Quintile family: Difference between revisions

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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).

== 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 {{nowrap| ''n'' {{=}} 5 }}, so it equates a [[256/243|Pythagorean limma]] with a stack of five [[81/80|syntonic commas]].

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.

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

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Badness (Smith): 0.047624

==== ~~Hemiquint~~ ====

==== Hemiquintilis ====

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* CTE: ~147/128 = 240.0000, ~~~140~~/81 = 950.~~6607~~ (~~~144~~/~~143~~ = 9.~~3393~~)

* CTE: ~147/128 = 240.0000, ~26/15 = 950.6775 (~176/175 = 9.3225)

* POTE: ~147/128 = 240.0000, ~~~140~~/81 = 950.~~6677~~ (~~~144~~/~~143~~ = 9.~~3323~~)

* POTE: ~147/128 = 240.0000, ~26/15 = 950.6593 (~176/175 = 9.3407)

{{Optimal ET sequence|legend=0| ~~125~~, 130, ~~255~~, ~~385, 515~~ }}

Badness (Smith): 0.~~041043~~

Badness (Smith): 0.033542

==== ~~Hemiquintilis~~ ====

==== Hemiquint ====

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* CTE: ~147/128 = 240.0000, ~140/81 = 950.~~6775~~ (~~~176~~/~~175~~ = 9.~~3225~~)

* CTE: ~147/128 = 240.0000, ~140/81 = 950.6607 (~144/143 = 9.3393)

* POTE: ~147/128 = 240.0000, ~26/15 = 950.~~6593~~ (~~~176~~/~~175~~ = 9.~~3407~~)

* POTE: ~147/128 = 240.0000, ~140/81 = 950.6677 (~144/143 = 9.3323)

{{Optimal ET sequence|legend=0| ~~125f~~, 130, ~~255f~~, ~~385f~~ }}

Badness (Smith): 0.~~033542~~

Badness (Smith): 0.041043

== Decile ==

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: mapping generators: ~15/14, ~3

~~{{Multival|legend=1| -10 -50 -60 -56 -67 1 }}~~

[[Optimal tuning]]s:

@@ Line 1: / Line 1: @@
+{{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).
 == 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 {{nowrap| ''n'' {{=}} 5 }}, so it equates a [[256/243|Pythagorean limma]] with a stack of five [[81/80|syntonic commas]].
+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.
+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
@@ Line 109: / Line 110: @@
 Badness (Smith): 0.047624
-==== Hemiquint ====
+==== Hemiquintilis ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000
+Comma list: 351/350, 540/539, 676/675, 124215/123904
-Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 -13 }}
+Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 13 }}
 Optimal tunings:
-* CTE: ~147/128 = 240.0000, ~140/81 = 950.6607 (~144/143 = 9.3393)
+* CTE: ~147/128 = 240.0000, ~26/15 = 950.6775 (~176/175 = 9.3225)
-* POTE: ~147/128 = 240.0000, ~140/81 = 950.6677 (~144/143 = 9.3323)
+* POTE: ~147/128 = 240.0000, ~26/15 = 950.6593 (~176/175 = 9.3407)
-{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 515 }}
+{{Optimal ET sequence|legend=0| 125f, 130, 255f, 385f }}
-Badness (Smith): 0.041043
+Badness (Smith): 0.033542
-==== Hemiquintilis ====
+==== Hemiquint ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 351/350, 540/539, 676/675, 124215/123904
+Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000
-Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 13 }}
+Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 -13 }}
 Optimal tunings:
-* CTE: ~147/128 = 240.0000, ~140/81 = 950.6775 (~176/175 = 9.3225)
+* CTE: ~147/128 = 240.0000, ~140/81 = 950.6607 (~144/143 = 9.3393)
-* POTE: ~147/128 = 240.0000, ~26/15 = 950.6593 (~176/175 = 9.3407)
+* POTE: ~147/128 = 240.0000, ~140/81 = 950.6677 (~144/143 = 9.3323)
-{{Optimal ET sequence|legend=0| 125f, 130, 255f, 385f }}
+{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 515 }}
-Badness (Smith): 0.033542
+Badness (Smith): 0.041043
 == Decile ==
@@ Line 149: / Line 150: @@
 : mapping generators: ~15/14, ~3
-{{Multival|legend=1| -10 -50 -60 -56 -67 1 }}
 [[Optimal tuning]]s: