Quintile family: Difference between revisions

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+a short intro on the structure of this temp
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The '''pental family''' tempers out the [[pental comma]], 847288609443/838860800000 = {{monzo| -28 25 -5 }}.
The '''quintile family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[quintile comma]] ([[ratio]]: 847288609443/838860800000, {{monzo|legend=1| -28 25 -5 }}.


== Pental ==
== Quintile ==
The 5-limit version of pental reaches the interval class of 5 by 5 perfect fifths (i.e. a major seventh) plus two periods of 1/5-octave. This temperament was first introduced by [[Mike Battaglia]] in 2011 along with other temperaments in the [[syntonic-diatonic equivalence continuum]]<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 by [[Petr Pařízek]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104270.html Yahoo! Tuning Group | ''2D temperaments, part II -- new 5-limit temperaments'']</ref>.  
Quintile reaches the interval class of 5 by five perfect fifths (i.e. a major seventh) plus two periods of 1/5-octave. It 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.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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== Pentacloud ==
== Pentacloud ==
The pentacloud (formerly ''septimal pental'') temperament can be described as 5&amp;60 temperament, 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
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Badness: 0.067549
Badness: 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
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Badness: 0.047624
Badness: 0.047624


==== Hemipent ====
==== Hemiquint ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Badness: 0.041043
Badness: 0.041043


==== Hemipentalis ====
==== Hemiquintilis ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Badness: 0.033542
Badness: 0.033542


== 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 ([[2460375/2458624|breeze comma]], laquadru-atriyo), and the [[15/14 equal-step tuning|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 235298/234375 (triwellisma, tribizo-asepgu), 2460375/2458624 ([[2460375/2458624|breeze comma]], laquadru-atriyo), and the [[linus comma]], {{monzo| 11 -10 -10 10 }}.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pental family| ]] <!-- main article -->
[[Category:Quintile family| ]] <!-- main article -->
[[Category:Pental| ]] <!-- key article -->
[[Category:Quintile| ]] <!-- key article -->
[[Category:Rank 2]]

Revision as of 12:02, 19 February 2025

The quintile family of temperaments tempers out the quintile comma (ratio: 847288609443/838860800000, monzo: [-28 25 -5⟩.

Quintile

Quintile reaches the interval class of 5 by five perfect fifths (i.e. a major seventh) plus two periods of 1/5-octave. It is a member of the syntonic–diatonic equivalence continuum with n = 5, so it equates a Pythagorean limma with a stack of five syntonic commas.

The temperament was first introduced by Mike Battaglia in 2011 along with other temperaments in the continuum mentioned above[1]. It did not get named until 2012, when Petr Pařízek called it pental[2]. In 2024, the community has decided to rename it for fear of confusion.

Subgroup: 2.3.5

Comma list: 847288609443/838860800000

Mapping: [⟨5 0 -28], ⟨0 1 5]]

mapping generators: ~59049/51200, ~3

Optimal tuning (POTE): ~59049/51200 = 1\5, ~3/2 = 701.210 (~81/80 = 18.790)

Optimal ET sequence: 5, 60, 65, 190, 255, 575, 830b, 1405b

Badness: 0.240050

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

Mapping: [⟨5 0 -28 14], ⟨0 1 5 0]]

Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 700.548 (~81/80 = 19.452)

Optimal ET sequence: 5, 60, 65, 125d, 185cdd

Badness: 0.120942

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 3087/3025

Mapping: [⟨5 0 -28 14 49], ⟨0 1 5 0 -4]]

Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 701.377 (~81/80 = 18.623)

Optimal ET sequence: 5, 60, 65

Badness: 0.093248

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 245/243, 3087/3025

Mapping: [⟨5 0 -28 14 49 -29], ⟨0 1 5 0 -4 6]]

Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 700.996 (~81/80 = 19.004)

Optimal ET sequence: 5, 60, 65, 125de, 190ddef

Badness: 0.067549

Hemiquintile

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, satribiru-agu) and 5250987/5242880 (mitonisma, laquadzo-agu).

Subgroup: 2.3.5.7

Comma list: 19683/19600, 589824/588245

Mapping: [⟨5 0 -28 18], ⟨0 2 10 -1]]

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

Optimal tuning (POTE): ~147/128 = 1\5, ~140/81 = 950.6536 (~1029/1024 = 9.3464)

Optimal ET sequence: 125, 130, 255, 385

Badness: 0.104163

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 8019/8000, 180224/180075

Mapping: [⟨5 0 -28 18 -54], ⟨0 2 10 -1 18]]

Optimal tuning (POTE): ~147/128 = 1\5, ~140/81 = 950.6341 (~176/175 = 9.3659)

Optimal ET sequence: 125, 130, 255, 385, 640

Badness: 0.047624

Hemiquint

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨5 0 -28 18 -54 34], ⟨0 2 10 -1 18 -13]]

Optimal tuning (POTE): ~147/128 = 1\5, ~140/81 = 950.6677 (~144/143 = 9.3323)

Optimal ET sequence: 125, 130, 255, 385, 515

Badness: 0.041043

Hemiquintilis

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨5 0 -28 18 -54 34], ⟨0 2 10 -1 18 13]]

Optimal tuning (POTE): ~147/128 = 1\5, ~26/15 = 950.6593 (~176/175 = 9.3407)

Optimal ET sequence: 125f, 130, 255f, 385f

Badness: 0.033542

Decile

Decile (formerly decal) can be described as 130 & 190 and tempers out the varunisma, 321489/320000 in the 7-limit, as well as 235298/234375 (triwellisma, tribizo-asepgu), 2460375/2458624 (breeze comma, laquadru-atriyo), and the linus comma, [11 -10 -10 10⟩.

Subgroup: 2.3.5.7

Comma list: 235298/234375, 321489/320000

Mapping: [⟨10 0 -56 -67], ⟨0 1 5 6]]

mapping generators: ~15/14, ~3

Wedgie: ⟨⟨ -10 -50 -60 -56 -67 1 ]]

Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 701.303 (~81/80 = 18.697)

Optimal ET sequence: 60, 130, 320, 450, 770d

Badness: 0.104859

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 234375/234256

Mapping: [⟨10 0 -56 -67 -108], ⟨0 1 5 6 9]]

Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 701.240 (~99/98 = 18.760)

Optimal ET sequence: 60e, 130, 190, 320

Badness: 0.040633

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 729/728, 1001/1000, 4225/4224

Mapping: [⟨10 0 -56 -67 -108 37], ⟨0 1 5 6 9 0]]

Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 701.252 (~91/90 = 18.748)

Optimal ET sequence: 60e, 130, 190, 320

Badness: 0.023948

Notes

  1. ↑ Yahoo! Tuning Group | Some new 5-limit microtemperaments
  2. ↑ Yahoo! Tuning Group | 2D temperaments, part II -- new 5-limit temperaments
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