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2.3.5.7.11.13.19.29 subgroup: Difference between revisions

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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.


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|intervals]] such that the highest [[prime factor]] in all [[Ratio|ratios]] 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 a [[Rank and codimension|rank-8]] 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]], [[19/1|19]] and [[29/1|29]]<nowiki/>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 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.  


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]].
== Regular temperaments ==
=== Rank-1 temperaments (edos) ===
[[Edo]]s which represents the subgroup better ([[monotonic]] in the [[tonality diamond]] of {1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 27, 29} and decreasing [[TE error]]): {{EDOs|'''41''', 53j, '''58h''', 72, 77, 80, 94, 99ef, 113, 118f, 118, 121, 130, 140, '''152fj''', 159, 183, 190, 198, 217, 224, 243e, '''270''', 422, 472, 494h, 552, … }} and so on. Bold edos are records of [[TE relative error]].
 
{{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.


== Regular temperaments ==
=== 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.


=== Rank-1 temperaments (edos) ===
[[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:
[[Edo|Edos]] which represents the subgroup better ([[monotonic]] in the subgroup and decreasing [[TE error]]): {{EDOs|'''41''', 53*, '''58h''', 72, 77, 80, 87* (94), 99ef, '''103h'''* (113, 118f, 118), 121, 130, 140, '''152fj''', 159, 183, 190, 198, 217, 224, 243e, '''270''', 422, 472, 494h, 552}}... and so on. Edos in parenthesis are inconsistent in the subgroup.
* 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.  


<nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 27, 29} (omission of 25 is intentional)
[[Category:Just intonation subgroups|#]]
[[Category:29-limit|#]]
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