Tour of regular temperaments: Difference between revisions

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===[[Pythagorean family]]===  

The Pythagorean family tempers out the Pythagorean comma, |-19 12 0&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. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate chains of 12-equal, offset from one another justly tuned 5/4.

===[[Apotome family]]===  

===[[Minortonic family]]===  

This tempers out the minortone comma, |-16 35 -17&gt;. The head of the family is minortonic temperament, with a generator of a minor tone (~10/9).

===[[Sycamore family]]===  

This tempers out the vishnuzma, |23 6 -14&gt;, or the amount by which seven chromatic semitones (25/24) fall short of a perfect fourth (4/), or (4/3)/(25/24)^7.

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

===[[Gamelismic clan]]===  

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

===[[Sensamagic clan]]===  

  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. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate chains of 12-equal, offset from one another justly tuned 5/4.&lt;br /&gt;

&lt;br /&gt;

  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 a generator of a minor tone (~10/9).&lt;br /&gt;

&lt;br /&gt;

  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 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;

&lt;br /&gt;

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

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc32"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Sensamagic clan"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt;&lt;a class="wiki_link" href="/Sensamagic%20clan"&gt;Sensamagic clan&lt;/a&gt;&lt;/h3&gt;