Mercator's comma: Difference between revisions

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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 21:27:57 UTC</tt>.<br>
| Name = Mercator's comma, 53-comma
: The original revision id was <tt>248845937</tt>.<br>
| Color name = Wa-53 comma
: 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>
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]].
<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]], [[795edo]], [[954edo]], [[1749edo]].</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;a class="wiki_link" href="/795edo"&gt;795edo&lt;/a&gt;, &lt;a class="wiki_link" href="/954edo"&gt;954edo&lt;/a&gt;, &lt;a class="wiki_link" href="/1749edo"&gt;1749edo&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
== Temperament ==
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 ==
* [[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]]
 
[[Category:Commas named after mathematicians]]
[[Category:Commas named after music theorists]]

Latest revision as of 12:28, 12 June 2025

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

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

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