Complexity, Time and Music

The formulation of the concept of complexity can be traced back to the late 19th century when the Austrian physicist Ludwig Boltzmann studied the evolution of physical systems. Boltzmann gave a microscopic content to the thermodynamic entropy in connection with the irreversible time evolution towards equilibrium where systems exhibit maximum disorder or complexity. It was another Austrian physicist, Erwin Schroedinger (also born in Vienna) who in the mid 20th century suggested that under non-equilibrium conditions, temporal evolution could produce ordered states with less complexity than the equilibrium state, or rather with a different kind of complexity. Time then acquires a new status: it is an evolutionary factor the arrow of time by which natural phenomena can make systems become organized, which organization materializes in the emergence of spaceand time-dependent structures with various degrees of complexity. The concept of emergence was at the core of new insights that were highlighted in a 1972 article entitled More is different [1] in which physicist P.W. Anderson emphasized that an ensemble of simple elements is more than merely their sum, and that the the whole can exhibit remarkable properties that cannot be predicted from the (even exact) knowledge of the constituting elements. Complexity appears as a characteristic of the ensemble of these elements, which, in physics, are often called the degrees of freedom of the system. Complexity in art pieces may be viewed as the materialization of the many degrees of freedom involved in artistic creation, and this is probably one of the many reasons why art resists theoretical analysis, theory being understood as a set of principles and methods from which logical computation and analysis can be performed on sets of data obtained from measurements carried on a system (here the piece of art). Considered from this point of view, music may be more easily amenable to scientific analysis because of its underlying mathematical structure and because, when performed, it steers a one-dimensional course, the dimension of time. The arrow of time is intrinsic to musical expression: music emerges from silence and returns to silence. And, in contrast with other forms of art, it is not possible to take a snapshot of a piece of music; if time were to be stopped, the music simply vanishes. In practical terms, a