2.3.5.7.11.13.19.29 subgroup: Difference between revisions

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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 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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{{Note|[[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "53j" means taking the second closest approximation of harmonic 29.}}

=== Rank-2 temperaments ===

[[Hemififths]], through the 41 & 58h extension, provides a fairly simple but efficient way to approach the subgroup through a dicot structure, finding ~19/16 and ~29/16 to be a pythagorean comma above ~7/6 and ~16/9 respectively, 29/16 here is approximated to within a cent of accuracy.

[[Newt]] is arguably the most efficient rank-2 microtemperament of the subgroup despite its apparent great complexity, and much like hemififths, has a dicot structure. It has a strong extension for ~29/16, though its mapping is proportionally less accurate than the rest, being a cent off instead of tenths of a cent off.

[[Cassandra]] provides a chain-of-fifths framework for approaching the subgroup, and has a very strong extension for 19/16 as a minor third, but has two plausible extensions for ~29/16:

* 41 & 53 is simpler but less accurate, which equates it with ~9/5.

* 41 & 94 is more accurate but more complex, which equates it with ~20/11.

=== Rank-3 temperaments ===

[[Freyr]] detempers the hemififths extension and massively improves upon all primes in the subgroup, finding ~19/16 an aberschisma sharper except for ~29/16 which has the same mapping as hemififths but tuned less accurately because the fifth is closer to pure.

[[Cassaschismic]] detempers cassandra by readily including 19/16 as a minor third plus an aberschisma with incredible accuracy. Coincidentally like cassandra, it has two plausible extensions for ~29/16:

* 41 & 94 & 270 is simpler, with the same exact mapping as 41 & 94 but less accurately tuned because the fifth is sharper.

* 41 & 53 & 270 is lower-badness, finding ~29/16 half a cent off as 2 aberschismas sharper than ~9/5 half a cent off pure.

[[Category:Just intonation subgroups|#]]

[[Category:29-limit|#]]

~~{{Todo|expand}}~~

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 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 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 10: / Line 10: @@
 {{Note|[[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "53j" means taking the second closest approximation of harmonic 29.}}
+=== Rank-2 temperaments ===
+[[Hemififths]], through the 41 & 58h extension, provides a fairly simple but efficient way to approach the subgroup through a dicot structure, finding ~19/16 and ~29/16 to be a pythagorean comma above ~7/6 and ~16/9 respectively, 29/16 here is approximated to within a cent of accuracy.
+[[Newt]] is arguably the most efficient rank-2 microtemperament of the subgroup despite its apparent great complexity, and much like hemififths, has a dicot structure. It has a strong extension for ~29/16, though its mapping is proportionally less accurate than the rest, being a cent off instead of tenths of a cent off.
+[[Cassandra]] provides a chain-of-fifths framework for approaching the subgroup, and has a very strong extension for 19/16 as a minor third, but has two plausible extensions for ~29/16:
+* 41 & 53 is simpler but less accurate, which equates it with ~9/5.
+* 41 & 94 is more accurate but more complex, which equates it with ~20/11.
+=== Rank-3 temperaments ===
+[[Freyr]] detempers the hemififths extension and massively improves upon all primes in the subgroup, finding ~19/16 an aberschisma sharper except for ~29/16 which has the same mapping as hemififths but tuned less accurately because the fifth is closer to pure.
+[[Cassaschismic]] detempers cassandra by readily including 19/16 as a minor third plus an aberschisma with incredible accuracy. Coincidentally like cassandra, it has two plausible extensions for ~29/16:
+* 41 & 94 & 270 is simpler, with the same exact mapping as 41 & 94 but less accurately tuned because the fifth is sharper.
+* 41 & 53 & 270 is lower-badness, finding ~29/16 half a cent off as 2 aberschismas sharper than ~9/5 half a cent off pure.
 [[Category:Just intonation subgroups|#]]
 [[Category:29-limit|#]]
-{{Todo|expand}}