49,223 research outputs found

    Voicing Transformations and a Linear Representation of Uniform Triadic Transformations (Preprint name)

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    Motivated by analytical methods in mathematical music theory, we determine the structure of the subgroup J\mathcal{J} of GL(3,Z12)GL(3,\mathbb{Z}_{12}) generated by the three voicing reflections. We determine the centralizer of J\mathcal{J} in both GL(3,Z12)GL(3,\mathbb{Z}_{12}) and the monoid Aff(3,Z12){Aff}(3,\mathbb{Z}_{12}) of affine transformations, and recover a Lewinian duality for trichords containing a generator of Z12\mathbb{Z}_{12}. We present a variety of musical examples, including Wagner's hexatonic Grail motive and the diatonic falling fifths as cyclic orbits, an elaboration of our earlier work with Satyendra on Schoenberg, String Quartet in DD minor, op. 7, and an affine musical map of Joseph Schillinger. Finally, we observe, perhaps unexpectedly, that the retrograde inversion enchaining operation RICH (for arbitrary 3-tuples) belongs to the setwise stabilizer H\mathcal{H} in Σ3⋉J\Sigma_3 \ltimes \mathcal{J} of root position triads. This allows a more economical description of a passage in Webern, Concerto for Nine Instruments, op. 24 in terms of a morphism of group actions. Some of the proofs are located in the Supplementary Material file, so that this main article can focus on the applications

    Higher order matching polynomials and d-orthogonality

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    We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials -- the Chebyshev, Hermite, and Laguerre polynomials -- can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogonality for polynomials and we use sign-reversing involutions to show that the higher-order Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are d-orthogonal. We also investigate the moments and find generating functions of those polynomials.Comment: 21 pages, many TikZ figures; v2: minor clarifications and addition

    Normalization, probability distribution, and impulse responses

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    When impulse responses in dynamic multivariate models such as identified VARs are given economic interpretations, it is important that reliable statistical inferences be provided. Before probability assessments are provided, however, the model must be normalized. Contrary to the conventional wisdom, this paper argues that normalization, a rule of reversing signs of coefficients in equations in a particular way, could considerably affect the shape of the likelihood and thus probability bands for impulse responses. A new concept called ML distance normalization is introduced to avoid distorting the shape of the likelihood. Moreover, this paper develops a Monte Carlo simulation technique for implementing ML distance normalization.Econometric models ; Monetary policy

    The syntax of polarity emphasis

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    A crosslinguistic survey of the expression of polarity emphasis reveals that some such expressions are subject to the distributional constraints typical of main clause phenomena, while others are not. The former have received a fairly homogeneous syntactic analysis, implicating specific left peripheral projections. The non-restricted variety, however, is not analysed uniformly with some phenomena receiving a fully syntactic account and others being accounted for in terms of semantics and pragmatics. (C) 2013 Elsevier B.V. All rights reserved
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