2.3.5.7.11.13.19 subgroup: Difference between revisions
Explain the significance |
A little clarification & deduplication (I don't think we need to say it's close to the minor third twice) |
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This subgroup is a [[rank and codimension|rank-7]] 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]] and [[19/1|19]] 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 a [[rank and codimension|rank-7]] 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]] and [[19/1|19]] 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 19 mirrors [[21/1|21]] in the 16::24 [[harmonic series segment]], and 21 is already present in the 13-limit; thus any fifth-bounded chord involving 21 over the [[root]] has a harmonic (i.e. frequency-scale) inverse involving 19 in this subgroup. Meanwhile, primes [[17/1|17]] and [[23/1|23]] may be considered to clash with the fundamental, being a semitone and a tritone when [[octave reduction|octave reduced]], so people may wish to exclude them. Therefore | This subgroup is significant because 19 mirrors [[21/1|21]] in the 16::24 [[harmonic series segment]], and 21 is already present in the 13-limit; thus any fifth-bounded chord involving 21 over the [[root]] has a harmonic (i.e. frequency-scale) inverse involving 19 in this subgroup. Because [[19/16]] is very close to the [[32/27|Pythagorean minor third]], it can bring a "pyth" flavor to the [[otonal]] chord. 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 [[2.3.5.7.11.13.19.29 subgroup]] as an [[expansion]]. | ||
== Regular temperaments == | == Regular temperaments == | ||
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[[Cassandra|Cassandra (41 & 53)]] provides a very intuitive approximation to this subgroup using the [[chain of fifths]], naturally mapping 19/16 to the minor third, and equating together the [[Pythagorean comma|pythagorean]], [[Septimal comma|septimal]], and [[syntonic comma]] into one generic comma, that doubled approximates 33/32~1053/1024. It is well represented with [[41edo]] and [[53edo]], though [[94edo]] is more optimized. | [[Cassandra|Cassandra (41 & 53)]] provides a very intuitive approximation to this subgroup using the [[chain of fifths]], naturally mapping 19/16 to the minor third, and equating together the [[Pythagorean comma|pythagorean]], [[Septimal comma|septimal]], and [[syntonic comma]] into one generic comma, that doubled approximates 33/32~1053/1024. It is well represented with [[41edo]] and [[53edo]], though [[94edo]] is more optimized. | ||
For those searching higher-accuracy temperaments, [[cotoneum|cotoneum (41 & 217)]] keeps the chain of fifths and a pyth-septimal comma, but does not temper out the [[schisma]], instead equating it with the [[41-comma]]. [[newt|Newt (41 & 270)]] halves the fifth (tempering out [[2401/2400]]) and finds the [[aberschisma]] -41 hemififths away with much more efficiency. | For those searching higher-accuracy temperaments, [[cotoneum|cotoneum (41 & 217)]] keeps the chain of fifths and a pyth-septimal comma, but does not temper out the [[schisma]], instead equating it with the [[41-comma]]. Another similar temperament is [[gariwizmic|gariwizmic (94 & 270)]], which instead halves the octave and finds the [[aberschisma]] at +53 fifths -1/2 pythcomma. | ||
[[newt|Newt (41 & 270)]] halves the fifth (tempering out [[2401/2400]]) and finds the [[aberschisma]] -41 hemififths away with much more efficiency. It is notable as one of the most efficient 13-limit and 2.3.5.7.11.13.19 temperaments, with errors down to tenths of a cent while being based on a [[Ploidacot/Dicot|dicot]] chain-of-fifths. | |||
Other non-chain-of-fifths temperaments that are good candidates for the subgroup include [[vulture|vulture (53 & 217)]], [[satin|satin (94 & 217)]], and [[paramity|paramity (53 & 311)]]. | Other non-chain-of-fifths temperaments that are good candidates for the subgroup include [[vulture|vulture (53 & 217)]], [[satin|satin (94 & 217)]], and [[paramity|paramity (53 & 311)]]. | ||
=== Rank-3 temperaments === | === Rank-3 temperaments === | ||
[[Cassaschismic]] is the union of all the rank-2 temperaments discussed above, relates several [[formal comma]]s in this subgroup to reduce them to essentially a generic comma and a generic aberschisma, making it significant for notation systems based on | [[Cassaschismic]] is the union of all the rank-2 temperaments discussed above, relates several [[formal comma]]s in this subgroup to reduce them to essentially a generic comma and a generic aberschisma, making it significant for notation systems based on a [[Ploidacot/Monocot|monocot]] chain of fifths. Other temperaments that achieve a similar level of accuracy and efficiency include [[lif]] and [[eir]]. | ||
All of these temperaments are very close to newt, and in fact, newt is the intersection of these three; Cassaschismic observes 2401/2400, lif observes 3025/3024, eir observes 4096/4095. | |||
[[Category:Just intonation subgroups|#]] | [[Category:Just intonation subgroups|#]] | ||
[[Category:19-limit|#]] | [[Category:19-limit|#]] | ||
Latest revision as of 18:48, 3 August 2026
The 2.3.5.7.11.13.19 subgroup (a.k.a. yazalathana in color notation) consists of just intonation intervals such that the highest prime factor in all ratios is 19, but without 17. It is thus a subset of the 19-limit, or alternatively, it can be seen as the 13-limit with an extra prime 19.
This subgroup is a rank-7 system, and can be modeled in a 6-dimensional lattice, with the primes 3, 5, 7, 11, 13 and 19 represented by each dimension. The prime 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 19 mirrors 21 in the 16::24 harmonic series segment, and 21 is already present in the 13-limit; thus any fifth-bounded chord involving 21 over the root has a harmonic (i.e. frequency-scale) inverse involving 19 in this subgroup. Because 19/16 is very close to the Pythagorean minor third, it can bring a "pyth" flavor to the otonal chord. Meanwhile, primes 17 and 23 may be considered to clash with the fundamental, being close to a semitone and a tritone when 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 2.3.5.7.11.13.19.29 subgroup as an expansion.
Regular temperaments
Rank-1 temperaments (edos)
Edos which represents the subgroup better (monotonic in the no-17 19-odd-limit and decreasing TE error): 27e, 31, 34dh, 38df, 41f, 41, 50, 53, 58h, 72, 87, 94, 103h, 111, 121, 130, 152f, 190, 217, 224, 270, 552, 581, … 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, "27e" means taking the second closest approximation of harmonic 11. |
270edo is arguably one of the best equal temperaments for this subgroup, achieving a record of relative error that no other equal temperament of its grain comes close to achieving. The next best ones are in the thousands of divisions: 2190, 6079, 8269 and 8539. The last two coincidentally differ by 270 and are prime edos.
Rank-2 temperaments
Cassandra (41 & 53) provides a very intuitive approximation to this subgroup using the chain of fifths, naturally mapping 19/16 to the minor third, and equating together the pythagorean, septimal, and syntonic comma into one generic comma, that doubled approximates 33/32~1053/1024. It is well represented with 41edo and 53edo, though 94edo is more optimized.
For those searching higher-accuracy temperaments, cotoneum (41 & 217) keeps the chain of fifths and a pyth-septimal comma, but does not temper out the schisma, instead equating it with the 41-comma. Another similar temperament is gariwizmic (94 & 270), which instead halves the octave and finds the aberschisma at +53 fifths -1/2 pythcomma.
Newt (41 & 270) halves the fifth (tempering out 2401/2400) and finds the aberschisma -41 hemififths away with much more efficiency. It is notable as one of the most efficient 13-limit and 2.3.5.7.11.13.19 temperaments, with errors down to tenths of a cent while being based on a dicot chain-of-fifths.
Other non-chain-of-fifths temperaments that are good candidates for the subgroup include vulture (53 & 217), satin (94 & 217), and paramity (53 & 311).
Rank-3 temperaments
Cassaschismic is the union of all the rank-2 temperaments discussed above, relates several formal commas in this subgroup to reduce them to essentially a generic comma and a generic aberschisma, making it significant for notation systems based on a monocot chain of fifths. Other temperaments that achieve a similar level of accuracy and efficiency include lif and eir.
All of these temperaments are very close to newt, and in fact, newt is the intersection of these three; Cassaschismic observes 2401/2400, lif observes 3025/3024, eir observes 4096/4095.