Würschmidt comma: Difference between revisions

Line 10:

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 also equal to the difference between the lesser diesis and the magic comma, <math>\frac{128}{125}/\frac{3125}{3072}</math>.

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>.

~~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.

@@ Line 10: / Line 10: @@
 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 also equal to the difference between the lesser diesis and the magic comma, <math>\frac{128}{125}/\frac{3125}{3072}</math>.
+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>.
-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.