2.3.5.13 subgroup: Difference between revisions

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The '''2.3.5.13 subgroup''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1|2]], [[3/1|3]], [[5/1|5]], and [[13/1|13]] are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 13. This is an infinite set, and is still infinite even if we [[Octave reduction|octave-reduce]] every interval in it. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[13/8]], [[13/10]], [[39/32]] and so on.

The '''2.3.5.13 subgroup''' (AKA ''yatha'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1|2]], [[3/1|3]], [[5/1|5]], and [[13/1|13]] are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 13. This is an infinite set, and is still infinite even if we [[Octave reduction|octave-reduce]] every interval in it. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[13/8]], [[13/10]], [[39/32]] and so on.

It can be thought out as an extension of the familiar [[5-limit]] with a tridecimal xenharmonic touch, or as a retraction of the 13-limit obtained by removing [[7/1|7]] and [[11/1|11]]. It shares some qualities with the [[2.3.5.11 subgroup]], specifically considering [[Neutral (interval quality)|neutral]] interval pairs such as [[39/32]]~[[11/9]] and [[16/13]]~[[27/22]], which differ by the small comma of [[352/351]].

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	The '''2.3.5.13 subgroup''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[13/1\|13]] are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 13. This is an infinite set, and is still infinite even if we [[Octave reduction\|octave-reduce]] every interval in it. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[13/8]], [[13/10]], [[39/32]] and so on.		The '''2.3.5.13 subgroup''' (AKA ''yatha'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[13/1\|13]] are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 13. This is an infinite set, and is still infinite even if we [[Octave reduction\|octave-reduce]] every interval in it. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[13/8]], [[13/10]], [[39/32]] and so on.

	It can be thought out as an extension of the familiar [[5-limit]] with a tridecimal xenharmonic touch, or as a retraction of the 13-limit obtained by removing [[7/1\|7]] and [[11/1\|11]]. It shares some qualities with the [[2.3.5.11 subgroup]], specifically considering [[Neutral (interval quality)\|neutral]] interval pairs such as [[39/32]]~[[11/9]] and [[16/13]]~[[27/22]], which differ by the small comma of [[352/351]].		It can be thought out as an extension of the familiar [[5-limit]] with a tridecimal xenharmonic touch, or as a retraction of the 13-limit obtained by removing [[7/1\|7]] and [[11/1\|11]]. It shares some qualities with the [[2.3.5.11 subgroup]], specifically considering [[Neutral (interval quality)\|neutral]] interval pairs such as [[39/32]]~[[11/9]] and [[16/13]]~[[27/22]], which differ by the small comma of [[352/351]].