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Bird's eye view of temperaments by accuracy: Difference between revisions

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m should remove this word as per changing sentence
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m →Garibaldi: the "respectively" refers to the 11-limit extension paths for cassandra and andromeda, which is why 27/22 is omitted. the info about interseptimals is important and relevant to prime 13. the damage to schismic via the slightly sharper fifth is already discussed.
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* Helenus (41ef & 53), tempering out 176/175 and 351/350, finds prime 11 at -30 fifths, and prime 13 at -33 fifths.
* Helenus (41ef & 53), tempering out 176/175 and 351/350, finds prime 11 at -30 fifths, and prime 13 at -33 fifths.


Which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about primes 7 and 11 or prime 5 and 13:
Which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about prime 11 or prime 13:
* For primes 7 and 11, [[41edo]] is better, as it finds [[11/9]][[~]][[27/22]] as [[Sqrt(3/2)|half of the fifth]] and as a comma above [[~]][[6/5]] or a comma below [[~]][[5/4]]; This also corresponds to being the intersection of [[cassandra]] and [[andromeda]] (respectively). Primes 5 and 13 are worse, as these mappings prefer an ever so slightly flat fifth.
* For prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as half of the fifth and as a comma above [[~]][[6/5]] or a comma below [[~]][[5/4]]. This corresponds to being [[cassandra]] + [[andromeda]] '''(respectively)'''.
* For primes 5 and 13, [[53edo]] is better, as it has a close to just fifth that benefits schismic, tempering out 325/324 too so that [[~]][[16/13]] is a comma below [[~]][[5/4]]. This corresponds to [[Schismatic family#Tridecaschismic (2.3.5.13)|tridecaschismic]]. Primes 7 and 11 are worse, as their cassandra mappings prefer a slightly sharper fifth.
* For prime 13, [[53edo]] is better, as it finds [[interseptimal interval]]s distinctly from adjacent [[septimal]] intervals so that [[~]][[15/13]] is half of a practically-just [[4/3]] (tempering out [[676/675|S13/S15]]) and is (resultantly) found as a comma above [[~]][[8/7]] or a comma below [[~]][[7/6]] (which reflects to [[~]][[13/10]] being made the midpoint of [[~]][[9/7]] and [[~]][[21/16]]). It also makes [[~]][[16/13]] a comma below [[~]][[5/4]] (by tempering out 325/324). This corresponds to a number of temperaments; the most relevant of which for [[#Schismic]] is the very accurate extension to prime 13 called [[Schismatic family#Tridecaschismic (2.3.5.13)|tridecaschismic]].


=== 11-limit focus ===
=== 11-limit focus ===
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