64edo: Difference between revisions
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64edo is a [[zeta valley edo]] and is very bad at approximating [[JI]] for its size. It has two options of [[3/2|fifth]] almost equally far from [[just]]. The sharp fifth from the 64b [[val]] is inherited from [[32edo]] and produces a hard [[superpythagorean]] scale, while the slightly more accurate flat fifth from the [[patent val]] is within the [[meantone]]/[[flattone]] range. However bizarrely, the flat fifth does not [[support]] meantone or flattone in its [[patent val]], and instead supports the obscure {{nowrap|7c & 12c}} (or {{nowrap|19 & 64}}) temperament (named [[User:Jerdle/The_Ocean|meso]]{{idio}} by [[User:Jerdle|Jerdle]]) which reaches [[5/4]] as a double-diminished fourth. In order to interpret it as flattone, the 64cd val must be used. | 64edo is a [[zeta valley edo]] and is very bad at approximating [[JI]] for its size. It has two options of [[3/2|fifth]] almost equally far from [[just]]. The sharp fifth from the 64b [[val]] is inherited from [[32edo]] and produces a hard [[superpythagorean]] scale, while the slightly more accurate flat fifth from the [[patent val]] is within the [[meantone]]/[[flattone]] range. However bizarrely, the flat fifth does not [[support]] meantone or flattone in its [[patent val]], and instead supports the obscure {{nowrap|7c & 12c}} (or {{nowrap|19 & 64}}) temperament (named [[User:Jerdle/The_Ocean|meso]]{{idio}} by [[User:Jerdle|Jerdle]]) which reaches [[5/4]] as a double-diminished fourth. In order to interpret it as flattone, the 64cd val must be used. | ||
64edo does not just have dual fifths, it is a "dual-everything" system in which every prime harmonic up to 17, except for 13 | The best way to access 5-limit intervals in 64edo's patent val is probably [[diminished]] temperament, which is familiar from [[12edo]]. In this case the Diminished[12] ([[4L 8s]]) MOS actually becomes dual-fifth, because the zero-indexed 7th step can be either the flat or sharp fifth in different modes. | ||
64edo does not just have dual fifths, it is a "dual-everything" system in which every prime harmonic up to 17, except for 2 (obviously) and 13, has above 25 [[relative cent]]s of error and arguably has two possible approximations. The 13th harmonic is alone in having a fairly unambiguous approximation of 237 steps (octave-reduces to [[13/8]] being at 45 steps, or 843.7 cents). Also whether or 64edo is dual-7 or not is questionable, as the 975-cent [[7/4]] is significantly more accurate than the 956.3-cent 7/4. But still, 64edo is clearly a dual-3, dual-5, dual-11, dual-17 system. | |||
Still, the patent val [[tempering out|tempers out]] [[648/625]] in the 5-limit and [[225/224]] in the 7-limit, plus [[66/65]], [[121/120]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit. It provides the optimal patent val in the 7-, 11- and 13-limits for the 16&64 temperament. | Still, the patent val [[tempering out|tempers out]] [[648/625]] in the 5-limit and [[225/224]] in the 7-limit, plus [[66/65]], [[121/120]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit. It provides the optimal patent val in the 7-, 11- and 13-limits for the 16&64 temperament. | ||
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The patent val of 64edo is the first patent val to represent the intervals 5/4, 81/64, 14/11, 9/7, 13/10, 21/16, and 4/3 distinctly, although in 64edo they are not in the correct order. [[72edo]] represents all of these intervals distinctly and in the correct order. | The patent val of 64edo is the first patent val to represent the intervals 5/4, 81/64, 14/11, 9/7, 13/10, 21/16, and 4/3 distinctly, although in 64edo they are not in the correct order. [[72edo]] represents all of these intervals distinctly and in the correct order. | ||
=== Odd harmonics === | === Odd harmonics === | ||