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~~|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.

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.

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~~ do particularly well in this subgroup]: [[7edo|7]], [[15edo|15]], [[19edo|19]], [[24edo|24]], [[27edo|27]], [[31edo|31]] ~~([[62edo|62]]''',[[217edo|217]]''')~~, '''[[34edo|34]]''' ~~([[68edo|68]])~~, [[46edo|46]], [[50edo|50]], '''[[53edo|53]]''' ([[~~106edo~~|~~106~~]], [[~~159edo~~|~~159~~]]), [[77edo|77]], [[80edo|80]], '''[[87edo|87]]''', [[94edo|94]], [[96edo|96]], [[130edo|130]], [[137edo|137]], [[140edo|140]], '''[[171edo|171]]''', [[190edo|190]], [[224edo|224]], [[270edo|~~'''~~270''']], [[311edo|311]]~~...~~

It is relatively well approximated by the following edos (bold ones 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]]''', [[62edo|62]], [[68edo|68]], [[77edo|77]], [[80edo|80]], '''[[87edo|87]]''', [[94edo|94]], [[96edo|96]], [[106edo|106]], [[130edo|130]], [[137edo|137]], [[140edo|140]], [[159edo|159]], '''[[171edo|171]]''', [[190edo|190]], '''[[217edo|217]]''', [[224edo|224]], '''[[270edo|270]]''', [[311edo|311]], …

=== 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-[[~~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 -1.954{{c}} and +1.428{{c}} 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. [[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 [[~~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.

For those searching very high accuracy temperaments, the 2.3.5.13 extension of [[egads]] ({{nowrap| 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 ===

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-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.
+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.
 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 do particularly well in this subgroup]: [[7edo|7]], [[15edo|15]], [[19edo|19]], [[24edo|24]], [[27edo|27]], [[31edo|31]] ([[62edo|62]]''',[[217edo|217]]'''), '''[[34edo|34]]''' ([[68edo|68]]), [[46edo|46]], [[50edo|50]], '''[[53edo|53]]''' ([[106edo|106]], [[159edo|159]]), [[77edo|77]], [[80edo|80]], '''[[87edo|87]]''', [[94edo|94]], [[96edo|96]], [[130edo|130]], [[137edo|137]], [[140edo|140]], '''[[171edo|171]]''', [[190edo|190]], [[224edo|224]], [[270edo|'''270''']], [[311edo|311]]...
+It is relatively well approximated by the following edos (bold ones 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]]''', [[62edo|62]], [[68edo|68]], [[77edo|77]], [[80edo|80]], '''[[87edo|87]]''', [[94edo|94]], [[96edo|96]], [[106edo|106]], [[130edo|130]], [[137edo|137]], [[140edo|140]], [[159edo|159]], '''[[171edo|171]]''', [[190edo|190]], '''[[217edo|217]]''', [[224edo|224]], '''[[270edo|270]]''', [[311edo|311]], …
 === 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-[[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 -1.954{{c}} and +1.428{{c}} 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. [[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 [[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.
+For those searching very high accuracy temperaments, the 2.3.5.13 extension of [[egads]] ({{nowrap| 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 ===