Harmonic limit: Difference between revisions
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Wikispaces>xenwolf **Imported revision 237666661 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 237668123 - 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:xenwolf|xenwolf]] and made on <tt>2011-06-20 | : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-20 03:03:14 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>237668123</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> | ||
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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]] | * [[2-limit]] contains only multiples of the [[octave]] (2/1), see [[1edo]] | ||
* [[3-limit]] contains [[3_2|3/2]], the | * [[3-limit]] contains [[3_2|3/2]], the [[just perfect fifth]] | ||
* [[5-limit]] contains [[5_4|5/4]], the just major third | * [[5-limit]] contains [[5_4|5/4]], the just major third | ||
* [[7-limit]] contains [[7_4|7/4]], the harmonic seventh | * [[7-limit]] contains [[7_4|7/4]], the harmonic seventh | ||
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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> | ||
With increasing limits, the tonal space becomes more dense.<br /> | 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 | <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></body></html></pre></div> | ||
Revision as of 03:03, 20 June 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 xenwolf and made on 2011-06-20 03:03:14 UTC.
- The original revision id was 237668123.
- 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. == 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]]
Original HTML content:
<html><head><title>Harmonic Limit</title></head><body>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 <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></body></html>