Xenharmonic Wiki

4L 5s

Revision as of 13:29, 11 May 2011 by Wikispaces>keenanpepper (**Imported revision 227574018 - Original comment: **)

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This revision was by author keenanpepper and made on 2011-05-11 13:29:44 UTC.
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Original Wikitext content:

4L 5s refers to the structure of [[MOSScales|MOS Scales]] whose generator falls between 2\9 (two degrees of [[9edo]] = approx. 266.667¢) and 1\4 (one degree of [[4edo]] = 300¢). In the case of 9edo, L and s are the same size; in the case of 4edo, s is so small it disappears. The spectrum, then, goes something like:

||||||||||~ Generator ||~ Scale ||~ Generator in cents ||~ Comments ||
|| 2\9 ||   ||   ||   ||   || 1 1 1 1 1 1 1 1 1 || 266.667 ||=   ||
||   ||   ||   || 7\31 ||   || 3 4 3 4 3 4 3 4 3 || 270.968 ||=   ||
||   ||   ||   ||   || 12\53 || 5 7 5 7 5 7 5 7 5 || 271.698 ||= Orwell is around here ||
||   ||   || 5\22 ||   ||   || 2 3 2 3 2 3 2 3 2 || 272.727 ||=   ||
||   ||   ||   || 8\35 ||   || 3 5 3 5 3 5 3 5 3 || 274.286 ||=   ||
||   || 3\13 ||   ||   ||   || 1 2 1 2 1 2 1 2 1 || 276.923 ||= Boundary of propriety: generators
smaller than this are proper ||
||   ||   ||   || 7\30 ||   || 2 5 2 5 2 5 2 5 2 || 280.000 ||=   ||
||   ||   || 4\17 ||   ||   || 1 3 1 3 1 3 1 3 1 || 282.353 ||=   ||
||   ||   ||   || 5\21 ||   || 1 4 1 4 1 4 1 4 1 || 285.714 ||=   ||
|| 1\4 ||   ||   ||   ||   || 0 1 0 1 0 1 0 1 0 || 300.000 ||=   ||

Note that between 7\31 and 5\22, g approximates frequency ratio 7:6, 2g approximates 16:11, and 3g approximates 8:5. This defines the range of Orwell Temperament, which is the only notable harmonic entropy minimum with this MOS pattern. 4L 5s scales outside of that range are not suitable for Orwell.

Original HTML content:

<html><head><title>4L 5s</title></head><body>4L 5s refers to the structure of <a class="wiki_link" href="/MOSScales">MOS Scales</a> whose generator falls between 2\9 (two degrees of <a class="wiki_link" href="/9edo">9edo</a> = approx. 266.667¢) and 1\4 (one degree of <a class="wiki_link" href="/4edo">4edo</a> = 300¢). In the case of 9edo, L and s are the same size; in the case of 4edo, s is so small it disappears. The spectrum, then, goes something like:<br />
<br />


<table class="wiki_table">
    <tr>
        <th colspan="5">Generator<br />
</th>
        <th>Scale<br />
</th>
        <th>Generator in cents<br />
</th>
        <th>Comments<br />
</th>
    </tr>
    <tr>
        <td>2\9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1 1 1 1 1 1 1 1 1<br />
</td>
        <td>266.667<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7\31<br />
</td>
        <td><br />
</td>
        <td>3 4 3 4 3 4 3 4 3<br />
</td>
        <td>270.968<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>12\53<br />
</td>
        <td>5 7 5 7 5 7 5 7 5<br />
</td>
        <td>271.698<br />
</td>
        <td style="text-align: center;">Orwell is around here<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>5\22<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2 3 2 3 2 3 2 3 2<br />
</td>
        <td>272.727<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>8\35<br />
</td>
        <td><br />
</td>
        <td>3 5 3 5 3 5 3 5 3<br />
</td>
        <td>274.286<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>3\13<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1 2 1 2 1 2 1 2 1<br />
</td>
        <td>276.923<br />
</td>
        <td style="text-align: center;">Boundary of propriety: generators<br />
smaller than this are proper<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7\30<br />
</td>
        <td><br />
</td>
        <td>2 5 2 5 2 5 2 5 2<br />
</td>
        <td>280.000<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>4\17<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1 3 1 3 1 3 1 3 1<br />
</td>
        <td>282.353<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5\21<br />
</td>
        <td><br />
</td>
        <td>1 4 1 4 1 4 1 4 1<br />
</td>
        <td>285.714<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td>1\4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>0 1 0 1 0 1 0 1 0<br />
</td>
        <td>300.000<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

<br />
Note that between 7\31 and 5\22, g approximates frequency ratio 7:6, 2g approximates 16:11, and 3g approximates 8:5. This defines the range of Orwell Temperament, which is the only notable harmonic entropy minimum with this MOS pattern. 4L 5s scales outside of that range are not suitable for Orwell.</body></html>
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