List of distinct EDO scales: Difference between revisions

Revision as of 22:05, 8 June 2015

IMPORTED REVISION FROM WIKISPACES

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This revision was by author Sarzadoce and made on 2015-06-08 22:05:49 UTC.

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Original Wikitext content:

Each [[Equal division of the octave|EDO]] has a finite number of distinct scales, assuming that the scales are equivalent up to cyclical permutation and that they are also irreducible. By irreducible is meant a scale that is not supported by a smaller EDO (e.g. 4424442, the diatonic scale in 24-EDO, is reducible because it is also contained in 12-EDO).

Below is a table which counts every possible scale for a given EDO (columns) and number of steps/notes (rows). Note that the total number of scales for each EDO is given by OEIS entries [[http://oeis.org/A059966|A059966]] and [[http://oeis.org/A001037|A001037]].

||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || EDO ||   ||   ||   ||   ||   ||   ||   ||
||   ||   || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 ||
||   || 1 || 1 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   || 2 ||   || 1 || 1 || 1 || 2 || 1 || 3 || 2 || 3 || 2 || 5 || 2 || 6 || 3 || 4 || 4 ||
||   || 3 ||   ||   || 1 || 1 || 2 || 3 || 5 || 6 || 9 || 10 || 15 || 14 || 22 || 21 || 28 || 28 ||
||   || 4 ||   ||   ||   || 1 || 1 || 3 || 5 || 9 || 14 || 21 || 30 || 39 || 55 || 68 || 90 || 106 ||
||   || 5 ||   ||   ||   ||   || 1 || 1 || 3 || 7 || 14 || 25 || 42 || 65 || 99 || 140 || 200 || 266 ||
||   || 6 ||   ||   ||   ||   ||   || 1 || 1 || 4 || 10 || 22 || 42 || 79 || 132 || 216 || 335 || 500 ||
||   || 7 ||   ||   ||   ||   ||   ||   || 1 || 1 || 4 || 12 || 30 || 66 || 132 || 245 || 429 || 714 ||
|| N || 8 ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 5 || 15 || 43 || 99 || 217 || 429 || 809 ||
||   || 9 ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 5 || 19 || 55 || 143 || 335 || 715 ||
||   || 10 ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 6 || 22 || 73 || 201 || 504 ||
||   || 11 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 6 || 26 || 91 || 273 ||
||   || 12 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 7 || 31 || 116 ||
||   || 13 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 7 || 35 ||
||   || 14 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 8 ||
||   || 15 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 ||
||   || 16 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   || Total || 1 || 1 || 2 || 3 || 6 || 9 || 18 || 30 || 56 || 99 || 186 || 335 || 630 || 1161 || 2182 || 4080 ||

(if someone could format this table a little better, it would be greatly appreciated)


==2-EDO Scales== 

11

==3-EDO Scales== 

21
111

==4-EDO Scales== 

31
211
1111

==5-EDO Scales== 

32
41
221
311
2111
11111

==6-EDO Scales== 

51
312
321
411
2121
2211
3111
21111
111111

==7-EDO Scales== 

43
52
61
322
331
412
421
511
2221
3112
3121
3211
4111
21211
22111
31111
211111
1111111

==8-EDO Scales== 

53
71
332
413
431
512
521
611
3122
3131
3212
3221
3311
4112
4121
4211
5111
22121
22211
31112
31121
31211
32111
41111
211211
212111
221111
311111
2111111
11111111

Original HTML content:

<html><head><title>Distinct EDO Scales</title></head><body>Each <a class="wiki_link" href="/Equal%20division%20of%20the%20octave">EDO</a> has a finite number of distinct scales, assuming that the scales are equivalent up to cyclical permutation and that they are also irreducible. By irreducible is meant a scale that is not supported by a smaller EDO (e.g. 4424442, the diatonic scale in 24-EDO, is reducible because it is also contained in 12-EDO).<br />
<br />
Below is a table which counts every possible scale for a given EDO (columns) and number of steps/notes (rows). Note that the total number of scales for each EDO is given by OEIS entries <a class="wiki_link_ext" href="http://oeis.org/A059966" rel="nofollow">A059966</a> and <a class="wiki_link_ext" href="http://oeis.org/A001037" rel="nofollow">A001037</a>.<br />
<br />


<table class="wiki_table">
    <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>EDO<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td>3<br />
</td>
        <td>4<br />
</td>
        <td>5<br />
</td>
        <td>6<br />
</td>
        <td>7<br />
</td>
        <td>8<br />
</td>
        <td>9<br />
</td>
        <td>10<br />
</td>
        <td>11<br />
</td>
        <td>12<br />
</td>
        <td>13<br />
</td>
        <td>14<br />
</td>
        <td>15<br />
</td>
        <td>16<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td>2<br />
</td>
        <td>3<br />
</td>
        <td>2<br />
</td>
        <td>5<br />
</td>
        <td>2<br />
</td>
        <td>6<br />
</td>
        <td>3<br />
</td>
        <td>4<br />
</td>
        <td>4<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td>3<br />
</td>
        <td>5<br />
</td>
        <td>6<br />
</td>
        <td>9<br />
</td>
        <td>10<br />
</td>
        <td>15<br />
</td>
        <td>14<br />
</td>
        <td>22<br />
</td>
        <td>21<br />
</td>
        <td>28<br />
</td>
        <td>28<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td>5<br />
</td>
        <td>9<br />
</td>
        <td>14<br />
</td>
        <td>21<br />
</td>
        <td>30<br />
</td>
        <td>39<br />
</td>
        <td>55<br />
</td>
        <td>68<br />
</td>
        <td>90<br />
</td>
        <td>106<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td>7<br />
</td>
        <td>14<br />
</td>
        <td>25<br />
</td>
        <td>42<br />
</td>
        <td>65<br />
</td>
        <td>99<br />
</td>
        <td>140<br />
</td>
        <td>200<br />
</td>
        <td>266<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td>10<br />
</td>
        <td>22<br />
</td>
        <td>42<br />
</td>
        <td>79<br />
</td>
        <td>132<br />
</td>
        <td>216<br />
</td>
        <td>335<br />
</td>
        <td>500<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td>12<br />
</td>
        <td>30<br />
</td>
        <td>66<br />
</td>
        <td>132<br />
</td>
        <td>245<br />
</td>
        <td>429<br />
</td>
        <td>714<br />
</td>
    </tr>
    <tr>
        <td>N<br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>5<br />
</td>
        <td>15<br />
</td>
        <td>43<br />
</td>
        <td>99<br />
</td>
        <td>217<br />
</td>
        <td>429<br />
</td>
        <td>809<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>9<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>1<br />
</td>
        <td>1<br />
</td>
        <td>5<br />
</td>
        <td>19<br />
</td>
        <td>55<br />
</td>
        <td>143<br />
</td>
        <td>335<br />
</td>
        <td>715<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>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>1<br />
</td>
        <td>1<br />
</td>
        <td>6<br />
</td>
        <td>22<br />
</td>
        <td>73<br />
</td>
        <td>201<br />
</td>
        <td>504<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>11<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>1<br />
</td>
        <td>1<br />
</td>
        <td>6<br />
</td>
        <td>26<br />
</td>
        <td>91<br />
</td>
        <td>273<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>7<br />
</td>
        <td>31<br />
</td>
        <td>116<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>13<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><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>7<br />
</td>
        <td>35<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>8<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>15<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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>Total<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td>3<br />
</td>
        <td>6<br />
</td>
        <td>9<br />
</td>
        <td>18<br />
</td>
        <td>30<br />
</td>
        <td>56<br />
</td>
        <td>99<br />
</td>
        <td>186<br />
</td>
        <td>335<br />
</td>
        <td>630<br />
</td>
        <td>1161<br />
</td>
        <td>2182<br />
</td>
        <td>4080<br />
</td>
    </tr>
</table>

<br />
(if someone could format this table a little better, it would be greatly appreciated)<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-2-EDO Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->2-EDO Scales</h2>
 <br />
11<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-3-EDO Scales"></a><!-- ws:end:WikiTextHeadingRule:2 -->3-EDO Scales</h2>
 <br />
21<br />
111<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-4-EDO Scales"></a><!-- ws:end:WikiTextHeadingRule:4 -->4-EDO Scales</h2>
 <br />
31<br />
211<br />
1111<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x-5-EDO Scales"></a><!-- ws:end:WikiTextHeadingRule:6 -->5-EDO Scales</h2>
 <br />
32<br />
41<br />
221<br />
311<br />
2111<br />
11111<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x-6-EDO Scales"></a><!-- ws:end:WikiTextHeadingRule:8 -->6-EDO Scales</h2>
 <br />
51<br />
312<br />
321<br />
411<br />
2121<br />
2211<br />
3111<br />
21111<br />
111111<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x-7-EDO Scales"></a><!-- ws:end:WikiTextHeadingRule:10 -->7-EDO Scales</h2>
 <br />
43<br />
52<br />
61<br />
322<br />
331<br />
412<br />
421<br />
511<br />
2221<br />
3112<br />
3121<br />
3211<br />
4111<br />
21211<br />
22111<br />
31111<br />
211111<br />
1111111<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="x-8-EDO Scales"></a><!-- ws:end:WikiTextHeadingRule:12 -->8-EDO Scales</h2>
 <br />
53<br />
71<br />
332<br />
413<br />
431<br />
512<br />
521<br />
611<br />
3122<br />
3131<br />
3212<br />
3221<br />
3311<br />
4112<br />
4121<br />
4211<br />
5111<br />
22121<br />
22211<br />
31112<br />
31121<br />
31211<br />
32111<br />
41111<br />
211211<br />
212111<br />
221111<br />
311111<br />
2111111<br />
11111111</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:Sarzadoce|Sarzadoce]] and made on <tt>2015-06-08 21:42:07 UTC</tt>.<br>
+: This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2015-06-08 22:05:49 UTC</tt>.<br>
-: The original revision id was <tt>553443752</tt>.<br>
+: The original revision id was <tt>553445150</tt>.<br>
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 ||   || Total || 1 || 1 || 2 || 3 || 6 || 9 || 18 || 30 || 56 || 99 || 186 || 335 || 630 || 1161 || 2182 || 4080 ||
-(if someone could format this table a little better, it would be greatly appreciated)</pre></div>
+(if someone could format this table a little better, it would be greatly appreciated)
+==2-EDO Scales==
+==3-EDO Scales==
+==4-EDO Scales==
+==5-EDO Scales==
+==6-EDO Scales==
+==7-EDO Scales==
+1111111
+==8-EDO Scales==
+2111111
+11111111</pre></div>
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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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Distinct EDO Scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Each &lt;a class="wiki_link" href="/Equal%20division%20of%20the%20octave"&gt;EDO&lt;/a&gt; has a finite number of distinct scales, assuming that the scales are equivalent up to cyclical permutation and that they are also irreducible. By irreducible is meant a scale that is not supported by a smaller EDO (e.g. 4424442, the diatonic scale in 24-EDO, is reducible because it is also contained in 12-EDO).&lt;br /&gt;
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-(if someone could format this table a little better, it would be greatly appreciated)&lt;/body&gt;&lt;/html&gt;</pre></div>
+(if someone could format this table a little better, it would be greatly appreciated)&lt;br /&gt;
+&lt;br /&gt;
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+ &lt;br /&gt;
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+ &lt;br /&gt;
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+ &lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-5-EDO Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;5-EDO Scales&lt;/h2&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-6-EDO Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;6-EDO Scales&lt;/h2&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x-7-EDO Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;7-EDO Scales&lt;/h2&gt;
+ &lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x-8-EDO Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;8-EDO Scales&lt;/h2&gt;
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+2111111&lt;br /&gt;
+11111111&lt;/body&gt;&lt;/html&gt;</pre></div>

List of distinct EDO scales: Difference between revisions

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