Quince clan
The quince clan tempers out the quince comma, [-15 0 -2 7⟩ = 823543/819200. Quince temperaments include:
- Casablanca (+126/125) → Starling temperaments
- Miracle (+225/224) → Gamelismic clan
- Quincy (+4375/4374) → Ragismic microtemperaments
- Birds (+3136/3125) → 31st-octave temperaments
- Octowerck (+321489/320000 or 420175/419904) → Varunismic temperaments
- Cotoneum (+10976/10935) → Garischismic clan
- Countermiracle (+6144/6125) → Porwell temperaments
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]
Optimal tuning (POTE): ~2 = 1\1, ~343/320 = 116.291
Optimal ET sequence: 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd
2.5.7.13 subgroup
Subgroup: 2.5.7.13
Comma list: 343/338, 640/637
Sval mapping: [⟨1 3 3 4], ⟨0 -7 -2 -3]]
Gencom mapping: [⟨1 0 3 3 0 4], ⟨0 0 -7 -2 0 -3]]
- gencom: [2 14/13; 343/338 640/637]
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 116.094
Optimal ET sequence: 10, 21, 31
2.5.7.13.17 subgroup
Subgroup: 2.5.7.13.17
Comma list: 170/169, 224/221, 640/637
Sval mapping: [⟨1 3 3 4 4], ⟨0 -7 -2 -3 1]]
Gencom mapping: [⟨1 0 3 3 0 4 4], ⟨0 0 -7 -2 0 -3 1]]
- gencom: [2 14/13; 170/169 224/221 640/637]
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 115.769
Optimal ET sequence: 10, 21, 31
2.5.7.13.17.19 subgroup
Subgroup: 2.5.7.13.17.19
Comma list: 170/169, 343/338, 640/637, 16384/16055
Sval mapping: [⟨1 3 3 4 4 3], ⟨0 -7 -2 -3 1 13]]
Gencom mapping: [⟨1 0 3 3 0 4 4 3], ⟨0 0 -7 -2 0 -3 1 13]]
- gencom: [2 14/13; 170/169 343/338 640/637 16384/16055]
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 115.716
Optimal ET sequence: 10, 21, 31, 52f