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Quintosec family: Difference between revisions

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The '''quintosec family''' tempers out the quintosec comma, 140737488355328/140126044921875 = {{monzo| 47 -15 -10 }}.
{{Technical data page}}
The '''quintosec family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[quintosec comma]] ({{monzo|legend=1| 47 -15 -10 }}, [[ratio]]: 140 737 488 355 328 / 140 126 044 921 875).


== Quintosec ==
== 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<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.  
Quintosec has a 1/5-octave period and a generator of around 831 [[cent]]s such that two of them plus a period make the [[3/1|3rd harmonic]]. However, an equivalent generator is the neutral third, in addition to the minimal generator, [[~]][[16/15]]. The [[ploidacot]] of this temperament is pentaploid dicot.
 
Quintosec is naturally a [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament. It was documented as ''qintosec'' on the [https://x31eq.com/temper/ 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 restored here, and the distinctive no-''u'' form is relegated to the lower-accuracy 7-limit extension considered below.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 9: Line 12:


{{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)
* [[WE]]: ~524288/455625 = 239.9786{{c}}, ~16384/10125 = 831.0310{{c}} (~16/15 = 111.0952{{c}})
* [[POTE]]: ~524288/455625 = 1\5, ~16384/10125 = 831.105 (~16/15 = 111.105)
: [[error map]]: {{val| -0.107 +0.086 +0.123 }}
* [[CWE]]: ~524288/455625 = 240.0000{{c}}, ~16384/10125 = 831.1054{{c}} (~16/15 = 111.1054{{c}})
: error map: {{val| 0.000 +0.256 +0.370 }}


{{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 }}


[[Badness]]: 0.139191
[[Badness]] (Sintel): 3.27


=== 2.3.5.11 subgroup ===
=== 2.3.5.11 subgroup ===
Line 25: Line 29:
Comma list: 5632/5625, 26214400/26198073
Comma list: 5632/5625, 26214400/26198073


Mapping: {{mapping| 5 1 22 45 | 0 2 -3 -8 }}
{{Mapping|legend=0| 5 1 22 45 | 0 2 -3 -8 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~1024/891 = 1\5, ~160/99 = 831.0932 (~16/15 = 111.0932)
* WE: ~1024/891 = 239.9820{{c}}, ~160/99 = 831.0062{{c}} (~16/15 = 111.0603{{c}})
* CWE: ~1024/891 = 1\5, ~160/99 = 831.0743 (~16/15 = 111.0743)
* CWE: ~1024/891 = 240.0000{{c}}, ~160/99 = 831.0743{{c}} (~16/15 = 111.0743{{c}})


Optimal ET sequence: {{Optimal ET sequence| 10e, 55e, 65, 205, 270, 335, 605, 940, 1545c }}
{{Optimal ET sequence|legend=0| 10e, 55e, 65, 205, 270, 335, 605, 940, 1545c }}


Badness: 0.0195
Badness (Sintel): 0.608


== Qintosec ==
== Qintosec ==
Qintosec can be described as the 10 &amp; 65d temperament, tempering out the [[marvel comma]], 225/224 in the 7-limit.  
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
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 5 1 22 21 | 0 2 -3 -2 }}
{{Mapping|legend=1| 5 1 22 21 | 0 2 -3 -2 }}


{{Multival|legend=1| 10 -15 -10 -47 -44 19 }}
[[Optimal tuning]]s:  
 
* [[WE]]: ~400/343 = 239.8397{{c}}, ~80/49 = 830.9971{{c}} (~16/15 = 111.7095{{c}})
Optimal tunings:  
: [[error map]]: {{val| -0.802 -0.121 -2.832 +5.813 }}
* CTE: ~400/343 = 1\5, ~80/49 = 831.7095 (~15/14 = 111.7095)
* [[CWE]]: ~400/343 = 240.0000{{c}}, ~80/49 = 831.5851{{c}} (~16/15 = 111.5851{{c}})
* POTE: ~400/343 = 1\5, ~80/49 = 831.553 (~15/14 = 111.553)
: error map: {{val| 0.000 +1.215 -1.069 +8.004 }}


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


[[Badness]]: 0.156764
[[Badness]] (Sintel): 3.97


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


Mapping: {{mapping| 5 1 22 21 0 | 0 2 -3 -2 5 }}
{{Mapping|legend=0| 5 1 22 21 0 | 0 2 -3 -2 5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/48 = 1\5, ~80/49 = 831.1939 (~15/14 = 111.1939)
* WE: ~55/48 = 239.8038{{c}}, ~80/49 = 830.6313{{c}} (~16/15 = 111.2200{{c}})
* POTE: ~55/48 = 1\5, ~80/49 = 831.311 (~15/14 = 111.311)
* CWE: ~55/48 = 240.0000{{c}}, ~80/49 = 831.2911{{c}} (~16/15 = 111.2911{{c}})


Optimal ET sequence: {{Optimal ET sequence| 10, 55d, 65d, 140de }}
{{Optimal ET sequence|legend=0| 10, 55d, 65d, 140de }}


Badness: 0.077907
Badness (Sintel): 2.58


=== Qinto ===
=== Qinto ===
This alternative extension can be tuned to [[75edo]] and [[85edo]], both in [[patent val]]s.
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 332750/321489
Comma list: 225/224, 385/384, 332750/321489


Mapping: {{mapping| 5 1 22 21 -7 | 0 2 -3 -2 7 }}
{{Mapping|legend=0| 5 1 22 21 -7 | 0 2 -3 -2 7 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~220/189 = 1\5, ~44/27 = 832.4051 (~15/14 = 112.4051)
* WE: ~220/189 = 239.9453{{c}}, ~44/27 = 832.2583{{c}} (~16/15 = 112.4224{{c}})
* POTE: ~220/189 = 1\5, ~44/27 = 832.448 (~15/14 = 112.448)
* CWE: ~220/189 = 240.0000{{c}}, ~44/27 = 832.4403{{c}} (~16/15 = 112.4403{{c}})


Optimal ET sequence: {{Optimal ET sequence| 10, 75 }}
{{Optimal ET sequence|legend=0| 10, 75 }}


Badness: 0.127732
Badness (Sintel): 4.22


== Sengasec ==
== 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.
Sengasec can be described as the 10 & 65 temperament, tempering out the [[cloudy comma]], 16807/16384 and the sengic comma, 686/675 in the 7-limit. It was named by [[Xenllium]] in 2021.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 5 1 22 14 | 0 2 -3 0 }}
{{Mapping|legend=1| 5 1 22 14 | 0 2 -3 0 }}
{{Multival|legend=1| 10 -15 0 -47 -28 42 }}


[[Optimal tuning]]:  
[[Optimal tuning]]:  
* [[CTE]]: ~8/7 = 1\5, ~45/28 = 831.1061 (~16/15 = 111.1061)
* [[WE]]: ~8/7 = 240.1396{{c}}, ~45/28 = 831.5960{{c}} (~16/15 = 111.1772{{c}})
* [[POTE]]: ~8/7 = 1\5, ~45/28 = 831.113 (~16/15 = 111.113)
: [[error map]]: {{val| +0.698 +1.377 +1.969 -6.872 }}
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~45/28 = 831.1113{{c}} (~16/15 = 111.1113{{c}})
: error map: {{val| 0.000 +0.268 +0.352 -8.826 }}


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


[[Badness]]: 0.156259
[[Badness]] (Sintel): 3.95


=== 11-limit ===
=== 11-limit ===
Line 108: Line 114:
Comma list: 243/242, 385/384, 686/675
Comma list: 243/242, 385/384, 686/675


Mapping: {{mapping| 5 1 22 14 0 | 0 2 -3 0 5 }}
{{Mapping|legend=0| 5 1 22 14 0 | 0 2 -3 0 5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~8/7 = 1\5, ~45/28 = 830.7772 (~16/15 = 110.7772)
* WE: ~8/7 = 240.0876{{c}}, ~45/28 = 830.9623{{c}} (~16/15 = 110.7772{{c}})
* POTE: ~8/7 = 1\5, ~45/28 = 830.659 (~16/15 = 110.659)
* CWE: ~8/7 = 240.0000{{c}}, ~45/28 = 830.6798{{c}} (~16/15 = 110.6798{{c}})


Optimal ET sequence: {{Optimal ET sequence| 10, 55, 65 }}
{{Optimal ET sequence|legend=0| 10, 55, 65 }}


Badness: 0.081607
Badness (Sintel): 2.70


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


Comma list: 105/104, 144/143, 243/242, 686/675
Comma list: 91/90, 169/168, 243/242, 385/384


Mapping: {{mapping| 5 1 22 14 0 22 | 0 2 -3 0 5 -1 }}
{{Mapping|legend=0| 5 1 22 14 0 15 | 0 2 -3 0 5 1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~8/7 = 1\5, ~45/28 = 830.8942 (~16/15 = 110.8942)
* WE: ~8/7 = 240.1611{{c}}, ~13/8 = 831.2123{{c}} (~16/15 = 110.7289{{c}})
* POTE: ~8/7 = 1\5, ~45/28 = 830.910 (~16/15 = 110.910)
* CWE: ~8/7 = 240.0000{{c}}, ~13/8 = 830.6933{{c}} (~16/15 = 110.6933{{c}})


Optimal ET sequence: {{Optimal ET sequence| 10, 55, 65 }}
{{Optimal ET sequence|legend=0| 10, 55f, 65f }}


Badness: 0.056390
Badness (Sintel): 2.08


== Decoid ==
== Decoid ==
{{See also| Breedsmic temperaments #Decoid }}
Decoid tempers out 2401/2400, the [[breedsma]], and 67108864/66976875, the [[decovulture comma]], as well as {{monzo| 11 -10 -10 10 }}, the [[linus comma]]. It may be described as the 130 & 270 temperament, and its [[ploidacot]] is decaploid dicot. The hemitwelfth or neutral third can be used as a generator, but also ~[[8/7]], ~[[16/15]], or the [[marvel comma]]. 181\940 or 233\1210 makes for an excellent tuning choice.  
 
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
[[Subgroup]]: 2.3.5.7
Line 143: Line 147:


{{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 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~15/14 = 1\10, ~8192/4725 = 951.1086 (~16/15 = 111.1086, or ~225/224 = 8.8914)
* [[WE]]: ~15/14 = 119.9913{{c}}, ~8192/4725 = 951.0298{{c}} (~225/224 = 8.9006{{c}})
* [[POTE]]: ~15/14 = 1\10, ~8192/4725 = 951.099 (~16/15 = 111.099, or ~225/224 = 8.901)
: [[error map]]: {{val| -0.087 +0.105 +0.188 -0.169 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~8192/4725 = 951.1007{{c}} (~225/224 = 8.8993{{c}})
: error map: {{val| 0.000 +0.246 +0.384 +0.073 }}


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


[[Badness]]: 0.033902
[[Badness]] (Sintel): 0.858


=== 11-limit ===
=== 11-limit ===
Line 161: Line 164:
Comma list: 2401/2400, 5632/5625, 9801/9800
Comma list: 2401/2400, 5632/5625, 9801/9800


Mapping: {{mapping| 10 0 47 36 98 | 0 2 -3 -1 -8 }}
{{Mapping|legend=0| 10 0 47 36 98 | 0 2 -3 -1 -8 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~15/14 = 1\10, ~400/231 = 951.0943 (~16/15 = 111.0943, or ~225/224 = 8.9057)
* WE: ~15/14 = 120.9921{{c}}, ~400/231 = 951.0074{{c}} (~225/224 = 8.9294{{c}})
* POTE: ~15/14 = 1\10, ~400/231 = 951.070 (~16/15 = 111.070, or ~225/224 = 8.930)
* CWE: ~15/14 = 120.0000{{c}}, ~400/231 = 951.0747{{c}} (~225/224 = 8.9253{{c}})


Optimal ET sequence: {{Optimal ET sequence| 10e, …, 130, 270, 670, 940, 1210, 2150c }}
{{Optimal ET sequence|legend=0| 130, 270, 670, 940, 1210, 2150c }}


Badness: 0.018735
Badness (Sintel): 0.619


=== 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: {{mapping| 10 0 47 36 98 37 | 0 2 -3 -1 -8 0 }}
{{Mapping|legend=0| 10 0 47 36 98 37 | 0 2 -3 -1 -8 0 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~15/14 = 1\10, ~26/15 = 951.0943 (~16/15 = 111.0943, or ~196/195 = 8.9057)
* WE: ~15/14 = 119.9964{{c}}, ~26/15 = 951.0546{{c}} (~196/195 = 8.9165{{c}})
* POTE: ~15/14 = 1\10, ~26/15 = 951.083 (~16/15 = 111.083, or ~196/195 = 8.917)
* CWE: ~15/14 = 120.0000{{c}}, ~26/15 = 951.0850{{c}} (~196/195 = 8.9150{{c}})


Optimal ET sequence: {{Optimal ET sequence| 10e, …, 130, 270, 940, 1210f, 1480cf }}
{{Optimal ET sequence|legend=0| 130, 270, 940, 1210f, 1480cf }}


Badness: 0.013475
Badness (Sintel): 0.557


== Notes ==
=== Triacontoid ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 42592/42525, 1771561/1769472
 
{{Mapping|legend=0| 30 0 141 108 80 | 0 2 -3 -1 1 }}
: mapping generators: ~45/44, ~8192/4725
 
Optimal tunings:
* WE: ~45/44 = 39.9981{{c}}, ~8192/4725 = 951.0672{{c}} (~225/224 = 8.8875{{c}})
* CWE: ~45/44 = 40.0000{{c}}, ~8192/4725 = 951.1124{{c}} (~225/224 = 8.8876{{c}})
 
{{Optimal ET sequence|legend=0| 120, 150, 270 }}
 
Badness (Sintel): 2.51
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 2401/2400, 4096/4095, 6656/6655
 
{{Mapping|legend=0| 30 0 141 108 80 111 | 0 2 -3 -1 1 0 }}
 
Optimal tunings:
* WE: ~45/44 = 39.9992{{c}}, ~26/15 = 951.0944{{c}} (~196/195 = 8.8868{{c}})
* CWE: ~45/44 = 40.0000{{c}}, ~26/15 = 951.1131{{c}} (~196/195 = 8.8869{{c}})
 
{{Optimal ET sequence|legend=0| 120, 150, 270 }}
 
Badness (Sintel): 1.69
 
== References ==


[[Category:Temperament families]]
[[Category:Quintosec family| ]] <!-- main article -->
[[Category:Quintosec family| ]] <!-- main article -->
[[Category:Quintosec| ]] <!-- key article -->
[[Category:Quintosec| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Quintosec_family"