2.3.5.13 subgroup: Difference between revisions

Line 1:

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.

The '''2.3.5.13 subgroup''' (a.k.a. ''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]].

== Regular temperaments ==

Line 14:

Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.

For those searching [[~~Very~~ high accuracy temperaments|very high-accuracy temperaments]], the 2.3.5.13 extension of [[egads]] ({{nowrap| 19 & 422 }}) provides an extremely complex, though insanely accurate representation of the subgroup, with lower badness than cata and with an almost just ~6/5 as a generator, finding 5/4 at -51 gens, 3/2 at -52 gens, and 13/8 at -138 gens, of which [[1342edo]] offers practically perfect approximations.

For those searching [[very high accuracy temperaments|very high-accuracy temperaments]], the 2.3.5.13 extension of [[egads]] ({{nowrap| 19 & 422 }}) provides an extremely complex, though insanely accurate representation of the subgroup, with lower badness than cata and with an almost just ~6/5 as a generator, finding 5/4 at -51 gens, 3/2 at -52 gens, and 13/8 at -138 gens, of which [[1342edo]] offers practically perfect approximations.

=== Rank-3 temperaments ===

[[Marveltwin]] offers a very low-complexity approximation to the subgroup, reaching [[16/13]] through ([[10/9]])<sup>2</sup>, and condensing the subgroup into a 5-limit [[planar temperament]].

[[Catasma|{140625/140608}]], the temperament that tempers the catasma alone, is an extremely accurate temperament, which also appears in the same Egads extension (catabolic). Non-cata edos at the boundary of usability are, [[407edo]], [[441edo]], [[494edo]], [[901edo]], and of course [[1342edo]].

{[[Catasma|140625/140608]]}, the temperament that tempers the catasma alone, is an extremely accurate temperament, which also appears in the same Egads extension (catabolic). Non-cata edos at the boundary of usability are, [[407edo]], [[441edo]], [[494edo]], [[901edo]], and of course [[1342edo]].

[[Category:Just intonation subgroups|#]]

[[Category:Rank-4 temperaments|#]]

[[Category:13-limit|#]]

@@ Line 1: / Line 1: @@
-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.
+The '''2.3.5.13 subgroup''' (a.k.a. ''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]].
 == Regular temperaments ==
@@ Line 14: / Line 14: @@
 Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.
-For those searching [[Very high accuracy temperaments|very high-accuracy temperaments]], the 2.3.5.13 extension of [[egads]] ({{nowrap| 19 & 422 }}) provides an extremely complex, though insanely accurate representation of the subgroup, with lower badness than cata and with an almost just ~6/5 as a generator, finding 5/4 at -51 gens, 3/2 at -52 gens, and 13/8 at -138 gens, of which [[1342edo]] offers practically perfect approximations.
+For those searching [[very high accuracy temperaments|very high-accuracy temperaments]], the 2.3.5.13 extension of [[egads]] ({{nowrap| 19 & 422 }}) provides an extremely complex, though insanely accurate representation of the subgroup, with lower badness than cata and with an almost just ~6/5 as a generator, finding 5/4 at -51 gens, 3/2 at -52 gens, and 13/8 at -138 gens, of which [[1342edo]] offers practically perfect approximations.
 === Rank-3 temperaments ===
 [[Marveltwin]] offers a very low-complexity approximation to the subgroup, reaching [[16/13]] through ([[10/9]])<sup>2</sup>, and condensing the subgroup into a 5-limit [[planar temperament]].
-[[Catasma|{140625/140608}]], the temperament that tempers the catasma alone, is an extremely accurate temperament, which also appears in the same Egads extension (catabolic). Non-cata edos at the boundary of usability are, [[407edo]], [[441edo]], [[494edo]], [[901edo]], and of course [[1342edo]].
+{[[Catasma|140625/140608]]}, the temperament that tempers the catasma alone, is an extremely accurate temperament, which also appears in the same Egads extension (catabolic). Non-cata edos at the boundary of usability are, [[407edo]], [[441edo]], [[494edo]], [[901edo]], and of course [[1342edo]].
 [[Category:Just intonation subgroups|#]]
 [[Category:Rank-4 temperaments|#]]
 [[Category:13-limit|#]]