Division- and Addition-Based Models of Rhythm in a Computer-Assisted Composition System

Abstract

Reading descriptions of different musical traditions or of the practice of different individual composers, it is not uncommon to see a rhythmic idiom described as additive/7 meaning in simple terms that it consists of durations formed by combining, or adding together, the units provided by a rapid underlying pulse (Koetting 1970; Pratt 1987; Clayton 2000). Such a characterization distinguishes the rhythmic idiom from others involving the division of the units of some referential pulse into variable numbers of smaller parts, an important feature of tonal Western classical music (Lerdahl and Jackendoff 1983). My present concern is not the ability of additionor division-based models of rhythm to accurately describe existing musical traditions, but the value of these models as a basis for generating rhythms in an algorithmic-composition system, in which potentially novel rhythmic idioms may be sought. In particular, I will describe a series of algorithms I have used in my own work as a composer, and I will investigate some of the advantages and disadvantages each of these algorithms inherits from the particular model of rhythm on which it is based. The development of these algorithms was shaped by at least two compositional goals to create music for human performance, and to express it using common-practice Western music notation (Byrd 1994). My investigations will be made in the context of the same goals. For the sake of making the distinction between the two types of models clearer, Figure 1 presents simple illustrations of rhythms generated by purely addition-based and purely division-based means. In Figure la, the upper row of the diagram represents a stream of timespans constituting the units of some isochronous pulse. The lower row of this diagram shows these timespans grouped into durations of 3, 5, 3, 2, 5, and 2 units to constitute the resulting rhythm. (These values are labeled in the diagram, but of course we could also count unit pulses to determine them.) Although the resulting rhythm lacks the isochrony of the original unit pulse, we can treat it like another pulse and combine adjacent timespans to form larger durations. In this way, the additive process is capable of forming hierarchies of timespans deeper than the two levels illustrated here.