Ptolemismic clan

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:

• Calliope (+45/44 or 81/80) → Didymus rank-3 family
• Jubilee (+50/49 or 99/98) → Jubilismic family
• Sonic (+55/54) → Porcupine rank-3 family
• Nakika (+245/242) → Octagar family
• Octarod (+245/243) → Sensamagic family
• Androboh (+1375/1372) → Gariboh family

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:

• 11-odd-limit

[[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:

• 11-odd-limit

[[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