Related papers: On the mathematical structure of Tonal Harmony
Given a dilation matrix M, a so-called space of M-positive vectors in the Euclidean space is introduced and studied. An algebraic structure of this space is similar to the positive half-line equipped with the termwise addition modulo 2,…
A theory of holomorphic extension of eigenfunctions on homogeneous harmonic spaces is developed.
The harmonic Lagrange top is the Lagrange top plus a quadratic (harmonic) potential term. We describe the top in the space fixed frame using a global description with a Poisson structure on $T^*S^3$. This global description naturally leads…
We analyze the relation of the notion of a pluri-Lagrangian system, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether. We treat classical mechanical systems and…
We present an algebraic construction of music notes and show how to associate them inseveral ways to construct music ranges. Then a family of ranges emerge with a fixed number of notes: two, three, five, seven, twelve, seventeen, etc. A…
Expressive variations of tempo and dynamics are an important aspect of music performances, involving a variety of underlying factors. Previous work has showed a relation between such expressive variations (in particular expressive tempo)…
This is an expository article which describes one approach to the construction and classification of harmonic tori "of finite type", namely, via their ring of polynomial Killing fields. To keep the discussion focussed, the first section is…
These informal notes deal with Fourier series in one or more variables, Fourier transforms in one variable, and related matters.
Recently, Lobb and Nijhoff initiated the study of variational (Lagrangian) structure of discrete integrable systems from the perspective of multi-dimensional consistency. In the present work, we follow this line of research and develop a…
In the present paper, we study harmonic mappings of complete Riemannian manifolds, as well as minimal and stable minimal submanifolds of complete Riemannian manifolds. We examine classical theorems in the theory of these manifolds from the…
We establish a regularity theorem for the Harmonic - Einstein Equation. As a byproduct of the local regularity, we also have a compactness theorem on Harmonic - Einstein equation. The method is mainly the Moser iteration technique which has…
In this article we solve this ancient problem of perfect tuning in all keys and present a system were all harmonies are conserved at once. It will become clear, when we expose our solution, why this solution could not be found in the way in…
We propose a continuous measure of tonal ambiguity that extends the established concept of uniqueness. While uniqueness is widely regarded as necessary for tonality, it cannot (i) discriminate among sets that possess it, (ii) capture…
The study of harmonicity for almost contact metric structures was initiated by Vergara-D\'iaz and Wood and continued by Gonz\'alez-D\'avila and the present author. By using the intrinsic torsion and some restriction on the type of almost…
We generalize recent work of Andrews, Just, and Simay on modular palindromic compositions and anti-palindromic compositions by viewing all compositions partially (modular) palindromic or anti-palindromic. More precisely, we enumerate…
An introduction to the basic ideas and methods of Chiral Perturbation Theory is presented. Several phenomenological applications of the effective Lagrangian technique to strong, electromagnetic and weak interactions are discussed.
We analyze the relation of the notion of pluri-Lagrangian systems, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether.
We prove some basic properties of quasinearly subharmonic functions and quasinearly subharmonic functions in the narrow sense.
The purpose of this book is to provide an introduction to one of the fundamental tools of abstract harmonic analysis, namely the Selberg trace formula.
Direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. We show that every linear Euler sum can be decomposed into a…