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The pental family tempers out the pental comma, 847288609443/838860800000 = |-28 25 -5>.
{{Technical data page}}
The '''quintile family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[quintile comma]] ({{monzo|legend=1| -28 25 -5 }}, [[ratio]]: 847 288 609 443 / 838 860 800 000).  


=Pental=
== Quintile ==
Comma: 847288609443/838860800000
Quintile reaches the interval class of 5 by five [[3/2|perfect fifths]] (i.e. a major seventh) plus two periods of 1/5-octave; this two-period interval represents a grave fourth of [[320/243]], that is, a [[4/3|perfect fourth]] minus a [[81/80|syntonic comma]]. Quintile is a member of the [[syntonic–limmic equivalence continuum]] with {{nowrap| ''n'' {{=}} 5 }}, so it equates a [[256/243|Pythagorean limma]] with a stack of five [[81/80|syntonic commas]].


[[POTE tuning|POTE generator]]: ~3/2 = 701.210
The temperament was first introduced by [[Mike Battaglia]] in 2011 along with other temperaments in the continuum mentioned above<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_99315.html#99323 Yahoo! Tuning Group | ''Some new 5-limit microtemperaments'']</ref>. It did not get named until 2012, when [[Petr Pařízek]] called it ''pental''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104270.html Yahoo! Tuning Group | ''2D temperaments, part II -- new 5-limit temperaments'']</ref>. In 2024, the community has decided to rename it for fear of confusion with the more common usage of the term ''pental'' to refer to the [[5-limit]].  


[[Map]]: [&lt;5 0 -28|, &lt;0 1 5|]
[[Subgroup]]: 2.3.5


{{Vals|legend=1| 5, 60, 65, 190, 255, 575, 830b, 1405b }}
[[Comma list]]: 847288609443/838860800000


[[Badness]]: 0.240050
{{Mapping|legend=1| 5 0 -28 | 0 1 5 }}
: mapping generators: ~59049/51200, ~3


=Pentacloud=
[[Optimal tuning]]s:
The ''pentacloud'' temperament (5&amp;60, named by [[User:Xenllium|Xenllium]]) tempers out the [[cloudy comma]], 16807/16384 and the [[245/243|sensamagic comma]], 245/243 in the 7-limit.
* [[WE]]: ~59049/51200 = 240.0389{{c}}, ~3/2 = 701.3235{{c}} (~81/80 = 18.7932{{c}})
: [[error map]]: {{val| +0.195 -0.437 +0.187 }}
* [[CWE]]: ~59049/51200 = 240.0000{{c}}, ~3/2 = 701.2536{{c}} (~81/80 = 18.7464{{c}})
: error map: {{val| 0.000 -0.701 -0.046 }}
 
{{Optimal ET sequence|legend=1| 60, 65, 190, 255, 575, 830b, 1405b }}
 
[[Badness]] (Sintel): 5.63
 
== Pentacloud ==
Pentacloud can be described as the 5 & 60 temperament. It identifies the period as [[~]][[8/7]], tempering out the [[cloudy comma]] 16807/16384 and the [[sensamagic comma]] 245/243 in the 7-limit.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 16807/16384
[[Comma list]]: 245/243, 16807/16384


[[POTE tuning|POTE generator]]: ~3/2 = 700.548
{{Mapping|legend=1| 5 0 -28 14 | 0 1 5 0 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~8/7 = 240.2778{{c}}, ~3/2 = 701.3587{{c}} (~81/80 = 19.4746{{c}})
: [[error map]]: {{val| +1.389 +0.793 -0.353 -4.937 }}
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~3/2 = 700.7696{{c}} (~81/80 = 19.2304{{c}})
: error map: {{val| 0.000 -1.185 -2.466 -8.826 }}


[[Map]]: [&lt;5 0 -28 14|, &lt;0 1 5 0|]
{{Optimal ET sequence|legend=1| 60, 125d, 185cdd }}


{{Vals|legend=1| 5, 60, 65, 125d, 185cdd }}
[[Badness]] (Sintel): 3.06


[[Badness]]: 0.120942
=== 11-limit ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 245/243, 385/384, 3087/3025
Comma list: 245/243, 385/384, 3087/3025


POTE generator: ~3/2 = 701.377
Mapping: {{mapping| 5 0 -28 14 49 | 0 1 5 0 -4 }}
 
Optimal tunings:
* WE: ~8/7 = 240.0919{{c}}, ~3/2 = 701.6455{{c}} (~81/80 = 18.6301{{c}})
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 701.3981{{c}} (~81/80 = 18.6019{{c}})


Map: [&lt;5 0 -28 14 49|, &lt;0 1 5 0 -4|]
{{Optimal ET sequence|legend=0| 60, 65 }}


Vals: {{Vals| 5, 60, 65 }}
Badness (Sintel): 3.08


Badness: 0.093248
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 105/104, 144/143, 245/243, 3087/3025
Comma list: 105/104, 144/143, 245/243, 3087/3025


POTE generator: ~3/2 = 700.996
Mapping: {{mapping| 5 0 -28 14 49 -29 | 0 1 5 0 -4 6 }}


Map: [&lt;5 0 -28 14 49 -29|, &lt;0 1 5 0 -4 6|]
Optimal tunings:  
* WE: ~8/7 = 240.0470{{c}}, ~3/2 = 701.1333{{c}} (~81/80 = 19.0078{{c}})
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 701.0105{{c}} (~81/80 = 18.9895{{c}})


Vals: {{Vals| 5, 60, 65, 125de, 190ddef }}
{{Optimal ET sequence|legend=0| 60, 65 }}


Badness: 0.067549
Badness (Sintel): 2.79


=Hemipental=
== Hemiquintile ==
The ''hemipental'' temperament (125&amp;130, named by [[User:Xenllium|Xenllium]]) tempers out the cataharry comma, 19683/19600 in the 7-limit, as well as 589824/588245 (satribiru-agu) and 5250987/5242880 (laquadzo-agu).
Hemiquintile (formerly ''hemipental'') can be described as 125 & 130 and tempers out the [[cataharry comma]], 19683/19600 in the 7-limit, as well as 589824/588245 ([[hewuermera comma]]) and 5250987/5242880 ([[mitonisma]]).
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 19683/19600, 589824/588245
[[Comma list]]: 19683/19600, 589824/588245


[[POTE tuning|POTE generator]]: ~8/7 = 230.654
{{Mapping|legend=1| 5 0 -28 18 | 0 2 10 -1 }}
: mapping generators: ~147/128, ~140/81


[[Map]]: [&lt;5 8 12 14|, &lt;0 -2 -10 1|]
[[Optimal tuning]]s:  
* [[WE]]: ~147/128 = 240.0063{{c}}, ~140/81 = 950.6787{{c}} (~1029/1024 = 9.3464{{c}})
: [[error map]]: {{val| +0.031 -0.598 +0.298 +0.608 }}
* [[CWE]]: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6561{{c}} (~1029/1024 = 9.3439{{c}})
: error map: {{val| 0.000 -0.643 +0.247 +0.518 }}


{{Vals|legend=1| 5, 125, 130, 255, 385 }}
{{Optimal ET sequence|legend=1| 125, 130, 255, 385 }}


[[Badness]]: 0.104163
[[Badness]] (Sintel): 2.64
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 540/539, 8019/8000, 180224/180075
Comma list: 540/539, 8019/8000, 180224/180075


POTE generator: ~8/7 = 230.634
Mapping: {{mapping| 5 0 -28 18 -54 | 0 2 10 -1 18 }}
 
Optimal tunings:
* WE: ~147/128 = 240.0080{{c}}, ~140/81 = 950.6658{{c}} (~176/175 = 9.3662{{c}})
* CWE: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6365{{c}} (~176/175 = 9.3635{{c}})
 
{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 640 }}
 
Badness (Sintel): 1.57
 
==== Hemiquintilis ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 540/539, 676/675, 124215/123904
 
Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 13 }}
 
Optimal tunings:
* WE: ~147/128 = 240.0155{{c}}, ~26/15 = 950.7207{{c}} (~176/175 = 9.3413{{c}})
* CWE: ~147/128 = 240.0000{{c}}, ~26/15 = 950.6641{{c}} (~176/175 = 9.3359{{c}})


Map: [&lt;5 8 12 14 18|, &lt;0 -2 -10 1 -18|]
{{Optimal ET sequence|legend=0| 125f, 130, 255f, 385f }}


Vals: {{Vals| 5e, 125, 130, 255, 385, 640, 1025b }}
Badness (Sintel): 1.39


Badness: 0.047624
==== Hemiquint ====
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000
Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000


POTE generator: ~8/7 = 230.668
Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 -13 }}
 
Optimal tunings:
* WE: ~147/128 = 239.9879{{c}}, ~140/81 = 950.6196{{c}} (~176/175 = 9.3318{{c}})
* CWE: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6666{{c}} (~176/175 = 9.3334{{c}})


Map: [&lt;5 8 12 14 18 18|, &lt;0 -2 -10 1 -18 13|]
{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 515 }}


Vals: {{Vals| 5ef, 125, 130, 255, 385, 515, 900ef }}
Badness (Sintel): 1.70


Badness: 0.041043
== Decile ==
Decile (formerly ''decal'') can be described as 130 & 190 and tempers out the [[triwellisma]] 235298/234375 in the 7-limit, as well as the [[varunisma]] 321489/320000, the [[breeze comma]] 2460375/2458624, and the [[linus comma]] {{monzo| 11 -10 -10 10 }}.
 
[[Subgroup]]: 2.3.5.7


=Decal=
[[Comma list]]: 235298/234375, 321489/320000
[[Comma list]]: 235298/234375, 321489/320000


[[POTE tuning|POTE generator]]: ~3/2 = 701.303
{{Mapping|legend=1| 10 0 -56 -67 | 0 1 5 6 }}
: mapping generators: ~15/14, ~3


[[Map]]: [&lt;10 0 -56 -67|, &lt;0 1 5 6|]
[[Optimal tuning]]s:  
* [[WE]]: ~15/14 = 120.0171{{c}}, ~3/2 = 701.4027{{c}} (~81/80 = 18.6999{{c}})
: [[error map]]: {{val| +0.171 -0.381 +0.597 -0.529 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 701.3381{{c}} (~81/80 = 18.6619{{c}})
: error map: {{val| 0.000 -0.617 +0.377 -0.797 }}


{{Vals|legend=1| 60, 130, 320, 450, 770d }}
{{Optimal ET sequence|legend=1| 60, 130, 320, 450, 770d }}


[[Badness]]: 0.104859
[[Badness]] (Sintel): 2.65
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 441/440, 8019/8000, 234375/234256
Comma list: 441/440, 8019/8000, 234375/234256


POTE generator: ~3/2 = 701.240
Mapping: {{mapping| 10 0 -56 -67 -108 | 0 1 5 6 9 }}


Map: [&lt;10 0 -56 -67 -108|, &lt;0 1 5 6 9|]
Optimal tunings:  
* WE: ~15/14 = 120.0208{{c}}, ~3/2 = 701.3614{{c}} (~99/98 = 18.7632{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 701.2776{{c}} (~99/98 = 18.7224{{c}})


Vals: {{Vals| 130, 190, 320 }}
{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }}


Badness: 0.040633
Badness (Sintel): 1.34
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 441/440, 729/728, 1001/1000, 4225/4224
Comma list: 441/440, 729/728, 1001/1000, 4225/4224


POTE generator: ~3/2 = 701.252
Mapping: {{mapping| 10 0 -56 -67 -108 37 | 0 1 5 6 9 0 }}
 
Optimal tunings:
* WE: ~15/14 = 120.0182{{c}}, ~3/2 = 701.3582{{c}} (~91/90 = 18.7510{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 701.2754{{c}} (~91/90 = 18.7246{{c}})


Map: [&lt;10 0 -56 -67 -108 37|, &lt;0 1 5 6 9 0|]
{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }}


Vals: {{Vals| 130, 190, 320 }}
Badness (Sintel): 0.990


Badness: 0.023948
== References ==


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