Magic comma

Interval information

Ratio	3125/3072
Factorization	2-10 × 3-1 × 55
Monzo	[-10 -1 5⟩
Size in cents	29.61357¢
Names	magic comma, small diesis
Color name	Ly5-2, Laquinyo comma
FJS name	[math]\displaystyle{ \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
Comma size	small
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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, and is also the amount by which two chromatic semitones of 25/24 exceed a diatonic semitone of 16/15.

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).

Temperaments

Tempering it out leads to 5-limit magic temperament.

See also

• Magic family, where it is tempered out
• Small comma