47edo: Difference between revisions
Wikispaces>xenwolf **Imported revision 602901266 - Original comment: ** |
Wikispaces>TallKite **Imported revision 617184721 - Original comment: ** |
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| 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: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-09-05 03:57:05 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>617184721</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> | ||
| Line 11: | Line 11: | ||
47edo is the 15th [[prime numbers|prime]] edo, following [[43edo]] and preceding [[53edo]]. | 47edo is the 15th [[prime numbers|prime]] edo, following [[43edo]] and preceding [[53edo]]. | ||
47edo is a regular edo because its 5th falls between 4\7 = 686¢ and 3\5 = 720¢. (Its alternate 5th does as well.) 47edo is the most difficult regular edo to notate, because no other regular edo's 5th is as far off from a just 3/2. | |||
A notation using the best 5th has major and minor 2nds of 7 and 6 edosteps respectively, with the naturals creating a very 7edo-ish scale: | |||
D * * * * * * E * * * * * F * * * * * * G * * * * * * A * * * * * * B * * * * * C * * * * * * D | |||
This notation requires triple, quadruple and even quintuple sharps and flats. | |||
Using the 2nd best 5th is even more awkward. The major 2nd is 9 edosteps and the minor is only one. The naturals create a very 5edo-ish scale, with 2 of the notes inflected by a comma: | |||
D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D | |||
This notation requires quintuple, sextuple, and even septuple ups and downs. | |||
==Intervals of 47edo== | ==Intervals of 47edo== | ||
||~ Degree ||~ Size (Cents) || | ||~ Degree ||~ Size (Cents) ||~ relative notation || | ||
||= | ||= 0 ||> 0.0000 ||= P1 || | ||
||= | ||= 1 ||> 25.5319 || A1 || | ||
||= | ||= 2 ||> 51.0638 || AA1 || | ||
||= | ||= 3 ||> 76.5957 || A31, d32 || | ||
||= | ||= 4 ||> 102.1277 || dd2 || | ||
||= | ||= 5 ||> 127.6596 || d2 || | ||
||= | ||= 6 ||> 153.1915 || m2 || | ||
||= | ||= 7 ||> 178.7234 || M2 || | ||
||= | ||= 8 ||> 204.2553 || || | ||
||= | ||= 9 ||> 229.7872 || || | ||
||= 10 ||> | ||= 10 ||> 255.3191 || || | ||
||= 11 ||> | ||= 11 ||> 280.8511 || || | ||
||= 12 ||> | ||= 12 ||> 306.3830 || || | ||
||= 13 ||> | ||= 13 ||> 331.9149 || || | ||
||= 14 ||> | ||= 14 ||> 357.4468 || M3 || | ||
||= 15 ||> | ||= 15 ||> 382.9787 || || | ||
||= 16 ||> | ||= 16 ||> 408.5106 || || | ||
||= 17 ||> | ||= 17 ||> 434.0426 || || | ||
||= 18 ||> | ||= 18 ||> 459.5745 || || | ||
||= 19 ||> | ||= 19 ||> 485.1064 || || | ||
||= 20 ||> | ||= 20 ||> 510.6383 || || | ||
||= 21 ||> | ||= 21 ||> 536.1702 || || | ||
||= 22 ||> | ||= 22 ||> 561.7021 || || | ||
||= 23 ||> | ||= 23 ||> 587.2340 || || | ||
||= 24 ||> | ||= 24 ||> 612.7660 || || | ||
||= 25 ||> | ||= 25 ||> 638.2979 || || | ||
||= 26 ||> | ||= 26 ||> 663.8298 || || | ||
||= 27 ||> | ||= 27 ||> 689.3617 || || | ||
||= 28 ||> | ||= 28 ||> 714.8936 || || | ||
||= 29 ||> | ||= 29 ||> 740.4255 || || | ||
||= 30 ||> | ||= 30 ||> 765.9574 || || | ||
||= 31 ||> | ||= 31 ||> 791.4894 || || | ||
||= 32 ||> | ||= 32 ||> 817.0213 || || | ||
||= 33 ||> | ||= 33 ||> 842.5532 || || | ||
||= 34 ||> | ||= 34 ||> 868.0851 || || | ||
||= 35 ||> | ||= 35 ||> 893.6170 || || | ||
||= 36 ||> | ||= 36 ||> 919.1489 || || | ||
||= 37 ||> | ||= 37 ||> 944.6809 || || | ||
||= 38 ||> | ||= 38 ||> 970.2128 || || | ||
||= 39 ||> | ||= 39 ||> 995.7447 || || | ||
||= 40 ||> 1021.2766 || | ||= 40 ||> 1021.2766 || || | ||
||= 41 ||> 1046.8085 || | ||= 41 ||> 1046.8085 || || | ||
||= 42 ||> 1072.3404 || | ||= 42 ||> 1072.3404 || || | ||
||= 43 ||> 1097.8723 || | ||= 43 ||> 1097.8723 || || | ||
||= 44 ||> 1123.4043 || | ||= 44 ||> 1123.4043 || || | ||
||= 45 ||> 1148.9362 || | ||= 45 ||> 1148.9362 || || | ||
||= 46 ||> 1174.4681 || | ||= 46 ||> 1174.4681 || || | ||
||= 47 ||> 1200.0000 ||</pre></div> | ||= 47 ||> 1200.0000 || ||</pre></div> | ||
<h4>Original HTML content:</h4> | <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>47edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x47 tone Equal Temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #616161; font-size: 99%;">47 tone Equal Temperament</span></h1> | <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>47edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x47 tone Equal Temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #616161; font-size: 99%;">47 tone Equal Temperament</span></h1> | ||
| Line 69: | Line 80: | ||
<br /> | <br /> | ||
47edo is the 15th <a class="wiki_link" href="/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="/43edo">43edo</a> and preceding <a class="wiki_link" href="/53edo">53edo</a>.<br /> | 47edo is the 15th <a class="wiki_link" href="/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="/43edo">43edo</a> and preceding <a class="wiki_link" href="/53edo">53edo</a>.<br /> | ||
<br /> | |||
47edo is a regular edo because its 5th falls between 4\7 = 686¢ and 3\5 = 720¢. (Its alternate 5th does as well.) 47edo is the most difficult regular edo to notate, because no other regular edo's 5th is as far off from a just 3/2.<br /> | |||
<br /> | |||
A notation using the best 5th has major and minor 2nds of 7 and 6 edosteps respectively, with the naturals creating a very 7edo-ish scale: <br /> | |||
D * * * * * * E * * * * * F * * * * * * G * * * * * * A * * * * * * B * * * * * C * * * * * * D<br /> | |||
This notation requires triple, quadruple and even quintuple sharps and flats.<br /> | |||
<br /> | |||
Using the 2nd best 5th is even more awkward. The major 2nd is 9 edosteps and the minor is only one. The naturals create a very 5edo-ish scale, with 2 of the notes inflected by a comma: <br /> | |||
D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D<br /> | |||
This notation requires quintuple, sextuple, and even septuple ups and downs.<br /> | |||
<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x47 tone Equal Temperament-Intervals of 47edo"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals of 47edo</h2> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x47 tone Equal Temperament-Intervals of 47edo"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals of 47edo</h2> | ||
| Line 79: | Line 101: | ||
</th> | </th> | ||
<th>Size (Cents)<br /> | <th>Size (Cents)<br /> | ||
</th> | |||
<th>relative notation<br /> | |||
</th> | </th> | ||
</tr> | </tr> | ||
| Line 85: | Line 109: | ||
</td> | </td> | ||
<td style="text-align: right;">0.0000<br /> | <td style="text-align: right;">0.0000<br /> | ||
</td> | |||
<td style="text-align: center;">P1<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 91: | Line 117: | ||
</td> | </td> | ||
<td style="text-align: right;">25.5319<br /> | <td style="text-align: right;">25.5319<br /> | ||
</td> | |||
<td>A1<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 97: | Line 125: | ||
</td> | </td> | ||
<td style="text-align: right;">51.0638<br /> | <td style="text-align: right;">51.0638<br /> | ||
</td> | |||
<td>AA1<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 103: | Line 133: | ||
</td> | </td> | ||
<td style="text-align: right;">76.5957<br /> | <td style="text-align: right;">76.5957<br /> | ||
</td> | |||
<td>A31, d32<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 109: | Line 141: | ||
</td> | </td> | ||
<td style="text-align: right;">102.1277<br /> | <td style="text-align: right;">102.1277<br /> | ||
</td> | |||
<td>dd2<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 115: | Line 149: | ||
</td> | </td> | ||
<td style="text-align: right;">127.6596<br /> | <td style="text-align: right;">127.6596<br /> | ||
</td> | |||
<td>d2<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 121: | Line 157: | ||
</td> | </td> | ||
<td style="text-align: right;">153.1915<br /> | <td style="text-align: right;">153.1915<br /> | ||
</td> | |||
<td>m2<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 127: | Line 165: | ||
</td> | </td> | ||
<td style="text-align: right;">178.7234<br /> | <td style="text-align: right;">178.7234<br /> | ||
</td> | |||
<td>M2<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 133: | Line 173: | ||
</td> | </td> | ||
<td style="text-align: right;">204.2553<br /> | <td style="text-align: right;">204.2553<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 139: | Line 181: | ||
</td> | </td> | ||
<td style="text-align: right;">229.7872<br /> | <td style="text-align: right;">229.7872<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 145: | Line 189: | ||
</td> | </td> | ||
<td style="text-align: right;">255.3191<br /> | <td style="text-align: right;">255.3191<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 151: | Line 197: | ||
</td> | </td> | ||
<td style="text-align: right;">280.8511<br /> | <td style="text-align: right;">280.8511<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 157: | Line 205: | ||
</td> | </td> | ||
<td style="text-align: right;">306.3830<br /> | <td style="text-align: right;">306.3830<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 163: | Line 213: | ||
</td> | </td> | ||
<td style="text-align: right;">331.9149<br /> | <td style="text-align: right;">331.9149<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 169: | Line 221: | ||
</td> | </td> | ||
<td style="text-align: right;">357.4468<br /> | <td style="text-align: right;">357.4468<br /> | ||
</td> | |||
<td>M3<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 175: | Line 229: | ||
</td> | </td> | ||
<td style="text-align: right;">382.9787<br /> | <td style="text-align: right;">382.9787<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 181: | Line 237: | ||
</td> | </td> | ||
<td style="text-align: right;">408.5106<br /> | <td style="text-align: right;">408.5106<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 187: | Line 245: | ||
</td> | </td> | ||
<td style="text-align: right;">434.0426<br /> | <td style="text-align: right;">434.0426<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 193: | Line 253: | ||
</td> | </td> | ||
<td style="text-align: right;">459.5745<br /> | <td style="text-align: right;">459.5745<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 199: | Line 261: | ||
</td> | </td> | ||
<td style="text-align: right;">485.1064<br /> | <td style="text-align: right;">485.1064<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 205: | Line 269: | ||
</td> | </td> | ||
<td style="text-align: right;">510.6383<br /> | <td style="text-align: right;">510.6383<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 211: | Line 277: | ||
</td> | </td> | ||
<td style="text-align: right;">536.1702<br /> | <td style="text-align: right;">536.1702<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 217: | Line 285: | ||
</td> | </td> | ||
<td style="text-align: right;">561.7021<br /> | <td style="text-align: right;">561.7021<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 223: | Line 293: | ||
</td> | </td> | ||
<td style="text-align: right;">587.2340<br /> | <td style="text-align: right;">587.2340<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 229: | Line 301: | ||
</td> | </td> | ||
<td style="text-align: right;">612.7660<br /> | <td style="text-align: right;">612.7660<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 235: | Line 309: | ||
</td> | </td> | ||
<td style="text-align: right;">638.2979<br /> | <td style="text-align: right;">638.2979<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 241: | Line 317: | ||
</td> | </td> | ||
<td style="text-align: right;">663.8298<br /> | <td style="text-align: right;">663.8298<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 247: | Line 325: | ||
</td> | </td> | ||
<td style="text-align: right;">689.3617<br /> | <td style="text-align: right;">689.3617<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 253: | Line 333: | ||
</td> | </td> | ||
<td style="text-align: right;">714.8936<br /> | <td style="text-align: right;">714.8936<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 259: | Line 341: | ||
</td> | </td> | ||
<td style="text-align: right;">740.4255<br /> | <td style="text-align: right;">740.4255<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 265: | Line 349: | ||
</td> | </td> | ||
<td style="text-align: right;">765.9574<br /> | <td style="text-align: right;">765.9574<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 271: | Line 357: | ||
</td> | </td> | ||
<td style="text-align: right;">791.4894<br /> | <td style="text-align: right;">791.4894<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 277: | Line 365: | ||
</td> | </td> | ||
<td style="text-align: right;">817.0213<br /> | <td style="text-align: right;">817.0213<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 283: | Line 373: | ||
</td> | </td> | ||
<td style="text-align: right;">842.5532<br /> | <td style="text-align: right;">842.5532<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 289: | Line 381: | ||
</td> | </td> | ||
<td style="text-align: right;">868.0851<br /> | <td style="text-align: right;">868.0851<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 295: | Line 389: | ||
</td> | </td> | ||
<td style="text-align: right;">893.6170<br /> | <td style="text-align: right;">893.6170<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 301: | Line 397: | ||
</td> | </td> | ||
<td style="text-align: right;">919.1489<br /> | <td style="text-align: right;">919.1489<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 307: | Line 405: | ||
</td> | </td> | ||
<td style="text-align: right;">944.6809<br /> | <td style="text-align: right;">944.6809<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 313: | Line 413: | ||
</td> | </td> | ||
<td style="text-align: right;">970.2128<br /> | <td style="text-align: right;">970.2128<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 319: | Line 421: | ||
</td> | </td> | ||
<td style="text-align: right;">995.7447<br /> | <td style="text-align: right;">995.7447<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 325: | Line 429: | ||
</td> | </td> | ||
<td style="text-align: right;">1021.2766<br /> | <td style="text-align: right;">1021.2766<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 331: | Line 437: | ||
</td> | </td> | ||
<td style="text-align: right;">1046.8085<br /> | <td style="text-align: right;">1046.8085<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 337: | Line 445: | ||
</td> | </td> | ||
<td style="text-align: right;">1072.3404<br /> | <td style="text-align: right;">1072.3404<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 343: | Line 453: | ||
</td> | </td> | ||
<td style="text-align: right;">1097.8723<br /> | <td style="text-align: right;">1097.8723<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 349: | Line 461: | ||
</td> | </td> | ||
<td style="text-align: right;">1123.4043<br /> | <td style="text-align: right;">1123.4043<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 355: | Line 469: | ||
</td> | </td> | ||
<td style="text-align: right;">1148.9362<br /> | <td style="text-align: right;">1148.9362<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 361: | Line 477: | ||
</td> | </td> | ||
<td style="text-align: right;">1174.4681<br /> | <td style="text-align: right;">1174.4681<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 367: | Line 485: | ||
</td> | </td> | ||
<td style="text-align: right;">1200.0000<br /> | <td style="text-align: right;">1200.0000<br /> | ||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||