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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | A positive rational number q belongs to the '''p-limit''', called the '''p harmonic''' or '''prime limit''', for a given [[prime_number|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|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. |
| 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:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-15 13:31:16 UTC</tt>.<br>
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| : The original revision id was <tt>602249856</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">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.
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| ==List of small p-limits== | | ==List of small p-limits== |
| With increasing limits, the tonal space becomes more dense. | | With increasing limits, the tonal space becomes more dense. |
| * [[2-limit]] contains only multiples of the [[octave]] (2/1), see [[1edo]]
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| * [[3-limit]] contains [[3_2|3/2]], the [[just perfect fifth]]
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| * [[5-limit]] contains [[5_4|5/4]], the just major third
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| * [[7-limit]] contains [[7_4|7/4]], the harmonic seventh
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| * [[11-limit]] contains [[11_8|11/8]], the Alphorn-Fa
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| * [[13-limit]]
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| * [[17-limit]]
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| * [[19-limit]]
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| * [[23-limit]]
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| * [[29-limit]]
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| * [[31-limit]]
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| * [[37-limit]]
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| * [[41-limit]]
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| * [[43-limit]]
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| * [[47-limit]]
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| * [[53-limit]]
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| * [[59-limit]]
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| * [[61-limit]]
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| ==See also==
| | <ul><li>[[2-limit|2-limit]] contains only multiples of the [[Octave|octave]] (2/1), see [[1edo|1edo]]</li><li>[[3-limit|3-limit]] contains [[3/2|3/2]], the [[just_perfect_fifth|just perfect fifth]]</li><li>[[5-limit|5-limit]] contains [[5/4|5/4]], the just major third</li><li>[[7-limit|7-limit]] contains [[7/4|7/4]], the harmonic seventh</li><li>[[11-limit|11-limit]] contains [[11/8|11/8]], the Alphorn-Fa</li><li>[[13-limit|13-limit]]</li><li>[[17-limit|17-limit]]</li><li>[[19-limit|19-limit]]</li><li>[[23-limit|23-limit]]</li><li>[[29-limit|29-limit]]</li><li>[[31-limit|31-limit]]</li><li>[[37-limit|37-limit]]</li><li>[[41-limit|41-limit]]</li><li>[[43-limit|43-limit]]</li><li>[[47-limit|47-limit]]</li><li>[[53-limit|53-limit]]</li><li>[[59-limit|59-limit]]</li><li>[[61-limit|61-limit]]</li></ul> |
| * [[Odd limit]]
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| * [[Harmonic Class (HC)]]
| | ==See also== |
| * [[consistency]]
| | <ul><li>[[Odd_limit|Odd limit]]</li><li>[[Harmonic_Class_(HC)|Harmonic Class (HC)]]</li><li>[[consistency|consistency]]</li><li>[http://en.wikipedia.org/wiki/Limit_%28music%29 Limit (music) - Wikipedia] (covers also the distinction between odd-limit and prime-limit)</li><li>[http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem Størmer's theorem - Wikipedia]</li></ul> [[Category:definition]] |
| * [[http://en.wikipedia.org/wiki/Limit_%28music%29|Limit (music) - Wikipedia]] (covers also the distinction between odd-limit and prime-limit)
| | [[Category:limit]] |
| * [[http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem|Størmer's theorem - Wikipedia]]</pre></div>
| | [[Category:theory]] |
| <h4>Original HTML 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;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>
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| With increasing limits, the tonal space becomes more dense.<br />
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| <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> contains <a class="wiki_link" href="/11_8">11/8</a>, the Alphorn-Fa</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><li><a class="wiki_link" href="/29-limit">29-limit</a></li><li><a class="wiki_link" href="/31-limit">31-limit</a></li><li><a class="wiki_link" href="/37-limit">37-limit</a></li><li><a class="wiki_link" href="/41-limit">41-limit</a></li><li><a class="wiki_link" href="/43-limit">43-limit</a></li><li><a class="wiki_link" href="/47-limit">47-limit</a></li><li><a class="wiki_link" href="/53-limit">53-limit</a></li><li><a class="wiki_link" href="/59-limit">59-limit</a></li><li><a class="wiki_link" href="/61-limit">61-limit</a></li></ul><br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-See also"></a><!-- ws:end:WikiTextHeadingRule:2 -->See also</h2>
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| <ul><li><a class="wiki_link" href="/Odd%20limit">Odd limit</a></li><li><a class="wiki_link" href="/Harmonic%20Class%20%28HC%29">Harmonic Class (HC)</a></li><li><a class="wiki_link" href="/consistency">consistency</a></li><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Limit_%28music%29" rel="nofollow">Limit (music) - Wikipedia</a> (covers also the distinction between odd-limit and prime-limit)</li><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem" rel="nofollow">Størmer's theorem - Wikipedia</a></li></ul></body></html></pre></div>
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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 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 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.
See also