4L 2s: Difference between revisions

Revision as of 20:15, 28 August 2011

IMPORTED REVISION FROM WIKISPACES

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

This revision was by author keenanpepper and made on 2011-08-28 20:15:27 UTC.

The original revision id was 248977817.

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:

There are three scales with this [[MOSScales|MOS]] pattern that are significant minima of harmonic entropy.

The first is "[[antikythera]]", or no-3's [[Diaschismic family|srutal/pajara]], which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210-220 cents rather than 105-110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.

The second is [[Dicot family|decimal]], in which two generators make a 4/3, and the third is [[Jubilismic clan|Doublewide]], in which the generator is 7/6 so the period minus the generator is 6/5.

In addition to the true MOS with pattern LLsLLs, all these scales also come in a near-MOS version, LLLsLs, in which the period is the only generic interval that has more than two specific representatives. The true MOS is always proper, but the near-MOS is only proper if the generator is smaller than 2\10 of an octave (240 cents).

||||||||||||||||||||||~ Generator ||~ Cents ||~ Comments ||
|| 1\6 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 200 ||   ||
||   ||   ||   ||   ||   || 6\34 ||   ||   ||   ||   ||   || 211.76 ||   ||
||   ||   ||   ||   || 5\28 ||   ||   ||   ||   ||   ||   || 214.29 || Antikythera is around here ||
||   ||   ||   || 4\22 ||   ||   ||   ||   ||   ||   ||   || 218.18 ||   ||
||   ||   || 3\16 ||   ||   ||   ||   ||   ||   ||   ||   || 225 ||   ||
||   ||   ||   ||   || 8\42 ||   ||   ||   ||   ||   ||   || 228.57 ||   ||
||   ||   ||   ||   ||   ||   || 21\110 ||   ||   ||   ||   || 229.09 || Golden lemba ||
||   ||   ||   ||   ||   || 13\68 ||   ||   ||   ||   ||   || 229.41 ||   ||
||   ||   ||   || 5\26 ||   ||   ||   ||   ||   ||   ||   || 230.77 ||   ||
||   ||   ||   ||   || 7\36 ||   ||   ||   ||   ||   ||   || 233.33 || Lemba is around here ||
||   || 2\10 ||   ||   ||   ||   ||   ||   ||   ||   ||   || 240 || Boundary of propriety for near-MOS ||
||   ||   ||   || 5\24 ||   ||   ||   ||   ||   ||   ||   || 250 || Decimal is around here ||
||   ||   || 3\14 ||   ||   ||   ||   ||   ||   ||   ||   || 257.14 ||   ||
||   ||   ||   || 4\18 ||   ||   ||   ||   ||   ||   ||   || 266.67 ||   ||
||   ||   ||   ||   || 5\22 ||   ||   ||   ||   ||   ||   || 272.73 ||   ||
||   ||   ||   ||   ||   || 6\26 ||   ||   ||   ||   ||   || 276.92 ||   ||
||   ||   ||   ||   ||   ||   || 7\30 ||   ||   ||   ||   || 280 ||   ||
||   ||   ||   ||   ||   ||   ||   || 8\34 ||   ||   ||   || 282.35 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   || 9\38 ||   ||   || 284.21 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   || 10\42 ||   || 285.71 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 11\46 || 286.96 || Doublewide is around here ||
|| 1\4 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 300 ||   ||

Original HTML content:

<html><head><title>4L 2s</title></head><body>There are three scales with this <a class="wiki_link" href="/MOSScales">MOS</a> pattern that are significant minima of harmonic entropy.<br />
<br />
The first is &quot;<a class="wiki_link" href="/antikythera">antikythera</a>&quot;, or no-3's <a class="wiki_link" href="/Diaschismic%20family">srutal/pajara</a>, which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210-220 cents rather than 105-110), since odd numbers of generators are only needed for intervals with 3. So this is a basically &quot;whole tone&quot; scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.<br />
<br />
The second is <a class="wiki_link" href="/Dicot%20family">decimal</a>, in which two generators make a 4/3, and the third is <a class="wiki_link" href="/Jubilismic%20clan">Doublewide</a>, in which the generator is 7/6 so the period minus the generator is 6/5.<br />
<br />
In addition to the true MOS with pattern LLsLLs, all these scales also come in a near-MOS version, LLLsLs, in which the period is the only generic interval that has more than two specific representatives. The true MOS is always proper, but the near-MOS is only proper if the generator is smaller than 2\10 of an octave (240 cents).<br />
<br />


<table class="wiki_table">
    <tr>
        <th colspan="11">Generator<br />
</th>
        <th>Cents<br />
</th>
        <th>Comments<br />
</th>
    </tr>
    <tr>
        <td>1\6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>200<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>6\34<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>211.76<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5\28<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>214.29<br />
</td>
        <td>Antikythera is around here<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4\22<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>218.18<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>3\16<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>225<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>8\42<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>228.57<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>21\110<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>229.09<br />
</td>
        <td>Golden lemba<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>13\68<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>229.41<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5\26<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>230.77<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7\36<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>233.33<br />
</td>
        <td>Lemba is around here<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>2\10<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>240<br />
</td>
        <td>Boundary of propriety for near-MOS<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5\24<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>250<br />
</td>
        <td>Decimal is around here<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>3\14<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>257.14<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4\18<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>266.67<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5\22<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>272.73<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>6\26<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>276.92<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7\30<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>280<br />
</td>
        <td><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>8\34<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>282.35<br />
</td>
        <td><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><br />
</td>
        <td>9\38<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>284.21<br />
</td>
        <td><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><br />
</td>
        <td><br />
</td>
        <td>10\42<br />
</td>
        <td><br />
</td>
        <td>285.71<br />
</td>
        <td><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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>11\46<br />
</td>
        <td>286.96<br />
</td>
        <td>Doublewide is around here<br />
</td>
    </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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>300<br />
</td>
        <td><br />
</td>
    </tr>
</table>

</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:guest|guest]] and made on <tt>2011-08-28 00:41:27 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-08-28 20:15:27 UTC</tt>.<br>
-: The original revision id was <tt>248861579</tt>.<br>
+: The original revision id was <tt>248977817</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">There are three scales with this [[MOSScales|MOS]] pattern that are significant minima of harmonic entropy.
-The first is no-3's [[Diaschismic family|srutal/pajara]], which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210-220 cents rather than 105-110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.
+The first is "[[antikythera]]", or no-3's [[Diaschismic family|srutal/pajara]], which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210-220 cents rather than 105-110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.
 The second is [[Dicot family|decimal]], in which two generators make a 4/3, and the third is [[Jubilismic clan|Doublewide]], in which the generator is 7/6 so the period minus the generator is 6/5.
@@ Line 17: / Line 17: @@
 || 1\6 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 200 ||   ||
 ||   ||   ||   ||   ||   || 6\34 ||   ||   ||   ||   ||   || 211.76 ||   ||
-||   ||   ||   ||   || 5\28 ||   ||   ||   ||   ||   ||   || 214.29 || No-3's srutal/pajara is around here ||
+||   ||   ||   ||   || 5\28 ||   ||   ||   ||   ||   ||   || 214.29 || Antikythera is around here ||
 ||   ||   ||   || 4\22 ||   ||   ||   ||   ||   ||   ||   || 218.18 ||   ||
 ||   ||   || 3\16 ||   ||   ||   ||   ||   ||   ||   ||   || 225 ||   ||
@@ Line 40: / Line 40: @@
 <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 2s&lt;/title&gt;&lt;/head&gt;&lt;body&gt;There are three scales with this &lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt; pattern that are significant minima of harmonic entropy.&lt;br /&gt;
 &lt;br /&gt;
-The first is no-3's &lt;a class="wiki_link" href="/Diaschismic%20family"&gt;srutal/pajara&lt;/a&gt;, which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210-220 cents rather than 105-110), since odd numbers of generators are only needed for intervals with 3. So this is a basically &amp;quot;whole tone&amp;quot; scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.&lt;br /&gt;
+The first is &amp;quot;&lt;a class="wiki_link" href="/antikythera"&gt;antikythera&lt;/a&gt;&amp;quot;, or no-3's &lt;a class="wiki_link" href="/Diaschismic%20family"&gt;srutal/pajara&lt;/a&gt;, which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210-220 cents rather than 105-110), since odd numbers of generators are only needed for intervals with 3. So this is a basically &amp;quot;whole tone&amp;quot; scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.&lt;br /&gt;
 &lt;br /&gt;
 The second is &lt;a class="wiki_link" href="/Dicot%20family"&gt;decimal&lt;/a&gt;, in which two generators make a 4/3, and the third is &lt;a class="wiki_link" href="/Jubilismic%20clan"&gt;Doublewide&lt;/a&gt;, in which the generator is 7/6 so the period minus the generator is 6/5.&lt;br /&gt;
@@ Line 138: / Line 138: @@
          &lt;td&gt;214.29&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;No-3's srutal/pajara is around here&lt;br /&gt;
+         &lt;td&gt;Antikythera is around here&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;

4L 2s: Difference between revisions

Revision as of 20:15, 28 August 2011

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

Original HTML content:

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