Bird's eye view of temperaments by accuracy: Difference between revisions
m →Garibaldi: address a bunch of things discussed on my user page where it was brought up. i'll do a diff again to see if i missed anything, also cuz theres some repeat info i want to address |
m →Garibaldi: it does seem very subjective as far as i can tell as to which is a better 7-limit system |
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[[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a (barely) inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 41 = [[135edo]] and 94 + 53 = [[147edo]] also support it but with yet more inconsistencies due to the finer gamut, so it's worth checking the "Prime harmonics" tables to see if you're okay with the errors. | [[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a (barely) inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 41 = [[135edo]] and 94 + 53 = [[147edo]] also support it but with yet more inconsistencies due to the finer gamut, so it's worth checking the "Prime harmonics" tables to see if you're okay with the errors. | ||
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]]). 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. | * 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.) | ||
Both support [[cassandra]], a 13-limit extension which finds [[~]][[16/13]] as a comma below [[~]][[5/4]] and equates (3/2)/(16/13) = [[39/32]] with [[11/9]]. (This means that in 41edo, we have a single neutral third at the cost of damage to prime 13, while in 53edo we have two neutral thirds at the cost of damage to prime 11, hence 41 + 53 = [[94edo]] is a lot more characteristic of cassandra's tuning.) | Both support [[cassandra]], a 13-limit extension which finds [[~]][[16/13]] as a comma below [[~]][[5/4]] and equates (3/2)/(16/13) = [[39/32]] with [[11/9]]. (This means that in 41edo, we have a single neutral third at the cost of damage to prime 13, while in 53edo we have two neutral thirds at the cost of damage to prime 11, hence 41 + 53 = [[94edo]] is a lot more characteristic of cassandra's tuning.) | ||