Seventeen limit tetrads

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This revision was by author Andrew_Heathwaite and made on 2012-03-18 11:55:34 UTC.
The original revision id was 312094974.
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Original Wikitext content:

The 17th harmonic, octave reduced to the frequency ratio [[17_16|17/16]], is about 105¢. It is likely to sound dissonant against the fundamental, which is perhaps one reason why [[Just Intonation]] composers usually stop at the [[13-limit]] or lower. Another interval of 17 that can sound just as dissonant is [[18_17|18/17]], about 99¢. Thus, 17/16 also clashes with [[9_8|9/8]]. However, if we don't mind some clashing from semitone-sized harmonic intervals, we can make the following list of 56 17-limit tetrads.

||~ Simplest open position voicing with harmonics 2-17 ||
||~ subsets of 2:3:5:7:9:11:13:15:17 ||
|| 2:3:5:17 ||
|| 2:3:7:17 ||
|| 2:3:9:17 ||
|| 2:3:11:17 ||
|| 2:3:13:17 ||
|| 2:3:15:17 ||
|| 2:5:7:17 ||
|| 2:5:9:17 ||
|| 2:5:11:17 ||
|| 2:5:13:17 ||
|| 2:5:15:17 ||
|| 2:7:9:17 ||
|| 2:7:11:17 ||
|| 2:7:13:17 ||
|| 2:7:15:17 ||
|| 2:9:11:17 ||
|| 2:9:13:17 ||
|| 2:9:15:17 ||
|| 2:11:13:17 ||
|| 2:11:15:17 ||
|| 2:13:15:17 ||
|| 3:5:7:17 ||
|| 3:5:9:17 ||
|| 3:5:11:17 ||
|| 3:5:13:17 ||
|| 3:5:15:17 ||
|| 3:7:9:17 ||
|| 3:7:11:17 ||
|| 3:7:13:17 ||
|| 3:7:15:17 ||
|| 3:9:11:17 ||
|| 3:9:13:17 ||
|| 3:9:15:17 ||
|| 3:11:13:17 ||
|| 3:11:15:17 ||
|| 3:13:15:17 ||
|| 5:7:9:17 ||
|| 5:7:11:17 ||
|| 5:7:13:17 ||
|| 5:7:15:17 ||
|| 5:9:11:17 ||
|| 5:9:13:17 ||
|| 5:9:15:17 ||
|| 5:11:13:17 ||
|| 5:11:15:17 ||
|| 5:13:15:17 ||
|| 7:9:11:17 ||
|| 7:9:13:17 ||
|| 7:9:15:17 ||
|| 7:11:13:17 ||
|| 7:11:15:17 ||
|| 7:13:15:17 ||
|| 9:11:13:17 ||
|| 9:11:15:17 ||
|| 9:13:15:17 ||
|| 11:13:15:17 ||
||~   ||

Original HTML content:

<html><head><title>seventeen limit tetrads</title></head><body>The 17th harmonic, octave reduced to the frequency ratio <a class="wiki_link" href="/17_16">17/16</a>, is about 105¢. It is likely to sound dissonant against the fundamental, which is perhaps one reason why <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a> composers usually stop at the <a class="wiki_link" href="/13-limit">13-limit</a> or lower. Another interval of 17 that can sound just as dissonant is <a class="wiki_link" href="/18_17">18/17</a>, about 99¢. Thus, 17/16 also clashes with <a class="wiki_link" href="/9_8">9/8</a>. However, if we don't mind some clashing from semitone-sized harmonic intervals, we can make the following list of 56 17-limit tetrads.<br />
<br />


<table class="wiki_table">
    <tr>
        <th>Simplest open position voicing with harmonics 2-17<br />
</th>
    </tr>
    <tr>
        <th>subsets of 2:3:5:7:9:11:13:15:17<br />
</th>
    </tr>
    <tr>
        <td>2:3:5:17<br />
</td>
    </tr>
    <tr>
        <td>2:3:7:17<br />
</td>
    </tr>
    <tr>
        <td>2:3:9:17<br />
</td>
    </tr>
    <tr>
        <td>2:3:11:17<br />
</td>
    </tr>
    <tr>
        <td>2:3:13:17<br />
</td>
    </tr>
    <tr>
        <td>2:3:15:17<br />
</td>
    </tr>
    <tr>
        <td>2:5:7:17<br />
</td>
    </tr>
    <tr>
        <td>2:5:9:17<br />
</td>
    </tr>
    <tr>
        <td>2:5:11:17<br />
</td>
    </tr>
    <tr>
        <td>2:5:13:17<br />
</td>
    </tr>
    <tr>
        <td>2:5:15:17<br />
</td>
    </tr>
    <tr>
        <td>2:7:9:17<br />
</td>
    </tr>
    <tr>
        <td>2:7:11:17<br />
</td>
    </tr>
    <tr>
        <td>2:7:13:17<br />
</td>
    </tr>
    <tr>
        <td>2:7:15:17<br />
</td>
    </tr>
    <tr>
        <td>2:9:11:17<br />
</td>
    </tr>
    <tr>
        <td>2:9:13:17<br />
</td>
    </tr>
    <tr>
        <td>2:9:15:17<br />
</td>
    </tr>
    <tr>
        <td>2:11:13:17<br />
</td>
    </tr>
    <tr>
        <td>2:11:15:17<br />
</td>
    </tr>
    <tr>
        <td>2:13:15:17<br />
</td>
    </tr>
    <tr>
        <td>3:5:7:17<br />
</td>
    </tr>
    <tr>
        <td>3:5:9:17<br />
</td>
    </tr>
    <tr>
        <td>3:5:11:17<br />
</td>
    </tr>
    <tr>
        <td>3:5:13:17<br />
</td>
    </tr>
    <tr>
        <td>3:5:15:17<br />
</td>
    </tr>
    <tr>
        <td>3:7:9:17<br />
</td>
    </tr>
    <tr>
        <td>3:7:11:17<br />
</td>
    </tr>
    <tr>
        <td>3:7:13:17<br />
</td>
    </tr>
    <tr>
        <td>3:7:15:17<br />
</td>
    </tr>
    <tr>
        <td>3:9:11:17<br />
</td>
    </tr>
    <tr>
        <td>3:9:13:17<br />
</td>
    </tr>
    <tr>
        <td>3:9:15:17<br />
</td>
    </tr>
    <tr>
        <td>3:11:13:17<br />
</td>
    </tr>
    <tr>
        <td>3:11:15:17<br />
</td>
    </tr>
    <tr>
        <td>3:13:15:17<br />
</td>
    </tr>
    <tr>
        <td>5:7:9:17<br />
</td>
    </tr>
    <tr>
        <td>5:7:11:17<br />
</td>
    </tr>
    <tr>
        <td>5:7:13:17<br />
</td>
    </tr>
    <tr>
        <td>5:7:15:17<br />
</td>
    </tr>
    <tr>
        <td>5:9:11:17<br />
</td>
    </tr>
    <tr>
        <td>5:9:13:17<br />
</td>
    </tr>
    <tr>
        <td>5:9:15:17<br />
</td>
    </tr>
    <tr>
        <td>5:11:13:17<br />
</td>
    </tr>
    <tr>
        <td>5:11:15:17<br />
</td>
    </tr>
    <tr>
        <td>5:13:15:17<br />
</td>
    </tr>
    <tr>
        <td>7:9:11:17<br />
</td>
    </tr>
    <tr>
        <td>7:9:13:17<br />
</td>
    </tr>
    <tr>
        <td>7:9:15:17<br />
</td>
    </tr>
    <tr>
        <td>7:11:13:17<br />
</td>
    </tr>
    <tr>
        <td>7:11:15:17<br />
</td>
    </tr>
    <tr>
        <td>7:13:15:17<br />
</td>
    </tr>
    <tr>
        <td>9:11:13:17<br />
</td>
    </tr>
    <tr>
        <td>9:11:15:17<br />
</td>
    </tr>
    <tr>
        <td>9:13:15:17<br />
</td>
    </tr>
    <tr>
        <td>11:13:15:17<br />
</td>
    </tr>
    <tr>
        <th><br />
</th>
    </tr>
</table>

</body></html>
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