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, 3, 5, and 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 still infinite even if we restrict consideration to a single octave. 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''' 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 still infinite even if we restrict consideration to a single octave. 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 and 11. It can be similar to the 2.3.5.11 subgroup, specially considering neutral interval pairs such as 39/32 ~ 11/9 and 16/13 ~ 27/22, which are connected 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 and 11. It can be similar to the [[2.3.5.11 subgroup]], specially considering neutral interval pairs such as 39/32~11/9 and 16/13~27/22, which are connected by the small comma of [[352/351]]. | ||
== Regular temperaments == | == Regular temperaments == | ||
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=== Rank-2 temperaments === | === 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 34 | [[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 [[schismic]] extension via [[tempering out]] the [[schisma]] and the [[marveltwin comma]] provides a more complex temperament, well represented with [[41edo|41-]] 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 | ||
[[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 | |||
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. | 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, 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]] | For those searching 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 === | === 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]]. | [[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:Subgroup]] | [[Category:Subgroup]] | ||
Revision as of 13:07, 22 January 2026
The 2.3.5.13 subgroup is a just intonation subgroup consisting of rational intervals where 2, 3, 5, and 13 are the only allowable prime factors, 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 still infinite even if we restrict consideration to a single octave. 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 and 11. It can be similar to the 2.3.5.11 subgroup, specially considering neutral interval pairs such as 39/32~11/9 and 16/13~27/22, which are connected by the small comma of 352/351.
Regular temperaments
Rank-1 temperaments (edos)
It is relatively well approximated by the following edos (bold ones do particularly well in this subgroup): 7, 15, 19, 24, 27, 31, 34, 46, 50, 53, 62, 68, 77, 80, 87, 94, 96, 106, 130, 137, 140, 159, 171, 190, 217, 224, 270, 311, …
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 34- and 53edo, with 87edo being an almost perfect approximation.
The schismic extension via tempering out the schisma and the marveltwin comma provides a more complex temperament, well represented with 41- 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 ¢ and +1.428 ¢ 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, the 2.3.5.13 extension of egads (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)2, and condensing the subgroup into a 5-limit planar temperament.
{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.