Nearest just interval
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An irrational interval or ratio of frequencies given by a real number r has an infinite list of //nearest just intervals//; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call //best rational approximations//. A ratio of integers p/q with q > 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n < 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. 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. ==Examples== The 600-cent interval (6 steps of [[12edo]], "Tritone") approximates following ratios: || **freq. ratio** || **log([[Tenney Height]])** || **size** in cents || **"error"** in cents || || ... || ... || ... || ... || ||= 3 / 2 ||= 2.585 ||= 701.96 ||= 101.96 || ||= 7 / 5 ||= 5.129 ||= 582.51 ||= 17.49 || ||= 17 / 12 ||= 7.672 ||= 603.00 ||= 3.000 || || ... || ... || ... || ... || The 300-cent interval (3 steps of [[12edo]], "minor third") approximates following ratios: || **freq. ratio** || **log([[Tenney Height]])** || **size** in cents || **"error"** in cents || || ... || ... || ... || ... || ||= 6 / 5 ||= 4.907 ||= 315.64 ||= 15.64 || ||= 19 / 16 ||= 8.248 ||= 297.51 ||= 2.49 || ||= 25 / 21 ||= 9.036 ||= 301.84 ||= 1.84 || || ... || ... || ... || ... ||
Original HTML content:
<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 > 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n < 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. <br />
<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 />
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 < q. <br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 -->Examples</h2>
The 600-cent interval (6 steps of <a class="wiki_link" href="/12edo">12edo</a>, "Tritone") approximates following ratios:<br />
<table class="wiki_table">
<tr>
<td><strong>freq. ratio</strong><br />
</td>
<td><strong>log(<a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)</strong><br />
</td>
<td><strong>size</strong> in cents<br />
</td>
<td><strong>"error"</strong> in cents<br />
</td>
</tr>
<tr>
<td>...<br />
</td>
<td>...<br />
</td>
<td>...<br />
</td>
<td>...<br />
</td>
</tr>
<tr>
<td style="text-align: center;">3 / 2<br />
</td>
<td style="text-align: center;">2.585<br />
</td>
<td style="text-align: center;">701.96<br />
</td>
<td style="text-align: center;">101.96<br />
</td>
</tr>
<tr>
<td style="text-align: center;">7 / 5<br />
</td>
<td style="text-align: center;">5.129<br />
</td>
<td style="text-align: center;">582.51<br />
</td>
<td style="text-align: center;">17.49<br />
</td>
</tr>
<tr>
<td style="text-align: center;">17 / 12<br />
</td>
<td style="text-align: center;">7.672<br />
</td>
<td style="text-align: center;">603.00<br />
</td>
<td style="text-align: center;">3.000<br />
</td>
</tr>
<tr>
<td>...<br />
</td>
<td>...<br />
</td>
<td>...<br />
</td>
<td>...<br />
</td>
</tr>
</table>
<br />
The 300-cent interval (3 steps of <a class="wiki_link" href="/12edo">12edo</a>, "minor third") approximates following ratios:<br />
<table class="wiki_table">
<tr>
<td><strong>freq. ratio</strong><br />
</td>
<td><strong>log(<a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)</strong><br />
</td>
<td><strong>size</strong> in cents<br />
</td>
<td><strong>"error"</strong> in cents<br />
</td>
</tr>
<tr>
<td>...<br />
</td>
<td>...<br />
</td>
<td>...<br />
</td>
<td>...<br />
</td>
</tr>
<tr>
<td style="text-align: center;">6 / 5<br />
</td>
<td style="text-align: center;">4.907<br />
</td>
<td style="text-align: center;">315.64<br />
</td>
<td style="text-align: center;">15.64<br />
</td>
</tr>
<tr>
<td style="text-align: center;">19 / 16<br />
</td>
<td style="text-align: center;">8.248<br />
</td>
<td style="text-align: center;">297.51<br />
</td>
<td style="text-align: center;">2.49<br />
</td>
</tr>
<tr>
<td style="text-align: center;">25 / 21<br />
</td>
<td style="text-align: center;">9.036<br />
</td>
<td style="text-align: center;">301.84<br />
</td>
<td style="text-align: center;">1.84<br />
</td>
</tr>
<tr>
<td>...<br />
</td>
<td>...<br />
</td>
<td>...<br />
</td>
<td>...<br />
</td>
</tr>
</table>
</body></html>