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Abc, high quality commas, and epimericity: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-09-07 11:44:28 UTC</tt>.<br>
: The original revision id was <tt>362842988</tt>.<br>
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<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">If n/d &gt; 1 is a rational number with positive integers n and d relatively prime, we may define the //epimericity// of n/d as log(n-d)/log(d). Which logarithm we use is irrelevant; we can if we like use cents and so the peimericity is also cents(n-d)/cents(d). Then it appears to be true that [[http://en.wikipedia.org/wiki/St%C3%B8rmer's_theorem|Størmer's theorem]] generalizes to a claim that for any prime p, only finitely many rational numbers in the p-limit exist with epimericity less than any constant c less than one. Hence "interesting" commas in any p-limit can be defined as those below a given epimericity, such as the 7-limit commas under 0.5 in epimericity, or the 11-limit commas under 0.3.


This conjecture is related to the [[http://en.wikipedia.org/wiki/Abc_conjecture|abc conjecture]], and a related claim is in fact precisely the abc conjecture, which defines what we may call a //high quality comma//. Define the //radical// rad(n/d) of n/d as the product of all the primes dividing n, d, and n-d; so that rad(128/125) = 2*3*5 = 30. Then define the //quality// q(n/d) of n/d as log(n-d)/log(rad(n/d)). Then the abc conjecture, a very powerful conjecture, says that for any ϵ&gt;0 there are only finitely many commas such that q(n/d) &gt; 1+ϵ. Any comma with q(n/d)&gt;1 we may call "high quality".</pre></div>
{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}
<h4>Original HTML content:</h4>
== Epimericity ==
<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;ABC, High Quality Commas, and Epimericity&lt;/title&gt;&lt;/head&gt;&lt;body&gt;If n/d &amp;gt; 1 is a rational number with positive integers n and d relatively prime, we may define the &lt;em&gt;epimericity&lt;/em&gt; of n/d as log(n-d)/log(d). Which logarithm we use is irrelevant; we can if we like use cents and so the peimericity is also cents(n-d)/cents(d). Then it appears to be true that &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/St%C3%B8rmer's_theorem" rel="nofollow"&gt;Størmer's theorem&lt;/a&gt; generalizes to a claim that for any prime p, only finitely many rational numbers in the p-limit exist with epimericity less than any constant c less than one. Hence &amp;quot;interesting&amp;quot; commas in any p-limit can be defined as those below a given epimericity, such as the 7-limit commas under 0.5 in epimericity, or the 11-limit commas under 0.3.&lt;br /&gt;
If ''n''/''d'' &gt; 1 is a rational number in simplest form, we may define the ''epimericity'' of ''n''/''d'' in terms of cents as cents(''n'' - ''d'')/cents(''d'') - that is, for an example 9/7, n-d is 2 and d is 7, so we end up with roughly 1200/3369 or 0.356. (Note that using other logarithms is possible, but the result is still the same due to the bases cancelling out). Then it appears to be true that [[wikipedia: størmer's theorem|Størmer's theorem]] generalizes to a claim that for any prime ''p'', only finitely many rational numbers in the ''p''-limit exist with epimericity less than or equal to any constant ''c'' less than one. Hence a particular finite subset of commas in any ''p''-limit can be defined as those below a given epimericity, such as the 7-limit commas under 0.5 in epimericity, or the 11-limit commas under 0.3; these are considered "interesting" according to a value judgment whose nature is not known.
&lt;br /&gt;
 
This conjecture is related to the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abc_conjecture" rel="nofollow"&gt;abc conjecture&lt;/a&gt;, and a related claim is in fact precisely the abc conjecture, which defines what we may call a &lt;em&gt;high quality comma&lt;/em&gt;. Define the &lt;em&gt;radical&lt;/em&gt; rad(n/d) of n/d as the product of all the primes dividing n, d, and n-d; so that rad(128/125) = 2*3*5 = 30. Then define the &lt;em&gt;quality&lt;/em&gt; q(n/d) of n/d as log(n-d)/log(rad(n/d)). Then the abc conjecture, a very powerful conjecture, says that for any ϵ&amp;gt;0 there are only finitely many commas such that q(n/d) &amp;gt; 1+ϵ. Any comma with q(n/d)&amp;gt;1 we may call &amp;quot;high quality&amp;quot;.&lt;/body&gt;&lt;/html&gt;</pre></div>
== ''abc'' conjecture ==
This conjecture is related to the [[Wikipedia: abc conjecture|''abc'' conjecture]], and a related claim is in fact precisely the ''abc'' conjecture, which defines what we may call a ''high quality comma''. Define the ''radical'' rad (''n''/''d'') of ''n''/''d'' as the product of all the primes dividing ''n'', ''d'', and ''n'' - ''d''; so that rad (128/125) = 2×3×5 = 30. Then define the ''quality'' q (''n''/''d'') of ''n''/''d'' as log (''n'')/log (rad (''n''/''d'')). Then the ''abc'' conjecture, a very powerful conjecture, says that for any ϵ &gt; 0 there are only finitely many commas such that q (''n''/''d'') &gt; 1 + ϵ, where we may assume without loss of generality that ''n''/''d'' &lt; 2 so that it is an actual comma. Any comma with q (''n''/''d'') &gt; 1 we may call "high quality"; there are an infinite number of these, but the conjecture is that there are only finitely many above any value greater than 1. High quality commas in the 5-limit include 250/243, 128/125, 3125/3072, 81/80 and 2048/2025. It should be noted that while every superparticular ratio has an epimericity of 0, merely being superparticular is by no means enough to make a comma high quality; the list of high-quality superparticulars starts 9/8, 49/48, 64/63, 81/80, 225/224, 243/242 ... .
 
== DoReMi conjecture ==
Since not much musical meaning seems to attach to the commas dividing ''n'' - ''d'', it makes sense for our purposes to modify the definition of quality. Let doremi (''n''/''d'') = log (''n'')/log ((''n'' - ''d'') radical(''nd'')), where radical (''nd'') is the product of the primes dividing ''nd''. Then q (''n''/''d'') ≤ doremi (''n''/''d''), so that the condition that doremi (''n''/''d'') &gt; 1 + ϵ is stronger than q (''n''/''d'') &gt; 1 + ϵ, and there will be fewer intervals which qualify. This means that if the list of q (''n''/''d'') &gt; 1 + ϵ is finite, so is the list of doremi (''n''/''d'') &gt; 1 + ϵ. So ''abc'' implies DoReMi but the converse is not true; DoReMi is a slightly weaker conjecture that is also unproven (according to Noam Elkies and [http://math.stackexchange.com/a/235373 Stack Overflow]). Aside from its clearer music theory implications, DoReMi is enormously easier to compute if ''n''/''d'' is in some small ''p''-limit, as then the computation of radical (''nd'') involves only small primes. A comma ''n''/''d'' with doremi (''n''/''d'') &gt; 1 may be called a DoReMi comma.
 
The DoReMi commas with numerator less than 1000 are 32/27, 9/8, 256/243, 250/243, 128/125, 49/48, 64/63, 81/80, 512/507, 245/243, 225/224, 243/242, 289/288, 513/512, 625/624, 676/675, 729/728 and 961/960. Note that large powers of small primes are favored, so that commas such as 512/507 are on the list.
 
== See also ==
* [[Superpartient]]
 
== External links ==
* [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_4458.html Seven and eleven limit comma lists]
* [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5556.html An 11-limit linear temperament top 100 list]
 
[[Category:Comma]]
[[Category:Math]]
 
{{Todo| intro }}
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