Bird's eye view of temperaments by accuracy: Difference between revisions
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== 7-limit focus == | == 7-limit focus == | ||
=== [[Garibaldi]] === | |||
Note count: 18 for {3, 5, 7, 9, 15, 21, 27, 35, 45} ([[12L 5s]] or [[12L 17s]]) | |||
Bound-violating intervals: [[7/5]], [[21/20]], [[15/14]] (all derived from contrasting odd 7 (sharp) and 5 (flat)) | |||
Garibaldi is a very natural and very efficient (for its accuracy) way of extending [[#Schismic]] to prime 7, at the cost of some accuracy 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 distance being a conveniently general "comma"-sized interval that simultaneously represents not only [[64/63|S8]] and [[81/80|S9]] but also the [[Pythagorean comma]] (as per schismic). [[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has an inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 53 = [[147edo]] also supports it but with yet more inconsistencies, showing a slight preference to 53edo, though 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]] 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]]), and so that [[~]][[16/13]] is a comma below [[~]][[5/4]]. This corresponds to a number of temperaments; the most relevant of which for [[#Schismic]] is the very accurate extension to prime 13 called [[tridecapyth]]. | |||
== 11-limit focus == | == 11-limit focus == | ||
=== [[Miracle]] === | === [[Miracle]] === | ||