Mercator's comma: Difference between revisions

Interval information
Factorization	2-84 × 353
Monzo	[-84 53⟩
Size in cents	3.615046 ¢
Names	Mercator's comma,; 53-comma
Color name	53wM, fithewama
FJS name	[math]\displaystyle{ \text{7d}{-6} }[/math]
Special properties	reduced,; reduced harmonic
Tenney norm (log2 nd)	168.003
Weil norm (log2 max(n, d))	168.006
Wilson norm (sopfr(nd))	327
Comma size	small
	Open this interval in xen-calc

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VisualWikitext

Latest revision as of 06:57, 10 September 2026

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.

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-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Infobox Interval
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+| Monzo = -84 53
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-27 21:26:47 UTC</tt>.<br>
+| Name = Mercator's comma, 53-comma
-: The original revision id was <tt>248845799</tt>.<br>
+| Color name = 53wM, fithewama
-: The revision comment was: <tt></tt><br>
+| Comma = yes
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
+}}
-<h4>Original Wikitext content:</h4>
+'''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]].
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">|-84 53&gt;, 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)^53/2^31. 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]].</pre></div>
-<h4>Original HTML content:</h4>
+The comma is named for [[Nicholas Mercator]], who first took note of it as a part of his study of [[53edo]].
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Mercator's comma&lt;/title&gt;&lt;/head&gt;&lt;body&gt;|-84 53&amp;gt;, 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)^53/2^31. For equal divisions N up to 8745, the comma is tempered out if and only if 53 divides N. Examples of such EDOs include &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt;, &lt;a class="wiki_link" href="/159edo"&gt;159edo&lt;/a&gt;, &lt;a class="wiki_link" href="/212edo"&gt;212edo&lt;/a&gt;, &lt;a class="wiki_link" href="/265edo"&gt;265edo&lt;/a&gt;, &lt;a class="wiki_link" href="/742edo"&gt;742edo&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
+== Temperaments ==
+[[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]]
+* [[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

Latest revision as of 06:57, 10 September 2026

Temperaments

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