Mercator's comma: Difference between revisions
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→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.) |
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{{monzo|-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 fifth]]s 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 [[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 with the monzo of {{monzo|-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 fifth]]s 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 [[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]]. | ||
== Temperament == | == 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]]. | Tempering out this comma leads to the [[Mercator family]] of temperaments. 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 also == | == See also == | ||
* [[Holdrian 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]] | ||
[[Category:Commas named after mathematicians]] | [[Category:Commas named after mathematicians]] | ||
[[Category:Commas named after music theorists]] | [[Category:Commas named after music theorists]] | ||
Latest revision as of 12:28, 12 June 2025
| Interval information |
53-comma
reduced harmonic
The comma with the monzo of [-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