4L 3s: Difference between revisions

Revision as of 14:06, 22 May 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 2015-05-22 14:06:00 UTC.

The original revision id was 551910764.

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 3s refers to the structure of moment of symmetry scales with generators ranging from 1\4edo (one degree of [[4edo]], 300¢) to 2\7edo (two degrees of [[7edo]], or approx. 342.857¢). The spectrum looks like this:
||||||||||||||~ Generator ||~ Tetrachord ||~ g in cents ||~ 2g ||~ 3g ||~ 4g ||~ Comments ||
|| 1\4 ||   ||   ||   ||   ||   ||   || 1 0 1 || 300.000 || 600.000 || 900.000 || 0.000 ||=   ||
||   ||   ||   ||   ||   ||   || 8\31 || 7 1 7 || 309.677 || 619.355 || 929.023 || 38.71 ||= Myna is around here ||
||   ||   ||   ||   ||   || 7\27 ||   || 6 1 6 || 311.111 || 622.222 || 933.333 || 44.444 ||=   ||
||   ||   ||   ||   || 6\23 ||   ||   || 5 1 5 || 313.043 || 626.087 || 939.13 || 52.174 ||=   ||
||   ||   ||   || 5\19 ||   ||   ||   || 4 1 4 || 315.789 || 631.579 || 947.368 || 63.158 ||= L/s = 4 ||
||   ||   ||   ||   || 9\34 ||   ||   || 7 2 7 || 317.647 || 634.294 || 951.941 || 70.588 ||= Hanson/Keemun is around here ||
||   ||   ||   ||   ||   ||   ||   || pi 1 pi || 319.272 || 638.545 || 957.817 || 77.089 ||= <span style="display: block; text-align: center;">L/s = pi</span> ||
||   ||   || 4\15 ||   ||   ||   ||   || 3 1 3 || 320.000 || 640.000 || 960.000 || 80.000 ||= L/s = 3 ||
||   ||   ||   ||   ||   ||   ||   || e 1 e || 321.6245 || 641.249 || 964.874 || 86.498 || <span style="display: block; text-align: center;">L/s = e</span> ||
||   ||   ||   || 7\26 ||   ||   ||   || 5 2 5 || 323.077 || 646.154 || 969.231 || 92.308 ||= Orgone is around here ||
||   || 3\11 ||   ||   ||   ||   ||   || 2 1 2 || 327.273 || 654.545 || 981.818 || 109.091 ||= Boundary of propriety (generators
larger than this are proper) ||
||   ||   ||   || 8\29 ||   ||   ||   || 5 3 5 || 331.034 || 662.069 || 993.013 || 124.138 ||=   ||
||   ||   ||   ||   ||   || 21\76 ||   || 13 8 13 || 331.579 || 663.158 || 994.739 || 126.316 ||=   ||
||   ||   ||   ||   ||   ||   || 34\123 || 21 13 21 || 331.707 || 663.415 || 995.122 || 126.829 ||= Unnamed golden temperament ||
||   ||   ||   ||   || 13\47 ||   ||   || 8 5 8 || 331.915 || 663.83 || 995.745 || 127.66 ||=   ||
||   ||   || 5\18 ||   ||   ||   ||   || 3 2 3 || 333.333 || 666.667 || 1000.000 || 133.333 ||= Optimum rank range (L/s=3/2) ||
||   ||   ||   || 7\25 ||   ||   ||   || 4 3 4 || 336.000 || 672.000 || 1008.000 || 144.000 ||=   ||
||   ||   ||   ||   || 9\32 ||   ||   || 5 4 5 || 337.5 || 675.000 || 1012.500 || 150.000 ||= Sixix ||
||   ||   ||   ||   ||   || 11\39 ||   || 6 5 6 || 338.462 || 676.923 || 1015.385 || 153.846 ||= Sixix ||
||   ||   ||   ||   ||   ||   || 13\46 || 7 6 7 || 339.130 || 678.261 || 1017.391 || 156.522 ||= (17/14)^3=9/5 ||
|| 2\7 ||   ||   ||   ||   ||   ||   || 1 1 1 || 342.857 || 685.714 || 1028.571 || 171.429 ||=   ||
There are two notable harmonic entropy minima: [[Kleismic family|hanson/keemun]], in which the generator is 6/5 and 6 of them make a 3/1, and [[Starling temperaments|myna]], in which the generator is also 6/5 but now **10** of them make a 6/1 (so no 4/3's or 3/2's appear in this scale).

Original HTML content:

<html><head><title>4L 3s</title></head><body>4L 3s refers to the structure of moment of symmetry scales with generators ranging from 1\4edo (one degree of <a class="wiki_link" href="/4edo">4edo</a>, 300¢) to 2\7edo (two degrees of <a class="wiki_link" href="/7edo">7edo</a>, or approx. 342.857¢). The spectrum looks like this:<br />


<table class="wiki_table">
    <tr>
        <th colspan="7">Generator<br />
</th>
        <th>Tetrachord<br />
</th>
        <th>g in cents<br />
</th>
        <th>2g<br />
</th>
        <th>3g<br />
</th>
        <th>4g<br />
</th>
        <th>Comments<br />
</th>
    </tr>
    <tr>
        <td>1\4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1 0 1<br />
</td>
        <td>300.000<br />
</td>
        <td>600.000<br />
</td>
        <td>900.000<br />
</td>
        <td>0.000<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>8\31<br />
</td>
        <td>7 1 7<br />
</td>
        <td>309.677<br />
</td>
        <td>619.355<br />
</td>
        <td>929.023<br />
</td>
        <td>38.71<br />
</td>
        <td style="text-align: center;">Myna is around here<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7\27<br />
</td>
        <td><br />
</td>
        <td>6 1 6<br />
</td>
        <td>311.111<br />
</td>
        <td>622.222<br />
</td>
        <td>933.333<br />
</td>
        <td>44.444<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>6\23<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5 1 5<br />
</td>
        <td>313.043<br />
</td>
        <td>626.087<br />
</td>
        <td>939.13<br />
</td>
        <td>52.174<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5\19<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4 1 4<br />
</td>
        <td>315.789<br />
</td>
        <td>631.579<br />
</td>
        <td>947.368<br />
</td>
        <td>63.158<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>9\34<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7 2 7<br />
</td>
        <td>317.647<br />
</td>
        <td>634.294<br />
</td>
        <td>951.941<br />
</td>
        <td>70.588<br />
</td>
        <td style="text-align: center;">Hanson/Keemun is around here<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>pi 1 pi<br />
</td>
        <td>319.272<br />
</td>
        <td>638.545<br />
</td>
        <td>957.817<br />
</td>
        <td>77.089<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>4\15<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3 1 3<br />
</td>
        <td>320.000<br />
</td>
        <td>640.000<br />
</td>
        <td>960.000<br />
</td>
        <td>80.000<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><br />
</td>
        <td><br />
</td>
        <td>e 1 e<br />
</td>
        <td>321.6245<br />
</td>
        <td>641.249<br />
</td>
        <td>964.874<br />
</td>
        <td>86.498<br />
</td>
        <td><span style="display: block; text-align: center;">L/s = e</span><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7\26<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5 2 5<br />
</td>
        <td>323.077<br />
</td>
        <td>646.154<br />
</td>
        <td>969.231<br />
</td>
        <td>92.308<br />
</td>
        <td style="text-align: center;">Orgone is around here<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>3\11<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2 1 2<br />
</td>
        <td>327.273<br />
</td>
        <td>654.545<br />
</td>
        <td>981.818<br />
</td>
        <td>109.091<br />
</td>
        <td style="text-align: center;">Boundary of propriety (generators<br />
larger than this are proper)<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>8\29<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5 3 5<br />
</td>
        <td>331.034<br />
</td>
        <td>662.069<br />
</td>
        <td>993.013<br />
</td>
        <td>124.138<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>21\76<br />
</td>
        <td><br />
</td>
        <td>13 8 13<br />
</td>
        <td>331.579<br />
</td>
        <td>663.158<br />
</td>
        <td>994.739<br />
</td>
        <td>126.316<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><br />
</td>
        <td>34\123<br />
</td>
        <td>21 13 21<br />
</td>
        <td>331.707<br />
</td>
        <td>663.415<br />
</td>
        <td>995.122<br />
</td>
        <td>126.829<br />
</td>
        <td style="text-align: center;">Unnamed golden temperament<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>13\47<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>8 5 8<br />
</td>
        <td>331.915<br />
</td>
        <td>663.83<br />
</td>
        <td>995.745<br />
</td>
        <td>127.66<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>5\18<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3 2 3<br />
</td>
        <td>333.333<br />
</td>
        <td>666.667<br />
</td>
        <td>1000.000<br />
</td>
        <td>133.333<br />
</td>
        <td style="text-align: center;">Optimum rank range (L/s=3/2)<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7\25<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4 3 4<br />
</td>
        <td>336.000<br />
</td>
        <td>672.000<br />
</td>
        <td>1008.000<br />
</td>
        <td>144.000<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>9\32<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5 4 5<br />
</td>
        <td>337.5<br />
</td>
        <td>675.000<br />
</td>
        <td>1012.500<br />
</td>
        <td>150.000<br />
</td>
        <td style="text-align: center;">Sixix<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>11\39<br />
</td>
        <td><br />
</td>
        <td>6 5 6<br />
</td>
        <td>338.462<br />
</td>
        <td>676.923<br />
</td>
        <td>1015.385<br />
</td>
        <td>153.846<br />
</td>
        <td style="text-align: center;">Sixix<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>13\46<br />
</td>
        <td>7 6 7<br />
</td>
        <td>339.130<br />
</td>
        <td>678.261<br />
</td>
        <td>1017.391<br />
</td>
        <td>156.522<br />
</td>
        <td style="text-align: center;">(17/14)^3=9/5<br />
</td>
    </tr>
    <tr>
        <td>2\7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1 1 1<br />
</td>
        <td>342.857<br />
</td>
        <td>685.714<br />
</td>
        <td>1028.571<br />
</td>
        <td>171.429<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

There are two notable harmonic entropy minima: <a class="wiki_link" href="/Kleismic%20family">hanson/keemun</a>, in which the generator is 6/5 and 6 of them make a 3/1, and <a class="wiki_link" href="/Starling%20temperaments">myna</a>, in which the generator is also 6/5 but now <strong>10</strong> of them make a 6/1 (so no 4/3's or 3/2's appear in this scale).</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2015-02-10 23:47:54 UTC</tt>.<br>
+: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2015-05-22 14:06:00 UTC</tt>.<br>
-: The original revision id was <tt>540552366</tt>.<br>
+: The original revision id was <tt>551910764</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>
@@ Line 14: / Line 14: @@
 ||   ||   ||   || 5\19 ||   ||   ||   || 4 1 4 || 315.789 || 631.579 || 947.368 || 63.158 ||= L/s = 4 ||
 ||   ||   ||   ||   || 9\34 ||   ||   || 7 2 7 || 317.647 || 634.294 || 951.941 || 70.588 ||= Hanson/Keemun is around here ||
-||   ||   ||   ||   ||   ||   ||   || 3.03 1 3,03 || 319.8525 || 639.705 || 959.558 || 79.410 ||= &lt;span style="display: block; text-align: center;"&gt;L/s = 3*2^(1/75)&lt;/span&gt; ||
+||   ||   ||   ||   ||   ||   ||   || pi 1 pi || 319.272 || 638.545 || 957.817 || 77.089 ||= &lt;span style="display: block; text-align: center;"&gt;L/s = pi&lt;/span&gt; ||
 ||   ||   || 4\15 ||   ||   ||   ||   || 3 1 3 || 320.000 || 640.000 || 960.000 || 80.000 ||= L/s = 3 ||
-||   ||   ||   ||   ||   ||   ||   || 2.97 1 2.97 || 320.148 || 640.297 || 960.445 || 80.593 || &lt;span style="display: block; text-align: center;"&gt;L/s = 3/2^(1/75)&lt;/span&gt; ||
+||   ||   ||   ||   ||   ||   ||   || e 1 e || 321.6245 || 641.249 || 964.874 || 86.498 || &lt;span style="display: block; text-align: center;"&gt;L/s = e&lt;/span&gt; ||
 ||   ||   ||   || 7\26 ||   ||   ||   || 5 2 5 || 323.077 || 646.154 || 969.231 || 92.308 ||= Orgone is around here ||
 ||   || 3\11 ||   ||   ||   ||   ||   || 2 1 2 || 327.273 || 654.545 || 981.818 || 109.091 ||= Boundary of propriety (generators
@@ Line 235: / Line 235: @@
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;3.03 1 3,03&lt;br /&gt;
+         &lt;td&gt;pi 1 pi&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;319.8525&lt;br /&gt;
+         &lt;td&gt;319.272&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;639.705&lt;br /&gt;
+         &lt;td&gt;638.545&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;959.558&lt;br /&gt;
+         &lt;td&gt;957.817&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;79.410&lt;br /&gt;
+         &lt;td&gt;77.089&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;span style="display: block; text-align: center;"&gt;L/s = 3*2^(1/75)&lt;/span&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;span style="display: block; text-align: center;"&gt;L/s = pi&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 291: / Line 291: @@
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;2.97 1 2.97&lt;br /&gt;
+         &lt;td&gt;e 1 e&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;320.148&lt;br /&gt;
+         &lt;td&gt;321.6245&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;640.297&lt;br /&gt;
+         &lt;td&gt;641.249&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;960.445&lt;br /&gt;
+         &lt;td&gt;964.874&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;80.593&lt;br /&gt;
+         &lt;td&gt;86.498&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="display: block; text-align: center;"&gt;L/s = 3/2^(1/75)&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="display: block; text-align: center;"&gt;L/s = e&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;

4L 3s: Difference between revisions

Revision as of 14:06, 22 May 2015

IMPORTED REVISION FROM WIKISPACES

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