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''' (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]].

This subgroup is notable for containing the simplest JI representations of [[interseptimal]] intervals, which are halfway between two interval categories, with [[15/13]] being an ultramajor second/inframinor third, [[13/10]] being an ultramajor third/infrafourth, [[20/13]] being an ultrafifth/inframinor sixth, and [[26/15]] being an ultramajor sixth/inframinor seventh. Importantly, 15/13 is close to half of the [[4/3|perfect fourth]], and 26/15 is close to half of the [[3/1|perfect twelfth]], with two intervals of 15/13 falling short of 4/3 by [[676/675]], the island comma.

== Regular temperaments ==

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=== Rank-2 temperaments ===

[[~~Cata~~]] provides a fairly low complexity approximation to the subgroup, using a slightly sharp ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. It is ~~well represented by~~ [[34edo~~|34-~~]] and [[53edo]], with [[87edo]] being ~~an almost perfect approximation~~.

The 2.3.5.13 version of [[kleismic]] (sometimes called ''cata'') provides a fairly low complexity approximation to the subgroup, using a slightly sharp [[~]]6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. Two generators reach ~[[13/9]], [[tempering out]] the marveltwin comma [[325/324]]. Then ~[[26/15]] is found at three generators, with two such intervals reaching ~3/1, tempering out [[676/675]]. The interval at +4 generators is a third of a [[9/8]] whole tone, representing all of [[25/24]], [[26/25]], and [[27/26]]. Good tunings of cata include [[34edo]] and especially [[53edo]], with other tunings such as [[87edo]] and [[140edo]] being usable as well.

The [[schismic]] extension that adds prime 13 via [[tempering out~~]] the~~ [[~~marveltwin comma~~]] provides a more complex temperament, well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and +1.428{{c}} of error respectively.

The [[schismic]] extension that adds prime 13 via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and -1.428{{c}} of error respectively.

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|#]]

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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''' (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]].
+This subgroup is notable for containing the simplest JI representations of [[interseptimal]] intervals, which are halfway between two interval categories, with [[15/13]] being an ultramajor second/inframinor third, [[13/10]] being an ultramajor third/infrafourth, [[20/13]] being an ultrafifth/inframinor sixth, and [[26/15]] being an ultramajor sixth/inframinor seventh. Importantly, 15/13 is close to half of the [[4/3|perfect fourth]], and 26/15 is close to half of the [[3/1|perfect twelfth]], with two intervals of 15/13 falling short of 4/3 by [[676/675]], the island comma.
 == Regular temperaments ==
@@ Line 8: / Line 10: @@
 === Rank-2 temperaments ===
-[[Cata]] provides a fairly low complexity approximation to the subgroup, using a slightly sharp ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. It is well represented by [[34edo|34-]] and [[53edo]], with [[87edo]] being an almost perfect approximation.
+The 2.3.5.13 version of [[kleismic]] (sometimes called ''cata'') provides a fairly low complexity approximation to the subgroup, using a slightly sharp [[~]]6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. Two generators reach ~[[13/9]], [[tempering out]] the marveltwin comma [[325/324]]. Then ~[[26/15]] is found at three generators, with two such intervals reaching ~3/1, tempering out [[676/675]]. The interval at +4 generators is a third of a [[9/8]] whole tone, representing all of [[25/24]], [[26/25]], and [[27/26]]. Good tunings of cata include [[34edo]] and especially [[53edo]], with other tunings such as [[87edo]] and [[140edo]] being usable as well.
-The [[schismic]] extension that adds prime 13 via [[tempering out]] the [[marveltwin comma]] provides a more complex temperament, well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and +1.428{{c}} of error respectively.
+The [[schismic]] extension that adds prime 13 via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and -1.428{{c}} of error respectively.
 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|#]]