Multiple Viewpoint Systems: Time Complexity and the Construction of Domains for Complex Musical Viewpoints in the Harmonization Problem

We discuss the problem of automatic four-part harmonization: given a soprano part, add alto, tenor and bass in accordance with the compositional practices of a particular musical era. In particular, we focus on the development of representational and modelling techniques, within the framework of multiple viewpoint systems and Prediction by Partial Match (PPM), for the creation of statistical models of four-part harmony by machine learning. Our ultimate goal is to create better models, according to the information theoretic measure cross-entropy, than have yet been produced. We use multiple viewpoint because of their ability to represent both surface and underlying musical structure, and because they have already been successfully applied to melodic modelling. To allow for the complexities of harmony, however, the framework must be extended; for example, we begin by predicting complete chords, and then extend the framework to allow part by part prediction. As the framework is extended and generalized, the viewpoints become more complex. This article discusses matters related to viewpoint domains (alphabets), such as their size and consequent effect on run time; and presents methods for their reliable construction. We also present an empirical analysis of the time complexity of our computer implementation.