50edo: Difference between revisions

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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.

=Relations=  

=Commas=  

||~ Monzo ||~ Cents ||~ Ratio ||~ Name 1 ||~ Name 2 ||

|| | -4 4 -1 &gt; ||&gt; 21.51 ||= 81/80 || Syntonic comma || Didymus comma ||

|| | -6 -8 2 5 &gt; ||&gt; 1.12 ||=  || Wizma ||  ||

|| |-11 2 7 -3 &gt; ||&gt; 1.63 ||=  || Meter ||  ||

|| |-13 10 0 -1 &gt; ||&gt; 50.72 ||= 59049/57344 || Harrison's comma ||  ||

|| | 2 3 1 -2 -1 &gt; ||&gt; 3.21 ||= 540/539 || Swets' comma || Swetisma ||

&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;em&gt;50edo&lt;/em&gt; divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 50 equal parts of precisely 24 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. In the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;, it tempers out 81/80, making it a &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; system, and in that capacity has historically has drawn some notice. In &lt;a class="wiki_link_ext" href="http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf" rel="nofollow"&gt;&amp;quot;Harmonics or the Philosophy of Musical Sounds&amp;quot;&lt;/a&gt; (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 &lt;a class="wiki_link" href="/Target%20tunings"&gt;least squares&lt;/a&gt; tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher limit point of view. While &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; extends meantone with a &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; which is nearly pure, 50 has a flat 7/4 but both &lt;a class="wiki_link" href="/11_8"&gt;11/8&lt;/a&gt; and &lt;a class="wiki_link" href="/13_8"&gt;13/8&lt;/a&gt; are nearly pure.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Relations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Relations&lt;/h1&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Commas&lt;/h1&gt;

     &lt;/tr&gt;

     &lt;tr&gt;

&lt;/td&gt;

         &lt;td style="text-align: right;"&gt;5.57&lt;br /&gt;