Würschmidt comma: Difference between revisions

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

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-reduced stack of eight major thirds falls short of a perfect fifth: ~~<math>\frac{1}{4}\left~~(~~\frac{~~5}{4~~}\right~~)^{8~~}\left~~(~~\frac{~~393216}{390625~~}\right~~)=~~\frac{~~3}{2~~}</math>~~, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>.

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>. It is also equal to the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves). In other words, (5/4)<sup>7</sup>(393216/390625)/4 = 6/5.

It is ~~also equal to~~ the difference between the lesser diesis and the magic comma, ~~<math>\frac{~~128}{125}/~~\frac{~~3125}{3072~~}</math>, and the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves~~)~~. In other words, <math>\frac{1}{4}\left(\frac{5}{4}\right)^{7}\left(\frac{393216}{390625}\right)=\frac{6}{5}</math>~~.

In terms of commas it is the difference between the lesser diesis and the magic comma, (128/125)/(3125/3072).

Tempering it out leads to the [[würschmidt family]] of temperaments. As in [[meantone]], it implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat.

~~== See also ==~~

* [[Würschmidt family]]

* [[Small comma]]

[[Category:Würschmidt|#]]

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-The '''Würschmidt comma''' ({{monzo| 17 1 -8 }} = '''393216/390625''') is a [[5-limit]] [[comma]] of 11.4 cents.
+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-reduced stack of eight major thirds falls short of a perfect fifth: <math>\frac{1}{4}\left(\frac{5}{4}\right)^{8}\left(\frac{393216}{390625}\right)=\frac{3}{2}</math>, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>.
+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>. It is also equal to the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves). In other words, (5/4)<sup>7</sup>(393216/390625)/4 = 6/5.
-It is also equal to the difference between the lesser diesis and the magic comma, <math>\frac{128}{125}/\frac{3125}{3072}</math>, and the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves). In other words, <math>\frac{1}{4}\left(\frac{5}{4}\right)^{7}\left(\frac{393216}{390625}\right)=\frac{6}{5}</math>.
+In terms of commas it is the difference between the lesser diesis and the magic comma, (128/125)/(3125/3072).
 Tempering it out leads to the [[würschmidt family]] of temperaments. As in [[meantone]], it implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat.
-== See also ==
-* [[Würschmidt family]]
-* [[Small comma]]
 [[Category:Würschmidt|#]] <!-- list on top of cat -->