Würschmidt comma: Difference between revisions

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It is the amount by which an [[octave reduction|octave-reduced]] stack of eight [[5/4|classical major thirds]] falls short of a [[3/2|perfect fifth]]: (5/4)8(393216/390625)/4 = 3/2, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>. (Therefore, it is also equal to the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves), that is, (5/4)7(393216/390625)/4 = 6/5.)

~~In terms of commas, it~~ is:

It is also the difference between"

* the difference between the [[syntonic comma]] and the [[semicomma]], ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]].

* the [[syntonic comma]] and the [[semicomma]], ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]]

* ~~the difference between~~ the [[diesis]] and the [[magic comma]], ([[128/125]])/([[3125/3072]]); tempering out both leads to the trivial tuning [[3edo]].

* the [[diesis]] and the [[magic comma]], ([[128/125]])/([[3125/3072]]); tempering out both leads to the trivial tuning [[3edo]]

* ~~the difference between~~ two [[diaschisma]]s and the [[tetracot comma]], ([[2048/2025]])2/([[20000/19683]]); tempering out both leads to [[34edo]].

* two classic diatonic semitones and three classic chromatic semitones, ([[16/15]])2/([[25/24]])3

* equivalently, ~~between~~ one diaschisma and the [[kleisma]], ([[2048/2025]])/([[15625/15552]]); tempering out both thus also corresponds to [[34edo]].

* two [[diaschisma]]s and the [[tetracot comma]], ([[2048/2025]])2/([[20000/19683]]); tempering out both leads to [[34edo]]

* finally, between two dieses and the just chromatic semitone, ([[128/125]])2/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]].

** equivalently, one diaschisma and the [[kleisma]], ([[2048/2025]])/([[15625/15552]]); tempering out both thus also corresponds to [[34edo]]

* finally, between two dieses and the just chromatic semitone, ([[128/125]])2/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]]

The last expression means tempering it out in any nontrivial tuning (that is, not 3edo), there is an exact neutral third between 5/4 and 6/5, which usually represents ~[[11/9]] (or more accurately [[49/40]], tempering out [[2401/2400]] instead of or in addition to [[243/242]]).

Notice that [[magic]] is a simpler analogue of würschmidt, reaching [[3/1]] with ([[5/4]])5 which exceeds 3/1 by the magic comma, and a even simpler analogue of würschmidt is [[dicot]], where [[3/2]] is reached by ([[5/4]])2. More interesting is that there is a lower-accuracy but more complex analogue of würschmidt if we look at the pattern; the powers of [[5/4]] go 2 (dicot), 5 (magic), 8 (würschmidt), corresponding to increasingly sharp tunings of 5 where each additional three 5's represent a lowering of [[25/16]] by another [[128/125]]; finally, at ([[5/4]])11 / ([[12/1]]), we get [[magus]], a sharp-major-third analogue of würschmidt, which is in some sense the logical dual of magic, which tunes 5/4 flat. There is little reason to use magus unless you want a sharp [[5/4]] and/or want to use a temperament that happens to support it, a notable tuning of which is [[46edo]].

Notice that [[magic]] is a simpler analogue of würschmidt, reaching [[3/1]] with ([[5/4]])5 which exceeds 3/1 by the magic comma, and a even simpler analogue of würschmidt is [[dicot]], where [[3/2]] is reached by ([[5/4]])2. More interesting is that there is a lower-accuracy but more complex analogue of würschmidt if we look at the pattern; the powers of [[5/4]] go 2 (dicot), 5 (magic), 8 (würschmidt), corresponding to increasingly sharp tunings of 5 where each additional three 5's represent a lowering of [[25/16]] by another [[128/125]]; finally, at ([[5/4]])11 / ([[12/1]]), we get [[magus]], a sharp-major-third analogue of Würschmidt.

== Temperaments ==

@@ Line 10: / Line 10: @@
 It is the amount by which an [[octave reduction|octave-reduced]] stack of eight [[5/4|classical major thirds]] falls short of a [[3/2|perfect fifth]]: (5/4)<sup>8</sup>(393216/390625)/4 = 3/2, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>. (Therefore, it is also equal to the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves), that is, (5/4)<sup>7</sup>(393216/390625)/4 = 6/5.)
-In terms of commas, it is:
+It is also the difference between"
-* the difference between the [[syntonic comma]] and the [[semicomma]], ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]].
+* the [[syntonic comma]] and the [[semicomma]], ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]]
-* the difference between the [[diesis]] and the [[magic comma]], ([[128/125]])/([[3125/3072]]); tempering out both leads to the trivial tuning [[3edo]].
+* the [[diesis]] and the [[magic comma]], ([[128/125]])/([[3125/3072]]); tempering out both leads to the trivial tuning [[3edo]]
-* the difference between two [[diaschisma]]s and the [[tetracot comma]], ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]].
+* two classic diatonic semitones and three classic chromatic semitones, ([[16/15]])<sup>2</sup>/([[25/24]])<sup>3</sup>
-* equivalently, between one diaschisma and the [[kleisma]], ([[2048/2025]])/([[15625/15552]]); tempering out both thus also corresponds to [[34edo]].
+* two [[diaschisma]]s and the [[tetracot comma]], ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]]
-* finally, between two dieses and the just chromatic semitone, ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]].
+** equivalently, one diaschisma and the [[kleisma]], ([[2048/2025]])/([[15625/15552]]); tempering out both thus also corresponds to [[34edo]]
+* finally, between two dieses and the just chromatic semitone, ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]]
 The last expression means tempering it out in any nontrivial tuning (that is, not 3edo), there is an exact neutral third between 5/4 and 6/5, which usually represents ~[[11/9]] (or more accurately [[49/40]], tempering out [[2401/2400]] instead of or in addition to [[243/242]]).
-Notice that [[magic]] is a simpler analogue of würschmidt, reaching [[3/1]] with ([[5/4]])<sup>5</sup> which exceeds 3/1 by the magic comma, and a even simpler analogue of würschmidt is [[dicot]], where [[3/2]] is reached by ([[5/4]])<sup>2</sup>. More interesting is that there is a lower-accuracy but more complex analogue of würschmidt if we look at the pattern; the powers of [[5/4]] go 2 (dicot), 5 (magic), 8 (würschmidt), corresponding to increasingly sharp tunings of 5 where each additional three 5's represent a lowering of [[25/16]] by another [[128/125]]; finally, at ([[5/4]])<sup>11</sup> / ([[12/1]]), we get [[magus]], a sharp-major-third analogue of würschmidt, which is in some sense the logical dual of magic, which tunes 5/4 flat. There is little reason to use magus unless you want a sharp [[5/4]] and/or want to use a temperament that happens to support it, a notable tuning of which is [[46edo]].
+Notice that [[magic]] is a simpler analogue of würschmidt, reaching [[3/1]] with ([[5/4]])<sup>5</sup> which exceeds 3/1 by the magic comma, and a even simpler analogue of würschmidt is [[dicot]], where [[3/2]] is reached by ([[5/4]])<sup>2</sup>. More interesting is that there is a lower-accuracy but more complex analogue of würschmidt if we look at the pattern; the powers of [[5/4]] go 2 (dicot), 5 (magic), 8 (würschmidt), corresponding to increasingly sharp tunings of 5 where each additional three 5's represent a lowering of [[25/16]] by another [[128/125]]; finally, at ([[5/4]])<sup>11</sup> / ([[12/1]]), we get [[magus]], a sharp-major-third analogue of Würschmidt.
 == Temperaments ==