Mercator's comma: Difference between revisions

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~~The comma with the monzo of {{monzo|-84 53}}, known as~~ '''Mercator's comma''', '''mercator 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)53/231. 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]].

'''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)53/231. 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]].

== ~~Temperament~~ ==

== Temperaments ==

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]].

[[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 ==

* [[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 #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]]

* [[Unnoticeable comma]] (~~Despite not being an unnoticeable comma by itself,~~ it~~'s mentioned in the intro of this page~~.)

* [[Unnoticeable comma]] (its size often considered to be a threshold for it.)

[[Category:Mercator]]

[[Category:Commas named after mathematicians]]

[[Category:Commas named after music theorists]]

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-The comma with the monzo of {{monzo|-84 53}}, known as '''Mercator's comma''', '''mercator 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]].
+'''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]].
-== Temperament ==
+== Temperaments ==
-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]].
+[[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 ==
-* [[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 #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]]
-* [[Unnoticeable comma]] (Despite not being an unnoticeable comma by itself, it's mentioned in the intro of this page.)
+* [[Unnoticeable comma]] (its size often considered to be a threshold for it.)
+[[Category:Mercator]]
 [[Category:Commas named after mathematicians]]
 [[Category:Commas named after music theorists]]