Xenharmonic Wiki

4L 5s

Revision as of 15:34, 23 May 2015 by Wikispaces>JosephRuhf (**Imported revision 551972212 - Original comment: **)

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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 ||=   ||
||   ||   ||   ||   || 9\40 ||= 4 5 4 5 4 5 4 5 4 || 270 ||   ||
||   ||   ||   || 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 ||= Optimum rank range (L/s=3/2) orwell ||
||   ||   ||   ||   || 13\57 ||= 5 8 5 8 5 8 5 8 5 || 273.684 ||= Golden orwell (bad tuning) ||
||   ||   ||   || 8\35 ||   ||= 3 5 3 5 3 5 3 5 3 || 274.286 ||=   ||
||   ||   ||   ||   || 11\48 ||= 4 7 4 7 4 7 4 7 4 || 275 ||   ||
||   || 3\13 ||   ||   ||   ||= 1 2 1 2 1 2 1 2 1 || 276.923 ||= Boundary of propriety:
generators smaller than this are proper ||
||   ||   ||   ||   || 10\43 ||= 3 7 3 7 3 7 3 7 3 || 279.07 ||   ||
||   ||   ||   || 7\30 ||   ||= 2 5 2 5 2 5 2 5 2 || 280.000 ||=   ||
||   ||   ||   ||   || 11\47 ||= 3 8 3 8 3 8 3 8 3 || 280.851 ||   ||
||   ||   ||   ||   ||   ||= 1 e 1 e 1 e 1 e 1 || 281.100 ||= <span style="display: block; text-align: center;">L/s = e</span> ||
||   ||   || 4\17 ||   ||   ||= 1 3 1 3 1 3 1 3 1 || 282.353 ||= L/s = 3 ||
||   ||   ||   ||   ||   ||= 1 pi 1 pi 1 pi 1 pi 1 || 282.922 ||= <span style="display: block; text-align: center;">L/s = pi</span> ||
||   ||   ||   ||   || 9\38 ||= 2 7 2 <span style="font-size: 12.8000001907349px; line-height: 1.5;">7 </span><span style="font-size: 13px; line-height: 1.5;">2 7 2 7 2 </span> || 284.2105 ||   ||
||   ||   ||   || 5\21 ||   ||= 1 4 1 4 1 4 1 4 1 || 285.714 ||= L/s = 4 ||
||   ||   ||   ||   || 6\25 ||= 1 5 1 5 1 5 1 5 1 || 288 ||   ||
|| 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 11:8, 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 style="text-align: center;">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><br />
</td>
        <td>9\40<br />
</td>
        <td style="text-align: center;">4 5 4 5 4 5 4 5 4<br />
</td>
        <td>270<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7\31<br />
</td>
        <td><br />
</td>
        <td style="text-align: center;">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 style="text-align: center;">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 style="text-align: center;">2 3 2 3 2 3 2 3 2<br />
</td>
        <td>272.727<br />
</td>
        <td style="text-align: center;">Optimum rank range (L/s=3/2) orwell<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>13\57<br />
</td>
        <td style="text-align: center;">5 8 5 8 5 8 5 8 5<br />
</td>
        <td>273.684<br />
</td>
        <td style="text-align: center;">Golden orwell (bad tuning)<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>8\35<br />
</td>
        <td><br />
</td>
        <td style="text-align: center;">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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>11\48<br />
</td>
        <td style="text-align: center;">4 7 4 7 4 7 4 7 4<br />
</td>
        <td>275<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>3\13<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td style="text-align: center;">1 2 1 2 1 2 1 2 1<br />
</td>
        <td>276.923<br />
</td>
        <td style="text-align: center;">Boundary of propriety:<br />
generators smaller than this are proper<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>10\43<br />
</td>
        <td style="text-align: center;">3 7 3 7 3 7 3 7 3<br />
</td>
        <td>279.07<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7\30<br />
</td>
        <td><br />
</td>
        <td style="text-align: center;">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><br />
</td>
        <td><br />
</td>
        <td>11\47<br />
</td>
        <td style="text-align: center;">3 8 3 8 3 8 3 8 3<br />
</td>
        <td>280.851<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td style="text-align: center;">1 e 1 e 1 e 1 e 1<br />
</td>
        <td>281.100<br />
</td>
        <td style="text-align: center;"><span style="display: block; text-align: center;">L/s = e</span><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>4\17<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td style="text-align: center;">1 3 1 3 1 3 1 3 1<br />
</td>
        <td>282.353<br />
</td>
        <td style="text-align: center;">L/s = 3<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td style="text-align: center;">1 pi 1 pi 1 pi 1 pi 1<br />
</td>
        <td>282.922<br />
</td>
        <td style="text-align: center;"><span style="display: block; text-align: center;">L/s = pi</span><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>9\38<br />
</td>
        <td style="text-align: center;">2 7 2 <span style="font-size: 12.8000001907349px; line-height: 1.5;">7 </span><span style="font-size: 13px; line-height: 1.5;">2 7 2 7 2 </span><br />
</td>
        <td>284.2105<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5\21<br />
</td>
        <td><br />
</td>
        <td style="text-align: center;">1 4 1 4 1 4 1 4 1<br />
</td>
        <td>285.714<br />
</td>
        <td style="text-align: center;">L/s = 4<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>6\25<br />
</td>
        <td style="text-align: center;">1 5 1 5 1 5 1 5 1<br />
</td>
        <td>288<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1\4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td style="text-align: center;">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 11:8, 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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