Tour of regular temperaments: Difference between revisions

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===[[Sycamore family]]===  

The sycamore family tempers out the sycamore comma, |-16 -6 11&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.

===[[Escapade family]]===  

===[[Ragismic microtemperaments]]===  

A ragismic temperament is one which tempers out |-1 -7 4 1&gt; = 4375/4374. These are not by any means all microtemperaments, but those which are not highly accurate are probably best discussed under another heading. Accurate ones include ennealimmal, supermajor, enneadecal, amity, mitonic, parakleismic, gamera and vishnu. Pontiac belongs on the list but falls under the schismatic family rubric.

===[[Turkish maqam music temperaments]]===  

The hemimean family of rank three temperaments tempers out the hemimean comma, 3136/3125.

The dimcomp family of rank three temperaments tempers out the dimcomp comma, 390625/388962.

The quartonic temperament tempers out 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7.

These temper out the valinorsma, 176/175.

  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 escapade comma, |32 -7 -9&amp;gt;, which is the difference between nine just major thirds and seven just fourths.&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;, or the amount by which seven chromatic semitones (25/24) fall short of a perfect fourth (4/), or (4/3)/(25/24)^7.&lt;br /&gt;

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  This tempers out the Luna comma; |38 -2 -15&amp;gt; (274877906944/274658203125)&lt;br /&gt;

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  This tempers out the laconic comma, 2187/2000, which is the difference between three 10/9's and one 3/2. Laconic is supported by 16-EDO, 21-EDO, and 37b-EDO, among others.&lt;br /&gt;

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  If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;subgroup&lt;/a&gt; of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. We can modify the definition of &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.&lt;br /&gt;

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Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called &amp;quot;blackjack&amp;quot; and a 31-note scale called &amp;quot;canasta&amp;quot; have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.&lt;br /&gt;

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  This clan tempers out 245/243, the sensamagic comma.&lt;br /&gt;

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  This tempers out the jubilisma, 50/49, E.A. the difference between 10/7 and 7/5.&lt;br /&gt;

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  This tempers out the hemimean comma, 3136/3125, a no-threes comma.&lt;br /&gt;

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  This tempers out the mirkwai comma, |0 3 4 -5&amp;gt; = 16875/16807, a no-twos comma (ratio of odd numbers.)&lt;br /&gt;

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  These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35.&lt;br /&gt;

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  These temper out the avicennma, |-9 1 2 1&amp;gt; = 525/512, also known as Avicenna's enharmonic diesis.&lt;br /&gt;

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  Not a family or clan, but related by the fact that 126/125, the septimal semicomma or starling comma (&lt;span class="commentBody"&gt;The difference between three 6/5s, one 7/6, and an octave) &lt;/span&gt;is tempered out, are myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.&lt;br /&gt;

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  These temper out |-5 2 2 -1&amp;gt; = 225/224, the marvel comma, and include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.&lt;br /&gt;

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  The hemifamity temperaments temper out the hemifamity comma, |10 -6 1 -1&amp;gt; = 5120/5103.&lt;br /&gt;

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  The porwell temperaments temper out the porwell comma, |11 1 -3 -2&amp;gt; = 6144/6125.&lt;br /&gt;

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  A breedsmic temperament is one which tempers out the breedsma, |-5 -1 -2 4&amp;gt; = 2401/2400. Some which do not get discussed elsewhere are collected on a page &lt;a class="wiki_link" href="/Breedsmic%20temperaments"&gt;here&lt;/a&gt;.&lt;br /&gt;

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  A ragismic temperament is one which tempers out |-1 -7 4 1&amp;gt; = 4375/4374. These are not by any means all microtemperaments, but those which are not highly accurate are probably best discussed under another heading. Accurate ones include ennealimmal, supermajor, enneadecal, amity, mitonic, parakleismic, gamera and vishnu. Pontiac belongs on the list but falls under the schismatic family rubric.&lt;br /&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:90:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc45"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Turkish maqam music temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:90 --&gt;&lt;a class="wiki_link" href="/Turkish%20maqam%20music%20temperaments"&gt;Turkish maqam music temperaments&lt;/a&gt;&lt;/h3&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:128:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc64"&gt;&lt;a name="Rank 3 temperaments--Dimcomp family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:128 --&gt;&lt;a class="wiki_link" href="/Dimcomp%20family"&gt;Dimcomp family&lt;/a&gt;&lt;/h3&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:130:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc65"&gt;&lt;a name="Rank 3 temperaments--Kleismic rank three family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:130 --&gt;&lt;a class="wiki_link" href="/Kleismic%20rank%20three%20family"&gt;Kleismic rank three family&lt;/a&gt;&lt;/h3&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:146:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc73"&gt;&lt;a name="Rank 3 temperaments--Valinorismic temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:146 --&gt;&lt;a class="wiki_link" href="/Valinorismic%20temperaments"&gt;Valinorismic temperaments&lt;/a&gt;&lt;/h3&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:148:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc74"&gt;&lt;a name="Rank 3 temperaments--Werckismic temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:148 --&gt;&lt;a class="wiki_link" href="/Werckismic%20temperaments"&gt;Werckismic temperaments&lt;/a&gt;&lt;/h3&gt;