Xenharmonic Wiki

List of distinct EDO scales

Revision as of 02:29, 9 June 2015 by Wikispaces>Sarzadoce (**Imported revision 553456136 - Original comment: **)

IMPORTED REVISION FROM WIKISPACES

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

This revision was by author Sarzadoce and made on 2015-06-09 02:29:27 UTC.
The original revision id was 553456136.
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:

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

==9-EDO Scales== 

54
72
81
423
432
441
513
522
531
612
621
711
3222
3231
3312
3321
4113
4122
4131
4212
4221
4311
5112
5121
5211
6111
22221
31122
31212
31221
31311
32112
32121
32211
33111
41112
41121
41211
42111
51111
212121
221121
221211
222111
311112
311121
311211
312111
321111
411111
2112111
2121111
2211111
3111111
21111111
111111111

==10-EDO Scales== 

73
91
433
514
523
532
541
613
631
712
721
811
3232
3322
3331
4123
4132
4141
4213
4231
4312
4321
4411
5113
5122
5131
5212
5221
5311
6112
6121
6211
7111
31222
31312
32122
32131
32212
32221
32311
33112
33121
33211
41113
41122
41131
41212
41221
41311
42112
42121
42211
43111
51112
51121
51211
52111
61111
221221
222121
222211
311122
311212
311221
311311
312112
312121
312211
313111
321112
321121
321211
322111
331111
411112
411121
411211
412111
421111
511111
2121211
2211121
2211211
2212111
2221111
3111112
3111121
3111211
3112111
3121111
3211111
4111111
21112111
21121111
21211111
22111111
31111111
211111111
1111111111

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<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="x-9-EDO Scales"></a><!-- ws:end:WikiTextHeadingRule:14 -->9-EDO Scales</h2>
 <br />
54<br />
72<br />
81<br />
423<br />
432<br />
441<br />
513<br />
522<br />
531<br />
612<br />
621<br />
711<br />
3222<br />
3231<br />
3312<br />
3321<br />
4113<br />
4122<br />
4131<br />
4212<br />
4221<br />
4311<br />
5112<br />
5121<br />
5211<br />
6111<br />
22221<br />
31122<br />
31212<br />
31221<br />
31311<br />
32112<br />
32121<br />
32211<br />
33111<br />
41112<br />
41121<br />
41211<br />
42111<br />
51111<br />
212121<br />
221121<br />
221211<br />
222111<br />
311112<br />
311121<br />
311211<br />
312111<br />
321111<br />
411111<br />
2112111<br />
2121111<br />
2211111<br />
3111111<br />
21111111<br />
111111111<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="x-10-EDO Scales"></a><!-- ws:end:WikiTextHeadingRule:16 -->10-EDO Scales</h2>
 <br />
73<br />
91<br />
433<br />
514<br />
523<br />
532<br />
541<br />
613<br />
631<br />
712<br />
721<br />
811<br />
3232<br />
3322<br />
3331<br />
4123<br />
4132<br />
4141<br />
4213<br />
4231<br />
4312<br />
4321<br />
4411<br />
5113<br />
5122<br />
5131<br />
5212<br />
5221<br />
5311<br />
6112<br />
6121<br />
6211<br />
7111<br />
31222<br />
31312<br />
32122<br />
32131<br />
32212<br />
32221<br />
32311<br />
33112<br />
33121<br />
33211<br />
41113<br />
41122<br />
41131<br />
41212<br />
41221<br />
41311<br />
42112<br />
42121<br />
42211<br />
43111<br />
51112<br />
51121<br />
51211<br />
52111<br />
61111<br />
221221<br />
222121<br />
222211<br />
311122<br />
311212<br />
311221<br />
311311<br />
312112<br />
312121<br />
312211<br />
313111<br />
321112<br />
321121<br />
321211<br />
322111<br />
331111<br />
411112<br />
411121<br />
411211<br />
412111<br />
421111<br />
511111<br />
2121211<br />
2211121<br />
2211211<br />
2212111<br />
2221111<br />
3111112<br />
3111121<br />
3111211<br />
3112111<br />
3121111<br />
3211111<br />
4111111<br />
21112111<br />
21121111<br />
21211111<br />
22111111<br />
31111111<br />
211111111<br />
1111111111</body></html>
Retrieved from "https://en.xen.wiki/index.php?title=List_of_distinct_EDO_scales&oldid=14265"