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Page 78 ï~~Foot-tapping: a brief introduction to beat induction As an example, have a look at the following pattern of lines and dots: I..11..... 1.1..11".....1.1..11 Do you see any emergent structure? Probably not. When you would listen to it, though, (e.g., the pattern being played from left to right, with every line being a 16th note and every dot a 16th rest) you would quickly hear a regular pattern -the beat-, and could probably easily tap your foot along with it. This relatively simple cognitive task is called beat induction or foot-tapping. Beat induction is a fast process. Only after a few notes (5-10) a strong sense of beat can be induced (a "bottom-up" process). Once a beat is induced by the incoming material it sets up a persistent mental framework that guides the perception of new incoming material (a "top-down" process). This process, for example, facilitates the percept of syncopation, i.e., to "hear" a beat that is not carried by an event. However, this top-down processing is not rigidly adhering to a once established beat-percept, because when in a change of meter the evidence for the old percept becomes to meager, a new beat interpretation is induced. This duality, where a model needs to be able to infer a beat from scratch, but also to let an already induced beat percept guide the organization of more incoming material, is hard to model. There are a number of aspects that make beat induction an interesting and important process to model, as will be shown in this chapter which results from the joint effort of some researchers of this field. The individual papers will elaborate on the different aspects. Their diversity reflects the large body of work on the subject and the different computational formalisms used (Rule-based systems: LonguetHiggins & Lee (1982); Lee (1985); Miller, Scarborough & Jones (1992), Lee (1991). Optimization methods: Povel & Essens (1985); Parncutt (1994). Search: Longuet-Higgins (1976), Allen & Dannenberg (1990). Control theory: Dannenberg & Mont-Reynaud (1987); Dannenberg (1993). Distributed systems: Desain (1992); Minskian models: Chung (1989); Pennycook, Stammen, & Reynolds (1993); Rowe (1993); Rosenthal (1992); Neural nets: Miller, Scarborough & Jones (1992); Statistical models: Palmer & Krumhansl (1991), Brown (1993)). Interactive computer music systems Interactive computer music systems make some interesting additional demands with respect to those mentioned above (see, e.g., Boulanger, 1990). First, they have to perform in real-time, which means that they have to be efficient enough and that they have to deal with the musical material incrementally (i.e., while the input is processed). They also have to deal with real performance data (containing expressive timing, performance errors, etc.), these systems have to be robust (i.e., they should recover gracefully from errors), they have to deal with instruments that exhibit some response delay (like mechanically driven piano's or shoes), and to do this, they need careful temporal planning (i.e., scheduling). Most of these systems, though, have only been informally tested. It is unclear how and how well they deal with the characteristics mentioned above. However, the interactive performance situations for which these systems were designed, forced the designers to think of solutions on problems that are rarely touched in cognitive models, problems that can not be ignored when aiming at a realistic cognitive model of beat induction. Figure 1. The mechanical shoe. Unfortunately, neither cognitive or technological approaches have been able to arrive at a general, robust beat extraction method. The big challenge seems to lie in a unification and generalization of the existing, partially successful theories, since they all, apparently model at least one valid aspect of beat induction. We hope that the special ICMC 1994 paper session on foot-tapping can make a contribution towards this goal. To enable the audience to compare their own foot-tapping with that of the presented models, some of the computational models will be demonstrated with on-line computer implementations connected to a mechanical foot-tapper (see Figure 1). Peter Desain & Henkjan Honing References (shared by papers in this chapter) Allen, P. & R. Dannenberg (1990) Tracking musical beats in real time. Proceedings of the 1990 ICMC, 140-143. Bamberger, J. (1980) Cognitive structuring in the apprehension of simple rhythms. Archives de Psychologie, 48:171-199. Boulanger, R. (1990) Conducting the MIDI Orchestra, Part 1: Interviews with Max Mathews, Barry Vercoe and Roger Dannenberg. Computer Music Journal, 14(2). 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