46edo: Difference between revisions

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== Theory ==

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 [[support~~|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|41et]]. 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.

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-2 temperament]]s it [[support]]s include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]] and [[unidec]]. The [[11-odd-limit]] [[minimax tuning]] for valentine, (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|41et]]. In fact, while 41 is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak and 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 peak or 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.

~~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.

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=== Prime harmonics ===

=== Divisors ===

46edo can be treated as two [[23edo]]'s separated by an interval of 26.087 cents.

== Intervals ==

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 == Theory ==
-et 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 [[support|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|41et]]. 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.
+et 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-2 temperament]]s it [[support]]s include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]] and [[unidec]]. The [[11-odd-limit]] [[minimax tuning]] for valentine, (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|41et]]. In fact, while 41 is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak and 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 peak or 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.
-edo 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.
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 === Prime harmonics ===
 {{Harmonics in equal|46}}
+=== Divisors ===
+edo can be treated as two [[23edo]]'s separated by an interval of 26.087 cents.
 == Intervals ==