2.3.5.7.11.13.19.29 subgroup: Difference between revisions

Line 3:

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 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.

This subgroup is significant because it adds 19 as a counterpart of [[21/1|21]] in the 16::24 [[harmonic 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 ==

@@ Line 3: / Line 3: @@
 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 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.
+This subgroup is significant because it adds 19 as a counterpart of [[21/1|21]] in the 16::24 [[harmonic 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 ==