45 research outputs found
Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity
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
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)
Motivated by analytical methods in mathematical music theory, we determine the structure of the subgroup of generated by the three voicing reflections. We determine the centralizer of in both and the monoid of affine transformations, and recover a Lewinian duality for trichords containing a generator of . 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 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 in 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
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