2.3.5.7.11.13.19.29 subgroup: Difference between revisions

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~~{{Stub}}~~

The '''2.3.5.7.11.13.19.29 subgroup''' (a.k.a. ''yazalathanatwena'' in [[color notation]], hereon referred to as ''the subgroup'') consists of [[just intonation]] [[interval]]s such that the highest [[prime factor]] in all [[ratio]]s is 29, but without 17 or 23. It is thus a subset of the [[29-limit]], or alternatively, it can be seen as the [[13-limit]] with extra primes [[19/1|19]] and [[29/1|29]], or the [[2.3.5.7.11.13.19 subgroup]] with 29 added in.

This subgroup is a [[rank and codimension|rank-8]] system, and can be modeled in a 7-dimensional [[lattice]], with the primes [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]], [[13/1|13]], [[19/1|19]] and [[29/1|29]] 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, an eighth dimension is needed.

This subgroup is significant because ~~29 mirrors~~ [[27/1|27]] in the 24::32 [[~~harmonic~~ series segment]], ~~and 27 is already present in the [[3-limit]]; thus~~ any fifth-bounded chord involving 27 over the [[root]] has a harmonic (i.e. frequency-scale) inverse involving 29 in ~~this subgroup~~. 29/16 is close to ~~9/5,~~ a ~~kind of superminor~~ seventh, ~~which provides~~ a special "~~utonal~~" ~~flavor~~ to [[otonal]] chords without clashing ''too'' much with the fundamental. 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 subgroup]] as a [[retraction]]~~.

This subgroup is significant because it adds 19 as a counterpart of [[21/1|21]] in the 16::24 [[harmonics series segment]], so that any fifth-bounded chord involving 21 over the [[root]] has a harmonic (i.e. frequency-scale) inverse involving 19, and it adds 29 as a counterpart of [[27/1|27]] in 24::32, so that any fourth-bounded chord involving 27 over the root of 3 has a harmonic inverse involving 29. In addition, {19, 21} mirrors {27, 29} in 16::32, implying that any fifth-bounded chord involving 19 or 21 over the root has a fourth-bounded equivalent involving 27 or 29. 19/16 and 29/16 are close to a minor third and a supraminor seventh, respectively, so they can provide a special "minor" flavors to [[otonal]] chords without clashing ''too'' much with the fundamental. Meanwhile, harmonics [[17/1|17]], [[23/1|23]], [[25/1|25]], and [[31/1|31]] may be considered to clash with the fundamental, being close to tonic and fifth 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 fifth octave without the more difficult ones near the edges.

== Regular temperaments ==

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-{{Stub}}
 The '''2.3.5.7.11.13.19.29 subgroup''' (a.k.a. ''yazalathanatwena'' in [[color notation]], hereon referred to as ''the subgroup'') consists of [[just intonation]] [[interval]]s such that the highest [[prime factor]] in all [[ratio]]s is 29, but without 17 or 23. It is thus a subset of the [[29-limit]], or alternatively, it can be seen as the [[13-limit]] with extra primes [[19/1|19]] and [[29/1|29]], or the [[2.3.5.7.11.13.19 subgroup]] with 29 added in.
 This subgroup is a [[rank and codimension|rank-8]] system, and can be modeled in a 7-dimensional [[lattice]], with the primes [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]], [[13/1|13]], [[19/1|19]] and [[29/1|29]] 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, an eighth dimension is needed.
-This subgroup is significant because 29 mirrors [[27/1|27]] in the 24::32 [[harmonic series segment]], and 27 is already present in the [[3-limit]]; thus any fifth-bounded chord involving 27 over the [[root]] has a harmonic (i.e. frequency-scale) inverse involving 29 in this subgroup. 29/16 is close to 9/5, a kind of superminor seventh, which provides a special "utonal" flavor to [[otonal]] chords without clashing ''too'' much with the fundamental. 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 subgroup]] as a [[retraction]].
+This subgroup is significant because it adds 19 as a counterpart of [[21/1|21]] in the 16::24 [[harmonics series segment]], so that any fifth-bounded chord involving 21 over the [[root]] has a harmonic (i.e. frequency-scale) inverse involving 19, and it adds 29 as a counterpart of [[27/1|27]] in 24::32, so that any fourth-bounded chord involving 27 over the root of 3 has a harmonic inverse involving 29. In addition, {19, 21} mirrors {27, 29} in 16::32, implying that any fifth-bounded chord involving 19 or 21 over the root has a fourth-bounded equivalent involving 27 or 29. 19/16 and 29/16 are close to a minor third and a supraminor seventh, respectively, so they can provide a special "minor" flavors to [[otonal]] chords without clashing ''too'' much with the fundamental. Meanwhile, harmonics [[17/1|17]], [[23/1|23]], [[25/1|25]], and [[31/1|31]] may be considered to clash with the fundamental, being close to tonic and fifth 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 fifth octave without the more difficult ones near the edges.
 == Regular temperaments ==