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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 ===
In the 7-limit, the qintosec temperament (10&amp;65d) tempers out the [[225/224|marvel comma]], 225/224.
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


= Sengasec =
=== Qinto ===
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.
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 332750/321489
 
Mapping: {{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: {{Optimal ET sequence| 10, 75 }}
 
Badness: 0.127732
 
== Sengasec ==
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


= Decoid =
=== 13-limit ===
{{See also|Breedsmic temperaments #Decoid}}
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 243/242, 686/675
 
Mapping: {{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: {{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]], {{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 }}


[[Map]]: [&lt;10 0 47 36|, &lt;0 2 -3 -1|]
: Mapping generators: ~15/14, ~8192/4725


{{Vals|legend=1| 10, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}
[[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)
 
{{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]]
Retrieved from "https://en.xen.wiki/w/Quintosec_family"