Quintosec family: Difference between revisions

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 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

[[Comma list]]: {{monzo| 47 -15 -10 }}

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

 

 

 

=== 2.3.5.11 subgroup ===

 

 

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

 

 

 

 

== Qintosec ==

Qintosec can be described as the 10 & 65d temperament, tempering out the [[marvel comma]], 225/224 in the 7-limit.  

 

 

[[Optimal tuning]]s:

: [[error map]]: {{val| -0.802 -0.121 -2.832 +5.813 }}

* [[CWE]]: ~400/343 = 240.0000{{c}}, ~80/49 = 831.5851{{c}} (~16/15 = 111.5851{{c}})

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

Subgroup: 2.3.5.7.11

 

* WE: ~55/48 = 239.8038{{c}}, ~80/49 = 830.6313{{c}} (~16/15 = 111.2200{{c}})

* CWE: ~55/48 = 240.0000{{c}}, ~80/49 = 831.2911{{c}} (~16/15 = 111.2911{{c}})

 

Badness (Sintel): 2.58

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

Subgroup: 2.3.5.7.11

 

* WE: ~220/189 = 239.9453{{c}}, ~44/27 = 832.2583{{c}} (~16/15 = 112.4224{{c}})

* CWE: ~220/189 = 240.0000{{c}}, ~44/27 = 832.4403{{c}} (~16/15 = 112.4403{{c}})

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

Badness (Sintel): 4.22

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

 

[[Optimal tuning]]:

: [[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}})

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

Subgroup: 2.3.5.7.11

 

* WE: ~8/7 = 240.0876{{c}}, ~45/28 = 830.9623{{c}} (~16/15 = 110.7772{{c}})

 

Badness (Sintel): 2.70

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 169/168, 243/242, 385/384

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

Optimal tunings:  

* CWE: ~8/7 = 240.0000{{c}}, ~13/8 = 830.6933{{c}} (~16/15 = 110.6933{{c}})

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

Badness (Sintel): 2.08

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.  

 

[[Optimal tuning]]s:

: [[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| 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}

Subgroup: 2.3.5.7.11

 

Optimal tunings:  

* WE: ~15/14 = 120.9921{{c}}, ~400/231 = 951.0074{{c}} (~225/224 = 8.9294{{c}})

* CWE: ~15/14 = 120.0000{{c}}, ~400/231 = 951.0747{{c}} (~225/224 = 8.9253{{c}})

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

Badness (Sintel): 0.619

Subgroup: 2.3.5.7.11.13

 

* WE: ~15/14 = 119.9964{{c}}, ~26/15 = 951.0546{{c}} (~196/195 = 8.9165{{c}})

 

 

 

 

 

 

 

 

 

 

 

{{Mapping|legend=0| 30 0 141 108 80 111 | 0 2 -3 -1 1 0 }}

 

* 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

[[Category:Quintosec family| ]] <!-- main article -->

[[Category:Temperament families]]

[[Category:Catalogs of rank-2 temperaments]]