Mercator's comma: Difference between revisions
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'''Mercator's comma''', '''mercator comma''', or systematically the '''53-comma''', is a [[small comma|small]] [[3-limit]] [[comma]] of about 3.62 [[cent]]s. It is the amount by which 53 [[3/2|perfect fifths]] exceed 31 [[octave]]s, in other words (3/2)<sup>53</sup>/2<sup>31</sup>. It is also the amount by which a stack of four [[Pythagorean comma]]s exceeds a [[Pythagorean limma]], the amount by which a stack of eight [[Pythagorean apotome]]s exceeds a [[27/16]] major sixth, and the amount by which a stack of two [[Pythagorean countercomma]]s 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]]. | 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 [[ | [[Tempering out]] this comma leads to the [[mercator]] temperament. For edos ''N'' up to [[8745edo|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 == | == See also == | ||
* [[Holdrian comma#Mercator's comma, | * [[Holdrian comma #Mercator's comma, Mercator's old comma, and the Holdrian comma|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]] | * [[Holdrian comma]] | ||
* [[Unnoticeable comma]] ( | * [[Unnoticeable comma]] (its size often considered to be a threshold for it.) | ||
[[Category:Commas named after mathematicians]] | [[Category:Commas named after mathematicians]] | ||
[[Category:Commas named after music theorists]] | [[Category:Commas named after music theorists]] | ||
Revision as of 06:57, 10 September 2026
| Interval information |
53-comma
reduced harmonic
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.)