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

John Cage's Number Pieces as Stochastic Processes: a Large-Scale Analysis

The Number Pieces are a corpus of works by composer John Cage, which rely on a particular time-structure used for determining the temporal location of sounds, named the "time-bracket". The time-bracket system is an inherently stochastic process, which complicates the analysis of the Number Pieces as it leads to a large number of possibilities in terms of sonic content instead of one particular fixed performance. The purpose of this paper is to propose a statistical approach of the Number Pieces by assimilating them to stochastic processes. Two Number Pieces, "Four" and "Five", are studied here in terms of pitch-class set content: the stochastic processes at hand lead to a collection of random variables indexed over time giving the distribution of the possible pitch-class sets. This approach allows for a static and dynamic analysis of the score encompassing all the possible outcomes during the performance of these works.Comment: 25 pages, 9 figures, 5 tables; comments welcom

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

The model magnetic Laplacian on wedges

We study a model Schr\"odinger operator with constan tmagnetic field on an infinite wedge with natural boundary conditions. This problem is related to the semiclassical magnetic Laplacian on 3d domains with edges. We show that the ground energy is lower than the one coming from the regular part of the wedge and is continuous with respect to the geometry. We provide an upper bound for the ground energy on wedges of small opening. Numerical computations enlighten the theoretical approach