Tour of regular temperaments: Difference between revisions

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===[[Porcupine family]]===  

The porcupine family tempers out 250/243, 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.

===[[Gamelismic clan]]===  

&lt;a class="wiki_link" href="/Equal%20Temperaments"&gt; Equal temperaments&lt;/a&gt; (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. An EDO is simply a division of the octave into equal steps (specifically, steps of equal size in cents). An ET, on the other hand, is a temperament, an altered representation of some subset of the intervals of just intonation. The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents.&lt;br /&gt;

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&lt;a class="wiki_link" href="/Paul%20Erlich"&gt;Paul Erlich&lt;/a&gt; has given us a &lt;a class="wiki_link" href="/Catalog%20of%20five-limit%20rank%20two%20temperaments"&gt;catalog of 5-limit rank two temperaments&lt;/a&gt;. As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; of the various temperaments, where a &lt;strong&gt;comma&lt;/strong&gt; is a small interval, not a square or cube or other power, which is tempered out by the temperament.&lt;br /&gt;

  The porcupine family tempers out 250/243, 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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  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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So called because 19/84 (as a &lt;a class="wiki_link" href="/fraction%20of%20the%20octave"&gt;fraction of the octave&lt;/a&gt;) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with 22, 31 and 53-EDO. It's reasonable in the 7-limit and naturally extends into the 11-limit.&lt;br /&gt;

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Even less familiar than rank 2 temperaments are the &lt;a class="wiki_link" href="/Planar%20Temperament"&gt;rank 3 temperaments&lt;/a&gt;, based on a set of three intervals. Since these temperaments may be mapped in many different ways, it is more common to identify rank 3 temperaments by the commas they temper out.&lt;br /&gt;

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  Tempers out 225/224. An excellent tuning for marvel is &lt;a class="wiki_link" href="/240edo"&gt;240edo&lt;/a&gt;. It extends in a natural way to unidecimal marvel, which tempers out 385/384, for which the slightly less accurate tuning of &lt;a class="wiki_link" href="/72edo"&gt;72edo&lt;/a&gt; works well. Marvel is generated by 2, 3, and 5, the same generators as the &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;5-limit&lt;/a&gt;, which means a scale can be converted to marvel or unidecimal marvel simply by tempering it.&lt;br /&gt;

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  Starling tempers out 126/125, and like marvel it has the same generators as the 5-limit. An excellent tuning for starling is &lt;a class="wiki_link" href="/77edo"&gt;77edo&lt;/a&gt;, but 31, 46 or 58 also work nicely.&lt;br /&gt;

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Breed is a 7-limit microtemperament which tempers out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2578et will certainly do the trick. Breed has generators of a 49/40 neutral third, and 10/7.&lt;br /&gt;

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  Jove, formerly known as wonder, tempers out 243/242 and 441/440. Wonder has been depreciated as a name due to conflict with another temperament also given that name. Jove converts breed into an 11-limit temperament via 441/440, which equates 49/40 with 11/9, and 243/242, which tells us 11/9 can serve as a neutral third. While jove is no longer a super-accurate microtemperament like breed, it has the advantage of adjusting its tuning to deal with the 11-limit. 72 and 130 are good edos for jove, and if that doesn't suit there's 476edo.&lt;br /&gt;

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