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.
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 ==
== Regular temperaments ==
=== Rank-1 temperaments (edos) ===
=== 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.}}
{{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 ===
=== 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:Just intonation subgroups]]
[[Category:19-limit]]
[[Category:19-limit]]

Revision as of 11:47, 14 June 2026

The 2.3.5.7.11.13.19 subgroup (a.k.a. yazalathana in color notation) consists of just intonation intervals such that the highest prime factor in all 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-7 system, and can be modeled in a 6-dimensional lattice, with the primes 3, 5, 7, 11, 13 and 19 represented by each dimension. The prime 2 does not appear in typical lattices because octave equivalence is presumed. If octave equivalence is not presumed, a seventh dimension is needed.

Regular temperaments

Rank-1 temperaments (edos)

Edos which represents the subgroup better (monotonic in the no-17 19-odd-limit and decreasing TE error): 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 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 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: 2190, 6079, 8269 and 8539. The last two coincidentally differ by 270 and are prime edos.

Rank-2 temperaments

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.

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.

Other non-chain-of-fifths temperaments that meet in 270edo, and are thus great candidates for the subgroup, include vulture, satin, and paramity.

Rank-3 temperaments

Cassaschismic relates several formal commas 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.

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