List of distinct EDO scales
IMPORTED REVISION FROM WIKISPACES
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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)
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 />
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<td><br />
</td>
<td><br />
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<td><br />
</td>
<td><br />
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<td><br />
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<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 />
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<td><br />
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<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 />
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<td><br />
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<td><br />
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<td><br />
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<td><br />
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<td><br />
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<td><br />
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<td><br />
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<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)</body></html>