Würschmidt comma
(Redirected from 393216/390625)
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Ratio | 393216/390625 |
Factorization | 2^{17} × 3 × 5^{-8} |
Monzo | [17 1 -8⟩ |
Size in cents | 11.44529¢ |
Name | Würschmidt comma |
Color name | sg^{8}3, Saquadbigu comma |
FJS name | [math]\text{dddd3}_{5,5,5,5,5,5,5,5}[/math] |
Special properties | reduced |
Tenney height (log_{2} nd) | 37.1604 |
Weil height (log_{2} 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 5-limit comma of 11.4 cents.
It is the amount by which eight major thirds fall short of a perfect fifth, octave-reduced: ((5/4)^{8} × 393216/390625) / 4 = 3/2.
Therefore, it is also the amount by which seven major thirds fall short of 24/5 (i.e. 6/5 plus two octaves). In other words, ((5/4)^{7} × 393216/390625) / 4 = 6/5.
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.