Nearest just interval: Difference between revisions
Wikispaces>xenwolf **Imported revision 236918132 - Original comment: complementaries added to sqrt 2 approximation table** |
Wikispaces>xenwolf **Imported revision 236918684 - Original comment: links to some edos in table of approximations to 3:2** |
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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>2011-06-15 16: | : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-15 16:37:02 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>236918684</tt>.<br> | ||
: The revision comment was: <tt> | : The revision comment was: <tt>links to some edos in table of approximations to 3:2</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> | ||
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||= 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 \ [[5edo|5]] || 3.907 ||= 720.00 ||= 18.045 || | ||
||= 4 \ 7 || 4.807 ||= 685.7143 ||= -16.2407 || | ||= 4 \ [[7edo|7]] || 4.807 ||= 685.7143 ||= -16.2407 || | ||
||= 7 \ 12 || 6.392 ||= 700.00 ||= -1.955 || | ||= 7 \ [[12edo|12]] || 6.392 ||= 700.00 ||= -1.955 || | ||
||= 17 \ 29 || 8.945 ||= 703.4483 ||= 1.4933 || | ||= 17 \ [[29edo|29]] || 8.945 ||= 703.4483 ||= 1.4933 || | ||
||= 24 \ 41 || 9.943 ||= 702.43902 ||= 0.48402 ||</pre></div> | ||= 24 \ [[41edo|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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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">3 \ 5<br /> | <td style="text-align: center;">3 \ <a class="wiki_link" href="/5edo">5</a><br /> | ||
</td> | </td> | ||
<td>3.907<br /> | <td>3.907<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">4 \ 7<br /> | <td style="text-align: center;">4 \ <a class="wiki_link" href="/7edo">7</a><br /> | ||
</td> | </td> | ||
<td>4.807<br /> | <td>4.807<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">7 \ 12<br /> | <td style="text-align: center;">7 \ <a class="wiki_link" href="/12edo">12</a><br /> | ||
</td> | </td> | ||
<td>6.392<br /> | <td>6.392<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">17 \ 29<br /> | <td style="text-align: center;">17 \ <a class="wiki_link" href="/29edo">29</a><br /> | ||
</td> | </td> | ||
<td>8.945<br /> | <td>8.945<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">24 \ 41<br /> | <td style="text-align: center;">24 \ <a class="wiki_link" href="/41edo">41</a><br /> | ||
</td> | </td> | ||
<td>9.943<br /> | <td>9.943<br /> | ||