Quintosec family
The quintosec family tempers out the quintosec comma, 140737488355328/140126044921875 = [47 -15 -10⟩.
Quintosec
Quintosec is naturally a 2.3.5.11 subgroup temperament. It was documented as qintosec on the temperament finder, which was probably due to a typo[1]. The original, intended spelling is now restored, and the distinctive no-u form is relegated to the lower-accuracy 7-limit extension considered below.
Subgroup: 2.3.5
Comma list: [47 -15 -10⟩
Mapping: [⟨5 1 22], ⟨0 2 -3]]
- mapping generators: ~524288/455625, ~16384/10125
- CTE: ~524288/455625 = 1\5, ~16384/10125 = 831.1061 (~16/15 = 111.1061)
- POTE: ~524288/455625 = 1\5, ~16384/10125 = 831.105 (~16/15 = 111.105)
Optimal ET sequence: 10, 45c, 55, 65, 140, 205, 270, 1955c, 2225c, 2495bc, 2765bc, 3035bc, 3305bcc, 3575bcc
Badness: 0.139191
2.3.5.11 subgroup
Subgroup: 2.3.5.11
Comma list: 5632/5625, 26214400/26198073
Mapping: [⟨5 1 22 45], ⟨0 2 -3 -8]]
Optimal tuning (CTE): ~1024/891 = 1\5, ~160/99 = 831.0932 (~16/15 = 111.0932)
Optimal ET sequence: 10e, 55e, 65, 205, 270, 335, 605, 940, 1545c
Badness: 0.0195
Qintosec
Qintosec can be described as the 10 & 65d temperament, tempering out the marvel comma, 225/224 in the 7-limit.
Subgroup: 2.3.5.7
Comma list: 225/224, 2560000/2470629
Mapping: [⟨5 1 22 21], ⟨0 2 -3 -2]]
Wedgie: ⟨⟨10 -15 -10 -47 -44 19]]
Optimal tuning (POTE): ~400/343 = 1\5, ~80/49 = 831.553 (~15/14 = 111.553)
Optimal ET sequence: 10, 55d, 65d, 75, 140d, 215dd
Badness: 0.156764
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 243/242, 3840/3773
Mapping: [⟨5 1 22 21 0], ⟨0 2 -3 -2 5]]
Optimal tuning (POTE): ~55/48 = 1\5, ~80/49 = 831.311 (~15/14 = 111.311)
Optimal ET sequence: 10, 55d, 65d, 140de, 205dde
Badness: 0.077907
Qinto
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 332750/321489
Mapping: [⟨5 1 22 21 -7], ⟨0 2 -3 -2 7]]
Optimal tuning (POTE): ~220/189 = 1\5, ~44/27 = 832.448 (~15/14 = 112.448)
Optimal ET sequence: 10, 75, 160cde
Badness: 0.127732
Sengasec
Sengasec can be described as the 10 & 55 temperament, tempering out the cloudy comma, 16807/16384 and the sengic comma, 686/675 in the 7-limit.
Subgroup: 2.3.5.7
Comma list: 686/675, 16807/16384
Mapping: [⟨5 1 22 14], ⟨0 2 -3 0]]
Wedgie: ⟨⟨10 -15 0 -47 -28 42]]
Optimal tuning (POTE): ~8/7 = 1\5, ~45/28 = 831.113 (~16/15 = 111.113)
Optimal ET sequence: 10, 55, 65, 140dd, 205ddd, 270ddd
Badness: 0.156259
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 385/384, 686/675
Mapping: [⟨5 1 22 14 0], ⟨0 2 -3 0 5]]
Optimal tuning (POTE): ~8/7 = 1\5, ~45/28 = 830.659 (~16/15 = 110.659)
Optimal ET sequence: 10, 55, 65
Badness: 0.081607
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 144/143, 243/242, 686/675
Mapping: [⟨5 1 22 14 0 22], ⟨0 2 -3 0 5 -1]]
POTE generator: ~45/28 = 830.910 (~16/15 = 110.910)
Optimal ET sequence: 10, 55, 65
Badness: 0.056390
Decoid
- See also: Breedsmic temperaments #Decoid
Subgroup: 2.3.5.7
Comma list: 2401/2400, 67108864/66976875
Mapping: [⟨10 0 47 36], ⟨0 2 -3 -1]]
- mapping generators: ~15/14, ~8192/4725
Wedgie: ⟨⟨20 -30 -10 -94 -72 61]]
Optimal tuning (POTE): ~15/14 = 1\10, ~8192/4725 = 951.099 (~16/15 = 111.099)
Optimal ET sequence: 10, 120, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc
Badness: 0.033902
11-limit
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 5632/5625, 9801/9800
Mapping: [⟨10 0 47 36 98], ⟨0 2 -3 -1 -8]]
Optimal tuning (POTE): ~15/14 = 1\10, ~400/231 = 951.070 (~16/15 = 111.070)
Optimal ET sequence: 10e, 130, 270, 670, 940, 1210, 2150c
Badness: 0.018735
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095
Mapping: [⟨10 0 47 36 98 37], ⟨0 2 -3 -1 -8 0]]
Optimal tuning (POTE): ~15/14 = 1\10, ~26/15 = 951.083 (~16/15 = 111.083)
Optimal ET sequence: 10e, 130, 270, 940, 1210f, 1480cf
Badness: 0.013475