Nearest just interval: Difference between revisions

Revision as of 23:43, 8 November 2010

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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 sqrt(2) (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 2^(1/4) (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 &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. <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 &lt; q. <br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 -->Examples</h2>
The 600-cent interval sqrt(2) (6 steps of <a class="wiki_link" href="/12edo">12edo</a>, &quot;Tritone&quot;) 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>&quot;error&quot;</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 2^(1/4) (3 steps of <a class="wiki_link" href="/12edo">12edo</a>, &quot;minor third&quot;) 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>&quot;error&quot;</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>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11-08 23:41:26 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11-08 23:43:29 UTC</tt>.<br>
-: The original revision id was <tt>177732685</tt>.<br>
+: The original revision id was <tt>177732917</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>
@@ Line 13: / Line 13: @@
 ==Examples==
-The 600-cent interval (6 steps of [[12edo]], "Tritone") approximates following ratios:
+The 600-cent interval sqrt(2) (6 steps of [[12edo]], "Tritone") approximates following ratios:
 || **freq. ratio** || **log([[Tenney Height]])** || **size** in cents || **"error"** in cents ||
 || ... || ... || ... || ... ||
@@ Line 21: / Line 21: @@
 || ... || ... || ... || ... ||
-The 300-cent interval (3 steps of [[12edo]], "minor third") approximates following ratios:
+The 300-cent interval 2^(1/4) (3 steps of [[12edo]], "minor third") approximates following ratios:
 || **freq. ratio** || **log([[Tenney Height]])** || **size** in cents || **"error"** in cents ||
 || ... || ... || ... || ... ||
@@ Line 37: / Line 37: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Examples&lt;/h2&gt;
-The 600-cent interval (6 steps of &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &amp;quot;Tritone&amp;quot;) approximates following ratios:&lt;br /&gt;
+The 600-cent interval sqrt(2) (6 steps of &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &amp;quot;Tritone&amp;quot;) approximates following ratios:&lt;br /&gt;
@@ Line 104: / Line 104: @@
 &lt;br /&gt;
-The 300-cent interval (3 steps of &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &amp;quot;minor third&amp;quot;) approximates following ratios:&lt;br /&gt;
+The 300-cent interval 2^(1/4) (3 steps of &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &amp;quot;minor third&amp;quot;) approximates following ratios:&lt;br /&gt;

Nearest just interval: Difference between revisions

Revision as of 23:43, 8 November 2010

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