Würschmidt comma: Difference between revisions

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The '''Würschmidt comma''' ({{monzo| 17 1 -8 }} = '''393216/390625''') is a [[small comma|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s.

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.)~~

It is the difference between:

* an [[octave reduction|octave-reduced]] stack of eight [[5/4|classical major thirds]] and a [[3/2|perfect fifth]]: (5/4)8/6, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>

~~It is also 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 [[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]]

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

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

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

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

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

** 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]].

* 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]]).

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 The '''Würschmidt comma''' ({{monzo| 17 1 -8 }} = '''393216/390625''') is a [[small comma|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s.
-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.)
+It is the difference between:
+* an [[octave reduction|octave-reduced]] stack of eight [[5/4|classical major thirds]] and a [[3/2|perfect fifth]]: (5/4)<sup>8</sup>/6, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>
-It is also 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 [[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]]
+* two classic diatonic semitones and three classic chromatic semitones: ([[16/15]])<sup>2</sup>/([[25/24]])<sup>3</sup>
-* two classic diatonic semitones and three classic chromatic semitones, ([[16/15]])<sup>2</sup>/([[25/24]])<sup>3</sup>
+* two [[diaschisma]]s and the [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]]
-* two [[diaschisma]]s and the [[tetracot comma]], ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]]
+** equivalently, one diaschisma and the [[kleisma]]: ([[2048/2025]])/([[15625/15552]]); tempering out both thus also corresponds to [[34edo]]
-** 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]].
-* 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]]).