Würschmidt comma: Difference between revisions
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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 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 also the difference between | 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]] | ||