Meter as Mechanism : A Neural Network that Learns

One kind of prosodic structure that apparently underlies both music and some examples of speech production is meter. Yet detailed measurements of the timing of both music and speech show that the nested periodicities that deene metrical structure can be quite noisy in time. What kind of system could produce or perceive such variable metrical timing patterns? And what would it take to be able to store and reproduce particular metrical patterns from long-term memory? We have developed a network of coupled oscillators that both produces and perceives patterns of pulses that conform to particular meters. In addition, beginning with an initial state with no biases, it can learn to prefer the particular meter that it has been previously exposed to. Meter is an abstract structure in time based on the periodic recurrence of pulses, that is, on equal time intervals between distinct phase zeros. From this point of view, the simplest meter is a regular metronome pulse. But often there appear meters with two or three (or rarely even more) nested periodicities with integral frequency ratios. A hierarchy of such metrical structures is implied in standard Western musical notation, where diierent levels of the metrical hierarchy are indicated by kinds of notes (quarter notes, half notes, etc.) and by the bars separating measures with an equal number of beats. For example, in a basic waltz-time meter, there are individual beats, all with the same spacing, grouped into sets of three, with every third one receiving a stronger accent at its onset. In this meter there is a hierarchy consisting of both a faster periodic cycle (at the beat level) and a slower one (at the measure level) that is 1/3 as fast, with its onset (or zero phase angle) coinciding with the zero phase angle of every third beat. This essentially temporal view of meter contrasts with the traditional symbol-string theories (such as Hayes, 1981 for speech and Lerdahl and Jackendoo, 1983 for music). Metrical systems, however they are deened, seem to underlie most of what we call music. Indeed, an expanded version of European musical notation is found to be practical for transcribing most music from around the world. That is, most forms of music employ nested periodic temporal patterns (Titon, Fujie, & Locke, 1996). Musical notation has