Seventeen limit tetrads

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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]]. Therefore, one approach for 17-limit harmony is to filter the total list of chords to exclude these small steps, resulting in non-rooted harmonies.

Every otonal tetrad within the 17-limit is listed in the table below as the simplest possible subset of harmonics 2-17; there are 56 in total. The columns to the right of the chords provide different cutoffs for filtering out small steps. For example, chords with a 16/15 cutoff do not contain 17/16 or 18/17 among the dyads. Since this eliminates harmonics 2 and 9 right away, it also eliminates chords containing 10/9 and 9/8. (This is why there is a jump from 12/11 to 9/8 in the table below.)
||~   ||~   ||~   ||~   ||~   ||~   ||~   ||~   ||
||~ no cutoff
56 tetrads ||~ 16/15 cutoff
20 tetrads ||~ 15/14 cutoff
16 tetrads ||~ 14/13 cutoff
13 tetrads ||~ 13/12 cutoff
10 tetrads ||~ 12/11 cutoff
7 tetrads ||~ 9/8 cutoff
4 tetrads ||~ 17/15 cutoff
1 tetrad ||
|| 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 || x || x || x || x || x || x || x ||
|| 3:5:9:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 3:5:11:17 || x || x || x || x ||   ||   ||   ||
|| 3:5:13:17 || x || x || x ||   ||   ||   ||   ||
|| 3:5:15:17 || x || x || x || x || x || x ||   ||
|| 3:7:9:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 3:7:11:17 || x || x || x || x ||   ||   ||   ||
|| 3:7:13:17 || x || x ||   ||   ||   ||   ||   ||
|| 3:7:15:17 || x ||   ||   ||   ||   ||   ||   ||
|| 3:9:11:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 3:9:13:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 3:9:15:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 3:11:13:17 || x || x || x ||   ||   ||   ||   ||
|| 3:11:15:17 || x || x || x || x ||   ||   ||   ||
|| 3:13:15:17 || x || x || x ||   ||   ||   ||   ||
|| 5:7:9:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 5:7:11:17 || x || x || x || x || x ||   ||   ||
|| 5:7:13:17 || x || x ||   ||   ||   ||   ||   ||
|| 5:7:15:17 || x ||   ||   ||   ||   ||   ||   ||
|| 5:9:11:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 5:9:13:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 5:9:15:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 5:11:13:17 || x || x || x || x || x ||   ||   ||
|| 5:11:15:17 || x || x || x || x || x ||   ||   ||
|| 5:13:15:17 || x || x || x || x || x || x ||   ||
|| 7:9:11:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 7:9:13:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 7:9:15:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 7:11:13:17 || x || x ||   ||   ||   ||   ||   ||
|| 7:11:15:17 || x ||   ||   ||   ||   ||   ||   ||
|| 7:13:15:17 || x ||   ||   ||   ||   ||   ||   ||
|| 9:11:13:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 9:11:15:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 9:13:15:17 ||   ||   ||   ||   ||   ||   ||   ||
|| 11:13:15:17 || x || x || x || x || x || x ||   ||
||~   ||~   ||~   ||~   ||~   ||~   ||~   ||~   ||

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>. Therefore, one approach for 17-limit harmony is to filter the total list of chords to exclude these small steps, resulting in non-rooted harmonies.<br />
<br />
Every otonal tetrad within the 17-limit is listed in the table below as the simplest possible subset of harmonics 2-17; there are 56 in total. The columns to the right of the chords provide different cutoffs for filtering out small steps. For example, chords with a 16/15 cutoff do not contain 17/16 or 18/17 among the dyads. Since this eliminates harmonics 2 and 9 right away, it also eliminates chords containing 10/9 and 9/8. (This is why there is a jump from 12/11 to 9/8 in the table below.)<br />


<table class="wiki_table">
    <tr>
        <th><br />
</th>
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</th>
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</th>
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</th>
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</th>
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</th>
    </tr>
    <tr>
        <th>no cutoff<br />
56 tetrads<br />
</th>
        <th>16/15 cutoff<br />
20 tetrads<br />
</th>
        <th>15/14 cutoff<br />
16 tetrads<br />
</th>
        <th>14/13 cutoff<br />
13 tetrads<br />
</th>
        <th>13/12 cutoff<br />
10 tetrads<br />
</th>
        <th>12/11 cutoff<br />
7 tetrads<br />
</th>
        <th>9/8 cutoff<br />
4 tetrads<br />
</th>
        <th>17/15 cutoff<br />
1 tetrad<br />
</th>
    </tr>
    <tr>
        <td>2:3:5:17<br />
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</td>
        <td>x<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>5:11:15:17<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>5:13:15:17<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>7:9:11:17<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>7:9:13:17<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>7:9:15:17<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>7:11:13:17<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>7:11:15:17<br />
</td>
        <td>x<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>7:13:15:17<br />
</td>
        <td>x<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>9:11:13:17<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>9:11:15:17<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>9:13:15:17<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>11:13:15:17<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td>x<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th><br />
</th>
        <th><br />
</th>
        <th><br />
</th>
        <th><br />
</th>
        <th><br />
</th>
        <th><br />
</th>
        <th><br />
</th>
        <th><br />
</th>
    </tr>
</table>

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