Mercator's comma
| 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 |
| 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 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.