Mercator's comma: Difference between revisions

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| Monzo = -84 53

| Name = Mercator's comma, 53-comma

| Color name = ~~Wa-53 comma~~

| Color name = 53wM, fithewama

| Comma = yes

}}

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

'''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 [[~~Wikipedia: Nicholas Mercator|~~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~~ ==

== Temperaments ==

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]].

[[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 ==

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

[[Category:Mercator]]

[[Category:Commas named after mathematicians]]

[[Category:Commas named after music theorists]]

@@ Line 2: / Line 2: @@
 | Monzo = -84 53
 | Name = Mercator's comma, 53-comma
-| Color name = Wa-53 comma
+| Color name = 53wM, fithewama
 | Comma = yes
 }}
-{{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 fifths exceed 31 octaves, in other words (3/2)<sup>53</sup>/2<sup>31</sup>.
+'''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 [[Wikipedia: Nicholas Mercator|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 ==
+== Temperaments ==
-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]].
+[[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 ==
-* [[Unnoticeable 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]] (its size often considered to be a threshold for it.)
+[[Category:Mercator]]
+[[Category:Commas named after mathematicians]]
+[[Category:Commas named after music theorists]]

Mercator's comma: Difference between revisions

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