Tour of regular temperaments: Difference between revisions

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===[[Diaschismic family]]===  

The diaschismic family tempers out 2048/2025, the diaschisma. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. Using [[22edo]] as a tuning is associated with [[pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out.

===[[Porcupine family]]===  

This tempers out the vishnuzma, |23 6 -14&gt;.

===[[Bug family]]===  

  The diaschismic family tempers out 2048/2025, the diaschisma. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;, &lt;a class="wiki_link" href="/46edo"&gt;46edo&lt;/a&gt;, &lt;a class="wiki_link" href="/56edo"&gt;56edo&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58edo&lt;/a&gt; and &lt;a class="wiki_link" href="/80edo"&gt;80edo&lt;/a&gt;. Using &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; as a tuning is associated with &lt;a class="wiki_link" href="/pajara"&gt;pajara&lt;/a&gt; temperament, where the intervals 50/49 and 64/63 are tempered out.&lt;br /&gt;

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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 has a generator of a minor whole tone (10/9), three of which make up a fourth.&lt;br /&gt;

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  The wuerschmidt family tempers out Wuerschmidt's comma, 393216/390625 = |17 1 -8&amp;gt;. Wuerschmidt 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. Both &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; and &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt; can be used as wuerschmit tunings, as can &lt;a class="wiki_link" href="/65edo"&gt;65edo&lt;/a&gt;, which is quite accurate.&lt;br /&gt;

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  The augmented family tempers out the diesis of 128/125, the difference between three 5/4 major thirds and a 2/1 octave, and so identifies the major third with the third-octave. Hence it has the same 400-cent 5/4-approximations as &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, which is an excellent tuning for augmented.&lt;br /&gt;

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  The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, 25/24 (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2), and hence identify major and minor thirds. &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt; makes for a &amp;quot;good&amp;quot; dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the &amp;quot;neutral&amp;quot; dicot 3rds span a 3/2.&lt;br /&gt;

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  The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by 20000/19683, the minimal diesis or tetracot comma. 7edo can also be considered a tetracot tuning, as can 20edo, 27edo, 34edo, and 41edo.&lt;br /&gt;

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  This tempers out the sensipent comma, 78732/78125, also known as the medium semicomma.&lt;br /&gt;

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  The semicomma (also known as &lt;strong&gt;Fokker's comma)&lt;/strong&gt; 2109375/2097152 = |-21 3 7&amp;gt; is tempered out by the members of the semicomma family. It doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as 7/6, leading to orwell temperament.&lt;br /&gt;

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  The Pythagorean family tempers out the Pythagorean comma, |-19 12 0&amp;gt;. Since this is a 3-limit comma, it is also a 5-limit comma and can stand as parent to a 7-limit or higher family, in this case containing compton temperament and catler temperament.&lt;br /&gt;

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  This family tempers out the apotome, 2187/2048, which is a 3-limit comma.&lt;br /&gt;

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  The gammic family tempers out the gammic comma, |-29 -11 20&amp;gt;. The head of the family is 5-limit gammic, whose generator chain is &lt;a class="wiki_link" href="/Carlos%20Gamma"&gt;Carlos Gamma&lt;/a&gt;. Another member is Neptune temperament.&lt;br /&gt;

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  This tempers out the minortone comma, |-16 35 -17&amp;gt;. The head of the family is minortonic temperament, with generator a minor tone.&lt;br /&gt;

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  The sycamore family tempers out the sycamore comma, |-16 -6 11&amp;gt; = 48828125/47775744, which is the amount by which five stacked chromatic semitones, 25/24, exceed 6/5, and hence also the amount six exceeds 5/4.&lt;br /&gt;

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  This tempers out the mutt comma, |-44 -3 21&amp;gt;, leading to some strange properties.&lt;br /&gt;

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  This tempers out escapade, |32 -7 -9&amp;gt;.&lt;br /&gt;

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  This tempers out the vulture comma, |24 -21 4&amp;gt;.&lt;br /&gt;

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  This tempers out the vishnuzma, |23 6 -14&amp;gt;.&lt;br /&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc27"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Bug family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;&lt;a class="wiki_link" href="/Bug%20family"&gt;Bug family&lt;/a&gt;&lt;/h3&gt;