Bird's eye view of temperaments by accuracy: Difference between revisions
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=== [[Garibaldi]] === | === [[Garibaldi]] === | ||
Note count: 18 for {3, 5, 7, 9, 15, 21, 27, 35, 45} ([[12L 5s]] or [[12L 17s]]) | 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)) | |||
[[#Generator tunings|Generator tunings]]: 24\31, 31\53, 55\94 | [[#Generator tunings|Generator tunings]]: 24\31, 31\53, 55\94 | ||
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: | 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: | ||