Tour of regular temperaments: Difference between revisions
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-21 15:58:36 UTC</tt>.<br> | ||
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The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include [[15edo]], [[22edo]], [[37edo]], and [[59edo]]. | The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include [[15edo]], [[22edo]], [[37edo]], and [[59edo]]. | ||
===[[ | ===[[Würschmidt family]]=== | ||
The wuerschmidt family tempers out | The wuerschmidt family tempers out Würschmidt's comma, 393216/390625 = |17 1 -8>. Würschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a perfect 5th two octaves up); that is, (5/4)^8 * (393216/390625) = 6. It tends to generate the same MOS's as [[magic family|magic temperament]], but is tuned slightly more accurately. Both [[31edo]] and [[34edo]] can be used as würschmidt tunings, as can [[65edo]], which is quite accurate. | ||
===[[Augmented family]]=== | ===[[Augmented family]]=== | ||
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The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include <a class="wiki_link" href="/15edo">15edo</a>, <a class="wiki_link" href="/22edo">22edo</a>, <a class="wiki_link" href="/37edo">37edo</a>, and <a class="wiki_link" href="/59edo">59edo</a>.<br /> | The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include <a class="wiki_link" href="/15edo">15edo</a>, <a class="wiki_link" href="/22edo">22edo</a>, <a class="wiki_link" href="/37edo">37edo</a>, and <a class="wiki_link" href="/59edo">59edo</a>.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Rank 2 (including &quot;linear&quot;) temperaments-- | <!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Würschmidt family"></a><!-- ws:end:WikiTextHeadingRule:24 --><a class="wiki_link" href="/W%C3%BCrschmidt%20family">Würschmidt family</a></h3> | ||
The wuerschmidt family tempers out | The wuerschmidt family tempers out Würschmidt's comma, 393216/390625 = |17 1 -8&gt;. Würschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a perfect 5th two octaves up); that is, (5/4)^8 * (393216/390625) = 6. It tends to generate the same MOS's as <a class="wiki_link" href="/magic%20family">magic temperament</a>, but is tuned slightly more accurately. Both <a class="wiki_link" href="/31edo">31edo</a> and <a class="wiki_link" href="/34edo">34edo</a> can be used as würschmidt tunings, as can <a class="wiki_link" href="/65edo">65edo</a>, which is quite accurate.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Augmented family"></a><!-- ws:end:WikiTextHeadingRule:26 --><a class="wiki_link" href="/Augmented%20family">Augmented family</a></h3> | <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Augmented family"></a><!-- ws:end:WikiTextHeadingRule:26 --><a class="wiki_link" href="/Augmented%20family">Augmented family</a></h3> | ||