Quintosec family: Difference between revisions

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The '''qintosec family''' tempers out the qintosec comma, 140737488355328/140126044921875 = {{monzo|47 -15 -10}}.
{{Technical data page}}
The '''quintosec family''' tempers out the quintosec comma, 140737488355328/140126044921875 = {{monzo| 47 -15 -10 }}.


== Qintosec ==
== Quintosec ==
Subgroup: 2.3.5
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<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>. 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]]: 140737488355328/140126044921875
[[Comma list]]: {{monzo| 47 -15 -10 }}


[[Mapping]]: [{{val| 5 1 22 }}, {{val| 0 2 -3 }}]
{{Mapping|legend=1| 5 1 22 | 0 2 -3 }}


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


[[POTE generator]]: ~16/15 = 111.105
[[Optimal tuning]]s:
* [[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)


{{Val list|legend=1| 10, 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 }}


[[Badness]]: 0.139191
[[Badness]]: 0.139191


== Septimal qintosec ==
=== 2.3.5.11 subgroup ===
The qintosec temperament can be described as 10&amp;65d temperament, tempering out the [[marvel comma]], 225/224 in the 7-limit.
Subgroup: 2.3.5.11
 
Comma list: 5632/5625, 26214400/26198073
 
Mapping: {{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: {{Optimal ET sequence| 10e, 55e, 65, 205, 270, 335, 605, 940, 1545c }}
 
Badness: 0.0195
 
== Qintosec ==
Qintosec can be described as the 10 &amp; 65d temperament, tempering out the [[marvel comma]], 225/224 in the 7-limit.  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 2560000/2470629
[[Comma list]]: 225/224, 2560000/2470629


[[Mapping]]: [{{val| 5 1 22 21 }}, {{val| 0 2 -3 -2 }}]
{{Mapping|legend=1| 5 1 22 21 | 0 2 -3 -2 }}
 
{{Multival|legend=1| 10 -15 -10 -47 -44 19 }}


[[POTE generator]]: ~15/14 = 111.553
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)


{{Val list|legend=1| 10, 55d, 65d, 75, 140d, 215dd }}
{{Optimal ET sequence|legend=1| 10, …, 55d, 65d, 75, 140d }}


[[Badness]]: 0.156764
[[Badness]]: 0.156764
Line 38: Line 58:
Comma list: 225/224, 243/242, 3840/3773
Comma list: 225/224, 243/242, 3840/3773


Mapping: [{{val| 5 1 22 21 0 }}, {{val| 0 2 -3 -2 5 }}]
Mapping: {{mapping| 5 1 22 21 0 | 0 2 -3 -2 5 }}


POTE generator: ~15/14 = 111.311
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)


Vals: {{Val list| 10, 55d, 65d, 140de, 205dde }}
Optimal ET sequence: {{Optimal ET sequence| 10, 55d, 65d, 140de }}


Badness: 0.077907
Badness: 0.077907
Line 51: Line 73:
Comma list: 225/224, 385/384, 332750/321489
Comma list: 225/224, 385/384, 332750/321489


Mapping: [{{val| 5 1 22 21 -7 }}, {{val|0 2 -3 -2 7 }}]
Mapping: {{mapping| 5 1 22 21 -7 | 0 2 -3 -2 7 }}


POTE generator: ~15/14 = 112.448
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)


Vals: {{Val list| 10, 75, 160cde }}
Optimal ET sequence: {{Optimal ET sequence| 10, 75 }}


Badness: 0.127732
Badness: 0.127732


== Sengasec ==
== Sengasec ==
The ''sengasec'' temperament (10&amp;55) tempers out the [[cloudy comma]], 16807/16384 and the sengic comma, 686/675 in the 7-limit.
Sengasec can be described as the 10 &amp; 55 temperament, tempering out the [[cloudy comma]], 16807/16384 and the sengic comma, 686/675 in the 7-limit.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 686/675, 16807/16384
[[Comma list]]: 686/675, 16807/16384


[[Mapping]]: [{{val| 5 1 22 14 }}, {{val| 0 2 -3 0 }}]
{{Mapping|legend=1| 5 1 22 14 | 0 2 -3 0 }}


[[POTE generator]]: ~16/15 = 111.113
[[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)


{{Val list|legend=1| 10, 55, 65, 140dd, 205ddd, 270ddd }}
{{Optimal ET sequence|legend=1| 10, …, 55, 65, 140dd }}


[[Badness]]: 0.156259
[[Badness]]: 0.156259
Line 79: Line 105:
Comma list: 243/242, 385/384, 686/675
Comma list: 243/242, 385/384, 686/675


Mapping: [{{val| 5 1 22 14 0 }}, {{val| 0 2 -3 0 5 }}]
Mapping: {{mapping| 5 1 22 14 0 | 0 2 -3 0 5 }}


POTE generator: ~16/15 = 110.659
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)


Vals: {{Val list| 10, 55, 65 }}
Optimal ET sequence: {{Optimal ET sequence| 10, 55, 65 }}


Badness: 0.081607
Badness: 0.081607
Line 92: Line 120:
Comma list: 105/104, 144/143, 243/242, 686/675
Comma list: 105/104, 144/143, 243/242, 686/675


Mapping: [{{val| 5 1 22 14 0 22 }}, {{val| 0 2 -3 0 5 -1 }}]
Mapping: {{mapping| 5 1 22 14 0 22 | 0 2 -3 0 5 -1 }}


POTE generator: ~16/15 = 110.910
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)


Vals: {{Val list| 10, 55, 65 }}
Optimal ET sequence: {{Optimal ET sequence| 10, 55, 65 }}


Badness: 0.056390
Badness: 0.056390
Line 103: Line 133:
{{See also| Breedsmic temperaments #Decoid }}
{{See also| Breedsmic temperaments #Decoid }}


Subgroup: 2.3.5.7
Decoid tempers out 2401/2400 and 67108864/66976875, as well as the [[linus comma]], {{monzo| 11 -10 -10 10 }}. Either 8/7 or 16/15 can be used as its generator. It may be described as the 130 &amp; 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
[[Comma list]]: 2401/2400, 67108864/66976875


[[Mapping]]: [{{val| 10 0 47 36 }}, {{val| 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


[[POTE generator]]: ~16/15 = 111.099
[[Optimal tuning]]s:
* [[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)


{{Val list|legend=1| 10, 120, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}
{{Optimal ET sequence|legend=1| 10, …, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}


[[Badness]]: 0.033902
[[Badness]]: 0.033902
Line 122: Line 156:
Comma list: 2401/2400, 5632/5625, 9801/9800
Comma list: 2401/2400, 5632/5625, 9801/9800


Mapping: [{{val| 10 0 47 36 98 }}, {{val| 0 2 -3 -1 -8 }}]
Mapping: {{mapping| 10 0 47 36 98 | 0 2 -3 -1 -8 }}


POTE generator: ~16/15 = 111.070
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)


Vals: {{Val list| 10e, 130, 270, 670, 940, 1210, 2150c }}
Optimal ET sequence: {{Optimal ET sequence| 10e, …, 130, 270, 670, 940, 1210, 2150c }}


Badness: 0.018735
Badness: 0.018735


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


Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095
Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095


Mapping: [{{val| 10 0 47 36 98 37 }}, {{val| 0 2 -3 -1 -8 0 }}]
Mapping: {{mapping| 10 0 47 36 98 37 | 0 2 -3 -1 -8 0 }}


POTE generator: ~16/15 = 111.083
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)


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


Badness: 0.013475
Badness: 0.013475


[[Category:Regular temperament theory]]
=== Triacontoid ===
[[Category:Temperament family]]
Subgroup: 2.3.5.7.11
[[Category:Qintosec family| ]] <!-- main article -->
 
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 ==
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Quintosec family| ]] <!-- main article -->
[[Category:Quintosec| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:37, 24 June 2025

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

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]]

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]]

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

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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