Quintosec family: Difference between revisions

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{{Mapping|legend=1| 5 1 22 | 0 2 -3 }}
{{Mapping|legend=1| 5 1 22 | 0 2 -3 }}


: mapping generators: ~524288/455625, ~16384/10125
: Mapping generators: ~524288/455625, ~16384/10125


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~524288/455625 = 1\5, ~16384/10125 = 831.1061 (~16/15 = 111.1061)
* [[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)
* [[POTE]]: ~524288/455625 = 1＼5, ~16384/10125 = 831.1051 (~16/15 = 111.1051)


{{Optimal ET sequence|legend=1| 10, 45c, 55, 65, 140, 205, 270, 1955c, 2225c, 2495bc, 2765bc, 3035bc, 3305bcc, 3575bcc }}
{{Optimal ET sequence|legend=1| 10, 45c, 55, 65, 140, 205, 270, 1955c, 2225c, 2495bc, 2765bc, 3035bc, 3305bcc, 3575bcc }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~1024/891 = 1\5, ~160/99 = 831.0932 (~16/15 = 111.0932)
* 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)
* CWE: ~1024/891 = 1＼5, ~160/99 = 831.0743 (~16/15 = 111.0743)


Optimal ET sequence: {{Optimal ET sequence| 10e, 55e, 65, 205, 270, 335, 605, 940, 1545c }}
Optimal ET sequence: {{Optimal ET sequence| 10e, 55e, 65, 205, 270, 335, 605, 940, 1545c }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~400/343 = 1\5, ~80/49 = 831.7095 (~15/14 = 111.7095)
* CTE: ~400/343 = 1＼5, ~80/49 = 831.7095 (~15/14 = 111.7095)
* POTE: ~400/343 = 1\5, ~80/49 = 831.553 (~15/14 = 111.553)
* POTE: ~400/343 = 1＼5, ~80/49 = 831.5525 (~15/14 = 111.5525)


{{Optimal ET sequence|legend=1| 10, …, 55d, 65d, 75, 140d }}
{{Optimal ET sequence|legend=1| 10, …, 55d, 65d, 75, 140d }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/48 = 1\5, ~80/49 = 831.1939 (~15/14 = 111.1939)
* CTE: ~55/48 = 1＼5, ~80/49 = 831.1939 (~15/14 = 111.1939)
* POTE: ~55/48 = 1\5, ~80/49 = 831.311 (~15/14 = 111.311)
* POTE: ~55/48 = 1＼5, ~80/49 = 831.3110 (~15/14 = 111.3110)


Optimal ET sequence: {{Optimal ET sequence| 10, 55d, 65d, 140de }}
Optimal ET sequence: {{Optimal ET sequence| 10, 55d, 65d, 140de }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~220/189 = 1\5, ~44/27 = 832.4051 (~15/14 = 112.4051)
* CTE: ~220/189 = 1＼5, ~44/27 = 832.4051 (~15/14 = 112.4051)
* POTE: ~220/189 = 1\5, ~44/27 = 832.448 (~15/14 = 112.448)
* POTE: ~220/189 = 1＼5, ~44/27 = 832.4480 (~15/14 = 112.4480)


Optimal ET sequence: {{Optimal ET sequence| 10, 75 }}
Optimal ET sequence: {{Optimal ET sequence| 10, 75 }}
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[[Optimal tuning]]:  
[[Optimal tuning]]:  
* [[CTE]]: ~8/7 = 1\5, ~45/28 = 831.1061 (~16/15 = 111.1061)
* [[CTE]]: ~8/7 = 1＼5, ~45/28 = 831.1061 (~16/15 = 111.1061)
* [[POTE]]: ~8/7 = 1\5, ~45/28 = 831.113 (~16/15 = 111.113)
* [[POTE]]: ~8/7 = 1＼5, ~45/28 = 831.1126 (~16/15 = 111.1126)


{{Optimal ET sequence|legend=1| 10, …, 55, 65, 140dd }}
{{Optimal ET sequence|legend=1| 10, …, 55, 65, 140dd }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~8/7 = 1\5, ~45/28 = 830.7772 (~16/15 = 110.7772)
* CTE: ~8/7 = 1＼5, ~45/28 = 830.7772 (~16/15 = 110.7772)
* POTE: ~8/7 = 1\5, ~45/28 = 830.659 (~16/15 = 110.659)
* POTE: ~8/7 = 1＼5, ~45/28 = 830.6592 (~16/15 = 110.6592)


Optimal ET sequence: {{Optimal ET sequence| 10, 55, 65 }}
Optimal ET sequence: {{Optimal ET sequence| 10, 55, 65 }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~8/7 = 1\5, ~45/28 = 830.8942 (~16/15 = 110.8942)
* CTE: ~8/7 = 1＼5, ~45/28 = 830.8942 (~16/15 = 110.8942)
* POTE: ~8/7 = 1\5, ~45/28 = 830.910 (~16/15 = 110.910)
* POTE: ~8/7 = 1＼5, ~45/28 = 830.9099 (~16/15 = 110.9099)


Optimal ET sequence: {{Optimal ET sequence| 10, 55, 65 }}
Optimal ET sequence: {{Optimal ET sequence| 10, 55, 65 }}
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{{Mapping|legend=1| 10 0 47 36 | 0 2 -3 -1 }}
{{Mapping|legend=1| 10 0 47 36 | 0 2 -3 -1 }}


: mapping generators: ~15/14, ~8192/4725
: Mapping generators: ~15/14, ~8192/4725


{{Multival|legend=1| 20 -30 -10 -94 -72 61 }}
{{Multival|legend=1| 20 -30 -10 -94 -72 61 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~15/14 = 1\10, ~8192/4725 = 951.1086 (~16/15 = 111.1086, or ~225/224 = 8.8914)
* [[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 = 951.099 (~16/15 = 111.099, or ~225/224 = 8.901)
* [[POTE]]: ~15/14 = 1＼10, ~8192/4725 = 951.0987 (~16/15 = 111.0987, or ~225/224 = 8.9013)


{{Optimal ET sequence|legend=1| 10, …, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}
{{Optimal ET sequence|legend=1| 10, …, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~15/14 = 1\10, ~400/231 = 951.0943 (~16/15 = 111.0943, or ~225/224 = 8.9057)
* 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 = 951.070 (~16/15 = 111.070, or ~225/224 = 8.930)
* POTE: ~15/14 = 1＼10, ~400/231 = 951.0700 (~16/15 = 111.0700, or ~225/224 = 8.9300)


Optimal ET sequence: {{Optimal ET sequence| 10e, …, 130, 270, 670, 940, 1210, 2150c }}
Optimal ET sequence: {{Optimal ET sequence| 10e, …, 130, 270, 670, 940, 1210, 2150c }}
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Badness: 0.018735
Badness: 0.018735


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Optimal tunings:  
Optimal tunings:  
* CTE: ~15/14 = 1\10, ~26/15 = 951.0943 (~16/15 = 111.0943, or ~196/195 = 8.9057)
* 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 = 951.083 (~16/15 = 111.083, or ~196/195 = 8.917)
* POTE: ~15/14 = 1＼10, ~26/15 = 951.0832 (~16/15 = 111.0832, or ~196/195 = 8.9168)


Optimal ET sequence: {{Optimal ET sequence| 10e, …, 130, 270, 940, 1210f, 1480cf }}
Optimal ET sequence: {{Optimal ET sequence| 10e, …, 130, 270, 940, 1210f, 1480cf }}


Badness: 0.013475
Badness: 0.013475
=== Triacontoid ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 42592/42525, 1771561/1769472
Mapping: {{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: {{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: {{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: {{Optimal ET sequence| 120, 150, 270 }}
Badness: 0.040873


== Notes ==
== Notes ==

Revision as of 02:41, 1 January 2025

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

Optimal tunings:

  • 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]]

Wedgie: ⟨⟨ 10 -15 -10 -47 -44 19 ]]

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)

Optimal ET sequence: 10, 75

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:

  • 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

See also: Breedsmic temperaments § 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

Wedgie: ⟨⟨ 20 -30 -10 -94 -72 61 ]]

Optimal tunings:

  • 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

Notes

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