3L 3s: Difference between revisions

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Wikispaces>JosephRuhf
**Imported revision 551904666 - Original comment: **
Wikispaces>JosephRuhf
**Imported revision 564967897 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2015-05-22 13:14:22 UTC</tt>.<br>
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2015-11-03 12:51:23 UTC</tt>.<br>
: The original revision id was <tt>551904666</tt>.<br>
: The original revision id was <tt>564967897</tt>.<br>
: The revision comment was: <tt></tt><br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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||  ||  || 1\12 ||  ||  || 100 ||= L/s = 3 ||
||  ||  || 1\12 ||  ||  || 100 ||= L/s = 3 ||
||  ||  ||  ||  ||  || 400/(1+e) ||  ||
||  ||  ||  ||  ||  || 400/(1+e) ||  ||
||  ||  ||  ||  || 3\33 || 109.09 ||  ||
||  ||  ||  ||  ||  || 400/(2+phi) ||  ||
||  ||  ||  || 2\21 ||  || 114.29 ||  ||
||  || 1\9 ||  ||  ||  || 133.33 ||= Boundary of propriety for near-MOS
||  || 1\9 ||  ||  ||  || 133.33 ||= Boundary of propriety for near-MOS
Optimum rank range (L/s=2/1) for MOS ||
Optimum rank range (L/s=2/1) for MOS ||
||  ||  ||  ||  ||  || 400/(1+sqrt(3)) ||  ||
||  ||  ||  || 3\24 ||  || 150 ||=  ||
||  ||  ||  || 3\24 ||  || 150 ||=  ||
||  ||  ||  ||  ||  || 400/(1+phi) ||= Golden triforce ||
||  ||  ||  ||  ||  || 400/(1+phi) ||= Golden triforce ||
||  ||  ||  ||  || 5\39 || 153.85 ||=  ||
||  ||  ||  ||  || 5\39 || 153.85 ||=  ||
||  ||  ||  ||  ||  || 400/(1+pi/2) ||  ||
||  ||  || 2\15 ||  ||  || 160 ||=  ||
||  ||  || 2\15 ||  ||  || 160 ||=  ||
|| 1\6 ||  ||  ||  ||  || 200 ||=  ||
|| 1\6 ||  ||  ||  ||  || 200 ||=  ||
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&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;400/(1+e)&lt;br /&gt;
         &lt;td&gt;400/(1+e)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
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    &lt;/tr&gt;
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&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3\33&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;109.09&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
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    &lt;/tr&gt;
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        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;400/(2+phi)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
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        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2\21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;114.29&lt;br /&gt;
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         &lt;td style="text-align: center;"&gt;Boundary of propriety for near-MOS&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Boundary of propriety for near-MOS&lt;br /&gt;
Optimum rank range (L/s=2/1) for MOS&lt;br /&gt;
Optimum rank range (L/s=2/1) for MOS&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
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&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;400/(1+sqrt(3))&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
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&lt;/td&gt;
&lt;/td&gt;
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        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;400/(1+pi/2)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;

Revision as of 12:51, 3 November 2015

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 2015-11-03 12:51:23 UTC.
The original revision id was 564967897.
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:

Associated with the [[augmented family]], which comprises the only significant minimum of harmonic entropy for these scales.
||||||||||~ Generator ||~ Cents ||~ Comments ||
|| 0\3 ||   ||   ||   ||   || 0 ||=   ||
||   ||   ||   || 1\15 ||   || 80 ||= L/s = 4 ||
||   ||   ||   ||   || 2\27 || 88.88 ||= Optimal augmented is around here ||
||   ||   ||   ||   ||   || 400/(1+pi) ||   ||
||   ||   || 1\12 ||   ||   || 100 ||= L/s = 3 ||
||   ||   ||   ||   ||   || 400/(1+e) ||   ||
||   ||   ||   ||   || 3\33 || 109.09 ||   ||
||   ||   ||   ||   ||   || 400/(2+phi) ||   ||
||   ||   ||   || 2\21 ||   || 114.29 ||   ||
||   || 1\9 ||   ||   ||   || 133.33 ||= Boundary of propriety for near-MOS
Optimum rank range (L/s=2/1) for MOS ||
||   ||   ||   ||   ||   || 400/(1+sqrt(3)) ||   ||
||   ||   ||   || 3\24 ||   || 150 ||=   ||
||   ||   ||   ||   ||   || 400/(1+phi) ||= Golden triforce ||
||   ||   ||   ||   || 5\39 || 153.85 ||=   ||
||   ||   ||   ||   ||   || 400/(1+pi/2) ||   ||
||   ||   || 2\15 ||   ||   || 160 ||=   ||
|| 1\6 ||   ||   ||   ||   || 200 ||=   ||
The true MOS is LsLsLs but interesting near-MOSs include LLsLss and LLssLs. The MOS is always proper but the other forms are only proper if the generator is larger than 1\9 of an octave (so not augmented).

Out of all **[[Rothenberg propriety|proper]]** six-note MOS scales, this augmented scale probably has the lowest harmonic entropy.

Original HTML content:

<html><head><title>3L 3s</title></head><body>Associated with the <a class="wiki_link" href="/augmented%20family">augmented family</a>, which comprises the only significant minimum of harmonic entropy for these scales.<br />


<table class="wiki_table">
    <tr>
        <th colspan="5">Generator<br />
</th>
        <th>Cents<br />
</th>
        <th>Comments<br />
</th>
    </tr>
    <tr>
        <td>0\3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>0<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1\15<br />
</td>
        <td><br />
</td>
        <td>80<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>2\27<br />
</td>
        <td>88.88<br />
</td>
        <td style="text-align: center;">Optimal augmented is around here<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>400/(1+pi)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>1\12<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>100<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>400/(1+e)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3\33<br />
</td>
        <td>109.09<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>400/(2+phi)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2\21<br />
</td>
        <td><br />
</td>
        <td>114.29<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>1\9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>133.33<br />
</td>
        <td style="text-align: center;">Boundary of propriety for near-MOS<br />
Optimum rank range (L/s=2/1) for MOS<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>400/(1+sqrt(3))<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3\24<br />
</td>
        <td><br />
</td>
        <td>150<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>400/(1+phi)<br />
</td>
        <td style="text-align: center;">Golden triforce<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5\39<br />
</td>
        <td>153.85<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>400/(1+pi/2)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>2\15<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>160<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td>1\6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>200<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

The true MOS is LsLsLs but interesting near-MOSs include LLsLss and LLssLs. The MOS is always proper but the other forms are only proper if the generator is larger than 1\9 of an octave (so not augmented).<br />
<br />
Out of all <strong><a class="wiki_link" href="/Rothenberg%20propriety">proper</a></strong> six-note MOS scales, this augmented scale probably has the lowest harmonic entropy.</body></html>
Retrieved from "https://en.xen.wiki/index.php?title=3L_3s&oldid=8610"