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

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," 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 Jackend-off 1983). My present concern is not the ability of addition- or 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 1a, 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.

Figure 1b provides a contrasting illustration of purely division-based rhythm generation. In this case, the process begins with a single large times-pan, represented by the top row of the diagram. The second row shows a division of this timespan into three parts whose durations are related by the chain of proportions 3:5:3. Each of the resulting timespans is divided into smaller parts in the third row of the diagram, and portions of the structure extend to a fourth level according to the same principle.

Simple though they may be, both of these models are capable of generating a variety of rhythmic surfaces. Indeed, the question arises whether the range of possible outputs from one process is identical to that of the other and, if so, how meaningful the difference between the models might be.

Figure 2 transcribes the addition- and division-based results into conventional music notation. In each case, the transcription reflects the generative process. In Figure 2a, the transcription of the addition-based rhythm associates a simple notational unit (eighth notes) with the unit pulse, inviting performers to count that pulse to execute the rhythm accurately.

Transcription of the division-based rhythm taxes our notational resources slightly more, requiring various tuplets to show how the timespans expressed as measures are divided into various numbers of equal parts. Some of the operations involved in producing Figure 2b are addition-based: The top-level timespan is not assigned any special representation, but it is formed by the sum of the measure lengths, [End Page 59] and the tuplet-derived pulses within each measure are grouped into larger timespans that can be divided according to the next stage of the division-based model. For instance, the 2:3 division at the beginning of the third-largest level implies both a division into five parts and an additive grouping of those...

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