3-limit: Difference between revisions
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Wikispaces>xenwolf **Imported revision 215741072 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 232450846 - 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>2011-05-27 13:46:40 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>232450846</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">Some examples | <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 //3-limit// interval is either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. All 3-limit intervals can be written as 2^a 3^b, where a and b can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are [[3_2|3/2]], [[4_3|4/3]], [[9_8|9/8]]. Confining intervals to the 3-limit is known as [[http://en.wikipedia.org/wiki/Pythagorean_tuning|Pythagorean tuning]], and the Pythagorean tuning used in Europe during the Middle Ages is seed out of which grew the common-practice tradition of Western music. | ||
[[EDO]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the [[http://en.wikipedia.org/wiki/Continued_fraction|continued fraction]] for the logarithm of 3 base 2. These are 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306... . | |||
See [[Harmonic Limit]].</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>3-limit</title></head><body>Some examples | <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>3-limit</title></head><body>A <em>3-limit</em> interval is either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. All 3-limit intervals can be written as 2^a 3^b, where a and b can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are <a class="wiki_link" href="/3_2">3/2</a>, <a class="wiki_link" href="/4_3">4/3</a>, <a class="wiki_link" href="/9_8">9/8</a>. Confining intervals to the 3-limit is known as <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pythagorean_tuning" rel="nofollow">Pythagorean tuning</a>, and the Pythagorean tuning used in Europe during the Middle Ages is seed out of which grew the common-practice tradition of Western music.<br /> | ||
<br /> | |||
<a class="wiki_link" href="/EDO">EDO</a>s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction" rel="nofollow">continued fraction</a> for the logarithm of 3 base 2. These are 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306... .<br /> | |||
<br /> | <br /> | ||
See <a class="wiki_link" href="/Harmonic%20Limit">Harmonic Limit</a>.</body></html></pre></div> | |||
Revision as of 13:46, 27 May 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 2011-05-27 13:46:40 UTC.
- The original revision id was 232450846.
- 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 //3-limit// interval is either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. All 3-limit intervals can be written as 2^a 3^b, where a and b can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are [[3_2|3/2]], [[4_3|4/3]], [[9_8|9/8]]. Confining intervals to the 3-limit is known as [[http://en.wikipedia.org/wiki/Pythagorean_tuning|Pythagorean tuning]], and the Pythagorean tuning used in Europe during the Middle Ages is seed out of which grew the common-practice tradition of Western music. [[EDO]]s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the [[http://en.wikipedia.org/wiki/Continued_fraction|continued fraction]] for the logarithm of 3 base 2. These are 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306... . See [[Harmonic Limit]].
Original HTML content:
<html><head><title>3-limit</title></head><body>A <em>3-limit</em> interval is either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. All 3-limit intervals can be written as 2^a 3^b, where a and b can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are <a class="wiki_link" href="/3_2">3/2</a>, <a class="wiki_link" href="/4_3">4/3</a>, <a class="wiki_link" href="/9_8">9/8</a>. Confining intervals to the 3-limit is known as <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pythagorean_tuning" rel="nofollow">Pythagorean tuning</a>, and the Pythagorean tuning used in Europe during the Middle Ages is seed out of which grew the common-practice tradition of Western music.<br /> <br /> <a class="wiki_link" href="/EDO">EDO</a>s which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction" rel="nofollow">continued fraction</a> for the logarithm of 3 base 2. These are 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306... .<br /> <br /> See <a class="wiki_link" href="/Harmonic%20Limit">Harmonic Limit</a>.</body></html>