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 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. Meanwhile, primes [[17/1|17]] and [[23/1|23]] may be considered to clash with the fundamental, being 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 difficult ones near the edges. The same reasons will also give rise to the [[2.3.5.7.11.13.19.29 subgroup]] as an [[expansion]].

== Regular temperaments ==

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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 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. Meanwhile, primes [[17/1|17]] and [[23/1|23]] may be considered to clash with the fundamental, being 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 difficult ones near the edges. The same reasons will also give rise to the [[2.3.5.7.11.13.19.29 subgroup]] as an [[expansion]].
 == Regular temperaments ==