2.3.5.7.11.13.19 subgroup: Difference between revisions

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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]].

== Regular temperaments ==

=== Rank-1 temperaments (edos) ===

[[Edo]]s 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]].

[[Edo]]s which represents the subgroup better ([[monotonic]] in the no-17 [[19-odd-limit]] 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. Bold edos are records of [[Tenney–Euclidean temperament measures #TE simple badness|TE 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.

[[270edo]] is arguably one of the best equal temperaments for this subgroup, achieving a record of [[relative error]] that no other equal temperament of its grain comes close to achieving. The next best ones are in the thousands of divisions: [[2190edo|2190]], [[6079edo|6079]], [[8269edo|8269]] and [[8539edo|8539]]. The last two coincidentally 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]]~~.

[[Cassandra]] provides a very intuitive approximation to this subgroup using the [[chain of fifths]], naturally mapping 19/16 to the minor third, and is well represented with [[41edo]] and [[53edo]], though [[94edo]] is more optimized.

~~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, [[cotoneum]] keeps the chain of fifths, but does not temper out the [[schisma]]. However, [[newt]], which splits the perfect fifth in halves (tempering out [[2401/2400]]) and finding the [[aberschisma]] -41 hemififths away, is much more efficient. Another similar temperament is [[gariwizmic]], which instead of splitting the fifth, splits the octave in half. Newt and gariwizmic meet in [[270edo]].

~~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 meet in 270edo, and are thus great candidates for the subgroup, include [[vulture]], [[satin]], and [[paramity]].

~~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~~]].

=== Rank-3 temperaments ===

[[Cassaschismic]] relates several [[formal comma]]s in this subgroup to reduce them to essentially a generic comma a generic aberschisma, making it significant for notation systems based on the diatonic chain of fifths. Other temperaments that achieve a similar level of accuracy include [[lif]] and [[eir]].

[[Category:Just intonation subgroups]]

[[Category:19-limit]]

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 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]].
 == Regular temperaments ==
 === Rank-1 temperaments (edos) ===
-[[Edo]]s 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]].
+[[Edo]]s which represents the subgroup better ([[monotonic]] in the no-17 [[19-odd-limit]] 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. Bold edos are records of [[Tenney–Euclidean temperament measures #TE simple badness|TE 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 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.
+[[270edo]] is arguably one of the best equal temperaments for this subgroup, achieving a record of [[relative error]] that no other equal temperament of its grain comes close to achieving. The next best ones are in the thousands of divisions: [[2190edo|2190]], [[6079edo|6079]], [[8269edo|8269]] and [[8539edo|8539]]. The last two coincidentally 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]].
+[[Cassandra]] provides a very intuitive approximation to this subgroup using the [[chain of fifths]], naturally mapping 19/16 to the minor third, and is well represented with [[41edo]] and [[53edo]], though [[94edo]] is more optimized.
-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, [[cotoneum]] keeps the chain of fifths, but does not temper out the [[schisma]]. However, [[newt]], which splits the perfect fifth in halves (tempering out [[2401/2400]]) and finding the [[aberschisma]] -41 hemififths away, is much more efficient. Another similar temperament is [[gariwizmic]], which instead of splitting the fifth, splits the octave in half. Newt and gariwizmic meet in [[270edo]].
-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 meet in 270edo, and are thus great candidates for the subgroup, include [[vulture]], [[satin]], and [[paramity]].
-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]].
+=== Rank-3 temperaments ===
+[[Cassaschismic]] relates several [[formal comma]]s in this subgroup to reduce them to essentially a generic comma a generic aberschisma, making it significant for notation systems based on the diatonic chain of fifths. Other temperaments that achieve a similar level of accuracy include [[lif]] and [[eir]].
 [[Category:Just intonation subgroups]]
 [[Category:19-limit]]