Bird's eye view of temperaments by accuracy: Difference between revisions
Partial reversal on garibaldi: let's not beat around the bush it IS the simplest good 7-limit schismic extension; gracecordial is more complex and schism is not good; reduced bloat and acknowledged interseptimals of 53edo, relationship between primes 5,13 and 7,11 I consider to be very important in cassandra |
m →Sensipent: i dont intend on documenting 17-limit sensor as being too inaccurate for its complexity |
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* the most accurate is [[#Sendai]] which finds primes 23 and 29 | * the most accurate is [[#Sendai]] which finds primes 23 and 29 | ||
* the second most accurate is [[#Sensible]], which finds primes 11, 17 and 23 | * the second most accurate is [[#Sensible]], which finds primes 11, 17 and 23 | ||
* the simplest but least accurate is [[#Sensor]] (commonly just called "sensi"), which interprets it as a full 17-limit temperament. | * the simplest but least accurate is [[Sensipent family#Sensor|Sensor]] (commonly just called "sensi"), which interprets it as a full 17-limit temperament, for which the best tuning is [[46edo]]. | ||
==== [[Würschmidt]] ==== | ==== [[Würschmidt]] ==== | ||
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[[#Generator tunings|Generator tunings]]: 24\31, 31\53, 55\94 | [[#Generator tunings|Generator tunings]]: 24\31, 31\53, 55\94 | ||
Garibaldi is arguably the | Garibaldi is arguably the best 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 S8/S9 = [[5120/5103]] so that 8/7 and 10/9 are equidistant from 9/8, corresponding to equating S8 = [[64/63]] and S9 = [[81/80]] respectively. This results in a conveniently general tempered comma-sized interval that also represents the [[Pythagorean comma]], which is 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. | ||
[[41edo]] and [[ | 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]]), 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.) | |||
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.) | |||
=== 11-limit focus === | === 11-limit focus === | ||