Würschmidt comma
| Interval information |
The Würschmidt comma ([17 1 -8⟩ = 393216/390625) is a 5-limit comma of 11.4 cents.
It is the amount by which an octave-reduced stack of eight major thirds falls short of a perfect fifth: [math]\displaystyle{ \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]\displaystyle{ \sqrt[8]{6} }[/math].
It is also equal to the difference between the lesser diesis and the magic comma, [math]\displaystyle{ \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]\displaystyle{ \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.