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. Two generators reach ~[[13/9]], [[tempering out]] the marveltwin comma [[325/324]]. Then ~[[26/15]] is found at three generators, with two such intervals reaching ~3/1, tempering out [[676/675]]. The interval at +4 generators is a third of a [[9/8]] whole tone, representing all of [[25/24]], [[26/25]], and [[27/26]]. Good tunings of cata include [[34edo]] and especially [[53edo]], with other tunings such as [[87edo]] and [[140edo]] being usable as well.

The 2.3.5.13 version of [[kleismic]] (sometimes called ''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. Two generators reach ~[[13/9]], [[tempering out]] the marveltwin comma [[325/324]]. Then ~[[26/15]] is found at three generators, with two such intervals reaching ~3/1, tempering out [[676/675]]. The interval at +4 generators is a third of a [[9/8]] whole tone, representing all of [[25/24]], [[26/25]], and [[27/26]]. Good tunings of cata include [[34edo]] and especially [[53edo]], with other tunings such as [[87edo]] and [[140edo]] being usable as well.

The [[schismic]] extension that adds prime 13 via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] 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 mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. 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.

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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. Two generators reach ~[[13/9]], [[tempering out]] the marveltwin comma [[325/324]]. Then ~[[26/15]] is found at three generators, with two such intervals reaching ~3/1, tempering out [[676/675]]. The interval at +4 generators is a third of a [[9/8]] whole tone, representing all of [[25/24]], [[26/25]], and [[27/26]]. Good tunings of cata include [[34edo]] and especially [[53edo]], with other tunings such as [[87edo]] and [[140edo]] being usable as well.
+The 2.3.5.13 version of [[kleismic]] (sometimes called ''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. Two generators reach ~[[13/9]], [[tempering out]] the marveltwin comma [[325/324]]. Then ~[[26/15]] is found at three generators, with two such intervals reaching ~3/1, tempering out [[676/675]]. The interval at +4 generators is a third of a [[9/8]] whole tone, representing all of [[25/24]], [[26/25]], and [[27/26]]. Good tunings of cata include [[34edo]] and especially [[53edo]], with other tunings such as [[87edo]] and [[140edo]] being usable as well.
 The [[schismic]] extension that adds prime 13 via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] 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 mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. 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.