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 of temperaments tempers out the quintosec comma (monzo: [47 -15 -10⟩, ratio: 140 737 488 355 328 / 140 126 044 921 875).
Quintosec
Quintosec has a 1/5-octave period and a generator of around 831 cents such that two of them plus a period make the 3rd harmonic. However, an equivalent generator is the neutral third, in addition to the minimal generator, ~16/15. The ploidacot of this temperament is pentaploid dicot.
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 restored here, 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
- WE: ~524288/455625 = 239.9786 ¢, ~16384/10125 = 831.0310 ¢ (~16/15 = 111.0952 ¢)
- error map: ⟨-0.107 +0.086 +0.123]
- CWE: ~524288/455625 = 240.0000 ¢, ~16384/10125 = 831.1054 ¢ (~16/15 = 111.1054 ¢)
- error map: ⟨0.000 +0.256 +0.370]
Optimal ET sequence: 10, 45c, 55, 65, 140, 205, 270, 1955c, 2225c, 2495bc, 2765bc, 3035bc, 3305bcc, 3575bcc
Badness (Sintel): 3.27
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:
- WE: ~1024/891 = 239.9820 ¢, ~160/99 = 831.0062 ¢ (~16/15 = 111.0603 ¢)
- CWE: ~1024/891 = 240.0000 ¢, ~160/99 = 831.0743 ¢ (~16/15 = 111.0743 ¢)
Optimal ET sequence: 10e, 55e, 65, 205, 270, 335, 605, 940, 1545c
Badness (Sintel): 0.608
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]]
- WE: ~400/343 = 239.8397 ¢, ~80/49 = 830.9971 ¢ (~16/15 = 111.7095 ¢)
- error map: ⟨-0.802 -0.121 -2.832 +5.813]
- CWE: ~400/343 = 240.0000 ¢, ~80/49 = 831.5851 ¢ (~16/15 = 111.5851 ¢)
- error map: ⟨0.000 +1.215 -1.069 +8.004]
Optimal ET sequence: 10, …, 55d, 65d, 75, 140d
Badness (Sintel): 3.97
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:
- WE: ~55/48 = 239.8038 ¢, ~80/49 = 830.6313 ¢ (~16/15 = 111.2200 ¢)
- CWE: ~55/48 = 240.0000 ¢, ~80/49 = 831.2911 ¢ (~16/15 = 111.2911 ¢)
Optimal ET sequence: 10, 55d, 65d, 140de
Badness (Sintel): 2.58
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:
- WE: ~220/189 = 239.9453 ¢, ~44/27 = 832.2583 ¢ (~16/15 = 112.4224 ¢)
- CWE: ~220/189 = 240.0000 ¢, ~44/27 = 832.4403 ¢ (~16/15 = 112.4403 ¢)
Badness (Sintel): 4.22
Sengasec
Sengasec can be described as the 10 & 65 temperament, tempering out the cloudy comma, 16807/16384 and the sengic comma, 686/675 in the 7-limit. It was named by Xenllium in 2021.
Subgroup: 2.3.5.7
Comma list: 686/675, 16807/16384
Mapping: [⟨5 1 22 14], ⟨0 2 -3 0]]
- WE: ~8/7 = 240.1396 ¢, ~45/28 = 831.5960 ¢ (~16/15 = 111.1772 ¢)
- error map: ⟨+0.698 +1.377 +1.969 -6.872]
- CWE: ~8/7 = 240.0000 ¢, ~45/28 = 831.1113 ¢ (~16/15 = 111.1113 ¢)
- error map: ⟨0.000 +0.268 +0.352 -8.826]
Optimal ET sequence: 10, …, 55, 65, 140dd
Badness (Sintel): 3.95
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:
- WE: ~8/7 = 240.0876 ¢, ~45/28 = 830.9623 ¢ (~16/15 = 110.7772 ¢)
- CWE: ~8/7 = 240.0000 ¢, ~45/28 = 830.6798 ¢ (~16/15 = 110.6798 ¢)
Optimal ET sequence: 10, 55, 65
Badness (Sintel): 2.70
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:
- WE: ~8/7 = 239.9825 ¢, ~45/28 = 830.8492 ¢ (~16/15 = 110.9018 ¢)
- CWE: ~8/7 = 240.0000 ¢, ~45/28 = 830.9071 ¢ (~16/15 = 110.9071 ¢)
Optimal ET sequence: 10, 55, 65
Badness (Sintel): 2.33
Decoid
Decoid tempers out 2401/2400, the breedsma, and 67108864/66976875, the decovulture comma, as well as [11 -10 -10 10⟩, the linus comma. It may be described as the 130 & 270 temperament, and its ploidacot is decaploid dicot. The hemitwelfth or neutral third can be used as a generator, but also ~8/7, ~16/15, or the marvel comma. 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
- WE: ~15/14 = 119.9913 ¢, ~8192/4725 = 951.0298 ¢ (~225/224 = 8.9006 ¢)
- error map: ⟨-0.087 +0.105 +0.188 -0.169]
- CWE: ~15/14 = 120.0000 ¢, ~8192/4725 = 951.1007 ¢ (~225/224 = 8.8993 ¢)
- error map: ⟨0.000 +0.246 +0.384 +0.073]
Optimal ET sequence: 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc
Badness (Sintel): 0.858
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:
- WE: ~15/14 = 120.9921 ¢, ~400/231 = 951.0074 ¢ (~225/224 = 8.9294 ¢)
- CWE: ~15/14 = 120.0000 ¢, ~400/231 = 951.0747 ¢ (~225/224 = 8.9253 ¢)
Optimal ET sequence: 130, 270, 670, 940, 1210, 2150c
Badness (Sintel): 0.619
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:
- WE: ~15/14 = 119.9964 ¢, ~26/15 = 951.0546 ¢ (~196/195 = 8.9165 ¢)
- CWE: ~15/14 = 120.0000 ¢, ~26/15 = 951.0850 ¢ (~196/195 = 8.9150 ¢)
Optimal ET sequence: 130, 270, 940, 1210f, 1480cf
Badness (Sintel): 0.557
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:
- WE: ~45/44 = 39.9981 ¢, ~8192/4725 = 951.0672 ¢ (~225/224 = 8.8875 ¢)
- CWE: ~45/44 = 40.0000 ¢, ~8192/4725 = 951.1124 ¢ (~225/224 = 8.8876 ¢)
Optimal ET sequence: 120, 150, 270
Badness (Sintel): 2.51
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:
- WE: ~45/44 = 39.9992 ¢, ~26/15 = 951.0944 ¢ (~196/195 = 8.8868 ¢)
- CWE: ~45/44 = 40.0000 ¢, ~26/15 = 951.1131 ¢ (~196/195 = 8.8869 ¢)
Optimal ET sequence: 120, 150, 270
Badness (Sintel): 1.69