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The '''pental family''' tempers out the pental comma, 847288609443/838860800000 = {{Monzo|-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 ==
Subgroup: 2.3.5
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–diatonic equivalence continuum]] with {{nowrap| ''n'' {{=}} 5 }}, so it equates a [[256/243|Pythagorean limma]] with a stack of five [[81/80|syntonic commas]].
 
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]].
 
[[Subgroup]]: 2.3.5


[[Comma list]]: 847288609443/838860800000
[[Comma list]]: 847288609443/838860800000


[[Mapping]]: [{{val| 5 0 -28 }}, {{val| 0 1 5 }}]
{{Mapping|legend=1| 5 0 -28 | 0 1 5 }}


Mapping generators: ~59049/51200, ~3
: mapping generators: ~59049/51200, ~3


[[POTE generator]]: ~3/2 = 701.210
[[Optimal tuning]]s:
* [[CTE]]: ~59049/51200 = 240.000, ~3/2 = 701.317 (~81/80 = 18.683)
: [[error map]]: {{val| 0.000 -0.638 +0.274 }}
* [[POTE]]: ~59049/51200 = 240.000, ~3/2 = 701.210 (~81/80 = 18.790)
: error map: {{val| 0.000 -0.745 -0.265 }}


{{Vals|legend=1| 5, 60, 65, 190, 255, 575, 830b, 1405b }}
{{Optimal ET sequence|legend=1| 5, 60, 65, 190, 255, 575, 830b, 1405b }}


[[Badness]]: 0.240050
[[Badness]] (Smith): 0.240050


== Septimal pental ==
== Pentacloud ==
The pental temperament can be described as 5&amp;60 temperament (also known as ''pentacloud''), tempering out the [[cloudy comma]], 16807/16384 and the [[245/243|sensamagic comma]], 245/243 in the 7-limit.
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 [[245/243|sensamagic comma]] 245/243 in the 7-limit.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|5 0 -28 14}}, {{val|0 1 5 0}}]
{{Mapping|legend=1| 5 0 -28 14 | 0 1 5 0 }}


[[POTE generator]]: ~3/2 = 700.548
[[Optimal tuning]]s:
* [[CTE]]: ~8/7 = 240.000, ~3/2 = 701.317 (~81/80 = 18.683)
: [[error map]]: {{val| 0.000 -0.638 +0.274 -8.826 }}
* [[POTE]]: ~8/7 = 240.000, ~3/2 = 700.548 (~81/80 = 19.452)
: error map: {{val| 0.000 -1.407 -3.574 -8.826 }}


{{Val list|legend=1| 5, 60, 65, 125d, 185cdd }}
{{Optimal ET sequence|legend=1| 5, 60, 65, 125d, 185cdd }}


[[Badness]]: 0.120942
[[Badness]] (Smith): 0.120942


=== 11-limit ===
=== 11-limit ===
Line 36: Line 49:
Comma list: 245/243, 385/384, 3087/3025
Comma list: 245/243, 385/384, 3087/3025


Mapping: [{{val|5 0 -28 14 49}}, {{val|0 1 5 0 -4}}]
Mapping: {{mapping| 5 0 -28 14 49 | 0 1 5 0 -4 }}


POTE generator: ~3/2 = 701.377
Optimal tunings:
* CTE: ~8/7 = 240.000, ~3/2 = 701.496 (~81/80 = 18.304)
* POTE: ~8/7 = 240.000, ~3/2 = 701.377 (~81/80 = 18.623)


Optimal GPV sequence: {{Val list| 5, 60, 65 }}
{{Optimal ET sequence|legend=0| 5, 60, 65 }}


Badness: 0.093248
Badness (Smith): 0.093248


=== 13-limit ===
=== 13-limit ===
Line 49: Line 64:
Comma list: 105/104, 144/143, 245/243, 3087/3025
Comma list: 105/104, 144/143, 245/243, 3087/3025


Mapping: [{{val|5 0 -28 14 49 -29}}, {{val|0 1 5 0 -4 6}}]
Mapping: {{mapping| 5 0 -28 14 49 -29 | 0 1 5 0 -4 6 }}


POTE generator: ~3/2 = 700.996
Optimal tunings:
* CTE: ~8/7 = 240.000, ~3/2 = 701.085 (~81/80 = 18.915)
* POTE: ~8/7 = 240.000, ~3/2 = 700.996 (~81/80 = 19.004)


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


Badness: 0.067549
Badness (Smith): 0.067549


== Hemipental ==
== Hemiquintile ==
The ''hemipental'' temperament (125&amp;130) tempers out the cataharry comma, 19683/19600 in the 7-limit, as well as 589824/588245 ([[Hewuermera temperaments|hewuermera]], satribiru-agu) and 5250987/5242880 ([[Mitonismic temperaments|mitonisma]], 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 temperaments|hewuermera]], satribiru-agu) and 5250987/5242880 ([[mitonismic temperaments|mitonisma]], laquadzo-agu).


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 5 0 -28 18 }}, {{val| 0 2 10 -1 }}]
{{Mapping|legend=1| 5 0 -28 18 | 0 2 10 -1 }}


Mapping generators: ~147/128, ~140/81
: mapping generators: ~147/128, ~140/81


[[POTE generator]]: ~8/7 = 230.654
[[Optimal tuning]]s:
* [[CTE]]: ~147/128 = 240.0000, ~140/81 = 950.6620 (~1029/1024 = 9.3380)
: [[error map]]: {{val| 0.000 -0.6311 +0.3059 +0.5121 }}
* [[POTE]]: ~147/128 = 240.0000, ~140/81 = 950.6536 (~1029/1024 = 9.3464)
: error map: {{val| 0.000 -0.6473 +0.2249 +0.5202 }}


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


[[Badness]]: 0.104163
[[Badness]] (Smith): 0.104163


=== 11-limit ===
=== 11-limit ===
Line 79: Line 100:
Comma list: 540/539, 8019/8000, 180224/180075
Comma list: 540/539, 8019/8000, 180224/180075


Mapping: [{{val| 5 0 -28 18 -54 }}, {{val| 0 2 10 -1 18 }}]
Mapping: {{mapping| 5 0 -28 18 -54 | 0 2 10 -1 18 }}
 
Optimal tunings:
* CTE: ~147/128 = 240.0000, ~140/81 = 950.6430 (~176/175 = 9.3570)
* POTE: ~147/128 = 240.0000, ~140/81 = 950.6341 (~176/175 = 9.3659)
 
{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 640 }}


POTE generator: ~8/7 = 230.634
Badness (Smith): 0.047624


Optimal GPV sequence: {{Val list| 5e, 125, 130, 255, 385, 640, 1025b }}
==== Hemiquintilis ====
Subgroup: 2.3.5.7.11.13


Badness: 0.047624
Comma list: 351/350, 540/539, 676/675, 124215/123904


=== 13-limit ===
Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 13 }}
 
Optimal tunings:
* CTE: ~147/128 = 240.0000, ~26/15 = 950.6775 (~176/175 = 9.3225)
* POTE: ~147/128 = 240.0000, ~26/15 = 950.6593 (~176/175 = 9.3407)
 
{{Optimal ET sequence|legend=0| 125f, 130, 255f, 385f }}
 
Badness (Smith): 0.033542
 
==== Hemiquint ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000
Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000


Mapping: [{{val| 5 0 -28 18 -54 34 }}, {{val| 0 2 10 -1 18 -13 }}]
Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 -13 }}


POTE generator: ~8/7 = 230.668
Optimal tunings:
* CTE: ~147/128 = 240.0000, ~140/81 = 950.6607 (~144/143 = 9.3393)
* POTE: ~147/128 = 240.0000, ~140/81 = 950.6677 (~144/143 = 9.3323)


Optimal GPV sequence: {{Val list| 5ef, 125, 130, 255, 385, 515, 900ef }}
{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 515 }}


Badness: 0.041043
Badness (Smith): 0.041043


== Decal ==
== Decile ==
The decal temperament (130&amp;190) tempers out the varunisma, 321489/320000 in the 7-limit, as well as 235298/234375 (triwellisma, tribizo-asepgu), 2460375/2458624 (sesquiquartisma, laquadru-atriyo), and the [[15/14ths equal temperament|linus comma]], {{Monzo|11 -10 -10 10}}.
Decile (formerly ''decal'') can be described as 130 & 190 and tempers out the [[varunisma]] 321489/320000 in the 7-limit, as well as the [[triwellisma]] 235298/234375, the [[breeze comma]] 2460375/2458624, and the [[linus comma]] {{monzo| 11 -10 -10 10 }}.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 10 0 -56 -67 }}, {{val| 0 1 5 6 }}]
{{Mapping|legend=1| 10 0 -56 -67 | 0 1 5 6 }}


Mapping generators: ~15/14, ~3
: mapping generators: ~15/14, ~3


[[POTE generator]]: ~3/2 = 701.303
[[Optimal tuning]]s:
* [[CTE]]: ~15/14 = 120.000, ~3/2 = 701.390 (~81/80 = 18.610)
: [[error map]]: {{val| 0.000 -0.565 +0.639 -0.483 }}
* [[POTE]]: ~15/14 = 120.000, ~3/2 = 701.303 (~81/80 = 18.697)
: error map: {{val| 0.000 -0.652 +0.200 -1.009 }}


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


[[Badness]]: 0.104859
[[Badness]] (Smith): 0.104859


=== 11-limit ===
=== 11-limit ===
Line 122: Line 166:
Comma list: 441/440, 8019/8000, 234375/234256
Comma list: 441/440, 8019/8000, 234375/234256


Mapping: [{{val| 10 0 -56 -67 -108 }}, {{val| 0 1 5 6 9 }}]
Mapping: {{mapping| 10 0 -56 -67 -108 | 0 1 5 6 9 }}


POTE generator: ~3/2 = 701.240
Optimal tunings:
* CTE: ~15/14 = 120.000, ~3/2 = 701.336 (~99/98 = 18.664)
* POTE: ~15/14 = 120.000, ~3/2 = 701.240 (~99/98 = 18.760)


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


Badness: 0.040633
Badness (Smith): 0.040633


=== 13-limit ===
=== 13-limit ===
Line 135: Line 181:
Comma list: 441/440, 729/728, 1001/1000, 4225/4224
Comma list: 441/440, 729/728, 1001/1000, 4225/4224


Mapping: [{{val| 10 0 -56 -67 -108 37 }}, {{val| 0 1 5 6 9 0 }}]
Mapping: {{mapping| 10 0 -56 -67 -108 37 | 0 1 5 6 9 0 }}
 
Optimal tunings:
* CTE: ~15/14 = 120.000, ~3/2 = 701.336 (~91/90 = 18.664)
* POTE: ~15/14 = 120.000, ~3/2 = 701.252 (~91/90 = 18.748)


POTE generator: ~3/2 = 701.252
{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }}


Optimal GPV sequence: {{Val list| 60e, 130, 190, 320 }}
Badness (Smith): 0.023948


Badness: 0.023948
== Notes ==


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