Magic comma

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Interval information
Ratio 3125/3072
Factorization 2-10 × 3-1 × 55
Monzo [-10 -1 5
Size in cents 29.613568¢
Names magic comma,
small diesis
Color name Ly5-2, Laquinyo comma
FJS name [math]\text{dd}{-2}^{5,5,5,5,5}[/math]
Special properties reduced
Tenney height (log2 nd) 23.1946
Weil height (log2 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