Mercator's comma

Interval information

Factorization	2-84 × 353
Monzo	[-84 53⟩
Size in cents	3.615046 ¢
Names	Mercator's comma, 53-comma
Color name	53wM, fithewama
FJS name	[math]\displaystyle{ \text{7d}{-6} }[/math]
Special properties	reduced, reduced harmonic
Tenney norm (log2 nd)	168.003
Weil norm (log2 max(n, d))	168.006
Wilson norm (sopfr(nd))	327
Comma size	small
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Mercator's comma, mercator comma, or systematically the 53-comma, is a small 3-limit comma of about 3.62 cents. It is the amount by which 53 perfect fifths exceed 31 octaves, in other words (3/2)⁵³/2³¹. It is also the amount by which a stack of four Pythagorean commas exceeds a Pythagorean limma, the amount by which a stack of eight Pythagorean apotomes exceeds a 27/16 major sixth, and the amount by which a stack of two Pythagorean countercommas fall short of the mystery comma.

The comma is named for Nicholas Mercator, who first took note of it as a part of his study of 53edo.

Temperaments

Tempering out this comma leads to the mercator temperament. For edos N up to 8745, the comma is tempered out if and only if 53 divides N. Examples of such edos include 53edo, 159edo, 212edo, 265edo, 742edo, 954edo and 1749edo.

See Mercator family for the family of temperaments where it is tempered out.

See also

• Mercator's old comma (It is what Wikipedia calls "Mercator's comma", but it is not what most modern musicians or theorists mean by "Mercator's comma".)
• Holdrian comma
• Unnoticeable comma (its size often considered to be a threshold for it.)