Bird's eye view of temperaments by accuracy: Difference between revisions
m →Garibaldi: merge into one paragraph as the ratio appears on its own at some scales which feels weird/unstylistic |
m →Garibaldi: just noticed tetracot is conceived fairly naturally by this explanation |
||
| Line 227: | Line 227: | ||
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: | 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 prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as half of the fifth and as a comma above [[~]][[6/5]] (as in [[cassandra]]) or a comma below [[~]][[5/4]] (as in [[andromeda]]). However, there is significant damage to 15/13 and 13/10. | * For prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as half of the fifth and as a comma above [[~]][[6/5]] (as in [[cassandra]]) or a comma below [[~]][[5/4]] (as in [[andromeda]]), corresponding to [[tetracot]] in 2.3.5.11 (by tempering out (S9/S11 = [[243/242]],) S10 = [[100/99]] and S10/(S9/S11) = [[2200/2187]] respectively) which splits the halved fifth into two small major seconds of [[~]][[11/10]][[~]][[10/9]] around 175.6 cents. However, there is significant damage to 15/13 and 13/10. | ||
* For primes 5 and 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 (3/2)/(15/13) = [[~]][[13/10]] being made the midpoint of [[~]][[21/16]] and [[~]][[9/7]] respectively. It also makes [[~]][[16/13]] a comma below [[~]][[5/4]] (by tempering out ((5/4)/(16/13))/(81/80) = 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]], corresponding to reaching 13/4 through (9/8)<sup>10</sup> (tempering out the [[tridecapyth comma]]) and also corresponding to tempering out [[325/324]] = S25*S26 = S10/S12 as mentioned. However, there is significant damage to 14/11. (Also, 53edo's fifth is flatter so better tuned for schismic/for the 5-limit, as implicitly aforementioned.) | * For primes 5 and 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 (3/2)/(15/13) = [[~]][[13/10]] being made the midpoint of [[~]][[21/16]] and [[~]][[9/7]] respectively. It also makes [[~]][[16/13]] a comma below [[~]][[5/4]] (by tempering out ((5/4)/(16/13))/(81/80) = 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]], corresponding to reaching 13/4 through (9/8)<sup>10</sup> (tempering out the [[tridecapyth comma]]) and also corresponding to tempering out [[325/324]] = S25*S26 = S10/S12 as mentioned. However, there is significant damage to 14/11. (Also, 53edo's fifth is flatter so better tuned for schismic/for the 5-limit, as implicitly aforementioned.) | ||