2L 3s: Difference between revisions

Revision as of 21:08, 19 November 2012

IMPORTED REVISION FROM WIKISPACES

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This revision was by author JosephRuhf and made on 2012-11-19 21:08:39 UTC.

The original revision id was 384205648.

The revision comment was:

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"Classic" [[pentatonic]]. Perhaps the most common scale in the world.

The [[meantone]] pentatonic scale, in which the generator approximates 4/3 but other intervals in the scale approximate 6/5 and 5/4, has by far the lowest harmonic entropy of all 5-note MOS scales, which explains the worldwide popularity of these scales and their very long history of use. It is also strictly [[Rothenberg propriety|proper]].
||||||||||||~ Generator ||~ Cents ||~ s ||~ L-s ||~ |L-2s| ||~ Scale steps ||~ Comments ||
|| 2\5 ||   ||   ||   ||   ||   || 480 || 240 || 0 || 240 || 1 1 1 1 1 ||=   ||
||   ||   ||   ||   ||   || 11\27 || 488.89 || 222.22 || 44.44 || 177.78 || 6 5 5 6 5 ||= Slendro (insofar as it resembles a MOS)
would be in this region ||
||   ||   ||   ||   || 9\22 ||   || 490.91 || 218.18 || 54.55 || 163.64 || 5 4 4 5 4 ||=   ||
||   ||   ||   ||   ||   || 16\39 || 492.31 || 215.38 || 61.54 || 153.85 || 9 7 7 9 7 ||= No-5's superpyth/dominant is around here ||
||   ||   ||   || 7\17 ||   ||   || 494.12 || 211.76 || 70.59 || 141.18 || 4 3 3 4 3 ||=   ||
||   ||   ||   ||   || 12\29 ||   || 496.55 || 206.9 || 82.76 || 124.14 || 7 5 5 7 5 ||=   ||
||   ||   ||   ||   ||   || 17\41 || 497.56 || 204.88 || 87.8 || 117.07 || 10 7 7 10 7 ||= Pythagorean pentatonic is around here ||
||   ||   || 5\12 ||   ||   ||   || 500 || 200 || 100 || 100 || 3 2 2 3 2 ||= Familiar 12-equal pentatonic
(also optimum rank range: L/s=3/2) ||
||   ||   ||   ||   || 13\31 ||   || 503.23 || 193.55 || 116.13 || 77.42 || 8 5 5 8 5 ||= Optimal meantone pentatonic
is around here ||
||   ||   ||   ||   ||   ||   || 1200/(4-phi) || 192.43 || 118.93 || 73.50 || phi 1 1 phi 1 ||= Golden meantone ||
||   ||   ||   ||   ||   || 21\50 || 504 || 192 || 120 || 72 || 13 8 8 13 8 ||=   ||
||   ||   ||   || 8\19 ||   ||   || 505.26 || 189.47 || 126.32 || 63.16 || 5 3 3 5 3 ||=   ||
||   || 3\7 ||   ||   ||   ||   || 514.29 || 171.43 || 171.43 || 0 || 2 1 1 2 1 ||= (Boundary of propriety: smaller
generators than this are strictly proper) ||
||   ||   ||   || 7\16 ||   ||   || 525 || 150 || 225 || 75 || 5 2 2 5 2 ||= 5-note subset of pelog (insofar as it
resembles a MOS) would be in this region ||
||   ||   || 4\9 ||   ||   ||   || 533.33 || 133.33 || 266.67 || 133.33 || 3 1 1 3 1 ||= Boundary of "practicality" begins around here ||
||   ||   ||   || 5\11 ||   ||   || 545.45 || 109.09 || 327.27 || 218.18 || 4 1 1 4 1 ||= Boundary of "practicality" ends around here ||
|| 1\2 ||   ||   ||   ||   ||   || 600 || 0 || 600 || 600 || 1 0 0 1 0 ||=   ||

From a [[3-limit]] perspective, just make a chain of four 4/3's and octave-reduce, and you end up with pentatonic.

From a [[5-limit]] perspective, the most interesting temperaments with this kind of pentatonic scale are [[meantone]] and [[Pelogic famiy|mavila]].

There is also the interesting 2.3.7 temperament that tempers out [[64_63|64/63]] ("no-fives [[dominant]]").

Original HTML content:

<html><head><title>2L 3s</title></head><body>&quot;Classic&quot; <a class="wiki_link" href="/pentatonic">pentatonic</a>. Perhaps the most common scale in the world.<br />
<br />
The <a class="wiki_link" href="/meantone">meantone</a> pentatonic scale, in which the generator approximates 4/3 but other intervals in the scale approximate 6/5 and 5/4, has by far the lowest harmonic entropy of all 5-note MOS scales, which explains the worldwide popularity of these scales and their very long history of use. It is also strictly <a class="wiki_link" href="/Rothenberg%20propriety">proper</a>.<br />


<table class="wiki_table">
    <tr>
        <th colspan="6">Generator<br />
</th>
        <th>Cents<br />
</th>
        <th>s<br />
</th>
        <th>L-s<br />
</th>
        <th>|L-2s|<br />
</th>
        <th>Scale steps<br />
</th>
        <th>Comments<br />
</th>
    </tr>
    <tr>
        <td>2\5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>480<br />
</td>
        <td>240<br />
</td>
        <td>0<br />
</td>
        <td>240<br />
</td>
        <td>1 1 1 1 1<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>11\27<br />
</td>
        <td>488.89<br />
</td>
        <td>222.22<br />
</td>
        <td>44.44<br />
</td>
        <td>177.78<br />
</td>
        <td>6 5 5 6 5<br />
</td>
        <td style="text-align: center;">Slendro (insofar as it resembles a MOS)<br />
would be in this region<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>9\22<br />
</td>
        <td><br />
</td>
        <td>490.91<br />
</td>
        <td>218.18<br />
</td>
        <td>54.55<br />
</td>
        <td>163.64<br />
</td>
        <td>5 4 4 5 4<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>16\39<br />
</td>
        <td>492.31<br />
</td>
        <td>215.38<br />
</td>
        <td>61.54<br />
</td>
        <td>153.85<br />
</td>
        <td>9 7 7 9 7<br />
</td>
        <td style="text-align: center;">No-5's superpyth/dominant is around here<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7\17<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>494.12<br />
</td>
        <td>211.76<br />
</td>
        <td>70.59<br />
</td>
        <td>141.18<br />
</td>
        <td>4 3 3 4 3<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>12\29<br />
</td>
        <td><br />
</td>
        <td>496.55<br />
</td>
        <td>206.9<br />
</td>
        <td>82.76<br />
</td>
        <td>124.14<br />
</td>
        <td>7 5 5 7 5<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>17\41<br />
</td>
        <td>497.56<br />
</td>
        <td>204.88<br />
</td>
        <td>87.8<br />
</td>
        <td>117.07<br />
</td>
        <td>10 7 7 10 7<br />
</td>
        <td style="text-align: center;">Pythagorean pentatonic is around here<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>5\12<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>500<br />
</td>
        <td>200<br />
</td>
        <td>100<br />
</td>
        <td>100<br />
</td>
        <td>3 2 2 3 2<br />
</td>
        <td style="text-align: center;">Familiar 12-equal pentatonic<br />
(also optimum rank range: L/s=3/2)<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>13\31<br />
</td>
        <td><br />
</td>
        <td>503.23<br />
</td>
        <td>193.55<br />
</td>
        <td>116.13<br />
</td>
        <td>77.42<br />
</td>
        <td>8 5 5 8 5<br />
</td>
        <td style="text-align: center;">Optimal meantone pentatonic<br />
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>1200/(4-phi)<br />
</td>
        <td>192.43<br />
</td>
        <td>118.93<br />
</td>
        <td>73.50<br />
</td>
        <td>phi 1 1 phi 1<br />
</td>
        <td style="text-align: center;">Golden meantone<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>21\50<br />
</td>
        <td>504<br />
</td>
        <td>192<br />
</td>
        <td>120<br />
</td>
        <td>72<br />
</td>
        <td>13 8 8 13 8<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>8\19<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>505.26<br />
</td>
        <td>189.47<br />
</td>
        <td>126.32<br />
</td>
        <td>63.16<br />
</td>
        <td>5 3 3 5 3<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>3\7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>514.29<br />
</td>
        <td>171.43<br />
</td>
        <td>171.43<br />
</td>
        <td>0<br />
</td>
        <td>2 1 1 2 1<br />
</td>
        <td style="text-align: center;">(Boundary of propriety: smaller<br />
generators than this are strictly proper)<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7\16<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>525<br />
</td>
        <td>150<br />
</td>
        <td>225<br />
</td>
        <td>75<br />
</td>
        <td>5 2 2 5 2<br />
</td>
        <td style="text-align: center;">5-note subset of pelog (insofar as it<br />
resembles a MOS) would be in this region<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>4\9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>533.33<br />
</td>
        <td>133.33<br />
</td>
        <td>266.67<br />
</td>
        <td>133.33<br />
</td>
        <td>3 1 1 3 1<br />
</td>
        <td style="text-align: center;">Boundary of &quot;practicality&quot; begins around here<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5\11<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>545.45<br />
</td>
        <td>109.09<br />
</td>
        <td>327.27<br />
</td>
        <td>218.18<br />
</td>
        <td>4 1 1 4 1<br />
</td>
        <td style="text-align: center;">Boundary of &quot;practicality&quot; ends around here<br />
</td>
    </tr>
    <tr>
        <td>1\2<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>600<br />
</td>
        <td>0<br />
</td>
        <td>600<br />
</td>
        <td>600<br />
</td>
        <td>1 0 0 1 0<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

<br />
From a <a class="wiki_link" href="/3-limit">3-limit</a> perspective, just make a chain of four 4/3's and octave-reduce, and you end up with pentatonic.<br />
<br />
From a <a class="wiki_link" href="/5-limit">5-limit</a> perspective, the most interesting temperaments with this kind of pentatonic scale are <a class="wiki_link" href="/meantone">meantone</a> and <a class="wiki_link" href="/Pelogic%20famiy">mavila</a>.<br />
<br />
There is also the interesting 2.3.7 temperament that tempers out <a class="wiki_link" href="/64_63">64/63</a> (&quot;no-fives <a class="wiki_link" href="/dominant">dominant</a>&quot;).</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>2012-04-23 13:16:38 UTC</tt>.<br>
+: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2012-11-19 21:08:39 UTC</tt>.<br>
-: The original revision id was <tt>324240626</tt>.<br>
+: The original revision id was <tt>384205648</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 13: / Line 13: @@
 ||   ||   ||   ||   ||   || 11\27 || 488.89 || 222.22 || 44.44 || 177.78 || 6 5 5 6 5 ||= Slendro (insofar as it resembles a MOS)
 would be in this region ||
-||   ||   ||   ||   || 9\22 ||   || 490.91 || 218.18 || 54.55 || 163.64 || 5 4 4 5 4 ||   ||
+||   ||   ||   ||   || 9\22 ||   || 490.91 || 218.18 || 54.55 || 163.64 || 5 4 4 5 4 ||=   ||
 ||   ||   ||   ||   ||   || 16\39 || 492.31 || 215.38 || 61.54 || 153.85 || 9 7 7 9 7 ||= No-5's superpyth/dominant is around here ||
-||   ||   ||   || 7\17 ||   ||   || 494.12 || 211.76 || 70.59 || 141.18 || 4 3 3 4 3 ||   ||
+||   ||   ||   || 7\17 ||   ||   || 494.12 || 211.76 || 70.59 || 141.18 || 4 3 3 4 3 ||=   ||
-||   ||   ||   ||   || 12\29 ||   || 496.55 || 206.9 || 82.76 || 124.14 || 7 5 5 7 5 ||   ||
+||   ||   ||   ||   || 12\29 ||   || 496.55 || 206.9 || 82.76 || 124.14 || 7 5 5 7 5 ||=   ||
 ||   ||   ||   ||   ||   || 17\41 || 497.56 || 204.88 || 87.8 || 117.07 || 10 7 7 10 7 ||= Pythagorean pentatonic is around here ||
 ||   ||   || 5\12 ||   ||   ||   || 500 || 200 || 100 || 100 || 3 2 2 3 2 ||= Familiar 12-equal pentatonic
@@ Line 24: / Line 24: @@
 ||   ||   ||   ||   ||   ||   || 1200/(4-phi) || 192.43 || 118.93 || 73.50 || phi 1 1 phi 1 ||= Golden meantone ||
 ||   ||   ||   ||   ||   || 21\50 || 504 || 192 || 120 || 72 || 13 8 8 13 8 ||=   ||
-||   ||   ||   || 8\19 ||   ||   || 505.26 || 189.47 || 126.32 || 63.16 || 5 3 3 5 3 ||   ||
+||   ||   ||   || 8\19 ||   ||   || 505.26 || 189.47 || 126.32 || 63.16 || 5 3 3 5 3 ||=   ||
 ||   || 3\7 ||   ||   ||   ||   || 514.29 || 171.43 || 171.43 || 0 || 2 1 1 2 1 ||= (Boundary of propriety: smaller
 generators than this are strictly proper) ||
 ||   ||   ||   || 7\16 ||   ||   || 525 || 150 || 225 || 75 || 5 2 2 5 2 ||= 5-note subset of pelog (insofar as it
 resembles a MOS) would be in this region ||
-||   ||   || 4\9 ||   ||   ||   || 533.33 || 133.33 || 266.67 || 133.33 || 3 1 1 3 1 ||   ||
+||   ||   || 4\9 ||   ||   ||   || 533.33 || 133.33 || 266.67 || 133.33 || 3 1 1 3 1 ||= Boundary of "practicality" begins around here ||
-||   ||   ||   || 5\11 ||   ||   || 545.45 || 109.09 || 327.27 || 218.18 || 4 1 1 4 1 ||   ||
+||   ||   ||   || 5\11 ||   ||   || 545.45 || 109.09 || 327.27 || 218.18 || 4 1 1 4 1 ||= Boundary of "practicality" ends around here ||
-|| 1\2 ||   ||   ||   ||   ||   || 600 || 0 || 600 || 600 || 1 0 0 1 0 ||   ||
+|| 1\2 ||   ||   ||   ||   ||   || 600 || 0 || 600 || 600 || 1 0 0 1 0 ||=   ||
 From a [[3-limit]] perspective, just make a chain of four 4/3's and octave-reduce, and you end up with pentatonic.
@@ Line 137: / Line 137: @@
          &lt;td&gt;5 4 4 5 4&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 189: / Line 189: @@
          &lt;td&gt;4 3 3 4 3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 215: / Line 215: @@
          &lt;td&gt;7 5 5 7 5&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 373: / Line 373: @@
          &lt;td&gt;5 3 3 5 3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 453: / Line 453: @@
          &lt;td&gt;3 1 1 3 1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Boundary of &amp;quot;practicality&amp;quot; begins around here&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 479: / Line 479: @@
          &lt;td&gt;4 1 1 4 1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Boundary of &amp;quot;practicality&amp;quot; ends around here&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 505: / Line 505: @@
          &lt;td&gt;1 0 0 1 0&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;

2L 3s: Difference between revisions

Revision as of 21:08, 19 November 2012

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

Original HTML content:

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