2.3.5.7.11.13.19 subgroup: Difference between revisions

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=== Rank-2 temperaments ===

[[Cassandra]] provides a very intuitive approximation to this subgroup using the [[chain of fifths]], naturally mapping 19/16 to the minor third, and 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]] keeps the chain of fifths, but does not temper out the [[schisma]]. ~~However,~~ [[newt]]~~, which splits~~ the ~~perfect~~ fifth ~~in halves~~ (tempering out [[2401/2400]]) and ~~finding~~ the [[aberschisma]] -41 hemififths away~~, is~~ much more ~~efficient~~. Another similar temperament is [[gariwizmic]], which instead of ~~splitting~~ the fifth, ~~splits~~ the octave ~~in half. Newt~~ and ~~gariwizmic meet in~~ [[~~270edo~~]].

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.

Other non-chain-of-fifths temperaments that ~~meet in 270edo, and~~ are ~~thus great~~ candidates for the subgroup, include [[vulture]], [[satin]], and [[paramity]].

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]] relates several [[formal comma]]s in this subgroup to reduce them to essentially a generic comma 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 the diatonic chain of fifths. Other temperaments that achieve a similar level of accuracy include [[lif]] and [[eir]].

[[Category:Just intonation subgroups]]

[[Category:19-limit]]

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 === Rank-2 temperaments ===
-[[Cassandra]] provides a very intuitive approximation to this subgroup using the [[chain of fifths]], naturally mapping 19/16 to the minor third, and 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]] keeps the chain of fifths, but does not temper out the [[schisma]]. However, [[newt]], which splits the perfect fifth in halves (tempering out [[2401/2400]]) and finding the [[aberschisma]] -41 hemififths away, is much more efficient. Another similar temperament is [[gariwizmic]], which instead of splitting the fifth, splits the octave in half. Newt and gariwizmic meet in [[270edo]].
+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.
-Other non-chain-of-fifths temperaments that meet in 270edo, and are thus great candidates for the subgroup, include [[vulture]], [[satin]], and [[paramity]].
+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]] relates several [[formal comma]]s in this subgroup to reduce them to essentially a generic comma 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 the diatonic chain of fifths. Other temperaments that achieve a similar level of accuracy include [[lif]] and [[eir]].
 [[Category:Just intonation subgroups]]
 [[Category:19-limit]]