4L 5s: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:~~Andrew_Heathwaite~~|~~Andrew_Heathwaite~~]] and made on <tt>~~2010~~-09-~~05 17~~:59:09 UTC</tt>.

: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-05-11 13:29:44 UTC</tt>.

: The original revision id was <tt>~~160514487~~</tt>.

: The original revision id was <tt>227574018</tt>.

: The revision comment was: <tt></tt>

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

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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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:

|| || || || || ~~scale~~ || ~~generator~~ in cents ||

||||||||||~ Generator ||~ Scale ||~ Generator in cents ||~ Comments ||

|| 2\9 || || || || 1 1 1 1 1 1 1 1 1 || 266.667 ||

|| 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 ||

|| || || || 7\31 || || 3 4 3 4 3 4 3 4 3 || 270.968 ||= ||

|| || || 5\22 || || 2 3 2 3 2 3 2 3 2 || 272.727 ||

|| || || || || 12\53 || 5 7 5 7 5 7 5 7 5 || 271.698 ||= Orwell is around here ||

|| || || || 8\35 || 3 5 3 5 3 5 3 5 3 || 274.286 ||

|| || || 5\22 || || || 2 3 2 3 2 3 2 3 2 || 272.727 ||= ||

|| || 3\13 || || || 1 2 1 2 1 2 1 2 1 || 276.923 ||

|| || || || 8\35 || || 3 5 3 5 3 5 3 5 3 || 274.286 ||= ||

|| || || || 7\30 || 2 5 2 5 2 5 2 5 2 || 280.000 ||

|| || 3\13 || || || || 1 2 1 2 1 2 1 2 1 || 276.923 ||= Boundary of propriety: generators

|| || || 4\17 || || 1 3 1 3 1 3 1 3 1 || 282.353 ||

smaller than this are proper ||

|| || || || 5\21 || 1 4 1 4 1 4 1 4 1 || 285.714 ||

|| || || || 7\30 || || 2 5 2 5 2 5 2 5 2 || 280.000 ||= ||

|| 1\4 || || || || 0 1 0 1 0 1 0 1 0 || 300.000 ||

|| || || 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. 4L 5s scales outside of that range are not suitable for Orwell.</pre></div>

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.</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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:

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<tr>

<th colspan="5">Generator

</th>

<th>Scale

</th>

<th>Generator in cents

</th>

<th>Comments

</th>

</tr>

<tr>

<td>2\9

</td>

<td>

</td>

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<td>

</td>

<td>~~scale~~

<td>1 1 1 1 1 1 1 1 1

</td>

<td>~~generator in cents~~

<td>266.667

</td>

</td>

</tr>

<tr>

<td>~~2\9~~

<td>

</td>

<td>

</td>

<td>

</td>

<td>7\31

</td>

<td>

</td>

<td>~~1 1 1 1 1 1 1 1 1~~

<td>3 4 3 4 3 4 3 4 3

</td>

<td>270.968

</td>

<td>~~266.667~~

</td>

</tr>

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<td>

</td>

<td>7~~\31~~

<td>

</td>

<td>12\53

</td>

<td>5 7 5 7 5 7 5 7 5

</td>

<td>~~3 4 3 4 3 4 3 4 3~~

<td>271.698

</td>

<td>~~270.968~~

<td style="text-align: center;">Orwell is around here

</td>

</tr>

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</td>

<td>5\22

</td>

<td>

</td>

<td>

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</td>

<td>272.727

</td>

</td>

</tr>

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</td>

<td>8\35

</td>

<td>

</td>

<td>3 5 3 5 3 5 3 5 3

</td>

<td>274.286

</td>

</td>

</tr>

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</td>

<td>3\13

</td>

<td>

</td>

<td>

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</td>

<td>276.923

</td>

<td style="text-align: center;">Boundary of propriety: generators

smaller than this are proper

</td>

</tr>

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</td>

<td>7\30

</td>

<td>

</td>

<td>2 5 2 5 2 5 2 5 2

</td>

<td>280.000

</td>

</td>

</tr>

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</td>

<td>4\17

</td>

<td>

</td>

<td>

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</td>

<td>282.353

</td>

</td>

</tr>

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</td>

<td>5\21

</td>

<td>

</td>

<td>1 4 1 4 1 4 1 4 1

</td>

<td>285.714

</td>

</td>

</tr>

<tr>

<td>1\4

</td>

<td>

</td>

<td>

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</td>

<td>300.000

</td>

</td>

</tr>

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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. 4L 5s scales outside of that range are not suitable for Orwell.</body></html></pre></div>

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></pre></div>

@@ 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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-09-05 17:59:09 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-05-11 13:29:44 UTC</tt>.<br>
-: The original revision id was <tt>160514487</tt>.<br>
+: The original revision id was <tt>227574018</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 8: / Line 8: @@
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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:
-||   ||   ||   ||   || scale || generator in cents ||
+||||||||||~ Generator ||~ Scale ||~ Generator in cents ||~ Comments ||
-|| 2\9 ||   ||   ||   || 1 1 1 1 1 1 1 1 1 || 266.667 ||
+|| 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 ||
+||   ||   ||   || 7\31 ||   || 3 4 3 4 3 4 3 4 3 || 270.968 ||=   ||
-||   ||   || 5\22 ||   || 2 3 2 3 2 3 2 3 2 || 272.727 ||
+||   ||   ||   ||   || 12\53 || 5 7 5 7 5 7 5 7 5 || 271.698 ||= Orwell is around here ||
-||   ||   ||   || 8\35 || 3 5 3 5 3 5 3 5 3 || 274.286 ||
+||   ||   || 5\22 ||   ||   || 2 3 2 3 2 3 2 3 2 || 272.727 ||=   ||
-||   || 3\13 ||   ||   || 1 2 1 2 1 2 1 2 1 || 276.923 ||
+||   ||   ||   || 8\35 ||   || 3 5 3 5 3 5 3 5 3 || 274.286 ||=   ||
-||   ||   ||   || 7\30 || 2 5 2 5 2 5 2 5 2 || 280.000 ||
+||   || 3\13 ||   ||   ||   || 1 2 1 2 1 2 1 2 1 || 276.923 ||= Boundary of propriety: generators
-||   ||   || 4\17 ||   || 1 3 1 3 1 3 1 3 1 || 282.353 ||
+smaller than this are proper ||
-||   ||   ||   || 5\21 || 1 4 1 4 1 4 1 4 1 || 285.714 ||
+||   ||   ||   || 7\30 ||   || 2 5 2 5 2 5 2 5 2 || 280.000 ||=   ||
-|| 1\4 ||   ||   ||   || 0 1 0 1 0 1 0 1 0 || 300.000 ||
+||   ||   || 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. 4L 5s scales outside of that range are not suitable for Orwell.</pre></div>
+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.</pre></div>
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;4L 5s&lt;/title&gt;&lt;/head&gt;&lt;body&gt;4L 5s refers to the structure of &lt;a class="wiki_link" href="/MOSScales"&gt;MOS Scales&lt;/a&gt; whose generator falls between 2\9 (two degrees of &lt;a class="wiki_link" href="/9edo"&gt;9edo&lt;/a&gt; = approx. 266.667¢) and 1\4 (one degree of &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt; = 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:&lt;br /&gt;
@@ Line 27: / Line 29: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
+        &lt;th colspan="5"&gt;Generator&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Scale&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Generator in cents&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Comments&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;2\9&lt;br /&gt;
+&lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 35: / Line 49: @@
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;scale&lt;br /&gt;
+         &lt;td&gt;1 1 1 1 1 1 1 1 1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;generator in cents&lt;br /&gt;
+         &lt;td&gt;266.667&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;2\9&lt;br /&gt;
+         &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7\31&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1 1 1 1 1 1 1 1 1&lt;br /&gt;
+         &lt;td&gt;3 4 3 4 3 4 3 4 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;270.968&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;266.667&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 61: / Line 81: @@
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;7\31&lt;br /&gt;
+         &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;12\53&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;5 7 5 7 5 7 5 7 5&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;3 4 3 4 3 4 3 4 3&lt;br /&gt;
+         &lt;td&gt;271.698&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;270.968&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Orwell is around here&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 74: / Line 98: @@
 &lt;/td&gt;
          &lt;td&gt;5\22&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 80: / Line 106: @@
 &lt;/td&gt;
          &lt;td&gt;272.727&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 90: / Line 118: @@
 &lt;/td&gt;
          &lt;td&gt;8\35&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;3 5 3 5 3 5 3 5 3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;274.286&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 100: / Line 132: @@
 &lt;/td&gt;
          &lt;td&gt;3\13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 108: / Line 142: @@
 &lt;/td&gt;
          &lt;td&gt;276.923&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Boundary of propriety: generators&lt;br /&gt;
+smaller than this are proper&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 118: / Line 155: @@
 &lt;/td&gt;
          &lt;td&gt;7\30&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;2 5 2 5 2 5 2 5 2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;280.000&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 130: / Line 171: @@
 &lt;/td&gt;
          &lt;td&gt;4\17&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 136: / Line 179: @@
 &lt;/td&gt;
          &lt;td&gt;282.353&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 146: / Line 191: @@
 &lt;/td&gt;
          &lt;td&gt;5\21&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;1 4 1 4 1 4 1 4 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;285.714&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td&gt;1\4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 164: / Line 215: @@
 &lt;/td&gt;
          &lt;td&gt;300.000&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 169: / Line 222: @@
 &lt;br /&gt;
-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. 4L 5s scales outside of that range are not suitable for Orwell.&lt;/body&gt;&lt;/html&gt;</pre></div>
+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.&lt;/body&gt;&lt;/html&gt;</pre></div>