46edo: Difference between revisions
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The '''46 equal temperament''', often abbreviated '''46-tET''', '''46-EDO''', or '''46-ET''', is the scale derived by dividing the [[octave]] into 46 equally-sized steps. Each step represents a frequency ratio of 26.087 [[cent | The '''46 equal temperament''', often abbreviated '''46-tET''', '''46-EDO''', or '''46-ET''', is the scale derived by dividing the [[octave]] into 46 equally-sized steps. Each step represents a frequency ratio of 26.087 [[cent]]s, an interval close in size to [[66/65]], the interval from [[13/11]] to [[6/5]]. | ||
== Theory == | == Theory == | ||
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46et tempers out 507/500, 91/90, 686/675, 2048/2025, 121/120, 245/243, 126/125, 169/168, 176/175, 896/891, 196/195, 1029/1024, 5120/5103, 385/384, and 441/440 among other intervals, with various consequences. [[Rank two | 46et tempers out 507/500, 91/90, 686/675, 2048/2025, 121/120, 245/243, 126/125, 169/168, 176/175, 896/891, 196/195, 1029/1024, 5120/5103, 385/384, and 441/440 among other intervals, with various consequences. [[Rank two temperament]]s it supports include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]] and [[unidec]]. The [[11-limit]] [[minimax tuning]] for valentine temperament, (11/7)<sup>1/10</sup>, is only 0.01 cents flat of 3\46 octaves. In the opinion of some, 46et is the first equal division to deal adequately with the [[13-limit]], though others award that distinction to [[41edo]]. In fact, while 41 is a [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|zeta integral edo]] but not a [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|zeta gap edo]], 46 is zeta gap but not zeta integral. | ||
The fifth of 46 equal is 2.39 cents sharp, which some people (e.g. [[Margo Schulter]]) prefer, sometimes strongly, over both the [[3/2|just fifth]] and fifths of temperaments with flat fifths, such as meantone. It gives a characteristic bright sound to triads, distinct from the mellowness of a meantone triad. | The fifth of 46 equal is 2.39 cents sharp, which some people (e.g. [[Margo Schulter]]) prefer, sometimes strongly, over both the [[3/2|just fifth]] and fifths of temperaments with flat fifths, such as meantone. It gives a characteristic bright sound to triads, distinct from the mellowness of a meantone triad. | ||
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46edo can be treated as two [[23edo]]'s separated by an interval of 26.087 cents. | 46edo can be treated as two [[23edo]]'s separated by an interval of 26.087 cents. | ||
[[ | [[Magic22 as srutis #Shrutar.5B22.5D_as_srutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music. | ||
== Intervals == | == Intervals == | ||