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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]]: ~16384/10125 = 831.105 (~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


Subgroup: 2.3.5.7
== 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


[[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]]: ~80/49 = 831.553 (~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: ~80/49 = 831.311 (~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)


Optimal GPV sequence: {{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: ~44/27 = 832.448 (~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)


Optimal GPV sequence: {{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 }}
 
{{Multival|legend=1| 10 -15 0 -47 -28 42 }}


[[POTE generator]]: ~45/28 = 831.113 (~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 81: 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: ~45/28 = 830.659 (~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)


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


Badness: 0.081607
Badness: 0.081607
Line 94: 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: ~45/28 = 830.910 (~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)


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


Badness: 0.056390
Badness: 0.056390
Line 105: 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


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


[[POTE generator]]: ~8192/4725 = 951.099 (~16/15 = 111.099)
{{Optimal ET sequence|legend=1| 10, …, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}
 
{{Val list|legend=1| 10, 120, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}


[[Badness]]: 0.033902
[[Badness]]: 0.033902
Line 126: 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: ~400/231 = 951.070 (~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)


Optimal GPV sequence: {{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: ~26/15 = 951.083 (~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)


Optimal GPV sequence: {{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]]
Retrieved from "https://en.xen.wiki/w/Quintosec_family"