2.3.5.13 subgroup: Difference between revisions

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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.

[[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.

[[~~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~~.

[[Schismatic family#Maqamschismic (2.3.5.13)|Schismic]], via tempering 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.

~~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~~.

[[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

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~~.

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]] and its well known double [[2684edo]] offer 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]] / [[2684edo]].

[[Category:Subgroup]]

@@ Line 8: / Line 8: @@
 === 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.
+[[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.
-[[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.
+[[Schismatic family#Maqamschismic (2.3.5.13)|Schismic]], via tempering 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.
-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.
+[[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
-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.
+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]] and its well known double [[2684edo]] offer 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]] / [[2684edo]].
 [[Category:Subgroup]]