Harmonic limit
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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. == 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 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 <a class="wiki_link" href="/prime%20number">prime number</a> 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 /> <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>