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

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]\displaystyle{ \sqrt[8]{6} }[/math].

It is also the difference between a stack of two 16/15s and a stack of three 25/24s, and therefore belongs to the family of commas that denote a specific ratio between those two intervals. Among these, the würschmidt comma makes a rather accurate and rather intuitive equivalence, which can be seen by writing 25/24 as 50/48 and 16/15 as 48/45 = (48/46)×(46/45) where 50/48 and 48/46 differ by S24 = 576/575, and (46/45)² and 48/46 differ by S46²×S47 = 12167/12150. Thus it can also be seen that this comma's temperament extends to the 2.3.5.23 subgroup.

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, so that this comma is found in the syntonic–31 equivalence continuum.
• two dieses and a classic chromatic semitone: (128/125)²/(25/24); tempering out both leads to the trivial tuning 3edo, so that this comma is found in found in the augmented–dicot equivalence continuum.
• a diesis and a magic comma: (128/125)/(3125/3072); tempering out both also leads 3edo, because the magic comma is itself equal to (25/24)/(128/125), so that it's equivalent to the previous expression.
• two diaschismas and a tetracot comma: (2048/2025)²/(20000/19683); tempering out both leads to 34edo, so that this comma is found in the diaschismic–tetracot equivalence continuum.
• a diaschisma and a kleisma: (2048/2025)/(15625/15552); tempering out both also leads to 34edo, because the kleisma is itself equal to (2048/2025)/(20000/19683), so that it's equivalent to the previous expression.

Temperaments

Tempering out this comma leads to the würschmidt temperament and its extensions in the würschmidt family. 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.