Quince clan

The quince clan tempers out quince, the no-threes comma [-15 0 -2 7⟩ = 823543/819200. Quince temperaments include:

• Casablanca ({126/125, 589824/588245})
• Miracle ({225/224, 1029/1024})
• Quincy ({4375/4374, 823543/819200})
• Birds ({3136/3125, 823543/819200})
• Octowerck ({321489/320000, 420175/419904})
• Cotoneum ({10976/10935, 823543/819200})
• Countermiracle ({6144/6125, 50421/50000})

Mercy

Mercy is the no-3 version of miracle. It can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of 31edo. Two generators make an 8/7; five generators make a 7/5.

Subgroup: 2.5.7

Comma list: 823543/819200

Sval mapping: [⟨1 3 3], ⟨0 -7 -2]]

Mapping generators: ~2, ~343/320

Gencom mapping: [⟨1 0 3 3], ⟨0 0 -7 -2]] Gencom: [2 343/320; 823543/819200]

POTE generator: ~343/320 = 116.291

2.5.7.13

Subgroup: 2.5.7.13

Comma list: 343/338, 640/637

Gencom: [2 14/13; 343/338 640/637]

Gencom mapping: [⟨1 0 3 3 0 4], ⟨0 0 -7 -2 0 -3]]

Sval mapping: [⟨1 3 3 4], ⟨0 -7 -2 -3]]

POL2 generator: ~14/13 = 116.094

2.5.7.13.17

Subgroup: 2.5.7.13.17

Comma list: 170/169, 224/221, 640/637

Gencom: [2 14/13; 170/169 224/221 640/637]

Gencom mapping: [⟨1 0 3 3 0 4 4], ⟨0 0 -7 -2 0 -3 1]]

Sval mapping: [⟨1 3 3 4 4], ⟨0 -7 -2 -3 1]]

POL2 generator: ~14/13 = 115.769

2.5.7.13.17.19

Subgroup: 2.5.7.13.17.19

Comma list: 170/169, 343/338, 640/637, 16384/16055

Gencom: [2 14/13; 170/169 343/338 640/637 16384/16055]

Gencom mapping: [⟨1 0 3 3 0 4 4 3], ⟨0 0 -7 -2 0 -3 1 13]]

Sval mapping: [⟨1 3 3 4 4 3], ⟨0 -7 -2 -3 1 13]]

POL2 generator: ~14/13 = 115.716