2.3.5.7.13 subgroup: Difference between revisions

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The '''2.3.5.7.13 subgroup''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1|2]], [[3/1|3]], [[5/1|5]], [[7/1|7]], 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, 7 and 13; this makes it a rank-5 system. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[7/4]], [[13/8]], [[13/7]], [[13/10]], [[39/32]], and so on.

The '''2.3.5.7.13 subgroup''' (AKA ''yazatha'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1|2]], [[3/1|3]], [[5/1|5]], [[7/1|7]], 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, 7 and 13; this makes it a rank-5 system. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[7/4]], [[13/8]], [[13/7]], [[13/10]], [[39/32]], and so on.

It can be thought out as an extension of the [[7-limit]] with a tridecimal xenharmonic touch, or as a retraction of the full 13-limit obtained by removing 11. It can be similar to the [[11-limit]], specially considering neutral interval pairs such as {{nowrap|39/32~11/9}} and {{nowrap|16/13~27/22}}, which are connected by the small comma of [[352/351]].

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	The '''2.3.5.7.13 subgroup''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], 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, 7 and 13; this makes it a rank-5 system. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[7/4]], [[13/8]], [[13/7]], [[13/10]], [[39/32]], and so on.		The '''2.3.5.7.13 subgroup''' (AKA ''yazatha'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], 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, 7 and 13; this makes it a rank-5 system. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[7/4]], [[13/8]], [[13/7]], [[13/10]], [[39/32]], and so on.

	It can be thought out as an extension of the [[7-limit]] with a tridecimal xenharmonic touch, or as a retraction of the full 13-limit obtained by removing 11. It can be similar to the [[11-limit]], specially considering neutral interval pairs such as {{nowrap\|39/32~11/9}} and {{nowrap\|16/13~27/22}}, which are connected by the small comma of [[352/351]].		It can be thought out as an extension of the [[7-limit]] with a tridecimal xenharmonic touch, or as a retraction of the full 13-limit obtained by removing 11. It can be similar to the [[11-limit]], specially considering neutral interval pairs such as {{nowrap\|39/32~11/9}} and {{nowrap\|16/13~27/22}}, which are connected by the small comma of [[352/351]].