Würschmidt comma

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]\displaystyle{ \text{dddd3}_{5,5,5,5,5,5,5,5} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	37.1604
Weil norm (log2 max(n, d))	37.1699
Wilson norm (sopfr(nd))	77
Comma size	small
Open this interval in xen-calc

The Würschmidt comma ([17 1 -8⟩ = 393216/390625) is a small 5-limit comma of 11.4 cents.

It is the amount by which an octave-reduced stack of eight classical major thirds falls short of a perfect fifth: (5/4)⁸(393216/390625)/4 = 3/2, which comes from 5/4 being a convergent in the continued fraction of [math]\displaystyle{ \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)⁷(393216/390625)/4 = 6/5.)

In terms of commas, it is:

• the difference between the syntonic comma and the semicomma, (81/80)/(2109375/2097152); tempering out both leads to 31edo.
• the difference between the diesis and the magic comma, (128/125)/(3125/3072); tempering out both leads to the trivial tuning 3edo.
• the difference between two diaschismas and the tetracot comma, (2048/2025)²/(20000/19683); tempering out both leads to 34edo.
• equivalently, between one diaschisma and the kleisma, (2048/2025)/(15625/15552); tempering out both thus also corresponds to 34edo.
• finally, between two dieses and the just chromatic semitone, (128/125)²/(25/24); tempering out both leads to the trivial tuning 3edo.

The last expression means tempering it out in any nontrivial tuning (that is, not 3edo), there is an exact neutral third between 5/4 and 6/5, which usually represents ~11/9 (or more accurately 49/40, tempering out 2401/2400 instead of or in addition to 243/242).

Notice that 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, which is in some sense the logical dual of magic, which tunes 5/4 flat. There is little reason to use magus unless you want a sharp 5/4 and/or want to use a temperament that happens to support it, a notable tuning of which is 46edo.

Temperaments

Tempering out this comma leads to the würschmidt family of temperaments.