46edo: Difference between revisions

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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]] or [[53edo]]. 46edo's fifth is slightly sharp of just, 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. Many say that sharp fifths give a characteristic bright sound to 5-limit triads, and consider the sound of meantone triads to be more mellow in comparison. 46edo is not a [[zeta peak edo]], but it is a [[zeta gap edo]]. It 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]].

46edo is also notable for being the smallest ~~equal division~~ to approximate harmonics ~~3, 5, 7, 11, and 13~~ with less than 25% [[relative interval error]].

46edo is also notable for being the smallest edo to approximate odd harmonics 1–13 with less than 25% [[relative interval error|relative error]].

[[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|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 46edo (continued)}}

=== As a tuning of other temperaments ===

46edo is a superb tuning for [[sensi]] and [[leapday]]. It also [[support]]s but tunes less optimally [[valentine]], [[shrutar]], [[rodan]], 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, but [[77edo]] has a lower average error overall.

=== Subsets and supersets ===

@@ Line 11: / Line 11: @@
 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]] or [[53edo]]. 46edo's fifth is slightly sharp of just, 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. Many say that sharp fifths give a characteristic bright sound to 5-limit triads, and consider the sound of meantone triads to be more mellow in comparison. 46edo is not a [[zeta peak edo]], but it is a [[zeta gap edo]]. It 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]].
-edo is also notable for being the smallest equal division to approximate harmonics 3, 5, 7, 11, and 13 with less than 25% [[relative interval error]].
+edo is also notable for being the smallest edo to approximate odd harmonics 1–13 with less than 25% [[relative interval error|relative error]].
-[[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.
@@ Line 20: / Line 18: @@
 {{Harmonics in equal|46|columns=11}}
 {{Harmonics in equal|46|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 46edo (continued)}}
+=== As a tuning of other temperaments ===
+edo is a superb tuning for [[sensi]] and [[leapday]]. It also [[support]]s but tunes less optimally [[valentine]], [[shrutar]], [[rodan]], 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, but [[77edo]] has a lower average error overall.
 === Subsets and supersets ===