2.3.5.13 subgroup: Difference between revisions
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{{stub}}The 2.3.5.13 subgroup is a [[just intonation subgroup]] consisting of [[Rational interval|rational intervals]] where 2, 3, 5, and 13 are the only allowable [[Prime factor|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. | {{stub}}The 2.3.5.13 subgroup is a [[just intonation subgroup]] consisting of [[Rational interval|rational intervals]] where 2, 3, 5, and 13 are the only allowable [[Prime factor|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 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 == | ||
=== Rank-1 temperaments (edos) === | === Rank-1 temperaments (edos) === | ||
It is relatively well approximated by the following edos: [[7edo|7]], [[15edo|15]], [[19edo|19]], [[24edo|24]], [[27edo|27]], [[31edo|31]], '''[[34edo|34]]''', [[50edo|50]], '''[[53edo|53]]''', [[80edo|80]], '''[[87edo|87]]''', [[94edo|94]], [[96edo|96]], [[130edo|130]], [[140edo|140]], '''[[217edo|217]]''', [[270edo|270]]... | It is relatively well approximated by the following edos [bold ones edos that do particularly well in this subgroup]: [[7edo|7]], [[15edo|15]], [[19edo|19]], [[24edo|24]], [[27edo|27]], [[31edo|31]], '''[[34edo|34]]''', [[46edo|46]], [[50edo|50]], '''[[53edo|53]]''', [[80edo|80]], '''[[87edo|87]]''', [[94edo|94]], [[96edo|96]], [[130edo|130]], [[140edo|140]], '''[[171edo|171]]''', '''[[217edo|217]]''', [[224edo|224]], [[270edo|270]]... | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
[[Cata]] provides a fairly low complexity approximation to the subgroup, using ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. | [[Cata]] provides a fairly low complexity approximation to the subgroup, using a slightly flat ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. | ||
[[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 sound extremely close to 5/4 and 13/8 respectively, wherein tempering the [[schisma]] and [[tridecapyth comma]] provide a fairly more complex but 3/2-[[Telicity|telic]] microtemperament, of which 53edo offers an almost perfect approximation. Pure fifths and octaves on the other hand, offer 5 and 13 with -1.954c and +1.428c of error. | [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 sound extremely close to 5/4 and 13/8 respectively, wherein tempering the [[schisma]] and [[tridecapyth comma]] provide a fairly more complex but 3/2-[[Telicity|telic]] microtemperament, of which 53edo offers an almost perfect approximation. Pure fifths and octaves on the other hand, offer 5 and 13 with -1.954c and +1.428c of error. | ||
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. | 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. | ||
For those searching very high accuracy temperaments, the 2.3.5.13 extension of [[Very high accuracy temperaments#Catabolic|Egads]] (19&422) provides a highly complex, but 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 a practically perfect approximation. | |||
=== 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]]. | ||
Revision as of 13:52, 27 October 2025
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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 edos that do particularly well in this subgroup]: 7, 15, 19, 24, 27, 31, 34, 46, 50, 53, 80, 87, 94, 96, 130, 140, 171, 217, 224, 270...
Rank-2 temperaments
Cata provides a fairly low complexity approximation to the subgroup, using a slightly flat ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens.
Pythagorean tuning also works surprisingly well, where the diminished fourth (-8 fifths) 8192/6561 and the triple augmented fourth (+20 fifths) 3486784401/2147483648 sound extremely close to 5/4 and 13/8 respectively, wherein tempering the schisma and tridecapyth comma provide a fairly more complex but 3/2-telic microtemperament, of which 53edo offers an almost perfect approximation. Pure fifths and octaves on the other hand, offer 5 and 13 with -1.954c and +1.428c of error.
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.
For those searching very high accuracy temperaments, the 2.3.5.13 extension of Egads (19&422) provides a highly complex, but 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 a practically perfect approximation.
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.