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.

It is the amount by which eight major thirds ~~fall~~ short of a perfect fifth~~, octave-reduced~~: ((5/4)~~<sup>~~8~~</sup> ×~~ 393216/390625) ~~/ 4~~ = 3/~~2. It is also equal to~~ the ~~difference between the lesser diesis and the magic comma, (~~[~~[128/125~~]])/~~([[3125/3072]])~~.

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>.

Therefore, it is also the amount by which seven major thirds fall short of 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>.

Therefore, it is also the amount by which seven major thirds fall short of 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>

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.

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 The '''Würschmidt comma''' ({{monzo| 17 1 -8 }} = '''393216/390625''') is a [[5-limit]] [[comma]] of 11.4 cents.
-It is the amount by which eight major thirds fall short of a perfect fifth, octave-reduced: ((5/4)<sup>8</sup> × 393216/390625) / 4 = 3/2. It is also equal to the difference between the lesser diesis and the magic comma, ([[128/125]])/([[3125/3072]]).
+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>.
-Therefore, it is also the amount by which seven major thirds fall short of 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>.
+Therefore, it is also the amount by which seven major thirds fall short of 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>
 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.