Mercator's comma

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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 n⋅d) 168.003
Harmonic entropy
(Shannon, [math]\sqrt{n\cdot d}[/math])
~2.45434 bits
Comma size small
open this interval in xen-calc

[-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 fifths exceed 31 octaves, in other words (3/2)53/231.

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 equal divisions 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