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

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 [[#Generator tunings|Generator tunings]]: (24\41,) 31\53, 69\118, 100\171, 131\224
-Schismic is a very accurate and efficient [[5-limit]] temperament which is almost identical to [[Pythagorean tuning]] except that it tempers the perfect fifth very slightly flat so as to find [[8/5]] accurately at ([[9/8]])<sup>4</sup>, that is, as the [[Pythagorean augmented fifth]], or equivalently, finding [[5/4]] as the [[Pythagorean diminished fourth]]. Note that the smallest edo that validates its status as a microtemperament is [[118edo]], as [[53edo]], though a tone-efficient tuning, doesn't temper the fifth flat enough, as it is practically a relabeling of the [[3-limit]]. 41edo arguably qualifies as the coarsest equal temperament to support schismic well enough, but it is way too tempered for a 5-limit tuning, as it also it is [[magic]].
+Schismic is an extremely accurate and efficient [[5-limit]] temperament which is almost identical to [[Pythagorean tuning]] except that it tempers the perfect fifth very slightly flat so as to find [[8/5]] accurately at ([[9/8]])<sup>4</sup>, that is, as the [[Pythagorean augmented fifth]], or equivalently, finding [[5/4]] as the [[Pythagorean diminished fourth]]. Note that the smallest edo that validates its status as a microtemperament is [[118edo]], as [[53edo]], though a tone-efficient tuning, doesn't temper the fifth flat enough, as it is practically a relabeling of the [[3-limit]]. 41edo arguably qualifies as the coarsest equal temperament to support schismic well enough, but it is way too tempered for a 5-limit tuning, as it also it is [[magic]].
 In schismic, (9/8)<sup>6</sup> overshoots the octave by [[~]][[81/80]] so that the syntonic comma and the [[Pythagorean comma]] are equated.
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 [[#Generator tunings|Generator tunings]]: 24\31, 31\53, 55\94
-Garibaldi is a very natural and very efficient (for its accuracy) way of bestowing prime 7 upon [[#Schismic]], at the cost of accuracy as needing a slightly sharper fifth tunes the 5-limit worse so that it is no longer a microtemperament. This is done by interpreting ([[9/8]])<sup>3</sup> as [[~]][[10/7]] by tempering out [[5120/5103|S8/S9]] so that 8/7 and 10/9 are equidistant from 9/8, with the step being a conveniently general tempered comma-sized interval that simultaneously not only represents not only [[64/63]] = S8 and [[81/80]] = S9 but also the [[Pythagorean comma]] (as per schismic), equal to (9/8)<sup>6</sup> / (2/1). [[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.
+Garibaldi is arguably the simplest way to bestow prime 7 upon [[#Schismic]] effectively, at the cost of some accuracy. It uses a slightly sharper fifth that tunes the 5-limit worse, making it no longer a microtemperament. This is done by interpreting (9/8)<sup>3</sup> as [[~]][[10/7]] by tempering out [[5120/5103]] so that 8/7 and 10/9 are equidistant from 9/8. The distance is a tempered [[Pythagorean comma]] that also represents [[64/63]] and [[81/80]], which necessarily results in [[225/224]] tempered out as 5120/5103 * 32805/32768 = 225/224.
-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:
+[[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.
-* 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 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]], 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.
+The choice of 41edo and 53edo for the 7-limit is hard to determine, so a better way of choosing is choosing between primes 7,11 and primes 5,13. Both support [[cassandra]], a 13-limit extension which tempers out [[352/351]] and [[325/324]], so that [[~]][[16/13]] is a comma below [[~]][[5/4]] and [[39/32]] is equated with [[11/9]].
+* For prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as a hemififth and as a comma above [[~]][[6/5]] (as in [[cassandra]]) or a comma below [[~]][[5/4]] (as in [[andromeda]]). Primes 5 and 13 are worse. This is reflected in the [[tetracot comma]] also being tempered out, which necessarily tunes 16/13 and 5/4 flatter, and [[~]][[16/13]] also being a hemififth.
+* For primes 5 and 13, [[53edo]] is better, as it has a close to just fifth that benefits [[Schismatic family#Tridecaschismic (2.3.5.13)|tridecaschismic]], finding very well tuned ~5/4 and ~16/13. It also has half-fourths corresponding to [[15/13]], tempering out [[676/675]], telling apart [[Interseptimal interval|Interseptimal intervals]] from adjacent [[septimal]] intervals; 15/13 is here the midpoint between 8/7 and 7/6, making it part of [[The Archipelago]]. Primes 7 and 11 are worse. This is reflected in 14/11 being equated with 9/7, and 11/9 being equated with a flat 39/32, inflating the [[rastma]] to a whole step.
 === 11-limit focus ===