Würschmidt comma
| 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{n\cdot d}[/math]) |
~2.91565 bits |
| 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]\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.