50edo
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//50edo// divides the [[octave]] into 50 equal parts of precisely 24 [[cent]]s each. In the [[5-limit]], it tempers out 81/80, making it a [[meantone]] system, and in that capacity has historically has drawn some notice. In "Harmonics or the Philosophy of Musical Sounds" (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts - 50edo, in one word. Later W. S. B. Woolhouse noted it was fairly close to the [[Target tunings|least squares]] tuning for 5-limit meantone. 50, however, is especially interesting from a higher limit point of view. While [[31edo]] extends meantone with a [[7_4|7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11_8|11/8]] and [[13_8|13/8]] are nearly pure. 50 tempers out 126/125 in the [[7-limit]], indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the [[11-limit]] and 105/104, 144/143 and 196/195 in the [[13-limit]], and can be used for even higher limits. Aside from meantone, it can be used to advantage for the 15&50 temperament. It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma]], 6115295232/6103515625 = |23 6 -14>, so that in 50et seven chromatic semitones are a perfect fourth. In 12et by comparison this gives a fifth, in 31et a doubly diminished fifth, and in 19et a diminished fourth. [[http://www.archive.org/details/harmonicsorphilo00smit|Robert Smith's book online]] [[http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html|More information about Robert Smith's temperament]] ==Relations== The 50-edo system is related to [[7edo]], [[12edo]], [[19edo]], [[31edo]] as the next approximation to the "Golden Tone System" ([[Das Goldene Tonsystem]]) of Thorvald Kornerup. ==Intervals== || Degrees of 50-EDO || Cents value || || 0 || 0 || || 1 || 24 || || 2 || 48 || || 3 || 72 || || 4 || 96 || || 5 || 120 || || 6 || 144 || || 7 || 168 || || 8 || 192 || || 9 || 216 || || 10 || 240 || || 11 || 264 || || 12 || 288 || || 13 || 312 || || 14 || 336 || || 15 || 360 || || 16 || 384 || || 17 || 408 || || 18 || 432 || || 19 || 456 || || 20 || 480 || || 21 || 504 || || 22 || 528 || || 23 || 552 || || 24 || 576 || || 25 || 600 || || 26 || 624 || || 27 || 648 || || 28 || 672 || || 29 || 696 || || 30 || 720 || || 31 || 744 || || 32 || 768 || || 33 || 792 || || 34 || 816 || || 35 || 840 || || 36 || 864 || || 37 || 888 || || 38 || 912 || || 39 || 936 || || 40 || 960 || || 41 || 984 || || 42 || 1008 || || 43 || 1032 || || 44 || 1056 || || 45 || 1080 || || 46 || 1104 || || 47 || 1128 || || 48 || 1152 || || 49 || 1176 ||
Original HTML content:
<html><head><title>50edo</title></head><body><em>50edo</em> divides the <a class="wiki_link" href="/octave">octave</a> into 50 equal parts of precisely 24 <a class="wiki_link" href="/cent">cent</a>s each. In the <a class="wiki_link" href="/5-limit">5-limit</a>, it tempers out 81/80, making it a <a class="wiki_link" href="/meantone">meantone</a> system, and in that capacity has historically has drawn some notice. In "Harmonics or the Philosophy of Musical Sounds" (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts - 50edo, in one word. Later W. S. B. Woolhouse noted it was fairly close to the <a class="wiki_link" href="/Target%20tunings">least squares</a> tuning for 5-limit meantone. 50, however, is especially interesting from a higher limit point of view. While <a class="wiki_link" href="/31edo">31edo</a> extends meantone with a <a class="wiki_link" href="/7_4">7/4</a> which is nearly pure, 50 has a flat 7/4 but both <a class="wiki_link" href="/11_8">11/8</a> and <a class="wiki_link" href="/13_8">13/8</a> are nearly pure. <br />
<br />
50 tempers out 126/125 in the <a class="wiki_link" href="/7-limit">7-limit</a>, indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the <a class="wiki_link" href="/11-limit">11-limit</a> and 105/104, 144/143 and 196/195 in the <a class="wiki_link" href="/13-limit">13-limit</a>, and can be used for even higher limits. Aside from meantone, it can be used to advantage for the 15&50 temperament. It is also the unique equal temperament tempering out both 81/80 and the <a class="wiki_link" href="/vishnuzma">vishnuzma</a>, 6115295232/6103515625 = |23 6 -14>, so that in 50et seven chromatic semitones are a perfect fourth. In 12et by comparison this gives a fifth, in 31et a doubly diminished fifth, and in 19et a diminished fourth.<br />
<br />
<a class="wiki_link_ext" href="http://www.archive.org/details/harmonicsorphilo00smit" rel="nofollow">Robert Smith's book online</a><br />
<a class="wiki_link_ext" href="http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html" rel="nofollow">More information about Robert Smith's temperament</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Relations"></a><!-- ws:end:WikiTextHeadingRule:0 -->Relations</h2>
The 50-edo system is related to <a class="wiki_link" href="/7edo">7edo</a>, <a class="wiki_link" href="/12edo">12edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/31edo">31edo</a> as the next approximation to the "Golden Tone System" (<a class="wiki_link" href="/Das%20Goldene%20Tonsystem">Das Goldene Tonsystem</a>) of Thorvald Kornerup.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h2>
<br />
<table class="wiki_table">
<tr>
<td>Degrees of 50-EDO<br />
</td>
<td>Cents value<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>24<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>48<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>72<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>96<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>120<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>144<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>168<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>192<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>216<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>240<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>264<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>288<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>312<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>336<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>360<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>384<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>408<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>432<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>456<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>480<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>504<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>528<br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td>552<br />
</td>
</tr>
<tr>
<td>24<br />
</td>
<td>576<br />
</td>
</tr>
<tr>
<td>25<br />
</td>
<td>600<br />
</td>
</tr>
<tr>
<td>26<br />
</td>
<td>624<br />
</td>
</tr>
<tr>
<td>27<br />
</td>
<td>648<br />
</td>
</tr>
<tr>
<td>28<br />
</td>
<td>672<br />
</td>
</tr>
<tr>
<td>29<br />
</td>
<td>696<br />
</td>
</tr>
<tr>
<td>30<br />
</td>
<td>720<br />
</td>
</tr>
<tr>
<td>31<br />
</td>
<td>744<br />
</td>
</tr>
<tr>
<td>32<br />
</td>
<td>768<br />
</td>
</tr>
<tr>
<td>33<br />
</td>
<td>792<br />
</td>
</tr>
<tr>
<td>34<br />
</td>
<td>816<br />
</td>
</tr>
<tr>
<td>35<br />
</td>
<td>840<br />
</td>
</tr>
<tr>
<td>36<br />
</td>
<td>864<br />
</td>
</tr>
<tr>
<td>37<br />
</td>
<td>888<br />
</td>
</tr>
<tr>
<td>38<br />
</td>
<td>912<br />
</td>
</tr>
<tr>
<td>39<br />
</td>
<td>936<br />
</td>
</tr>
<tr>
<td>40<br />
</td>
<td>960<br />
</td>
</tr>
<tr>
<td>41<br />
</td>
<td>984<br />
</td>
</tr>
<tr>
<td>42<br />
</td>
<td>1008<br />
</td>
</tr>
<tr>
<td>43<br />
</td>
<td>1032<br />
</td>
</tr>
<tr>
<td>44<br />
</td>
<td>1056<br />
</td>
</tr>
<tr>
<td>45<br />
</td>
<td>1080<br />
</td>
</tr>
<tr>
<td>46<br />
</td>
<td>1104<br />
</td>
</tr>
<tr>
<td>47<br />
</td>
<td>1128<br />
</td>
</tr>
<tr>
<td>48<br />
</td>
<td>1152<br />
</td>
</tr>
<tr>
<td>49<br />
</td>
<td>1176<br />
</td>
</tr>
</table>
</body></html>