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-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.

In the opinion of some, 46edo 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 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. Like 41, 46 is distinctly [[consistent]] in the [[9-odd-limit]], and it is consistent to the [[13-odd-limit]] or the no-15 no-19 [[23-odd-limit]]. The fifth of 46edo 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~~.

The equal temperament tempers out [[2048/2025]] in the 5-limit; [[126/125]], [[245/243]], [[686/675]], [[1029/1024]], [[5120/5103]] in the 7-limit; [[121/120]], [[176/175]], [[385/384]], [[441/440]], [[896/891]] in the 11-limit; [[91/90]], [[169/168]], [[196/195]], [[507/500]] in the 13-limit. [[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.

[[Magic22 as srutis #Shrutar.5B22.5D_as_srutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music.

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=== ~~Divisors~~ ===

=== Subsets and supersets ===

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

46edo can be treated as two circles of [[23edo]] 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-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.
+In the opinion of some, 46edo 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 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. Like 41, 46 is distinctly [[consistent]] in the [[9-odd-limit]], and it is consistent to the [[13-odd-limit]] or the no-15 no-19 [[23-odd-limit]]. The fifth of 46edo 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.
+The equal temperament tempers out [[2048/2025]] in the 5-limit; [[126/125]], [[245/243]], [[686/675]], [[1029/1024]], [[5120/5103]] in the 7-limit; [[121/120]], [[176/175]], [[385/384]], [[441/440]], [[896/891]] in the 11-limit; [[91/90]], [[169/168]], [[196/195]], [[507/500]] in the 13-limit. [[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.
 [[Magic22 as srutis #Shrutar.5B22.5D_as_srutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music.
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 {{Harmonics in equal|46}}
-=== Divisors ===
+=== Subsets and supersets ===
-edo can be treated as two [[23edo]]'s separated by an interval of 26.087 cents.
+edo can be treated as two circles of [[23edo]] separated by an interval of 26.087 cents.
 == Intervals ==