2L 8s: Difference between revisions

Revision as of 04:17, 29 August 2011

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Original Wikitext content:

This is the MOS pattern of the [[decatonic]] scale of [[Paul Erlich]] and others.

The only significant harmonic entropy minimum that is [[Rothenberg propriety|proper]] is the decatonic scale itself ([[Diaschismic family|pajara]][10]), in which the period is 7/5~10/7 (tempered to be the same interval), one generator down from that makes [[4_3|4/3]], and another generator down makes [[5_4|5/4]]. More than a few people think this is a beautiful scale that deserves a lot of investigation and use, with some going so far as to say it's the next step up from the diatonic scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found at either of these links:
http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf
http://www.lumma.org/tuning/erlich/

Improper harmonic entropy minima include [[Meantone family#Injera|injera]] (which is similar to pajara except that 5/4 is now four generators **up** and no periods) and [[Diaschismic family#Shrutar|shrutar]] (which is basically pajara with the generator divided in two).

In addition to the true MOS form, LssssLssss, these scales also exist in a near-MOS form, LsssssLsss, in which the period is the only interval class with more than two flavors. In the case of the decatonic scale, LssssLssss is called the "symmetric" scale and LsssssLsss is called the "pentachordal" scale (because it has two identical "pentachords" in the same way that the diatonic scale has two identical tetrachords).
||||||||||||||||||~ Generator ||~ Cents ||~ Comments ||
|| 0\1 ||   ||   ||   ||   ||   ||   ||   ||   || 0 ||=   ||
||   ||   ||   ||   ||   ||   ||   || 1\24 ||   || 50 ||=   ||
||   ||   ||   ||   ||   ||   ||   ||   || 2\46 || 52.17 ||= Shrutar is around here ||
||   ||   ||   ||   ||   ||   || 1\22 ||   ||   || 54.55 ||=   ||
||   ||   ||   ||   ||   || 1\20 ||   ||   ||   || 60 ||=   ||
||   ||   ||   ||   || 1\18 ||   ||   ||   ||   || 66.67 ||=   ||
||   ||   ||   || 1\16 ||   ||   ||   ||   ||   || 75 ||=   ||
||   ||   || 1\14 ||   ||   ||   ||   ||   ||   || 85.71 ||=   ||
||   ||   ||   || 2\26 ||   ||   ||   ||   ||   || 92.31 ||=   ||
||   ||   ||   ||   || 3\38 ||   ||   ||   ||   || 94.74 ||=   ||
||   ||   ||   ||   ||   || 4\50 ||   ||   ||   || 96 ||= Injera is around here ||
||   || 1\12 ||   ||   ||   ||   ||   ||   ||   || 100 ||= Boundary of propriety (generators
larger than this are proper) ||
||   ||   ||   ||   || 4\46 ||   ||   ||   ||   || 104.35 ||=   ||
||   ||   ||   || 3\34 ||   ||   ||   ||   ||   || 105.88 ||=   ||
||   ||   ||   ||   ||   || 8\90 ||   ||   ||   || 106.67 ||=   ||
||   ||   ||   ||   ||   ||   ||   || 21\236 ||   || 106.78 ||= Golden decatonic ||
||   ||   ||   ||   ||   ||   || 13\146 ||   ||   || 106.85 ||=   ||
||   ||   ||   ||   || 5\56 ||   ||   ||   ||   || 107.14 ||= Srutal/pajara decatonic is around here ||
||   ||   || 2\22 ||   ||   ||   ||   ||   ||   || 109.09 ||= Optimum rank range (L/s=3/2) decatonic ||
|| 1\10 ||   ||   ||   ||   ||   ||   ||   ||   || 120 ||=   ||

Original HTML content:

<html><head><title>2L 8s</title></head><body>This is the MOS pattern of the <a class="wiki_link" href="/decatonic">decatonic</a> scale of <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a> and others.<br />
<br />
The only significant harmonic entropy minimum that is <a class="wiki_link" href="/Rothenberg%20propriety">proper</a> is the decatonic scale itself (<a class="wiki_link" href="/Diaschismic%20family">pajara</a>[10]), in which the period is 7/5~10/7 (tempered to be the same interval), one generator down from that makes <a class="wiki_link" href="/4_3">4/3</a>, and another generator down makes <a class="wiki_link" href="/5_4">5/4</a>. More than a few people think this is a beautiful scale that deserves a lot of investigation and use, with some going so far as to say it's the next step up from the diatonic scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found at either of these links:<br />
<!-- ws:start:WikiTextUrlRule:731:http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf --><a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf" rel="nofollow">http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf</a><!-- ws:end:WikiTextUrlRule:731 --><br />
<!-- ws:start:WikiTextUrlRule:732:http://www.lumma.org/tuning/erlich/ --><a class="wiki_link_ext" href="http://www.lumma.org/tuning/erlich/" rel="nofollow">http://www.lumma.org/tuning/erlich/</a><!-- ws:end:WikiTextUrlRule:732 --><br />
<br />
Improper harmonic entropy minima include <a class="wiki_link" href="/Meantone%20family#Injera">injera</a> (which is similar to pajara except that 5/4 is now four generators <strong>up</strong> and no periods) and <a class="wiki_link" href="/Diaschismic%20family#Shrutar">shrutar</a> (which is basically pajara with the generator divided in two).<br />
<br />
In addition to the true MOS form, LssssLssss, these scales also exist in a near-MOS form, LsssssLsss, in which the period is the only interval class with more than two flavors. In the case of the decatonic scale, LssssLssss is called the &quot;symmetric&quot; scale and LsssssLsss is called the &quot;pentachordal&quot; scale (because it has two identical &quot;pentachords&quot; in the same way that the diatonic scale has two identical tetrachords).<br />


<table class="wiki_table">
    <tr>
        <th colspan="9">Generator<br />
</th>
        <th>Cents<br />
</th>
        <th>Comments<br />
</th>
    </tr>
    <tr>
        <td>0\1<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>0<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><br />
</td>
        <td>1\24<br />
</td>
        <td><br />
</td>
        <td>50<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><br />
</td>
        <td><br />
</td>
        <td>2\46<br />
</td>
        <td>52.17<br />
</td>
        <td style="text-align: center;">Shrutar 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>1\22<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>54.55<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>1\20<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>60<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1\18<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>66.67<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1\16<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>75<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>1\14<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>85.71<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2\26<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>92.31<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3\38<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>94.74<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>4\50<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>96<br />
</td>
        <td style="text-align: center;">Injera is around here<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>1\12<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>100<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><br />
</td>
        <td>4\46<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>104.35<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3\34<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>105.88<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>8\90<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>106.67<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><br />
</td>
        <td>21\236<br />
</td>
        <td><br />
</td>
        <td>106.78<br />
</td>
        <td style="text-align: center;">Golden decatonic<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>13\146<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>106.85<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5\56<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>107.14<br />
</td>
        <td style="text-align: center;">Srutal/pajara decatonic is around here<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>2\22<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>109.09<br />
</td>
        <td style="text-align: center;">Optimum rank range (L/s=3/2) decatonic<br />
</td>
    </tr>
    <tr>
        <td>1\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>120<br />
</td>
        <td style="text-align: center;"><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 04:03:25 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2011-08-29 04:17:26 UTC</tt>.<br>
-: The original revision id was <tt>248871529</tt>.<br>
+: The original revision id was <tt>249049713</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 32: / Line 32: @@
 ||   ||   ||   || 3\34 ||   ||   ||   ||   ||   || 105.88 ||=   ||
 ||   ||   ||   ||   ||   || 8\90 ||   ||   ||   || 106.67 ||=   ||
-||   ||   ||   ||   ||   ||   ||   || 21\236 ||   || 106.78 ||= Golden srutal/pajara ||
+||   ||   ||   ||   ||   ||   ||   || 21\236 ||   || 106.78 ||= Golden decatonic ||
 ||   ||   ||   ||   ||   ||   || 13\146 ||   ||   || 106.85 ||=   ||
 ||   ||   ||   ||   || 5\56 ||   ||   ||   ||   || 107.14 ||= Srutal/pajara decatonic is around here ||
-||   ||   || 2\22 ||   ||   ||   ||   ||   ||   || 109.09 ||=   ||
+||   ||   || 2\22 ||   ||   ||   ||   ||   ||   || 109.09 ||= Optimum rank range (L/s=3/2) decatonic ||
 || 1\10 ||   ||   ||   ||   ||   ||   ||   ||   || 120 ||=   ||</pre></div>
 <h4>Original HTML content:</h4>
@@ Line 440: / Line 440: @@
          &lt;td&gt;106.78&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;Golden srutal/pajara&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Golden decatonic&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 512: / Line 512: @@
          &lt;td&gt;109.09&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Optimum rank range (L/s=3/2) decatonic&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;

2L 8s: Difference between revisions

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IMPORTED REVISION FROM WIKISPACES

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