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 with the more common usage of the term ''pental'' to refer to the [[5-limit]].

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: mapping generators: ~15/14, ~3

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[[Optimal tuning]]s:

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+{{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 with the more common usage of the term ''pental'' to refer to the [[5-limit]].
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 : mapping generators: ~15/14, ~3
-{{Multival|legend=1| -10 -50 -60 -56 -67 1 }}
 [[Optimal tuning]]s: