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Nearest just interval

Revision as of 03:38, 10 November 2010 by Wikispaces>xenwolf (**Imported revision 178159803 - Original comment: approximation of fifth by prime deleted**)

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The revision comment was: approximation of fifth by prime deleted

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Original Wikitext content:

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. 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.

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** || **log2([[Tenney Height]])** || **size** in cents || **"error"** in cents ||
|| ... || ... || ... || ... ||
||= 1 / 1 ||= 0.0 ||= 0.0 ||= 600.0||
||= 3 / 2 ||= 2.585 ||= 701.96 ||= 101.96 ||
||= 4 / 3 ||= 3.585 ||= 498.04 ||= 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 ||
|| ... || ... || ... || ... ||
||= 1 / 1 ||= 0.0 ||= 0.0 ||= 300.0||
||= 6 / 5 ||= 4.907 ||= 315.64 ||= 15.64 ||
||= 13 / 11 ||= 7.160 ||= 289.21 ||= 10.79 ||
||= 19 / 16 ||= 8.248 ||= 297.51 ||= 2.49 ||
||= 25 / 21 ||= 9.036 ||= 301.84 ||= 1.84 ||
|| ... || ... || ... || ... ||

The best rational approximations to log2(3/2) define edos which have especially good approximations to the fifth (701,955000865... cents):
|| **freq. ratio** || **log([[Tenney Height]])** || **size** in cents || **"error"** in cents ||
|| ... || ... || ... || ... ||
||= 1 / 1 || 0.0 ||= 1200.0 ||= 498.04 ||
||= 1 / 2 || 1.0 ||= 600.00 ||= 101.96 ||
||= 2 / 3 || 2.585 ||= 800.00 ||= 98.045 ||
||= 3 / 5 || 3.907 ||= 720.00 ||= 18.045 ||
||= 4 / 7 || 4.807 ||= 685.71 ||= 16.241 ||
||= 7 / 12 || 6.392 ||= 700.00 ||= 1.9550 ||
||= 17 / 29 || 8.945 ||= 703.45 ||= 1.4933 ||
||= 24 / 41 || 9.943 ||= 702.44 ||= 0.48402 ||

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. 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 />
<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>log2(<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;">1 / 1<br />
</td>
        <td style="text-align: center;">0.0<br />
</td>
        <td style="text-align: center;">0.0<br />
</td>
        <td style="text-align: center;">600.0<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;">4 / 3<br />
</td>
        <td style="text-align: center;">3.585<br />
</td>
        <td style="text-align: center;">498.04<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;">1 / 1<br />
</td>
        <td style="text-align: center;">0.0<br />
</td>
        <td style="text-align: center;">0.0<br />
</td>
        <td style="text-align: center;">300.0<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;">13 / 11<br />
</td>
        <td style="text-align: center;">7.160<br />
</td>
        <td style="text-align: center;">289.21<br />
</td>
        <td style="text-align: center;">10.79<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>

<br />
The best rational approximations to log2(3/2) define edos which have especially good approximations to the fifth (701,955000865... cents):<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;">1 / 1<br />
</td>
        <td>0.0<br />
</td>
        <td style="text-align: center;">1200.0<br />
</td>
        <td style="text-align: center;">498.04<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1 / 2<br />
</td>
        <td>1.0<br />
</td>
        <td style="text-align: center;">600.00<br />
</td>
        <td style="text-align: center;">101.96<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">2 / 3<br />
</td>
        <td>2.585<br />
</td>
        <td style="text-align: center;">800.00<br />
</td>
        <td style="text-align: center;">98.045<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3 / 5<br />
</td>
        <td>3.907<br />
</td>
        <td style="text-align: center;">720.00<br />
</td>
        <td style="text-align: center;">18.045<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4 / 7<br />
</td>
        <td>4.807<br />
</td>
        <td style="text-align: center;">685.71<br />
</td>
        <td style="text-align: center;">16.241<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">7 / 12<br />
</td>
        <td>6.392<br />
</td>
        <td style="text-align: center;">700.00<br />
</td>
        <td style="text-align: center;">1.9550<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">17 / 29<br />
</td>
        <td>8.945<br />
</td>
        <td style="text-align: center;">703.45<br />
</td>
        <td style="text-align: center;">1.4933<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">24 / 41<br />
</td>
        <td>9.943<br />
</td>
        <td style="text-align: center;">702.44<br />
</td>
        <td style="text-align: center;">0.48402<br />
</td>
    </tr>
</table>

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