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]\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{nd}[/math])	~4.4397 bits
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, 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