From Time to Time: The Representation of Timing and Tempo

Timing plays an important role in the performance and appreciation of almost all types of music. It has been studied extensively in music perception and music performance research (see Palmer 1997 for a review). The most important outcome of this research is that a large part of the timing patterns found in music performance—commonly referred to as expressive timing—can be explained in terms of musical structure, such as the recurrent patterns associated with the metrical structure that are used in jazz swing, or the typical slowing down at the end of phrases in classical music from the Romantic period. These timing patterns help in communicating temporal structure (such as rhythm, meter, or phrase structure) to the listener. Furthermore, timing is adapted with regard to the global tempo: at different tempi, other structural levels of the music are emphasized, and the expressive timing is adapted accordingly. In short, in music performance there is a close relationship between expressive timing, global tempo, and temporal structure. One cannot be modeled without the other (see Figure 1). Existing computational models of expressive timing (e.g., Clarke 1999; Gabrielsson 1999) are primarily concerned with explaining tempo variations , using tempo curves (specifying tempo or the reciprocal of duration as a function of the position in the score) as the underlying representation. Although a useful means of measuring tempo patterns in a performance, tempo curves have been shown to fall short as an underlying representation of timing from a musical perspective (Desain and Honing 1991, 1993) and a psychological perspective (Desain and Honing 1994). For instance, some types of timing, like chord spread (the asynchrony in performing a chord), ornaments (like grace notes), or the timing between parallel voices simply cannot be measured or represented as tempo deviations. Furthermore, in some musical situations or styles of music, where the global tempo is mostly constant, event-shift (Bilmes 1993)—mea-sured as the deviation with respect to a fixed beat or pulse in a constant tempo—offers a more natural way of representing timing. First, this article will review existing representations of timing and tempo common in computational models of music cognition and in programming languages for music. Their differences are discussed, and some refinements will be proposed (referred to as time maps, or TMs). The second part presents an alternative representation and model for time transformation: so-called timing functions (TIFs, an acronym chosen to distinguish them from TFs, or time functions, …