3-limit
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- This revision was by author genewardsmith and made on 2011-05-27 13:46:40 UTC.
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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>