Bird's eye view of temperaments by accuracy: Difference between revisions
m →Garibaldi: this has erroneous information and the principles of extension are already discussed non-erroneously in the mention of cassandra and andromeda for 41edo. also, cassandra isnt necessarily the "best" as helenus is also a 23-limit temp judging by 53 & 65d |
m removal of word "general" makes this a bit confusing to read and harder to understand what was intending to be communicated |
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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 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 | 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. | ||
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: | 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: | ||