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-Euclidean 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 subgroups]] and [[Transversal|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. * **[[Normal lists|Normal comma lists]]** The normal comma list uniquely defines the abstract temperament, and has the advantage of showing family relationships even more clearly than the normal val list. Intervals of the temperament may be defined after computing another means of representing the temperament such as the normal val list. * **[[http://en.wikipedia.org/wiki/Row_echelon_form|Reduced row echelon form]]** If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix. For example, if we feed [<22 35 51 62|, <31 49 72 87|, <84 133 195 236|] into a reduced row echelon form routine, we obtain [<1 0 3 1|, <0 1 -3/7 8/7|, <0 0 0 0|]. Stripping off the zero val in the final row, we get E = [<1 0 3 1|, <0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1>, and |-1 -1 0 1>E* = [0 1/7]. Multiply by |1 0 0 0>, the val for 2, and the result is |1 0 0 0>E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6. ===The Geometry of Regular Temperaments=== Abstract regular temperaments can be identified with [[http://en.wikipedia.org/wiki/Rational_point|rational points]] on an [[http://en.wikipedia.org/wiki/Algebraic_variety|algebraic variety]] known as a [[http://en.wikipedia.org/wiki/Grassmannian|Grassmannian]]. In particular, if the number of primes in the p-limit is n, and the rank of the temperament is r, then the real Grassmannian **Gr**(r, n) has points identified with the r-dimensional subspaces of the n-dimensional real vector space **R**^n. This has an embedding into a real vector space known as the [[http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding|Plücker embedding]], which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank r in the p-limit may be defined as rational points on **Gr**(r, n), though we should note that most of these do not correspond to anything worth much as a temperament. Grassmannians have the structure of a smooth, homogenous [[http://en.wikipedia.org/wiki/Metric_space|metric space]], and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian **Gr**(2, 3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below. [[image:dual.jpg]] [[image:dualzoomer.svg]]
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><strong>The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a></strong></li></ul>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><strong><a class="wiki_link" href="/Normal%20lists">Normal val lists</a></strong></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><strong>The <a class="wiki_link" href="/Tenney-Euclidean%20Tuning">Frobenius projection map</a></strong></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><strong><a class="wiki_link" href="/Just%20intonation%20subgroups">Just intonation subgroups</a> and <a class="wiki_link" href="/Transversal">transversals</a></strong></li></ul>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.<br /> <br /> <ul><li><strong><a class="wiki_link" href="/Normal%20lists">Normal comma lists</a></strong></li></ul>The normal comma list uniquely defines the abstract temperament, and has the advantage of showing family relationships even more clearly than the normal val list. Intervals of the temperament may be defined after computing another means of representing the temperament such as the normal val list.<br /> <br /> <ul><li><strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Row_echelon_form" rel="nofollow">Reduced row echelon form</a></strong></li></ul>If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.<br /> <br /> For example, if we feed [<22 35 51 62|, <31 49 72 87|, <84 133 195 236|] into a reduced row echelon form routine, we obtain [<1 0 3 1|, <0 1 -3/7 8/7|, <0 0 0 0|]. Stripping off the zero val in the final row, we get E = [<1 0 3 1|, <0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1>, and |-1 -1 0 1>E* = [0 1/7]. Multiply by |1 0 0 0>, the val for 2, and the result is |1 0 0 0>E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h3> --><h3 id="toc0"><a name="x--The Geometry of Regular Temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->The Geometry of Regular Temperaments</h3> <br /> Abstract regular temperaments can be identified with <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_point" rel="nofollow">rational points</a> on an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Algebraic_variety" rel="nofollow">algebraic variety</a> known as a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Grassmannian" rel="nofollow">Grassmannian</a>. In particular, if the number of primes in the p-limit is n, and the rank of the temperament is r, then the real Grassmannian <strong>Gr</strong>(r, n) has points identified with the r-dimensional subspaces of the n-dimensional real vector space <strong>R</strong>^n. This has an embedding into a real vector space known as the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding" rel="nofollow">Plücker embedding</a>, which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank r in the p-limit may be defined as rational points on <strong>Gr</strong>(r, n), though we should note that most of these do not correspond to anything worth much as a temperament.<br /> <br /> Grassmannians have the structure of a smooth, homogenous <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Metric_space" rel="nofollow">metric space</a>, and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian <strong>Gr</strong>(2, 3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below.<br /> <br /> <!-- ws:start:WikiTextLocalImageRule:26:<img src="/file/view/dual.jpg/203359118/dual.jpg" alt="" title="" /> --><img src="/file/view/dual.jpg/203359118/dual.jpg" alt="dual.jpg" title="dual.jpg" /><!-- ws:end:WikiTextLocalImageRule:26 --><br /> <!-- ws:start:WikiTextLocalImageRule:27:<img src="/file/view/dualzoomer.svg/198894646/dualzoomer.svg" alt="" title="" /> --><img src="/file/view/dualzoomer.svg/198894646/dualzoomer.svg" alt="dualzoomer.svg" title="dualzoomer.svg" /><!-- ws:end:WikiTextLocalImageRule:27 --></body></html>