45 research outputs found

    Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity

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    Previous work on applications of Abstract Differential Geometry (ADG) to discrete Lorentzian quantum gravity is brought to its categorical climax by organizing the curved finitary spacetime sheaves of quantum causal sets involved therein, on which a finitary (:locally finite), singularity-free, background manifold independent and geometrically prequantized version of the gravitational vacuum Einstein field equations were seen to hold, into a topos structure. This topos is seen to be a finitary instance of both an elementary and a Grothendieck topos, generalizing in a differential geometric setting, as befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies. The paper closes with a thorough discussion of four future routes we could take in order to further develop our topos-theoretic perspective on ADG-gravity along certain categorical trends in current quantum gravity research.Comment: 49 pages, latest updated version (errata corrected, references polished) Submitted to the International Journal of Theoretical Physic

    Building Generalized Neo-Riemannian Groups of Musical Transformations as Extensions

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    Chords in musical harmony can be viewed as objects having shapes (major/minor/etc.) attached to base sets (pitch class sets). The base set and the shape set are usually given the structure of a group, more particularly a cyclic group. In a more general setting, any object could be defined by its position on a base set and by its internal shape or state. The goal of this paper is to determine the structure of simply transitive groups of transformations acting on such sets of objects with internal symmetries. In the main proposition, we state that, under simple axioms, these groups can be built as group extensions of the group associated to the base set by the group associated to the shape set, or the other way. By doing so, interesting groups of transformations are obtained, including the traditional ones such as the dihedral groups. The knowledge of the group structure and product allows to explicitly build group actions on the objects. In particular we differentiate between left and right group actions and we show how they are related to non-contextual and contextual transformations. Finally we show how group extensions can be used to build transformational models of time-spans and rhythms.Comment: 30 pages, 4 figures ; submitted to Journal of Mathematics and Music - v.4: corrected many errors, clarified some proposition

    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 Σ3J\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

    Using Monoidal Categories in the Transformational Study of Musical Time-Spans and Rhythms

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    Transformational musical theory has so far mainly focused on the study of groups acting on musical chords, one of the most famous example being the action of the dihedral group D24 on the set of major and minor chords. Comparatively less work has been devoted to the study of transformations of time-spans and rhythms. D. Lewin was the first to study group actions on time-spans by using a subgroup of the affine group in one dimension. In our previous work, the work of Lewin has been included in the more general framework of group extensions, and generalizations to time-spans on multiple timelines have been proposed. The goal of this paper is to show that such generalizations have a categorical background in free monoidal categories generated by a group-as-category. In particular, symmetric monoidal categories allow to deal with the possible interexchanges between timelines. We also show that more general time-spans can be considered, in which single time-spans are encapsulated in a "bracket" of time-spans, which allows for the description of complex rhythms.Comment: 17 pages; 7 figures - Minor corrections brought to the first versions; comments welcom
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