50edo: Difference between revisions
Wikispaces>xenwolf **Imported revision 575219197 - Original comment: ** |
Wikispaces>JosephRuhf **Imported revision 601556602 - Original comment: ** |
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<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:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-06 15:51:23 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>601556602</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> | ||
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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 [[http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf|"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. 50edo, 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. | //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 [[http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf|"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. 50edo, 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, 225/224 and 3136/3125 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 and its extension meanpop, it can be used to advantage for the 15&50 temperament ([[http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&limit=11|Coblack]]), and provides the optimal patent val for 11 and 13 limit [[Meantone family#Septimal | 50 tempers out 126/125, 225/224 and 3136/3125 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 and its extension meanpop, it can be used to advantage for the 15&50 temperament ([[http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&limit=11|Coblack]]), and provides the optimal patent val for 11 and 13 limit [[Meantone family#Septimal%20meantone-Bimeantone|bimeantone]]. It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma]], [[tel:6115295232|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. | ||
=Relations= | =Relations= | ||
| Line 71: | Line 71: | ||
The following table shows how [[Just-24|some prominent just intervals]] are represented in 50edo (ordered by absolute error). | The following table shows how [[Just-24|some prominent just intervals]] are represented in 50edo (ordered by absolute error). | ||
|| **Interval, complement** || **Error (abs., in [[cent|cents]])** || | || **Interval, complement** || **Error (abs., in [[cent|cents]])** || | ||
||= [[16_13|16/13]], [[13_8|13/8]] | ||= [[16_13|16/13]], [[13_8|13/8]] ||= 0.528 || | ||
||= [[15_14|15/14]], [[28_15|28/15]] ||= | ||= [[15_14|15/14]], [[28_15|28/15]] ||= 0.557 || | ||
||= [[11_8|11/8]], | ||= [[11_8|11/8]], [[16_11|16/11]] ||= 0.682 || | ||
||= [[13_11|13/11]], [[22_13|22/13]] ||= | ||= [[13_11|13/11]], [[22_13|22/13]] ||= 1.210 || | ||
||= [[13_10|13/10]], [[20_13|20/13]] ||= | ||= [[13_10|13/10]], [[20_13|20/13]] ||= 1.786 || | ||
||= [[5_4|5/4]], | ||= [[5_4|5/4]], [[8_5|8/5]] ||= 2.314 || | ||
||= [[7_6|7/6]], | ||= [[7_6|7/6]], [[12_7|12/7]] ||= 2.871 || | ||
||= [[11_10|11/10]], [[20_11|20/11]] ||= | ||= [[11_10|11/10]], [[20_11|20/11]] ||= 2.996 || | ||
||= [[9_7|9/7]], | ||= [[9_7|9/7]], [[14_9|14/9]] ||= 3.084 || | ||
||= [[6_5|6/5]], | ||= [[6_5|6/5]], [[5_3|5/3]] ||= 3.641 || | ||
||= [[13_12|13/12]], [[24_13|24/13]] ||= | ||= [[13_12|13/12]], [[24_13|24/13]] ||= 5.427 || | ||
||= [[4_3|4/3]], | ||= [[4_3|4/3]], [[3_2|3/2]] ||= 5.955 || | ||
||= [[7_5|7/5]], | ||= [[7_5|7/5]], [[10_7|10/7]] ||= 6.512 || | ||
||= [[12_11|12/11]], [[11_6|11/6]] | ||= [[12_11|12/11]], [[11_6|11/6]] ||= 6.637 || | ||
||= [[15_13|15/13]], [[26_15|26/15]] ||= | ||= [[15_13|15/13]], [[26_15|26/15]] ||= 7.741 || | ||
||= [[16_15|16/15]], [[15_8|15/8]] | ||= [[16_15|16/15]], [[15_8|15/8]] ||= 8.269 || | ||
||= [[14_13|14/13]], [[13_7|13/7]] | ||= [[14_13|14/13]], [[13_7|13/7]] ||= 8.298 || | ||
||= [[8_7|8/7]], | ||= [[8_7|8/7]], [[7_4|7/4]] ||= 8.826 || | ||
||= [[15_11|15/11]], [[22_15|22/15]] ||= | ||= [[15_11|15/11]], [[22_15|22/15]] ||= 8.951 || | ||
||= [[14_11|14/11]], [[11_7|11/7]] | ||= [[14_11|14/11]], [[11_7|11/7]] ||= 9.508 || | ||
||= [[10_9|10/9]], | ||= [[10_9|10/9]], [[9_5|9/5]] ||= 9.596 || | ||
||= [[18_13|18/13]], [[13_9|13/9]] | ||= [[18_13|18/13]], [[13_9|13/9]] ||= 11.382 || | ||
||= [[11_9|11/9]], | ||= [[11_9|11/9]], [[18_11|18/11]] ||= 11.408 || | ||
||= [[9_8|9/8]], | ||= [[9_8|9/8]], [[16_9|16/9]] ||= 11.910 || | ||
=Commas= | =Commas= | ||
50 EDO tempers out the following commas. (Note: This assumes the val < 50 79 116 140 173 185 204 212 226 |, comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2. | 50 EDO tempers out the following commas. (Note: This assumes the val < 50 79 116 140 173/1 185 204 212 226/1 |, comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2. | ||
||~ Monzo ||~ Cents ||~ Ratio ||~ Name 1 ||~ Name 2 || | ||~ Monzo ||~ Cents ||~ Ratio ||~ Name 1 ||~ Name 2 || | ||
|| | -4 4 -1 > ||> 21.51 ||= 81/80 || Syntonic comma || Didymus comma || | || | -4 4 -1 > ||> 21.51 ||= 81/80 || Syntonic comma || Didymus comma || | ||
|| | -27 -2 13 > ||> 18.17 ||= | || | -27 -2 13 > ||> 18.17 ||= || Ditonma || || | ||
|| | 23 6 -14 > ||> 3.34 ||= | || | 23 6 -14 > ||> 3.34 ||= || Vishnu comma || || | ||
|| | 1 2 -3 1 > ||> 13.79 ||= 126/125 || Starling comma || Small septimal comma || | || | 1 2 -3 1 > ||> 13.79 ||= 126/125 || Starling comma || Small septimal comma || | ||
|| | -5 2 2 -1 > ||> 7.71 ||= 225/224 || Septimal kleisma || Marvel comma || | || | -5 2 2 -1 > ||> 7.71 ||= 225/224 || Septimal kleisma || Marvel comma || | ||
|| | 6 0 -5 2 > ||> 6.08 ||= 3136/3125 || Hemimean || Middle second comma || | || | 6 0 -5 2 > ||> 6.08 ||= 3136/3125 || Hemimean || Middle second comma || | ||
|| | -6 -8 2 5 > ||> 1.12 ||= | || | -6 -8 2 5 > ||> 1.12 ||= || Wizma || || | ||
|| |-11 2 7 -3 > ||> 1.63 ||= | || |-11 2 7 -3 > ||> 1.63 ||= || Meter || || | ||
|| | 11 -10 -10 10 > ||> 5.57 ||= | || | 11 -10 -10 10/1 > ||> 5.57 ||= || Linus || || | ||
|| |-13 10 0 -1 > ||> 50.72 ||= 59049/57344 || Harrison's comma || || | || |-13 10 0 -1 > ||> 50.72 ||= 59049/57344 || Harrison's comma || || | ||
|| | 2 3 1 -2 -1 > ||> 3.21 ||= 540/539 || Swets' comma || Swetisma || | || | 2 3 1 -2 -1 > ||> 3.21 ||= 540/539 || Swets' comma || Swetisma || | ||
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<!-- ws:end:WikiTextTocRule:19 --><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 <a class="wiki_link_ext" href="http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf" rel="nofollow">&quot;Harmonics or the Philosophy of Musical Sounds&quot;</a> (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. 50edo, 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 /> | <!-- ws:end:WikiTextTocRule:19 --><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 <a class="wiki_link_ext" href="http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf" rel="nofollow">&quot;Harmonics or the Philosophy of Musical Sounds&quot;</a> (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. 50edo, 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 /> | <br /> | ||
50 tempers out 126/125, 225/224 and 3136/3125 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 and its extension meanpop, it can be used to advantage for the 15&amp;50 temperament (<a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&amp;limit=11" rel="nofollow">Coblack</a>), and provides the optimal patent val for 11 and 13 limit <a class="wiki_link" href="/Meantone%20family#Septimal | 50 tempers out 126/125, 225/224 and 3136/3125 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 and its extension meanpop, it can be used to advantage for the 15&amp;50 temperament (<a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&amp;limit=11" rel="nofollow">Coblack</a>), and provides the optimal patent val for 11 and 13 limit <a class="wiki_link" href="/Meantone%20family#Septimal%20meantone-Bimeantone">bimeantone</a>. It is also the unique equal temperament tempering out both 81/80 and the <a class="wiki_link" href="/vishnuzma">vishnuzma</a>, <a class="wiki_link" href="http://tel.wikispaces.com/6115295232">6115295232</a>/6103515625 = |23 6 -14&gt;, 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Relations"></a><!-- ws:end:WikiTextHeadingRule:0 -->Relations</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Relations"></a><!-- ws:end:WikiTextHeadingRule:0 -->Relations</h1> | ||
| Line 690: | Line 690: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/11_8">11/8</a>, | <td style="text-align: center;"><a class="wiki_link" href="/11_8">11/8</a>, <a class="wiki_link" href="/16_11">16/11</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">0.682<br /> | <td style="text-align: center;">0.682<br /> | ||
| Line 708: | Line 708: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/5_4">5/4</a>, | <td style="text-align: center;"><a class="wiki_link" href="/5_4">5/4</a>, <a class="wiki_link" href="/8_5">8/5</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">2.314<br /> | <td style="text-align: center;">2.314<br /> | ||
| Line 714: | Line 714: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/7_6">7/6</a>, | <td style="text-align: center;"><a class="wiki_link" href="/7_6">7/6</a>, <a class="wiki_link" href="/12_7">12/7</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">2.871<br /> | <td style="text-align: center;">2.871<br /> | ||
| Line 726: | Line 726: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/9_7">9/7</a>, | <td style="text-align: center;"><a class="wiki_link" href="/9_7">9/7</a>, <a class="wiki_link" href="/14_9">14/9</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">3.084<br /> | <td style="text-align: center;">3.084<br /> | ||
| Line 732: | Line 732: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/6_5">6/5</a>, | <td style="text-align: center;"><a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/5_3">5/3</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">3.641<br /> | <td style="text-align: center;">3.641<br /> | ||
| Line 744: | Line 744: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/4_3">4/3</a>, | <td style="text-align: center;"><a class="wiki_link" href="/4_3">4/3</a>, <a class="wiki_link" href="/3_2">3/2</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">5.955<br /> | <td style="text-align: center;">5.955<br /> | ||
| Line 750: | Line 750: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/7_5">7/5</a>, | <td style="text-align: center;"><a class="wiki_link" href="/7_5">7/5</a>, <a class="wiki_link" href="/10_7">10/7</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">6.512<br /> | <td style="text-align: center;">6.512<br /> | ||
| Line 780: | Line 780: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/8_7">8/7</a>, | <td style="text-align: center;"><a class="wiki_link" href="/8_7">8/7</a>, <a class="wiki_link" href="/7_4">7/4</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">8.826<br /> | <td style="text-align: center;">8.826<br /> | ||
| Line 798: | Line 798: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/10_9">10/9</a>, | <td style="text-align: center;"><a class="wiki_link" href="/10_9">10/9</a>, <a class="wiki_link" href="/9_5">9/5</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">9.596<br /> | <td style="text-align: center;">9.596<br /> | ||
| Line 810: | Line 810: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/11_9">11/9</a>, | <td style="text-align: center;"><a class="wiki_link" href="/11_9">11/9</a>, <a class="wiki_link" href="/18_11">18/11</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">11.408<br /> | <td style="text-align: center;">11.408<br /> | ||
| Line 816: | Line 816: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><a class="wiki_link" href="/9_8">9/8</a>, | <td style="text-align: center;"><a class="wiki_link" href="/9_8">9/8</a>, <a class="wiki_link" href="/16_9">16/9</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">11.910<br /> | <td style="text-align: center;">11.910<br /> | ||
| Line 825: | Line 825: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->Commas</h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->Commas</h1> | ||
50 EDO tempers out the following commas. (Note: This assumes the val &lt; 50 79 116 140 173 185 204 212 226 |, comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.<br /> | 50 EDO tempers out the following commas. (Note: This assumes the val &lt; 50 79 116 140 173/1 185 204 212 226/1 |, comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.<br /> | ||
| Line 858: | Line 858: | ||
<td style="text-align: right;">18.17<br /> | <td style="text-align: right;">18.17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td>Ditonma<br /> | <td>Ditonma<br /> | ||
| Line 870: | Line 870: | ||
<td style="text-align: right;">3.34<br /> | <td style="text-align: right;">3.34<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td>Vishnu comma<br /> | <td>Vishnu comma<br /> | ||
| Line 918: | Line 918: | ||
<td style="text-align: right;">1.12<br /> | <td style="text-align: right;">1.12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td>Wizma<br /> | <td>Wizma<br /> | ||
| Line 930: | Line 930: | ||
<td style="text-align: right;">1.63<br /> | <td style="text-align: right;">1.63<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td>Meter<br /> | <td>Meter<br /> | ||
| Line 938: | Line 938: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>| 11 -10 -10 10 &gt;<br /> | <td>| 11 -10 -10 10/1 &gt;<br /> | ||
</td> | </td> | ||
<td style="text-align: right;">5.57<br /> | <td style="text-align: right;">5.57<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td>Linus<br /> | <td>Linus<br /> | ||