4L 5s

From Xenharmonic Wiki
Revision as of 10:16, 17 May 2014 by Wikispaces>JosephRuhf (**Imported revision 509514758 - Original comment: **)
Jump to navigation Jump to search

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author JosephRuhf and made on 2014-05-17 10:16:46 UTC.
The original revision id was 509514758.
The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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 || 5 4 5 4 5 4 5 4 5 || 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 ||   ||
||   ||   || 4\17 ||   ||   || 1 3 1 3 1 3 1 3 1 || 282.353 ||= L/s = 3 ||
||   ||   ||   ||   || 9\38 || 2 7 2 2 7 2 7 2 7 || 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>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>5 4 5 4 5 4 5 4 5<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>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;">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>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>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>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>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>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>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>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>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;">L/s = 3<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>9\38<br />
</td>
        <td>2 7 2 2 7 2 7 2 7<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>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>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>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>
Retrieved from "https://en.xen.wiki/index.php?title=4L_5s&oldid=9151"