2.3.5.13 subgroup: Difference between revisions

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[[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 [[34edo|34-]] and [[53edo]], with [[87edo]] being an almost perfect approximation.

The [[schismic]] extension via [[tempering out]] ~~the [[schisma]] and~~ the [[marveltwin comma]] provides a more complex temperament, well represented with [[41edo~~|41-~~]] 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 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.

The [[schismic]] extension that adds prime 13 via [[tempering out]] the [[marveltwin comma]] 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 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.

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]] offers practically perfect approximations.

For those searching [[Very high accuracy temperaments|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]] offers practically perfect approximations.

=== Rank-3 temperaments ===

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 [[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 [[34edo|34-]] and [[53edo]], with [[87edo]] being an almost perfect approximation.
-The [[schismic]] extension via [[tempering out]] the [[schisma]] and the [[marveltwin comma]] provides a more complex temperament, well represented with [[41edo|41-]] 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 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.
+The [[schismic]] extension that adds prime 13 via [[tempering out]] the [[marveltwin comma]] 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 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.
 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]] offers practically perfect approximations.
+For those searching [[Very high accuracy temperaments|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]] offers practically perfect approximations.
 === Rank-3 temperaments ===