Mercator's comma

Interval information

Factorization	2-84 × 353
Monzo	[-84 53⟩
Size in cents	3.6150459¢
Names	Mercator's comma, 53-comma
Color name	Wa-53 comma
FJS name	[math]\text{7d}{-6}[/math]
Special properties	reduced, reduced harmonic
Tenney height (log2 nd)	168.003
Weil height (log2 max(n, d))	168.006
Wilson height (sopfr(nd))	327
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~1.3203 bits
Comma size	small
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[-84 53⟩, known as Mercator's comma or the 53-comma, is a small comma of 3.615 cents which 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 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.

Temperament

Tempering out this comma leads to the Mercator family of temperaments. 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 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
• Syntonic comma
• Unnoticeable comma