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, 5 | {{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. | ||
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[[Cata]] provides a fairly low complexity approximation to the subgroup, using ~6/5 as a generator, finding ~5/4 at +6 gens, ~3/2 at +7 gens, and ~13/8 at +14 gens. | [[Cata]] provides a fairly low complexity approximation to the subgroup, using ~6/5 as a generator, finding ~5/4 at +6 gens, ~3/2 at +7 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 | [[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. | ||
Revision as of 11:54, 7 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.
Regular temperaments
Rank-1 temperaments (edos)
It is relatively well approximated by the following edos: 7, 15, 19, 24, 27, 31, 34, 50, 53, 80, 87, 94, 96, 130, 140, 217, 270...
Rank-2 temperaments
Cata provides a fairly low complexity approximation to the subgroup, using ~6/5 as a generator, finding ~5/4 at +6 gens, ~3/2 at +7 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.
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.