Nearest just interval: Difference between revisions
Wikispaces>guest **Imported revision 210034874 - Original comment: ** |
Wikispaces>Osmiorisbendi **Imported revision 236910286 - 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:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2011-06-15 15:55:43 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>236910286</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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Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest. | Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest. | ||
The [[http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents|semiconvergents]] of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely [[http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations|best relative approximation]]. Here it is required that |qr - p| is less than |nr - m| for any n < q. | The [[http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents|semiconvergents]] of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely [[http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations|best relative approximation]]. Here it is required that |qr - p| is less than |nr - m| for any n < q. | ||
==Examples== | ==Examples== | ||
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||= 1 / 1 ||= 0.0 ||= 0.0 ||= 600.0 || | ||= 1 / 1 ||= 0.0 ||= 0.0 ||= 600.0 || | ||
||= 3 / 2 ||= 2.585 ||= 701.96 ||= 101.96 || | ||= 3 / 2 ||= 2.585 ||= 701.96 ||= 101.96 || | ||
||= 4 / 3 ||= 3.585 ||= 498.04 ||= 101.96 || | ||= 4 / 3 ||= 3.585 ||= 498.04 ||= -101.96 || | ||
||= 7 / 5 ||= 5.129 ||= 582.51 ||= 17.49 || | ||= 7 / 5 ||= 5.129 ||= 582.51 ||= -17.49 || | ||
||= 17 / 12 ||= 7.672 ||= 603.00 ||= 3.000 || | ||= 17 / 12 ||= 7.672 ||= 603.00 ||= 3.000 || | ||
|| 99 / 70 || || 600.09 || 0.09 || | |||
|| ... || ... || ... || ... || | || ... || ... || ... || ... || | ||
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||= 1 / 1 ||= 0.0 ||= 0.0 ||= 300.0 || | ||= 1 / 1 ||= 0.0 ||= 0.0 ||= 300.0 || | ||
||= 6 / 5 ||= 4.907 ||= 315.64 ||= 15.64 || | ||= 6 / 5 ||= 4.907 ||= 315.64 ||= 15.64 || | ||
||= 13 / 11 ||= 7.160 ||= 289.21 ||= 10.79 || | ||= 13 / 11 ||= 7.160 ||= 289.21 ||= -10.79 || | ||
||= 19 / 16 ||= 8.248 ||= 297.51 ||= 2.49 || | ||= 19 / 16 ||= 8.248 ||= 297.51 ||= -2.49 || | ||
||= 25 / 21 ||= 9.036 ||= 301.84 ||= 1.84 || | ||= 25 / 21 ||= 9.036 ||= 301.84 ||= 1.84 || | ||
|| ... || ... || ... || ... || | || ... || ... || ... || ... || | ||
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|| ... || ... || ... || ... || | || ... || ... || ... || ... || | ||
||= 1 \ 1 || 0.0 ||= 1200.0 ||= 498.04 || | ||= 1 \ 1 || 0.0 ||= 1200.0 ||= 498.04 || | ||
||= 1 \ 2 || 1.0 ||= 600.00 ||= 101.96 || | ||= 1 \ 2 || 1.0 ||= 600.00 ||= -101.96 || | ||
||= 2 \ 3 || 2.585 ||= 800.00 ||= 98.045 || | ||= 2 \ 3 || 2.585 ||= 800.00 ||= 98.045 || | ||
||= 3 \ 5 || 3.907 ||= 720.00 ||= 18.045 || | ||= 3 \ 5 || 3.907 ||= 720.00 ||= 18.045 || | ||
||= 4 \ 7 || 4.807 ||= 685. | ||= 4 \ 7 || 4.807 ||= 685.7143 ||= -16.2407 || | ||
||= 7 \ 12 || 6.392 ||= 700.00 ||= 1. | ||= 7 \ 12 || 6.392 ||= 700.00 ||= -1.955 || | ||
||= 17 \ 29 || 8.945 ||= 703. | ||= 17 \ 29 || 8.945 ||= 703.4483 ||= 1.4933 || | ||
||= 24 \ 41 || 9.943 ||= 702. | ||= 24 \ 41 || 9.943 ||= 702.43902 ||= 0.48402 ||</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>Nearest just interval</title></head><body>An irrational interval or ratio of frequencies given by a real number r has an infinite list of <em>nearest just intervals</em>; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call <em>best rational approximations</em>. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from sqrt(2) = 600 cents, but |4/3 - sqrt(2)| = .08088 whereas |3/2 - sqrt(2)| = 0.08479, which is larger.<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>Nearest just interval</title></head><body>An irrational interval or ratio of frequencies given by a real number r has an infinite list of <em>nearest just intervals</em>; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call <em>best rational approximations</em>. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from sqrt(2) = 600 cents, but |4/3 - sqrt(2)| = .08088 whereas |3/2 - sqrt(2)| = 0.08479, which is larger.<br /> | ||
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Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest.<br /> | Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest.<br /> | ||
<br /> | <br /> | ||
The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents" rel="nofollow">semiconvergents</a> of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations" rel="nofollow">best relative approximation</a>. Here it is required that |qr - p| is less than |nr - m| for any n &lt; q. <br /> | The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents" rel="nofollow">semiconvergents</a> of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations" rel="nofollow">best relative approximation</a>. Here it is required that |qr - p| is less than |nr - m| for any n &lt; q.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 -->Examples</h2> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 -->Examples</h2> | ||
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<td style="text-align: center;">498.04<br /> | <td style="text-align: center;">498.04<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">101.96<br /> | <td style="text-align: center;">-101.96<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td style="text-align: center;">582.51<br /> | <td style="text-align: center;">582.51<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">17.49<br /> | <td style="text-align: center;">-17.49<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">3.000<br /> | <td style="text-align: center;">3.000<br /> | ||
</td> | |||
</tr> | |||
<tr> | |||
<td>99 / 70<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
<td>600.09<br /> | |||
</td> | |||
<td>0.09<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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<td style="text-align: center;">289.21<br /> | <td style="text-align: center;">289.21<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">10.79<br /> | <td style="text-align: center;">-10.79<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td style="text-align: center;">297.51<br /> | <td style="text-align: center;">297.51<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">2.49<br /> | <td style="text-align: center;">-2.49<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td style="text-align: center;">600.00<br /> | <td style="text-align: center;">600.00<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">101.96<br /> | <td style="text-align: center;">-101.96<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>4.807<br /> | <td>4.807<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">685. | <td style="text-align: center;">685.7143<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">16. | <td style="text-align: center;">-16.2407<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td style="text-align: center;">700.00<br /> | <td style="text-align: center;">700.00<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">1. | <td style="text-align: center;">-1.955<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>8.945<br /> | <td>8.945<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">703. | <td style="text-align: center;">703.4483<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">1.4933<br /> | <td style="text-align: center;">1.4933<br /> | ||
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<td>9.943<br /> | <td>9.943<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">702. | <td style="text-align: center;">702.43902<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">0.48402<br /> | <td style="text-align: center;">0.48402<br /> | ||