Abc, high quality commas, and epimericity: Difference between revisions
Wikispaces>genewardsmith **Imported revision 362915370 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 362961204 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-09-07 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-09-07 20:05:02 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>362961204</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | 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> | <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"> | <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">[[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 ... . | 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. | |||
=Links= | =Links= | ||
| Line 15: | Line 22: | ||
[[Superpartient|Degree of Epimericity]]</pre></div> | [[Superpartient|Degree of Epimericity]]</pre></div> | ||
<h4>Original HTML content:</h4> | <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"><html><head><title>ABC, High Quality Commas, and Epimericity</title></head><body>If n/d &gt; 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 &quot;interesting&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.<br /> | <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"><html><head><title>ABC, High Quality Commas, and Epimericity</title></head><body><!-- ws:start:WikiTextTocRule:8:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- 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:&lt;h1&gt; --><h1 id="toc0"><a name="Epimericity"></a><!-- ws:end:WikiTextHeadingRule:0 -->Epimericity</h1> | |||
If n/d &gt; 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 &quot;interesting&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.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><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 &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 &quot;high quality&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 ... .<br /> | 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 &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 &quot;high quality&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 ... .<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><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) &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 (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.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><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&amp;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/4458?threaded=1&amp;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_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></pre></div> | <a class="wiki_link" href="/Superpartient">Degree of Epimericity</a></body></html></pre></div> | ||