Würschmidt comma: Difference between revisions

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The '''Würschmidt comma''' ({{monzo| 17 1 -8 }} = '''393216/390625''') is a [[5-limit]] [[comma]] of 11.4 cents.
The '''Würschmidt comma''' ({{monzo| 17 1 -8 }} = '''393216/390625''') is a [[small comma|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s.


It is the amount by which an octave-reduced stack of eight major thirds falls short of a perfect fifth: <math>\frac{1}{4}\left(\frac{5}{4}\right)^{8}\left(\frac{393216}{390625}\right)=\frac{3}{2}</math>, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>.
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>. It is also equal to the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves). In other words, (5/4)<sup>7</sup>(393216/390625)/4 = 6/5.


It is also equal to the difference between the lesser diesis and the magic comma, <math>\frac{128}{125}/\frac{3125}{3072}</math>, and the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves). In other words, <math>\frac{1}{4}\left(\frac{5}{4}\right)^{7}\left(\frac{393216}{390625}\right)=\frac{6}{5}</math>.
In terms of commas it is the difference between the lesser diesis and the magic comma, (128/125)/(3125/3072).


Tempering it out leads to the [[würschmidt family]] of temperaments. As in [[meantone]], it implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat.  
Tempering it out leads to the [[würschmidt family]] of temperaments. As in [[meantone]], it implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat.  
== See also ==
* [[Würschmidt family]]
* [[Small comma]]


[[Category:Würschmidt|#]] <!-- list on top of cat -->
[[Category:Würschmidt|#]] <!-- list on top of cat -->

Revision as of 15:40, 24 April 2024

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]. It is also equal to the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves). In other words, (5/4)7(393216/390625)/4 = 6/5.

In terms of commas it is the difference between the lesser diesis and the magic comma, (128/125)/(3125/3072).

Tempering it out leads to the würschmidt family of temperaments. As in meantone, it implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat.

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