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]], 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.}}

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

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|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 ===

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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 ==~~

[[Category:Just intonation subgroups]]

[[Category:19-limit]]

* [[~~Gallery of just intervals~~]]

* [[13-limit]]

* [[19-limit]]~~~~

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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]], 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.}}
-=== Rank-1 temperaments (edos) ===
+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.
-[[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 ===
@@ Line 19: / Line 22: @@
 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 ==
+[[Category:Just intonation subgroups]]
+[[Category:19-limit]]
-* [[Gallery of just intervals]]
-* [[13-limit]]
-* [[19-limit]]<!--main article-->