List of distinct EDO scales: Difference between revisions
Wikispaces>Sarzadoce **Imported revision 553441766 - Original comment: ** |
Wikispaces>Sarzadoce **Imported revision 553443752 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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: | : This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2015-06-08 21:42:07 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>553443752</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<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;white-space: pre-wrap ! important" class="old-revision-html">|| || || || || || || || || || || | <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;white-space: pre-wrap ! important" class="old-revision-html">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 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || | ||
|| || 1 || 1 || || || || || || || || || || || || || || || || | || || 1 || 1 || || || || || || || || || || || || || || || || | ||
|| || 2 || || 1 || 1 || 1 || 2 || 1 || 3 || 2 || 3 || 2 || 5 || 2 || 6 || 3 || 4 || 4 || | || || 2 || || 1 || 1 || 1 || 2 || 1 || 3 || 2 || 3 || 2 || 5 || 2 || 6 || 3 || 4 || 4 || | ||
|| || 3 || || || | || || 3 || || || 1 || 1 || 2 || 3 || 5 || 6 || 9 || 10 || 15 || 14 || 22 || 21 || 28 || 28 || | ||
|| || 4 || || || || | || || 4 || || || || 1 || 1 || 3 || 5 || 9 || 14 || 21 || 30 || 39 || 55 || 68 || 90 || 106 || | ||
|| || 5 || || || || || | || || 5 || || || || || 1 || 1 || 3 || 7 || 14 || 25 || 42 || 65 || 99 || 140 || 200 || 266 || | ||
|| || 6 || || || || || || | || || 6 || || || || || || 1 || 1 || 4 || 10 || 22 || 42 || 79 || 132 || 216 || 335 || 500 || | ||
|| || 7 || || || || || || || | || || 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)</pre></div> | |||
<h4>Original HTML content:</h4> | |||
<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"><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"> | <table class="wiki_table"> | ||
| Line 68: | Line 61: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>EDO<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
| Line 208: | Line 201: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>2<br /> | ||
</td> | </td> | ||
<td><br /> | <td>3<br /> | ||
</td> | </td> | ||
<td><br /> | <td>5<br /> | ||
</td> | </td> | ||
<td><br /> | <td>6<br /> | ||
</td> | </td> | ||
<td><br /> | <td>9<br /> | ||
</td> | </td> | ||
<td><br /> | <td>10<br /> | ||
</td> | </td> | ||
<td><br /> | <td>15<br /> | ||
</td> | </td> | ||
<td><br /> | <td>14<br /> | ||
</td> | </td> | ||
<td><br /> | <td>22<br /> | ||
</td> | </td> | ||
<td><br /> | <td>21<br /> | ||
</td> | </td> | ||
<td><br /> | <td>28<br /> | ||
</td> | </td> | ||
<td><br /> | <td>28<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 248: | Line 241: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>3<br /> | ||
</td> | </td> | ||
<td><br /> | <td>5<br /> | ||
</td> | </td> | ||
<td><br /> | <td>9<br /> | ||
</td> | </td> | ||
<td><br /> | <td>14<br /> | ||
</td> | </td> | ||
<td><br /> | <td>21<br /> | ||
</td> | </td> | ||
<td><br /> | <td>30<br /> | ||
</td> | </td> | ||
<td><br /> | <td>39<br /> | ||
</td> | </td> | ||
<td><br /> | <td>55<br /> | ||
</td> | </td> | ||
<td><br /> | <td>68<br /> | ||
</td> | </td> | ||
<td><br /> | <td>90<br /> | ||
</td> | </td> | ||
<td><br /> | <td>106<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 288: | Line 281: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>3<br /> | ||
</td> | </td> | ||
<td><br /> | <td>7<br /> | ||
</td> | </td> | ||
<td><br /> | <td>14<br /> | ||
</td> | </td> | ||
<td><br /> | <td>25<br /> | ||
</td> | </td> | ||
<td><br /> | <td>42<br /> | ||
</td> | </td> | ||
<td><br /> | <td>65<br /> | ||
</td> | </td> | ||
<td><br /> | <td>99<br /> | ||
</td> | </td> | ||
<td><br /> | <td>140<br /> | ||
</td> | </td> | ||
<td><br /> | <td>200<br /> | ||
</td> | </td> | ||
<td><br /> | <td>266<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 328: | Line 321: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>4<br /> | ||
</td> | </td> | ||
<td><br /> | <td>10<br /> | ||
</td> | </td> | ||
<td><br /> | <td>22<br /> | ||
</td> | </td> | ||
<td><br /> | <td>42<br /> | ||
</td> | </td> | ||
<td><br /> | <td>79<br /> | ||
</td> | </td> | ||
<td><br /> | <td>132<br /> | ||
</td> | </td> | ||
<td><br /> | <td>216<br /> | ||
</td> | </td> | ||
<td><br /> | <td>335<br /> | ||
</td> | </td> | ||
<td><br /> | <td>500<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 368: | Line 361: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>4<br /> | ||
</td> | </td> | ||
<td><br /> | <td>12<br /> | ||
</td> | </td> | ||
<td><br /> | <td>30<br /> | ||
</td> | </td> | ||
<td><br /> | <td>66<br /> | ||
</td> | </td> | ||
<td><br /> | <td>132<br /> | ||
</td> | </td> | ||
<td><br /> | <td>245<br /> | ||
</td> | </td> | ||
<td><br /> | <td>429<br /> | ||
</td> | </td> | ||
<td><br /> | <td>714<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td><br /> | <td>N<br /> | ||
</td> | </td> | ||
<td>8<br /> | <td>8<br /> | ||
| Line 408: | Line 401: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>5<br /> | ||
</td> | </td> | ||
<td><br /> | <td>15<br /> | ||
</td> | </td> | ||
<td><br /> | <td>43<br /> | ||
</td> | </td> | ||
<td><br /> | <td>99<br /> | ||
</td> | </td> | ||
<td><br /> | <td>217<br /> | ||
</td> | </td> | ||
<td><br /> | <td>429<br /> | ||
</td> | </td> | ||
<td><br /> | <td>809<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 448: | Line 441: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>5<br /> | ||
</td> | </td> | ||
<td><br /> | <td>19<br /> | ||
</td> | </td> | ||
<td><br /> | <td>55<br /> | ||
</td> | </td> | ||
<td><br /> | <td>143<br /> | ||
</td> | </td> | ||
<td><br /> | <td>335<br /> | ||
</td> | </td> | ||
<td><br /> | <td>715<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 488: | Line 481: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>6<br /> | ||
</td> | </td> | ||
<td><br /> | <td>22<br /> | ||
</td> | </td> | ||
<td><br /> | <td>73<br /> | ||
</td> | </td> | ||
<td><br /> | <td>201<br /> | ||
</td> | </td> | ||
<td><br /> | <td>504<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 528: | Line 521: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>6<br /> | ||
</td> | </td> | ||
<td><br /> | <td>26<br /> | ||
</td> | </td> | ||
<td><br /> | <td>91<br /> | ||
</td> | </td> | ||
<td><br /> | <td>273<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 568: | Line 561: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>7<br /> | ||
</td> | </td> | ||
<td><br /> | <td>31<br /> | ||
</td> | </td> | ||
<td><br /> | <td>116<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 608: | Line 601: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>7<br /> | ||
</td> | </td> | ||
<td><br /> | <td>35<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 648: | Line 641: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>8<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 688: | Line 681: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 728: | Line 721: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>1<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>Total<br /> | ||
</td> | </td> | ||
<td>1<br /> | <td>1<br /> | ||
| Line 822: | Line 772: | ||
</td> | </td> | ||
<td>2<br /> | <td>2<br /> | ||
</td> | </td> | ||
<td>3<br /> | <td>3<br /> | ||
</td> | </td> | ||
<td>6<br /> | <td>6<br /> | ||
</td> | </td> | ||
<td>9<br /> | <td>9<br /> | ||
</td> | </td> | ||
<td> | <td>18<br /> | ||
</td> | </td> | ||
<td>30<br /> | <td>30<br /> | ||
</td> | </td> | ||
<td> | <td>56<br /> | ||
</td> | </td> | ||
<td>99<br /> | <td>99<br /> | ||
</td> | </td> | ||
<td> | <td>186<br /> | ||
</td> | </td> | ||
<td>335<br /> | <td>335<br /> | ||
</td> | </td> | ||
<td> | <td>630<br /> | ||
</td> | </td> | ||
<td> | <td>1161<br /> | ||
</td> | </td> | ||
<td> | <td>2182<br /> | ||
</td> | </td> | ||
<td> | <td>4080<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
</body></html></pre></div> | <br /> | ||
(if someone could format this table a little better, it would be greatly appreciated)</body></html></pre></div> | |||