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41-comma: Difference between revisions

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**Imported revision 248843627 - Original comment: **
 
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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>
| Ratio = 36 893 488 147 419 103 232 / <br>36 472 996 377 170 786 403
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-27 21:02:47 UTC</tt>.<br>
| Monzo = 65 -41
: The original revision id was <tt>248843627</tt>.<br>
| Name = 41-comma, Pythagorean countercomma, countercomp comma
: The revision comment was: <tt></tt><br>
| Color name = 41wM, fowowama
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| Comma = yes
<h4>Original Wikitext content:</h4>
}}
<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">|65 -41&gt;, the 41-comma, is an interval of 19.845 cents, is the amount by which 41 fifths fall short of 24 octaves, in other words 2^24/(3/2)^41. For equal divisions N up to 1230, the comma is tempered out if and only if 41 divides N. Examples are [[41edo]], [[164edo]], [[205edo]], [[246edo]], [[328edo]] and [[369edo]].</pre></div>
The '''41-comma''', '''Pythagorean countercomma''', or '''countercomp comma''' ([[monzo]]: {{monzo| 65 -41 }}), is a [[3-limit]] interval of 19.845 [[cent]]s. It is the amount by which a stack of 41 [[3/2|perfect fifths (3/2)]] falls short of 24 [[octave]]s, in other words 2<sup>24</sup>/(3/2)<sup>41</sup>. It can also be described as the difference between a perfect fifth and a stack of six [[2187/2048|apotomes]], or equivalently as the small gap between the Pythagorean triple-augmented unison and the triple-diminished fifth.
<h4>Original HTML content:</h4>
 
<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;41-comma&lt;/title&gt;&lt;/head&gt;&lt;body&gt;|65 -41&amp;gt;, the 41-comma, is an interval of 19.845 cents, is the amount by which 41 fifths fall short of 24 octaves, in other words 2^24/(3/2)^41. For equal divisions N up to 1230, the comma is tempered out if and only if 41 divides N. Examples are &lt;a class="wiki_link" href="/41edo"&gt;41edo&lt;/a&gt;, &lt;a class="wiki_link" href="/164edo"&gt;164edo&lt;/a&gt;, &lt;a class="wiki_link" href="/205edo"&gt;205edo&lt;/a&gt;, &lt;a class="wiki_link" href="/246edo"&gt;246edo&lt;/a&gt;, &lt;a class="wiki_link" href="/328edo"&gt;328edo&lt;/a&gt; and &lt;a class="wiki_link" href="/369edo"&gt;369edo&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
== Temperaments ==
Tempering out this comma leads to the [[countercomp]] temperament, which splits the octave into 41 equal parts and maps the harmonic 3 to 24\41. For equal divisions ''N'' up to 1230, the comma is tempered out if and only if 41 divides ''N''. Examples are [[41edo]], [[164edo]], [[205edo]], [[246edo]], [[328edo]] and [[369edo]]. See [[countercomp family]] for a number of rank-2 temperaments where it is tempered out.
 
== Terminology ==
The names were given by [[Flora Canou]] in 2022. ''Pythagorean countercomma'' was derived by analogy to ''Pythagorean comma'' as part of a series of Pythagorean interval names. ''Countercomp comma'' was derived from the temperament name, ''countercomp'', which was changed from ''counterpyth'' as in earlier materials, where this comma was also called ''counterpyth comma''. It was renamed after the convention was established that the temperament of the [[Pythagorean comma]] should be [[compton]] and never ''Pythagorean'', for fear of confusion with [[Pythagorean tuning]].  
 
== See also ==
* [[Small comma]]
* [[Schismic–countercommatic equivalence continuum]]
 
[[Category:Countercomp]]
[[Category:Commas named systematically]]
[[Category:Commas named for their regular temperament properties]]
Retrieved from "https://en.xen.wiki/w/41-comma"