Circularity in Neural Computation and its Application to Musical Composition

Backpropagation neural networks and their simulations can be useful for composers and theorists. Yet, their usefulness and general applicability depends mainly on the nature of the training data presented to the network. Certain musical parameters are easier to formalize than others, and in the case of parameters that exhibit a circular characteristic like pitch or key space--which have been represented as a winding helix (Shepard) or as a torus (Krumhansl&Kessler)--trigonometric functions can provide the means to prepare the training sets. Introduction In this paper, I shall describe two environments in which the MAX multi-layer perceptron object, developed by Michael Lee at CNMAT, was used to control circular musical parameters: Firstly, an interactive model of key space in which a particular location on the torus is transformed into tonal hierarchies. These hierarchies, in turn, serve as the basis for a stochastic process generating melodies in a given key. Movement on the surface of the torus leads to a change of tonal hierarchies perceived as modulation. Secondly, a compositional environment I used in my opera in progress ÒDer Sprung.Ó Motivated by the text, a set of networks were trained to different melodies and used for the subsequent melodic and metric interpolation process. The perceptron training file format requires the configuration of the net to be specified as well as the number of training patterns. A typical pattern includes data for both the input and the desired output associated with it. Key Space The most convenient way to establish a graphical model of the toroidal key space is to project it onto the plane of the computer screen: