Harmonic limit: Difference between revisions
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Wikispaces>xenwolf **Imported revision 239308775 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 244906299 - Original comment: ** |
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<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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-08 17:23:51 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>244906299</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">A positive rational number q belongs to the **p-limit**, called the **p harmonic** or **prime limit**, for a given [[prime number]] p if and only if it can be factored into primes (with positive or negative integer exponents) of size less than or equal to p. For any prime number p, the set of all rational numbers in the p-limit defines a [[http://en.wikipedia.org/wiki/Free_abelian_group|finitely generated free abelian group]]. The rank of this group is equal to pi(p), the number of prime numbers less than or equal to p. Hence, for example, the rank of the [[7-limit]] is 4, as it is generated by 2, 3, 5 and 7. | <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">A positive rational number q belongs to the **p-limit**, called the **p harmonic** or **prime limit**, for a given [[prime number]] p if and only if it can be factored into primes (with positive or negative integer exponents) of size less than or equal to p. For any prime number p, the set of all rational numbers in the p-limit defines a [[http://en.wikipedia.org/wiki/Free_abelian_group|finitely generated free abelian group]]. The rank of this group is equal to pi(p), the number of prime numbers less than or equal to p. Hence, for example, the rank of the [[7-limit]] is 4, as it is generated by 2, 3, 5 and 7. Another way to express the p-limit is that it consists of the ratios of [[http://en.wikipedia.org/wiki/Smooth_number|p-smooth numbers]], where a p-smooth number is an integer with prime factors no larger than p. | ||
== List of small p-limits == | == List of small p-limits == | ||
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* [[consistency]]</pre></div> | * [[consistency]]</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>Harmonic Limit</title></head><body>A positive rational number q belongs to the <strong>p-limit</strong>, called the <strong>p harmonic</strong> or <strong>prime limit</strong>, for a given <a class="wiki_link" href="/prime%20number">prime number</a> p if and only if it can be factored into primes (with positive or negative integer exponents) of size less than or equal to p. For any prime number p, the set of all rational numbers in the p-limit defines a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">finitely generated free abelian group</a>. The rank of this group is equal to pi(p), the number of prime numbers less than or equal to p. Hence, for example, the rank of the <a class="wiki_link" href="/7-limit">7-limit</a> is 4, as it is generated by 2, 3, 5 and 7.<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>Harmonic Limit</title></head><body>A positive rational number q belongs to the <strong>p-limit</strong>, called the <strong>p harmonic</strong> or <strong>prime limit</strong>, for a given <a class="wiki_link" href="/prime%20number">prime number</a> p if and only if it can be factored into primes (with positive or negative integer exponents) of size less than or equal to p. For any prime number p, the set of all rational numbers in the p-limit defines a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">finitely generated free abelian group</a>. The rank of this group is equal to pi(p), the number of prime numbers less than or equal to p. Hence, for example, the rank of the <a class="wiki_link" href="/7-limit">7-limit</a> is 4, as it is generated by 2, 3, 5 and 7. Another way to express the p-limit is that it consists of the ratios of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Smooth_number" rel="nofollow">p-smooth numbers</a>, where a p-smooth number is an integer with prime factors no larger than p.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-List of small p-limits"></a><!-- ws:end:WikiTextHeadingRule:0 --> List of small p-limits </h2> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-List of small p-limits"></a><!-- ws:end:WikiTextHeadingRule:0 --> List of small p-limits </h2> | ||
Revision as of 17:23, 8 August 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-08-08 17:23:51 UTC.
- The original revision id was 244906299.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
A positive rational number q belongs to the **p-limit**, called the **p harmonic** or **prime limit**, for a given [[prime number]] p if and only if it can be factored into primes (with positive or negative integer exponents) of size less than or equal to p. For any prime number p, the set of all rational numbers in the p-limit defines a [[http://en.wikipedia.org/wiki/Free_abelian_group|finitely generated free abelian group]]. The rank of this group is equal to pi(p), the number of prime numbers less than or equal to p. Hence, for example, the rank of the [[7-limit]] is 4, as it is generated by 2, 3, 5 and 7. Another way to express the p-limit is that it consists of the ratios of [[http://en.wikipedia.org/wiki/Smooth_number|p-smooth numbers]], where a p-smooth number is an integer with prime factors no larger than p. == List of small p-limits == With increasing limits, the tonal space becomes more dense. * [[2-limit]] contains only multiples of the [[octave]] (2/1), see [[1edo]] * [[3-limit]] contains [[3_2|3/2]], the [[just perfect fifth]] * [[5-limit]] contains [[5_4|5/4]], the just major third * [[7-limit]] contains [[7_4|7/4]], the harmonic seventh * [[11-limit]] * [[13-limit]] * [[17-limit]] * [[19-limit]] * [[23-limit]] == See also == * [[Odd limit]] * [[consistency]]
Original HTML content:
<html><head><title>Harmonic Limit</title></head><body>A positive rational number q belongs to the <strong>p-limit</strong>, called the <strong>p harmonic</strong> or <strong>prime limit</strong>, for a given <a class="wiki_link" href="/prime%20number">prime number</a> p if and only if it can be factored into primes (with positive or negative integer exponents) of size less than or equal to p. For any prime number p, the set of all rational numbers in the p-limit defines a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">finitely generated free abelian group</a>. The rank of this group is equal to pi(p), the number of prime numbers less than or equal to p. Hence, for example, the rank of the <a class="wiki_link" href="/7-limit">7-limit</a> is 4, as it is generated by 2, 3, 5 and 7. Another way to express the p-limit is that it consists of the ratios of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Smooth_number" rel="nofollow">p-smooth numbers</a>, where a p-smooth number is an integer with prime factors no larger than p.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-List of small p-limits"></a><!-- ws:end:WikiTextHeadingRule:0 --> List of small p-limits </h2> With increasing limits, the tonal space becomes more dense.<br /> <ul><li><a class="wiki_link" href="/2-limit">2-limit</a> contains only multiples of the <a class="wiki_link" href="/octave">octave</a> (2/1), see <a class="wiki_link" href="/1edo">1edo</a></li><li><a class="wiki_link" href="/3-limit">3-limit</a> contains <a class="wiki_link" href="/3_2">3/2</a>, the <a class="wiki_link" href="/just%20perfect%20fifth">just perfect fifth</a></li><li><a class="wiki_link" href="/5-limit">5-limit</a> contains <a class="wiki_link" href="/5_4">5/4</a>, the just major third</li><li><a class="wiki_link" href="/7-limit">7-limit</a> contains <a class="wiki_link" href="/7_4">7/4</a>, the harmonic seventh</li><li><a class="wiki_link" href="/11-limit">11-limit</a></li><li><a class="wiki_link" href="/13-limit">13-limit</a></li><li><a class="wiki_link" href="/17-limit">17-limit</a></li><li><a class="wiki_link" href="/19-limit">19-limit</a></li><li><a class="wiki_link" href="/23-limit">23-limit</a></li></ul><br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-See also"></a><!-- ws:end:WikiTextHeadingRule:2 --> See also </h2> <ul><li><a class="wiki_link" href="/Odd%20limit">Odd limit</a></li><li><a class="wiki_link" href="/consistency">consistency</a></li></ul></body></html>