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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|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.
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.


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 ==
=== Rank-1 temperaments (edos) ===
[[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]].


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.
{{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 approximation ==
[[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.
[[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 ==
=== Rank-2 temperaments ===
[[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.


* [[Gallery of just intervals]]
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]]. [[newt|Newt (41 & 270)]] halves the fifth (tempering out [[2401/2400]]) and finds the [[aberschisma]] -41 hemififths away with much more efficiency. Another similar temperament is [[gariwizmic|gariwizmic (94 & 270)]], which instead of halving the fifth, halves the octave and finds the [[aberschisma]] at +53 fifths -1/2 pythcomma.
* [[13-limit]]
 
* [[19-limit]]<!--main article-->
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)]].
 
=== Rank-3 temperaments ===
[[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 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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