For the rank-4 temperament, see [[Rank-4 temperament #Ptolemismic (100/99)]]. |
|
| |
|
| == No-7 ptolemismic == | | == Ptolemic == |
| [[Subgroup]]: 2.3.5.11 | | [[Subgroup]]: 2.3.5.11 |
|
| |
|
| [[Comma list]]: 100/99 | | [[Comma list]]: 100/99 |
|
| |
|
| {{mapping|legend=2| 1 0 0 2 | 0 1 0 -2 | 0 0 1 2 }} | | {{Mapping|legend=2| 1 0 0 2 | 0 1 0 -2 | 0 0 1 2 }} |
| | : mapping generators: ~2, ~3, ~5 |
|
| |
|
| [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.9532, ~5/4 = 384.0675 | | [[Optimal tuning]]s: |
| | * [[WE]]: ~2 = 1199.2633{{c}}, ~3/2 = 704.5205{{c}}, ~5/4 = 383.8317{{c}} |
| | : [[error map]]: {{val| -0.737 +1.829 -3.955 +4.358 }} |
| | * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.4751{{c}}, ~5/4 = 383.0756{{c}} |
| | : error map: {{val| 0.000 +2.520 -3.238 +5.883 }} |
|
| |
|
| {{Optimal ET sequence|legend=1| 7, 12, 15, 19, 22, 34, 41, 56, 75e, 97, 131e }} | | {{Optimal ET sequence|legend=1| 7, 12, 15, 19, 22, 34, 41, 56, 75e, 97, 131e }} |
|
| |
|
| [[Badness]]: 0.0226 × 10<sup>-3</sup> | | [[Badness]] (Sintel): 0.187 |
|
| |
|
| == Apollo == | | == Apollo == |
| {{See also| Marvel family #Apollo }}
| | See [[Marvel family #Apollo]]. |
|
| |
|
| [[Subgroup]]: 2.3.5.7.11
| | == Keemic == |
| | | See [[Keemic family #Keemic]]. |
| [[Comma list]]: 100/99, 225/224
| |
| | |
| {{Mapping|legend=1| 1 0 0 -5 2 | 0 1 0 2 -2 | 0 0 1 2 2 }}
| |
| | |
| [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.4846, ~5/4 = 381.6033
| |
| | |
| [[Minimax tuning]]: [[11-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5.11/9
| |
| | |
| {{Optimal ET sequence|legend=1| 12, 19, 22, 34d, 41, 104, 157ce, 198ce, 220ce, 261ce }}
| |
| | |
| [[Projection pair]]s: 7 225/32 11 100/9
| |
| | |
| [[Associated temperament]]: [[magic]]
| |
| | |
| === 13-limit ===
| |
| Subgroup: 2.3.5.7.11.13
| |
| | |
| Comma list: 100/99, 225/224, 275/273
| |
| | |
| Mapping: {{mapping| 1 0 0 -5 2 7 | 0 1 0 2 -2 -5 | 0 0 1 2 2 2 }}
| |
| | |
| Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.9984, ~5/4 = 381.5352
| |
| | |
| Minimax tuning: 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.11/9.13/9
| |
| | |
| {{Optimal ET sequence|legend=1| 12f, 19f, 22, 29, 34d, 41, 63, 104, 179cef, 242cde, 283def, 346bcdef }}
| |
| | |
| Projection pairs: 7 225/32 11 100/9 13 3200/243
| |
| | |
| == Supermagic ==
| |
| {{See also| Keemic family #Supermagic }}
| |
| | |
| [[Subgroup]]: 2.3.5.7.11
| |
| | |
| [[Comma list]]: 100/99, 385/384
| |
| | |
| {{Mapping|legend=1| 1 0 0 5 2 | 0 1 0 3 -2 | 0 0 1 -3 2 }}
| |
| | |
| [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.1985, ~5/4 = 381.5394
| |
| | |
| {{Optimal ET sequence|legend=1| 7, 15, 19, 22, 37, 41, 104 }}
| |
| | |
| [[Badness]]: 0.641 × 10<sup>-3</sup>
| |
| | |
| Scales: [[supermagic15]]
| |
|
| |
|
| == Thrasher == | | == Thrasher == |
| {{See also| Starling family #Thrasher }}
| | See [[Starling family #Thrasher]]. |
| | |
| [[Subgroup]]: 2.3.5.7.11
| |
| | |
| [[Comma list]]: 56/55, 100/99
| |
| | |
| {{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}
| |
| | |
| Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]
| |
| | |
| Lattice basis:
| |
| : 6/5 length = 0.9089, 5/4 length = 1.2007
| |
| : Angle (6/5, 5/4) = 98.8447
| |
| | |
| [[Minimax tuning]]:
| |
| * [[11-odd-limit]]
| |
| : [{{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 }}]
| |
| : [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7.11/9
| |
| | |
| {{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} <nowiki>*</nowiki>
| |
| | |
| <nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
| |
| | |
| [[Badness]]: 0.480 × 10<sup>-3</sup>
| |
| | |
| === 13-limit ===
| |
| Subgroup: 2.3.5.7.11.13
| |
| | |
| Comma list: 56/55, 91/90, 100/99
| |
| | |
| Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}
| |
| | |
| {{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e, 69bceef }} <nowiki>*</nowiki>
| |
| | |
| <nowiki>*</nowiki> optimal patent val: [[34edo|34]]
| |
| | |
| Badness: 0.876 × 10<sup>-3</sup>
| |
| | |
| === Mockingbird ===
| |
| Subgroup: 2.3.5.7.11.13
| |
| | |
| Comma list: 40/39, 56/55, 100/99
| |
| | |
| Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}
| |
| | |
| {{Optimal ET sequence|legend=1| 7d, 8d, 12f, 15, 27eff }}
| |
| | |
| Badness: 0.859 × 10<sup>-3</sup>
| |
| | |
| === Catbird ===
| |
| Subgroup: 2.3.5.7.11.13
| |
| | |
| Comma list: 78/77, 100/99, 126/125
| |
| | |
| Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}
| |
| | |
| {{Optimal ET sequence|legend=1| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
| |
| | |
| Badness: 0.905 × 10<sup>-3</sup>
| |
|
| |
|
| == Ares == | | == Ares == |
| {{Main| Ares }} | | {{Main| Archytas and ares }} |
| {{See also| Archytas family #Ares }}
| | See [[Archytas family #Ares]]. |
| | |
| [[Subgroup]]: 2.3.5.7.11
| |
| | |
| [[Comma list]]: 64/63, 100/99
| |
| | |
| {{Mapping|legend=1| 1 0 0 6 2 | 0 1 0 -2 -2 | 0 0 1 0 2 }}
| |
| | |
| Mapping to lattice: [{{val| 0 1 1 -2 0 }}, {{val| 0 0 -1 0 -2 }}]
| |
| | |
| Lattice basis:
| |
| : 3/2 length = 0.9878, 6/5 length = 1.4312
| |
| : Angle (3/2, 6/5) = 75.0475 degrees
| |
| | |
| [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 709.6869, ~5/4 = 391.4507
| |
| | |
| [[Minimax tuning]]:
| |
| * [[11-odd-limit]]
| |
| : [{{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 }}]
| |
| : [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.11/9.9/7
| |
| | |
| {{Optimal ET sequence|legend=1| 12, 15, 22, 27e, 34d, 49, 181bbcddeeee }}
| |
| | |
| [[Badness]]: 0.469 × 10<sup>-3</sup>
| |
| | |
| Scales: [[ares12]]
| |
|
| |
|
| [[Category:Temperament clans]] | | [[Category:Temperament clans]] |