Xenharmonic Wiki

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

Optimal tunings:

  • 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]]

Optimal tunings:

  • 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 ¢)

Optimal ET sequence: 10, 75

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]]

Optimal tuning:

  • 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

Optimal tunings:

  • 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

References

  1. ↑ Yahoo Tuning Group | 2D temperament names, part I -- reclassified temperaments from message #101780
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