Ptolemismic clan: Difference between revisions

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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]]
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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]]

Latest revision as of 08:36, 8 June 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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

Strong extensions are apollo, keemic, thrasher, and ares, considered below. Discussed elsewhere are:

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

Ptolemic

Subgroup: 2.3.5.11

Comma list: 100/99

Subgroup-val mapping: [⟨1 0 0 2], ⟨0 1 0 -2], ⟨0 0 1 2]]

mapping generators: ~2, ~3, ~5

Optimal tunings:

  • WE: ~2 = 1199.2633 ¢, ~3/2 = 704.5205 ¢, ~5/4 = 383.8317 ¢
error map: ⟨-0.737 +1.829 -3.955 +4.358]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.4751 ¢, ~5/4 = 383.0756 ¢
error map: ⟨0.000 +2.520 -3.238 +5.883]

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

Badness (Sintel): 0.187

Apollo

See Marvel family #Apollo.

Keemic

See Keemic family #Keemic.

Thrasher

See Starling family #Thrasher.

Ares

Main article: Archytas and ares

See Archytas family #Ares.

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