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 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. ~~It is also significant because~~ [[19/16]] is very close to ~~a pythagorean~~ minor third, ~~thus~~ it can bring "pyth" ~~flavors while keeping otonality~~.

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]].

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. ~~This something prime 19 doesn't do on account of being close to a minor third octave reduced, 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 ==

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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 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. It is also significant because [[19/16]] is very close to a pythagorean minor third, thus it can bring "pyth" flavors while keeping otonality.
+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]].
-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. This something prime 19 doesn't do on account of being close to a minor third octave reduced, 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 ==