Abc, high quality commas, and epimericity
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[[toc|flat]] =Epimericity= If n/d > 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 epimericity 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 or equal to 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. =The ABC conjecture= 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)/log(rad(n/d)). Then the abc conjecture, a very powerful conjecture, says that for any ϵ > 0 there are only finitely many commas such that q(n/d) > 1+ϵ, where we may assume without loss of generality that n/d < 2 so that it is an actual comma. Any comma with q(n/d) > 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 ... . =The 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) > 1+ϵ is stronger than q(n/d) > 1+ϵ, and there will be fewer intervals which qualify. This means that if the list of q(n/d) > 1+ϵ is finite, so is the list of doremi(n/d) > 1+ϵ, and so ABC implies DoReMi but not conversely; DoReMi is a slightly weaker conjecture, but (according to Noam Elkies) still unproven also. Aside from making more musical sense, doremi has the further advantage of being 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) > 1 we may call 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. =Links= [[http://tech.groups.yahoo.com/group/tuning-math/message/4458?threaded=1&l=1|Seven and eleven limit comma lists]] [[http://tech.groups.yahoo.com/group/tuning-math/message/5556|An 11-limit linear temperament top 100 list]] [[Superpartient|Degree of Epimericity]]
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<html><head><title>ABC, High Quality Commas, and Epimericity</title></head><body><!-- ws:start:WikiTextTocRule:8:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#Epimericity">Epimericity</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#The ABC conjecture">The ABC conjecture</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#The DoReMi conjecture">The DoReMi conjecture</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> <!-- ws:end:WikiTextTocRule:13 --><br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Epimericity"></a><!-- ws:end:WikiTextHeadingRule:0 -->Epimericity</h1> If n/d > 1 is a rational number with positive integers n and d relatively prime, we may define the <em>epimericity</em> 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 epimericity is also cents(n-d)/cents(d). Then it appears to be true that <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/St%C3%B8rmer's_theorem" rel="nofollow">Størmer's theorem</a> 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 "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.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="The ABC conjecture"></a><!-- ws:end:WikiTextHeadingRule:2 -->The ABC conjecture</h1> This conjecture is related to the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abc_conjecture" rel="nofollow">abc conjecture</a>, and a related claim is in fact precisely the abc conjecture, which defines what we may call a <em>high quality comma</em>. Define the <em>radical</em> 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 <em>quality</em> 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 > 0 there are only finitely many commas such that q(n/d) > 1+ϵ, where we may assume without loss of generality that n/d < 2 so that it is an actual comma. Any comma with q(n/d) > 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 ... .<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="The DoReMi conjecture"></a><!-- ws:end:WikiTextHeadingRule:4 -->The DoReMi conjecture</h1> 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) > 1+ϵ is stronger than q(n/d) > 1+ϵ, and there will be fewer intervals which qualify. This means that if the list of q(n/d) > 1+ϵ is finite, so is the list of doremi(n/d) > 1+ϵ, and so ABC implies DoReMi but not conversely; DoReMi is a slightly weaker conjecture, but (according to Noam Elkies) still unproven also. Aside from making more musical sense, doremi has the further advantage of being 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) > 1 we may call a doremi comma.<br /> <br /> 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.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:6 -->Links</h1> <a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/4458?threaded=1&l=1" rel="nofollow">Seven and eleven limit comma lists</a><br /> <a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/5556" rel="nofollow">An 11-limit linear temperament top 100 list</a><br /> <a class="wiki_link" href="/Superpartient">Degree of Epimericity</a></body></html>