Quintosec family: Difference between revisions

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


== Qintosec  ==
== Quintosec ==
[[Comma]]: 140737488355328/140126044921875
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.


[[POTE tuning|POTE generator]]: ~16/15 = 111.105
[[Subgroup]]: 2.3.5


[[Map]]: [&lt;5 7 13|, &lt;0 2 -3|]
[[Comma list]]: {{monzo| 47 -15 -10 }}


{{Vals|legend=1| 10, 55, 65, 140, 205, 270, 1955c, 2225c, 2495bc, 2765bc, 3035bc, 3305bcc, 3575bcc }}
{{Mapping|legend=1| 5 1 22 | 0 2 -3 }}
 
: Mapping generators: ~524288/455625, ~16384/10125
 
[[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)
 
{{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


=== 7-limit  ===
=== 2.3.5.11 subgroup ===
The qintosec temperament can be described as 10&amp;65d temperament, tempering out the [[225/224|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


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


[[POTE tuning|POTE generator]]: ~15/14 = 111.553
{{Mapping|legend=1| 5 1 22 21 | 0 2 -3 -2 }}


[[Map]]: [&lt;5 7 13 15|, &lt;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)


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


[[Badness]]: 0.156764
[[Badness]]: 0.156764


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 243/242, 3840/3773
Comma list: 225/224, 243/242, 3840/3773


POTE generator: ~15/14 = 111.311
Mapping: {{mapping| 5 1 22 21 0 | 0 2 -3 -2 5 }}


Map: [&lt;5 7 13 15 15|, &lt;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)


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


Badness: 0.077907
Badness: 0.077907


=== Qinto ===
=== Qinto ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 332750/321489
Comma list: 225/224, 385/384, 332750/321489


POTE generator: ~15/14 = 112.448
Mapping: {{mapping| 5 1 22 21 -7 | 0 2 -3 -2 7 }}


Map: [&lt;5 7 13 15 14|, &lt;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)


Vals: {{Vals| 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, named by [[User:Xenllium|Xenllium]]) 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


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


[[POTE tuning|POTE generator]]: ~16/15 = 111.113
{{Mapping|legend=1| 5 1 22 14 | 0 2 -3 0 }}


[[Map]]: [&lt;5 7 13 14|, &lt;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)


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


[[Badness]]: 0.156259
[[Badness]]: 0.156259


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 385/384, 686/675
Comma list: 243/242, 385/384, 686/675


POTE generator: ~16/15 = 110.659
Mapping: {{mapping| 5 1 22 14 0 | 0 2 -3 0 5 }}


Map: [&lt;5 7 13 14 15|, &lt;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)


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


Badness: 0.081607
Badness: 0.081607


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 243/242, 686/675
Comma list: 105/104, 144/143, 243/242, 686/675


POTE generator: ~16/15 = 110.910
Mapping: {{mapping| 5 1 22 14 0 22 | 0 2 -3 0 5 -1 }}


Map: [&lt;5 7 13 14 15 19|, &lt;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)


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


Badness: 0.056390
Badness: 0.056390


== Decoid ==
== Decoid ==
{{See also|Breedsmic temperaments #Decoid}}
{{See also| Breedsmic temperaments #Decoid }}
 
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


[[POTE tuning|POTE generator]]: ~16/15 = 111.099
{{Mapping|legend=1| 10 0 47 36 | 0 2 -3 -1 }}
 
: Mapping generators: ~15/14, ~8192/4725


[[Map]]: [&lt;10 0 47 36|, &lt;0 2 -3 -1|]
[[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)


{{Vals|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


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 5632/5625, 9801/9800
Comma list: 2401/2400, 5632/5625, 9801/9800


POTE generator: ~16/15 = 111.070
Mapping: {{mapping| 10 0 47 36 98 | 0 2 -3 -1 -8 }}


Map: [&lt;10 0 47 36 98|, &lt;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)


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


POTE generator: ~16/15 = 111.083
Mapping: {{mapping| 10 0 47 36 98 37 | 0 2 -3 -1 -8 0 }}


Map: [&lt;10 0 47 36 98 37|, &lt;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)


Vals: {{Vals| 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:Theory]]
=== Triacontoid ===
[[Category:Temperament family]]
Subgroup: 2.3.5.7.11
[[Category:Qintosec]]
 
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]]

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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