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]\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:

• a syntonic comma and a semicomma: (81/80)/(2109375/2097152); tempering out both leads to 31edo.
• a diesis and a magic comma: (128/125)/(3125/3072); tempering out both leads to the trivial tuning 3edo.
• two dieses and a classic chromatic semitone: (128/125)²/(25/24); tempering out both leads to 3edo.
• two classic diatonic semitones and three classic chromatic semitones: (16/15)²/(25/24)³; tempering out both leads to 3edo.
• a diaschisma and a kleisma: (2048/2025)/(15625/15552); tempering out both leads to 34edo.
• two diaschismas and a tetracot comma: (2048/2025)²/(20000/19683); tempering out both also leads to 34edo.

Temperaments

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.