3-limit: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 232450846 - Original comment: ** |
Wikispaces>PiotrGrochowski **Imported revision 591574462 - 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:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-09-10 14:42:37 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>591574462</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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<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. | <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... . | [[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..., ... | ||
Another approach is to find EDOs which have more accurate 3 than all smaller EDOs. This results in 1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, ... | |||
See [[Harmonic Limit]].</pre></div> | See [[Harmonic Limit]].</pre></div> | ||
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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>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 /> | <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 /> | <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 /> | <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 /> | ||
Another approach is to find EDOs which have more accurate 3 than all smaller EDOs. This results in 1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, ...<br /> | |||
<br /> | <br /> | ||
See <a class="wiki_link" href="/Harmonic%20Limit">Harmonic Limit</a>.</body></html></pre></div> | See <a class="wiki_link" href="/Harmonic%20Limit">Harmonic Limit</a>.</body></html></pre></div> | ||
Revision as of 14:42, 10 September 2016
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author PiotrGrochowski and made on 2016-09-10 14:42:37 UTC.
- The original revision id was 591574462.
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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.
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..., ... Another approach is to find EDOs which have more accurate 3 than all smaller EDOs. This results in 1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, ... 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 /> Another approach is to find EDOs which have more accurate 3 than all smaller EDOs. This results in 1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, ...<br /> <br /> See <a class="wiki_link" href="/Harmonic%20Limit">Harmonic Limit</a>.</body></html>