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

: This revision was by author [[User:clumma|clumma]] and made on <tt>2012-11-20 15:53:30 UTC</tt>.<br>

: The original revision id was <tt>384496242</tt>.<br>

 

=Epimericity=

=The ABC conjecture=

=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) &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, and so ABC implies DoReMi but not conversely; DoReMi is a slightly weaker conjecture, but ([[@http://math.stackexchange.com/a/235373|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) &gt; 1 we may call a DoReMi comma.

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&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Epimericity"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Epimericity&lt;/h1&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 epimericity 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 or equal to 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;

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)/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, where we may assume without loss of generality that n/d &amp;lt; 2 so that it is an actual comma. Any comma with q(n/d) &amp;gt; 1 we may call &amp;quot;high quality&amp;quot;; 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 ... .&lt;br /&gt;

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) &amp;gt; 1+ϵ is stronger than q(n/d) &amp;gt; 1+ϵ, and there will be fewer intervals which qualify. This means that if the list of q(n/d) &amp;gt; 1+ϵ is finite, so is the list of doremi(n/d) &amp;gt; 1+ϵ, and so ABC implies DoReMi but not conversely; DoReMi is a slightly weaker conjecture, but (&lt;a class="wiki_link_ext" href="http://math.stackexchange.com/a/235373" rel="nofollow" target="_blank"&gt;according to Noam Elkies&lt;/a&gt; ) 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) &amp;gt; 1 we may call a DoReMi comma.&lt;br /&gt;