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

* 176 for {3, 5, 7, 9, 11, 13, 15, 21, 25}

* 41 for {3, 7, 9, 11, 21, 27, 33, 77, 81, 99} ([[12L 29s]])

* 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]].

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)³ 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)⁶ / (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.  

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)¹⁰ (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.)