Ptolemismic clan: Difference between revisions

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Copypaste apollo, supermagic, thrasher, and ares here as strong extensions
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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]] [[temperament]]s [[Tempering out|tempers out]] the ptolemisma, [[100/99]].  


Temperaments discussed elsewhere include:  
Temperaments considered below are apollo, supermagic, thrasher, and ares. Discussed elsewhere include:  
* ''[[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]]
* ''[[Sonic]]'' (+55/54) → [[Porcupine rank three family #Sonic|Porcupine rank-3 family]]
* ''[[Sonic]]'' (+55/54) → [[Porcupine rank three family #Sonic|Porcupine rank-3 family]]
* ''[[Thrasher]]'' (+56/55) → [[Starling family #Thrasher|Starling family]]
* [[Ares]] (+64/63) → [[Archytas family #Ares|Archytas family]]
* ''[[Apollo]]'' (+225/224) → [[Marvel family #Apollo|Marvel 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]]
* ''[[Supermagic]]'' (+385/384) → [[Keemic family #Supermagic|Keemic family]]
* ''[[Androboh]]'' (+1375/1372) → [[Gariboh family #Androboh|Gariboh family]]
* ''[[Androboh]]'' (+1375/1372) → [[Gariboh family #Androboh|Gariboh family]]
For the rank-4 temperament, see [[Rank-4 temperament #Ptolemismic (100/99)]].  
For the rank-4 temperament, see [[Rank-4 temperament #Ptolemismic (100/99)]].  
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[[Badness]]: 0.0226 × 10<sup>-3</sup>
[[Badness]]: 0.0226 × 10<sup>-3</sup>
== Apollo ==
{{See also| Marvel family #Apollo }}
[[Subgroup]]: 2.3.5.7.11
[[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 ==
{{See also| 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 ==
{{Main| Ares }}
{{See also| 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]]

Revision as of 06:06, 18 September 2023

The ptolemismic clan of rank-3 temperaments tempers out the ptolemisma, 100/99.

Temperaments considered below are apollo, supermagic, thrasher, and ares. Discussed elsewhere include:

For the rank-4 temperament, see Rank-4 temperament #Ptolemismic (100/99).

No-7 ptolemismic

Subgroup: 2.3.5.11

Comma list: 100/99

Sval mapping: [⟨1 0 0 2], ⟨0 1 0 -2], ⟨0 0 1 2]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.9532, ~5/4 = 384.0675

Optimal ET sequence: 7, 12, 15, 19, 22, 34, 41, 56, 75e, 97, 131e

Badness: 0.0226 × 10-3

Apollo

See also: Marvel family § Apollo

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224

Mapping: [⟨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 (unchanged-interval) basis: 2.7/5.11/9

Optimal ET sequence: 12, 19, 22, 34d, 41, 104, 157ce, 198ce, 220ce, 261ce

Projection pairs: 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: [⟨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: 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: [⟨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: 7, 15, 19, 22, 37, 41, 104

Badness: 0.641 × 10-3

Scales: supermagic15

Thrasher

See also: Starling family § Thrasher

Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99

Mapping: [⟨1 0 0 -1 2], ⟨0 1 0 -2 -2], ⟨0 0 1 3 2]]

Mapping to lattice: [⟨0 1 0 -2 -2], ⟨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:

[[1 0 0 0 0⟩, [1 3/4 0 1/4 -3/8⟩, [1 1/2 0 1/2 -1/4⟩, [0 0 0 1 0⟩, [2 -1/2 0 1/2 1/4⟩]
eigenmonzo (unchanged-interval) basis: 2.7.11/9

Optimal ET sequence: 7d, 8d, 12, 15, 19, 27e *

* optimal patent val: 34

Badness: 0.480 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 91/90, 100/99

Mapping: [⟨1 0 0 -1 2 2], ⟨0 1 0 -2 -2 4], ⟨0 0 1 3 2 -2]]

Optimal ET sequence: 7d, 8d, 12, 15, 19, 27e, 69bceef *

* optimal patent val: 34

Badness: 0.876 × 10-3

Mockingbird

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 56/55, 100/99

Mapping: [⟨1 0 0 -1 2 3], ⟨0 1 0 -2 -2 -1], ⟨0 0 1 3 2 1]]

Optimal ET sequence: 7d, 8d, 12f, 15, 27eff

Badness: 0.859 × 10-3

Catbird

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 100/99, 126/125

Mapping: [⟨1 0 0 -1 2 0], ⟨0 1 0 -2 -2 -5], ⟨0 0 1 3 2 5]]

Optimal ET sequence: 7df, 8d, 12f, 19, 27e, 66cdeeef

Badness: 0.905 × 10-3

Ares

Main article: Ares
See also: Archytas family § Ares

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99

Mapping: [⟨1 0 0 6 2], ⟨0 1 0 -2 -2], ⟨0 0 1 0 2]]

Mapping to lattice: [⟨0 1 1 -2 0], ⟨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:

[[1 0 0 0 0⟩, [3/2 1/2 0 -1/4 0⟩, [2 0 0 -1/2 1/2⟩, [3 -1 0 1/2 0⟩, [3 -1 0 -1/2 1⟩]
eigenmonzo (unchanged-interval) basis: 2.11/9.9/7

Optimal ET sequence: 12, 15, 22, 27e, 34d, 49, 181bbcddeeee

Badness: 0.469 × 10-3

Scales: ares12

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