Quintosec family
- 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 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 = 831.1051 (~16/15 = 111.1051)
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 tunings:
- CTE: ~1024/891 = 1＼5, ~160/99 = 831.0932 (~16/15 = 111.0932)
- CWE: ~1024/891 = 1＼5, ~160/99 = 831.0743 (~16/15 = 111.0743)
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]]
Optimal tunings:
- CTE: ~400/343 = 1＼5, ~80/49 = 831.7095 (~15/14 = 111.7095)
- POTE: ~400/343 = 1＼5, ~80/49 = 831.5525 (~15/14 = 111.5525)
Optimal ET sequence: 10, …, 55d, 65d, 75, 140d
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 tunings:
- CTE: ~55/48 = 1＼5, ~80/49 = 831.1939 (~15/14 = 111.1939)
- POTE: ~55/48 = 1＼5, ~80/49 = 831.3110 (~15/14 = 111.3110)
Optimal ET sequence: 10, 55d, 65d, 140de
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 tunings:
- CTE: ~220/189 = 1＼5, ~44/27 = 832.4051 (~15/14 = 112.4051)
- POTE: ~220/189 = 1＼5, ~44/27 = 832.4480 (~15/14 = 112.4480)
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]]
- CTE: ~8/7 = 1＼5, ~45/28 = 831.1061 (~16/15 = 111.1061)
- POTE: ~8/7 = 1＼5, ~45/28 = 831.1126 (~16/15 = 111.1126)
Optimal ET sequence: 10, …, 55, 65, 140dd
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 tunings:
- CTE: ~8/7 = 1＼5, ~45/28 = 830.7772 (~16/15 = 110.7772)
- POTE: ~8/7 = 1＼5, ~45/28 = 830.6592 (~16/15 = 110.6592)
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]]
Optimal tunings:
- CTE: ~8/7 = 1＼5, ~45/28 = 830.8942 (~16/15 = 110.8942)
- POTE: ~8/7 = 1＼5, ~45/28 = 830.9099 (~16/15 = 110.9099)
Optimal ET sequence: 10, 55, 65
Badness: 0.056390
Decoid
Decoid tempers out 2401/2400 and 67108864/66976875, as well as the linus comma, [11 -10 -10 10⟩. Either 8/7 or 16/15 can be used as its generator. It may be described as the 130 & 270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice.
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
- CTE: ~15/14 = 1＼10, ~8192/4725 = 951.1086 (~16/15 = 111.1086, or ~225/224 = 8.8914)
- POTE: ~15/14 = 1＼10, ~8192/4725 = 951.0987 (~16/15 = 111.0987, or ~225/224 = 8.9013)
Optimal ET sequence: 10, …, 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 tunings:
- CTE: ~15/14 = 1＼10, ~400/231 = 951.0943 (~16/15 = 111.0943, or ~225/224 = 8.9057)
- POTE: ~15/14 = 1＼10, ~400/231 = 951.0700 (~16/15 = 111.0700, or ~225/224 = 8.9300)
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 tunings:
- CTE: ~15/14 = 1＼10, ~26/15 = 951.0943 (~16/15 = 111.0943, or ~196/195 = 8.9057)
- POTE: ~15/14 = 1＼10, ~26/15 = 951.0832 (~16/15 = 111.0832, or ~196/195 = 8.9168)
Optimal ET sequence: 10e, …, 130, 270, 940, 1210f, 1480cf
Badness: 0.013475
Triacontoid
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 42592/42525, 1771561/1769472
Mapping: [⟨30 0 141 108 80], ⟨0 2 -3 -1 1]]
- Mapping generators: ~45/44, ~8192/4725
Optimal tunings:
- CTE: ~45/44 = 1＼30, ~8192/4725 = 951.1137 (~225/224 = 8.8863)
- POTE: ~45/44 = 1＼30, ~8192/4725 = 951.1121 (~225/224 = 8.8879)
Optimal ET sequence: 120, 150, 270
Badness: 0.075991
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 2401/2400, 4096/4095, 6656/6655
Mapping: [⟨30 0 141 108 80 111], ⟨0 2 -3 -1 1 0]]
Optimal tunings:
- CTE: ~45/44 = 1＼30, ~26/15 = 951.1137 (~196/195 = 8.8863)
- POTE: ~45/44 = 1＼30, ~26/15 = 951.1130 (~196/195 = 8.8870)
Optimal ET sequence: 120, 150, 270
Badness: 0.040873