Regular temperament
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An //abstract regular temperament// is a [[regular temperament]] considered apart from any consideration of tuning, which classifies all of its tunings as tunings of the single abstract temperament. Various methods have been proposed for uniquely characterizing an abstract regular temperament. Among these are * The [[Wedgies and Multivals|wedgie]] This uses [[http://en.wikipedia.org/wiki/Exterior_algebra|multilinear algebra]] to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the [[http://mathworld.wolfram.com/InteriorProduct.html|interior product]] of a [[Wedgies and Multivals|wedgie]] for a p-limit temperament with the p-limit monzos. For example, using "v" to represent the interior product, we have mir = <<6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0> is <0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1> is <1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0> is also <1 1 3 3|. * [[Normal lists|Normal val lists]] Given a list of vals, we may [[Saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [<1 1 3 3|, <0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1]. * The [[Tenney-Eucidean tuning|Frobenius projection map]] Given any list of monzos, or any list of vals, we may compute the associated Frobenius projection map. This corresponds uniquely with an abstract regular temperament. The intervals of the abstract temperament may be defined via multiplication by the projection map, leading to [[fractional monzos]] which are actually the tunings of these intervals in [[Fractional monzos|Frobenius tuning]]. However, using the Frobenius projection map to define the abstract temperament by no means commits us to Frobenius tuning. * [[Just intonation subgroup|Just intonation subgroups]] and [[transversals]] A relatively concrete approach, but one which is not canonically defined, is to define a [[transversal]] for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament. For example, for [[Gamelismic clan|miracle temperament]] [2, 15/14] defines a rank two 7-limit subgroup whose [[Normal lists|normal interval list]] is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.
Original HTML content:
<html><head><title>Regular temperament</title></head><body>An <em>abstract regular temperament</em> is a <a class="wiki_link" href="/regular%20temperament">regular temperament</a> considered apart from any consideration of tuning, which classifies all of its tunings as tunings of the single abstract temperament. Various methods have been proposed for uniquely characterizing an abstract regular temperament. Among these are<br /> <br /> <ul><li>The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a></li></ul><br /> This uses <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">multilinear algebra</a> to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/InteriorProduct.html" rel="nofollow">interior product</a> of a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for a p-limit temperament with the p-limit monzos. <br /> <br /> For example, using "v" to represent the interior product, we have mir = <<6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0> is <0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1> is <1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0> is also <1 1 3 3|. <br /> <br /> <ul><li><a class="wiki_link" href="/Normal%20lists">Normal val lists</a></li></ul>Given a list of vals, we may <a class="wiki_link" href="/Saturation">saturate</a> it and reduce it using the <a class="wiki_link" href="/Normal%20lists">Hermite normal form</a> to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [<1 1 3 3|, <0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1]. <br /> <br /> <ul><li>The <a class="wiki_link" href="/Tenney-Eucidean%20tuning">Frobenius projection map</a></li></ul>Given any list of monzos, or any list of vals, we may compute the associated Frobenius projection map. This corresponds uniquely with an abstract regular temperament. The intervals of the abstract temperament may be defined via multiplication by the projection map, leading to <a class="wiki_link" href="/fractional%20monzos">fractional monzos</a> which are actually the tunings of these intervals in <a class="wiki_link" href="/Fractional%20monzos">Frobenius tuning</a>. However, using the Frobenius projection map to define the abstract temperament by no means commits us to Frobenius tuning.<br /> <br /> <ul><li><a class="wiki_link" href="/Just%20intonation%20subgroup">Just intonation subgroups</a> and <a class="wiki_link" href="/transversals">transversals</a></li></ul><br /> A relatively concrete approach, but one which is not canonically defined, is to define a <a class="wiki_link" href="/transversal">transversal</a> for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament.<br /> <br /> For example, for <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a> [2, 15/14] defines a rank two 7-limit subgroup whose <a class="wiki_link" href="/Normal%20lists">normal interval list</a> is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.</body></html>