Magic comma
(Redirected from 3125/3072)
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Ratio | 3125/3072 |
Factorization | 2^{-10} × 3^{-1} × 5^{5} |
Monzo | [-10 -1 5⟩ |
Size in cents | 29.613568¢ |
Names | magic comma, small diesis |
Color name | Ly^{5}-2, Laquinyo comma |
FJS name | [math]\text{dd}{-2}^{5,5,5,5,5}[/math] |
Special properties | reduced |
Tenney height (log_{2} nd) | 23.1946 |
Weil height (log_{2} max(n, d)) | 23.2193 |
Wilson height (sopfr (nd)) | 48 |
Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~4.38674 bits |
Comma size | small |
open this interval in xen-calc |
The magic comma or small diesis is the interval 3125/3072. It is the amount by which five major thirds (5/4) exceeds the twelfth, or tritave; in other words (5/4)^{5}/3, the amount by which two chromatic semitones of 25/24 exceed a diatonic semitone of 16/15, and the amount by which 25/24 exceeds the lesser diesis of 128/125.
In 12edo, instead of being tempered out, it is inflated to a whole semitone—five 5/4s make a minor thirteenth (20 steps), while 3/1 is a perfect tweltfth (19 steps). Similarly, 25/24 and 16/15 are both mapped to one semitone, making a stack of two 25/24s one semitone larger than 16/15, and 25/24 is mapped to one step while 128/125 is tempered out.
Temperaments
Tempering it out leads to 5-limit magic temperament.
See also
- Magic family, where it is tempered out
- Small comma