Eufalesio: Shameless near-copypaste from other articles, things missing TBA

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Shameless near-copypaste from other articles, things missing TBA

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	The subgroup can be conveniently rank-reduced into the 5-limit without much loss in accuracy by tempering out [[2080/2079]] and [[4096/4095]] and [[1216/1215]], resulting in the [[cassaschismic]] temperament, which equates 36/35 with 1053/1024 and (64/63)<sup>2</sup> with 33/32, and 64/63 with the [[Pythagorean comma]]. Other notable rank-reductions include [[Garischismic clan#2.3.5.7.11.13.19 subgroup (neonewt)\|neonewt]] and [[garibaldi]]/[[cassandra]]; newt splits the fifth in half (tempering out [[2401/2400]]) and finding the aberschisma at -41 hemififths; and garibaldi combines the pythagorean comma, 64/63 and 81/80 into one general comma, that when doubled acts as ~33/32 and ~1053/1024; this tempers out [[225/224]] and [[352/351]].		The subgroup can be conveniently rank-reduced into the 5-limit without much loss in accuracy by tempering out [[2080/2079]] and [[4096/4095]] and [[1216/1215]], resulting in the [[cassaschismic]] temperament, which equates 36/35 with 1053/1024 and (64/63)<sup>2</sup> with 33/32, and 64/63 with the [[Pythagorean comma]]. Other notable rank-reductions include [[Garischismic clan#2.3.5.7.11.13.19 subgroup (neonewt)\|neonewt]] and [[garibaldi]]/[[cassandra]]; newt splits the fifth in half (tempering out [[2401/2400]]) and finding the aberschisma at -41 hemififths; and garibaldi combines the pythagorean comma, 64/63 and 81/80 into one general comma, that when doubled acts as ~33/32 and ~1053/1024; this tempers out [[225/224]] and [[352/351]].

	270edo is arguably the equal best temperament for this subgroup, achieving a record of absolute, relative error, and logflat badness that no other equal temperament of its grain comes close to achieving. The next best ones are in the thousands of divisions; [[8539edo]] and [[8269edo]], which concidentally differ by 270 and are prime edos.		== Regular temperaments ==

	== ~~Edo approximation~~ ==		=== Rank-1 temperaments (edos) ===
	[[Edo\|Edos]] which represents the subgroup better ([[monotonic]], and decreasing [[TE error]]): {{EDOs\|'''27e''', 31, 34dh, 38df, 41f*, '''41''', 50, '''53''', 58h, '''72''', 87, 94, 103h, 111, 121, '''130''', '''152f''', 190, 217, 224, '''270''', 552, 581…}} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]]. Bold ~~temperaments~~ are records of [[relative error]].{{Note\|[[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.}}		[[Edo\|Edos]] which represents the subgroup better ([[monotonic]], and decreasing [[TE error]]): {{EDOs\|'''27e''', 31, 34dh, 38df, 41f, '''41''', 50, '''53''', 58h, '''72''', 87, 94, 103h, 111, 121, '''130''', '''152f''', 190, 217, 224, '''270''', 552, 581…}} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]]. Bold edos are records of [[relative error]].{{Note\|[[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.}}270edo is arguably the equal best temperament for this subgroup, achieving a record of absolute, relative error, and [[logflat badness]] that no other equal temperament of its grain comes close to achieving. The next best ones are in the thousands of divisions; [[8539edo]] and [[8269edo]], which concidentally differ by 270 and are prime edos.

			=== Rank-2 temperaments ===
			[[Cassandra]] provides a very intuitive extension using the [[chain of fifths]], naturally extending 19/16 to the minor third. Well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups, and is specially prominent in 94edo, extending to the full [[23-limit]].

			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, 7/4 at +25 gens, and 13/8 at −13 gens, which is optimal in [[130edo]], albeit [[19/16]] is worsely tuned because the fifth is flatter.

			For those searching higher accuracy temperaments, [[gariwizmic]] also keeps the chain of fifths, spliting the octave in half, but does not temper out the schisma. It finds 5/4 at 39 fifths minus one [[semioctave]], 7/4 at −14 fifths, 11/8 at +23 fifths and 13/8 at −27 fifths plus a semioctave. This is a much worse mapping, but it ends at [[270edo]].

			Other non-chain-of-fifths temperaments that converge in 270edo, and are thus great candidates for the subgroup are [[vulture]], [[cotoneum]], [[newt]], and [[ennealimmal]]. Cotoneum, well represented by [[217edo]], has 31edo's 2.5.7 and vastly improves upon 3 and 13; 13 itself being a semiconvergent, albeit prime 11 is not that good, though prime 19 is decent. Ennealimmal is extremely accurate and well represented, as it can be naturally extended to the subgroup by adding the minisma, equating the [[36/35]] generator to the [[1053/1024]].

	== See also ==		== See also ==

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The '''2.3.5.7.11.13.19 subgroup''' (a.k.a. ''yazalathana'' in [[color notation]]) consists of [[just intonation]] [[Interval|intervals]] such that the highest [[prime factor]] in all [[Ratio|ratios]] is 19, but without 17. It is thus a subset of the [[19-limit]], or alternatively, it can be seen as the 13-limit with an extra prime 19.

This subgroup is a [[Rank and codimension|rank-7]] system, and can be modeled in a 6-dimensional [[lattice]], with the primes [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]], [[13/1|13]] and [[19/1|19]] represented by each dimension. The prime [[2/1|2]] does not appear in typical lattices because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a seventh dimension is needed.

The subgroup can be conveniently rank-reduced into the 5-limit without much loss in accuracy by tempering out [[2080/2079]] and [[4096/4095]] and [[1216/1215]], resulting in the [[cassaschismic]] temperament, which equates 36/35 with 1053/1024 and (64/63)<sup>2</sup> with 33/32, and 64/63 with the [[Pythagorean comma]]. Other notable rank-reductions include [[Garischismic clan#2.3.5.7.11.13.19 subgroup (neonewt)|neonewt]] and [[garibaldi]]/[[cassandra]]; newt splits the fifth in half (tempering out [[2401/2400]]) and finding the aberschisma at -41 hemififths; and garibaldi combines the pythagorean comma, 64/63 and 81/80 into one general comma, that when doubled acts as ~33/32 and ~1053/1024; this tempers out [[225/224]] and [[352/351]].

270edo is arguably the equal best temperament for this subgroup, achieving a record of absolute, relative error, and logflat badness that no other equal temperament of its grain comes close to achieving. The next best ones are in the thousands of divisions; [[8539edo]] and [[8269edo]], which concidentally differ by 270 and are prime edos.

== Edo approximation ==
[[Edo|Edos]] which represents the subgroup better ([[monotonic]], and decreasing [[TE error]]): {{EDOs|'''27e''', 31, 34dh, 38df, 41f*, '''41''', 50, '''53''', 58h, '''72''', 87, 94, 103h, 111, 121, '''130''', '''152f''', 190, 217, 224, '''270''', 552, 581…}} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]]. Bold temperaments are records of [[relative error]].{{Note|[[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.}}

== See also ==

* [[Gallery of just intervals]]
* [[13-limit]]
* [[19-limit]]

2.3.5.7.11.13.19 subgroup - Revision history

Eufalesio: Shameless near-copypaste from other articles, things missing TBA

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