Würschmidt comma

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Interval information
Ratio 393216/390625
Factorization 217 × 3 × 5-8
Monzo [17 1 -8
Size in cents 11.44529¢
Name würschmidt comma
Color name sg83, Saquadbigu comma
FJS name [math]\text{dddd3}_{5,5,5,5,5,5,5,5}[/math]
Special properties reduced
Tenney height (log2 nd) 37.1604
Weil height (log2 max(n, d)) 37.1699
Wilson height (sopfr (nd)) 77
Harmonic entropy
(Shannon, [math]\sqrt{nd}[/math])
~2.32402 bits
Comma size small
open this interval in xen-calc

Würschmidt's comma ([17 1 -8 = 393216/390625) is a small 5-limit comma of 11.4 cents. It is the difference between an octave-reduced stack of eight classical major thirds and a perfect fifth: (5/4)8/6, which comes from 5/4 being a convergent in the continued fraction of [math]\sqrt[8]{6}[/math].

In terms of commas, it is the difference between:


Tempering out this comma leads to the würschmidt family of temperaments. In any nontrivial tuning (that is, not 3edo), there is an exact neutral third between 5/4 and 6/5, which represents a tempering of 625/512~768/625 and can be used to represent 11/9~27/22 (or more accurately 49/40~60/49, tempering out 2401/2400 instead of or in addition to 243/242).

Magic is a simpler analogue of würschmidt, reaching 3/1 with (5/4)5 which exceeds 3/1 by the magic comma, and a even simpler analogue of würschmidt is dicot, where 3/2 is reached by (5/4)2. More interesting is that there is a lower-accuracy but more complex analogue of würschmidt if we look at the pattern; the powers of 5/4 go 2 (dicot), 5 (magic), 8 (würschmidt), corresponding to increasingly sharp tunings of 5 where each additional three 5's represent a lowering of 25/16 by another 128/125; finally, at (5/4)11 / (12/1), we get magus, a sharp-major-third analogue of würschmidt.