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 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 [[ | 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 == | == Regular temperaments == | ||
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[[Category:Just intonation subgroups|#]] | [[Category:Just intonation subgroups|#]] | ||
[[Category:29-limit|#]] | [[Category:29-limit|#]] | ||
=== 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 of [[Garischismic clan#2.3.5.7.11.13.19 subgroup (neonewt)|neonewt]] (itself an extension of newt) 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 three plausible extensions for ~29/16: | |||
* 41 & 53 is the simplest, lowest-badness but least accurate, which simply equates it with ~9/5. | |||
* 41 & 94 is more accurate and complex, which equates it with ~20/11. | |||
* 53 & 94 is is the most accurate but most complex, which equates it with a stack of ~169/128 and ~11/8. | |||
=== Rank-3 temperaments === | |||
[[Breed family#Freyr|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 three plausible extensions for ~29/16: | |||
* 41 & 94 & 270 is the simplest, with the same exact mapping as 41 & 94 but less accurately tuned because the fifth is sharper. | |||
* 41 & 53 & 270 is the lowest-badness, finding ~29/16 half a cent off as 2 aberschismas sharper than ~9/5 half a cent off pure. | |||
* 53 & 94 & 217 is the worst extension with the most complex mapping of ~29/16 with close to no increase in accuracy from the others, but its mapping can be detempered further down to thousands of a cent of error precision with [[insatinismic]]. | |||
{{Todo|expand}} | {{Todo|expand}} | ||
Latest revision as of 18:55, 6 August 2026
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 intervals such that the highest prime factor in all 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 and 29, or the 2.3.5.7.11.13.19 subgroup with 29 added in.
This subgroup is a rank-8 system, and can be modeled in a 7-dimensional lattice, with the primes 3, 5, 7, 11, 13, 19 and 29 represented by each dimension. The prime 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 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 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, 23, 25, and 31 may be considered to clash with the fundamental, being close to tonic and fifth when 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
Rank-1 temperaments (edos)
Edos 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): 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 of neonewt (itself an extension of newt) 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 three plausible extensions for ~29/16:
- 41 & 53 is the simplest, lowest-badness but least accurate, which simply equates it with ~9/5.
- 41 & 94 is more accurate and complex, which equates it with ~20/11.
- 53 & 94 is is the most accurate but most complex, which equates it with a stack of ~169/128 and ~11/8.
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 three plausible extensions for ~29/16:
- 41 & 94 & 270 is the simplest, with the same exact mapping as 41 & 94 but less accurately tuned because the fifth is sharper.
- 41 & 53 & 270 is the lowest-badness, finding ~29/16 half a cent off as 2 aberschismas sharper than ~9/5 half a cent off pure.
- 53 & 94 & 217 is the worst extension with the most complex mapping of ~29/16 with close to no increase in accuracy from the others, but its mapping can be detempered further down to thousands of a cent of error precision with insatinismic.