Musical Collaboration in Rhythmic Improvisation

A review of symbolic dynamics and symbolic reconstruction of dynamical systems

Discretizing a nonlinear time series enables us to calculate its statistics fast and rigorously. Before the turn of the century, the approach using partitions was dominant. In the last two decades, discretization via permutations has been developed to a powerful methodology, while recurrence plots have recently begun to be recognized as a method of discretization. In the meantime, horizontal visibility graphs have also been proposed to discretize time series. In this review, we summarize these methods and compare them from the viewpoint of symbolic dynamics, which is the right framework to study the symbolic representation of nonlinear time series and the inverse process: the symbolic reconstruction of dynamical systems. As we will show, symbolic dynamics is currently a very active research field with interesting applications.

Symbolic dynamics of music from Europe and Japan

After a brief introduction to the theory underlying block-entropy and its relation to the dynamics of complex systems as well as certain information theory aspects, we study musical texts coming from two distinct musical traditions, Japanese and Western European, encoded via symbolic dynamics. We quantify their information content, also known as the degree of "non-randomness" which essentially defines the complexity of the text. We analyze the departure of "total randomness" to the constraints underlying the dynamics of the symbol generating process. Following Shannon on his attribution of these constraints as the key factors of the emergence of complexity, we observe that it can be accurately assessed by the texts' block-entropy vs block-length scaling laws.