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2.3.5.7.11.13.19 subgroup: Difference between revisions

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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]]s such that the highest [[prime factor]] in all [[ratio]]s 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.
The '''2.3.5.7.11.13.19 subgroup''' (a.k.a. ''yazalathana'' in [[color notation]]) consists of [[just intonation]] [[interval]]s such that the highest [[prime factor]] in all [[ratio]]s 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 [[19/1|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.
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]].
This subgroup is significant because 19 mirrors [[21/1|21]] in the 16::24 [[harmonic series segment]], and 21 is already present in the 13-limit; thus any fifth-bounded chord involving 21 over the [[root]] has a harmonic (i.e. frequency-scale) inverse involving 19 in this subgroup. Because [[19/16]] is very close to the [[32/27|Pythagorean minor third]], it can bring a "pyth" flavor to the [[otonal]] chord. Meanwhile, primes [[17/1|17]] and [[23/1|23]] may be considered to clash with the fundamental, being close to a semitone and a tritone when [[octave reduction|octave reduced]], so people may wish to exclude them. Therefore the subgroup can be considered to complete the harmonic series up to the lower half of the fifth octave without the more difficult ones near the edges. The same reasons also give rise to the [[2.3.5.7.11.13.19.29 subgroup]] as an [[expansion]].  


== 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|Cassandra (41 & 53)]] provides a very intuitive approximation to this subgroup using the [[chain of fifths]], naturally mapping 19/16 to the minor third, and equating together the [[Pythagorean comma|pythagorean]], [[Septimal comma|septimal]], and [[syntonic comma]] into one generic comma, that doubled approximates 33/32~1053/1024. It 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|cotoneum (41 & 217)]] keeps the chain of fifths and a pyth-septimal comma, but does not temper out the [[schisma]], instead equating it with the [[41-comma]]. Another similar temperament is [[gariwizmic|gariwizmic (94 & 270)]], which instead halves the octave and finds the [[aberschisma]] at +53 fifths -1/2 pythcomma.  


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]].
[[newt|Newt (41 & 270)]] halves the fifth (tempering out [[2401/2400]]) and finds the [[aberschisma]] -41 hemififths away with much more efficiency. It is notable as one of the most efficient 13-limit and 2.3.5.7.11.13.19 temperaments, with errors down to tenths of a cent while being based on a [[Ploidacot/Dicot|dicot]] chain-of-fifths.


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]].
Other non-chain-of-fifths temperaments that are good candidates for the subgroup include [[vulture|vulture (53 & 217)]], [[satin|satin (94 & 217)]], and [[paramity|paramity (53 & 311)]].  


[[Category:Just intonation subgroups]]
=== Rank-3 temperaments ===
[[Category:19-limit]]
[[Cassaschismic]] is the union of all the rank-2 temperaments discussed above, relates several [[formal comma]]s in this subgroup to reduce them to essentially a generic comma and a generic aberschisma, making it significant for notation systems based on a [[Ploidacot/Monocot|monocot]] chain of fifths. Other temperaments that achieve a similar level of accuracy and efficiency include [[lif]] and [[eir]].
 
All of these temperaments are very close to newt, and in fact, newt is the intersection of these three; Cassaschismic observes 2401/2400, lif observes 3025/3024, eir observes 4096/4095.
 
[[Category:Just intonation subgroups|#]]
[[Category:19-limit|#]]
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