2.3.5.7.11.13.19 subgroup: Difference between revisions

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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 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~~, the subgroup can be considered to complete the harmonic series up to the lower half of the fifth octave without the difficult ones near the edges. The same reasons ~~will~~ also give rise to the [[2.3.5.7.11.13.19.29 subgroup]] as an [[expansion]].

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. It is also significant because [[19/16]] is very close to a pythagorean minor third, thus it can bring "pyth" flavors while keeping otonality.

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. This something prime 19 doesn't do on account of being close to a minor third octave reduced, 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 ==

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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.

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. ~~Another similar temperament~~ is ~~[[gariwizmic|gariwizmic (94 & 270)]], which instead~~ of ~~halving~~ the ~~fifth~~, ~~halves the octave and finds the~~ [[~~aberschisma~~]] ~~at +53~~ fifths ~~-1/2 pythcomma~~.

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)]].

=== 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 ~~the diatonic~~ chain of fifths. Other temperaments that achieve a similar level of accuracy include [[lif]] and [[eir]].

[[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:19-limit|#]]

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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 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, the subgroup can be considered to complete the harmonic series up to the lower half of the fifth octave without the difficult ones near the edges. The same reasons will also give rise to the [[2.3.5.7.11.13.19.29 subgroup]] as an [[expansion]].
+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. It is also significant because [[19/16]] is very close to a pythagorean minor third, thus it can bring "pyth" flavors while keeping otonality.
+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. This something prime 19 doesn't do on account of being close to a minor third octave reduced, 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 ==
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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.
-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. Another similar temperament is [[gariwizmic|gariwizmic (94 & 270)]], which instead of halving the fifth, halves the octave and finds the [[aberschisma]] at +53 fifths -1/2 pythcomma.
+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)]].
 === 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 the diatonic chain of fifths. Other temperaments that achieve a similar level of accuracy include [[lif]] and [[eir]].
+[[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:19-limit|#]]