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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>2012-09-08 10:43:20 UTC</tt>.<br>
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| : The original revision id was <tt>363020322</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <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]]
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| =Epimericity= | | {{DISPLAYTITLE:''abc'', high quality commas, and 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. | | == Epimericity == |
| | If ''n''/''d'' > 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. |
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| =The ABC conjecture= | | == ''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 [[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 > 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 ... . |
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| =The DoReMi conjecture= | | == 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. | | 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. 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'') > 1 may be called a DoReMi comma. |
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| 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 commas are favored, so that commas such as 512/507 are on the list. | | 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. |
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| =Links= | | == See also == |
| [[http://tech.groups.yahoo.com/group/tuning-math/message/4458?threaded=1&l=1|Seven and eleven limit comma lists]]
| | * [[Superpartient]] |
| [[http://tech.groups.yahoo.com/group/tuning-math/message/5556|An 11-limit linear temperament top 100 list]]
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| [[Superpartient|Degree of Epimericity]]</pre></div> | | == External links == |
| <h4>Original HTML content:</h4>
| | * [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_4458.html Seven and eleven limit comma lists] |
| <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: -->
| | * [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5556.html An 11-limit linear temperament top 100 list] |
| <!-- ws:end:WikiTextTocRule:13 --><br />
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Epimericity"></a><!-- ws:end:WikiTextHeadingRule:0 -->Epimericity</h1>
| | [[Category:Comma]] |
| 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 />
| | [[Category:Math]] |
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="The ABC conjecture"></a><!-- ws:end:WikiTextHeadingRule:2 -->The ABC conjecture</h1>
| | {{Todo| intro }} |
| 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 />
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="The DoReMi conjecture"></a><!-- ws:end:WikiTextHeadingRule:4 -->The DoReMi conjecture</h1>
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| 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 />
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| <br />
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| 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 commas are favored, so that commas such as 512/507 are on the list.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:6 -->Links</h1>
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| <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 />
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| <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 />
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| <a class="wiki_link" href="/Superpartient">Degree of Epimericity</a></body></html></pre></div>
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