Information dynamics: patterns of expectation and surprise in the perception of music

Abstract

Measures such as entropy and mutual information can be used to characterise random processes. In this paper, we propose the use of several time-varying information measures, computed in the context of a probabilistic model which evolves as a sample of the process unfolds, as a way to characterise temporal structure in music. One such measure is a novel predictive information rate which we conjecture may provide a conceptually simple explanation for the 'inverted-U' relationship often found between simple measures of randomness (e.g. entropy rate) and judgements of aesthetic value (1). We explore these ideas in the context of Markov chains using both artificially generated sequences and two pieces of minimalist music by Philip Glass, showing that even such a manifestly simplistic model (the Markov chain), when interpreted according to information dynamic principles, produces a structural analysis which largely agrees with that of an expert human listener. Thus, we propose that our approach could form the basis of a theoretically coherent yet computationally plausible model of human perception of formal structure, potentially including seemingly abstract qualities like interestingness and aesthetic goodness.Keywords: information theory; expectation; surprise; subjective probability; Bayesian inference; Markov chain; music. Expectation and surprise in musicOne of the more salient effects of listening to music is to create expectations of what is to come next, which may be fulfilled immediately, after some delay, or not at all as the case may be. This is the thesis put forward by, amongst others, music theorists L. B. Meyer (2) and Narmour (3). In fact, this insight predates Meyer quite considerably; for example, it was elegantly put by Hanslick (4) in the nineteenth century: 'The most important factor in the mental process which accompanies the act of listening to music, and which converts it to a source of pleasure, is frequently overlooked. We here refer to the intellectual satisfaction which the listener derives from continually following and anticipating the composer's intentions-now, to see his expectations fulfilled, and now, to find himself agreeably mistaken. It is a matter of course that this intellectual flux and reflux, this perpetual giving and receiving takes place unconsciously, and with the rapidity of lightning-flashes. ' An essential aspect of this is that music is experienced as a phenomenon that 'unfolds' in time, rather than being apprehended as a static object presented in its entirety. Meyer argued that musical experience depends on how we change and revise our conceptions as events happen, on how expectation and prediction interact with occurrence, and that, to a large degree, the way to understand the effect of music is to focus on this 'kinetics' of expectation and surprise. The business of making predictions and assessing surprise is essentially one of reasoning under conditions of uncertainty and manipulating degrees of belief about the various proposition which may or may not hold, an...

Time Series Analysis as a Method to Examine Acoustical Influences on Real-time Perception of Music

“…Information Dynamic approaches have so far treated events as successive and accretive, without taking regard for their relative timing, or directly addressing issues of temporal autocorrelation. Indeed, much of the Information Dynamics work has been done on monophonic isochronic music, such as Glass' Gradus (Abdallah & Plumbley, 2009;Potter, et al, 2007). The Information Dynamic approach can be enhanced further by superimposed time series analyses, as well as by awareness that notationally-or musicologically-defined features may or may not be perceptible and readily categorized, and hence may or may not contribute new information.…”

“…The time series analysis methodology developed in this paper can support ongoing work to understand the perception of music in terms of dynamically changing information content (Abdallah & Plumbley, 2009;Pearce & Wiggins, 2006;Potter, Wiggins, & Pearce, 2007; reviewed by Wiggins, Pearce, & Müllensiefen, 2009). In an information dynamics approach it is assumed that the predictive capacity and decisions of an observer concerning future events in the musical stream are continuously evolving as new information is assimilated into the observer's (potentially individualistic) data probability structure.…”

The Role of Expectation and Probabilistic Learning in Auditory Boundary Perception: A Model Comparison

“…It will also be useful to examine ways of tailoring IDyOM specifically for segmentation, including a metrically-based rather than an event-based representation of time, optimising the derived features that it uses to make event predictions, and using other information-theoretic measures such as entropy or predictive information (Abdallah and Plumbley 2009). One of the attractive features of the model is that such measures (and the learning models on which they rely) are in no sense domain-specific and so can be applied generally.…”

Predictive Information in Gaussian Processes with Application to Music Analysis

“…It also means that the sequence of surprisingness values is itself uncorrelated in time. This is in marked constrast with the situation for Markov chains [4], where, in general, the expected suprise depends on the previous observation and thus varies in time, reflecting the observer's varying levels of uncertainty about the next observation. In a Gaussian processes, this predictive uncertainty is constant and therefore does not provide any useful structural analysis of the sequence.…”

“…Addressing the observation that some processes with long-range dependencies have infinite excess entropy [2], Bialek et al [3] introduced the predictive information as the mutual information between a finite segment of a process and the infinite future following it, and studied its behaviour, especially in relation to learning in statistical models. In previous work [4], we defined the predictive information rate (PIR) of a random process as the average information in one observation about future observations yet to be made given the observations made so far; thus, it quantifies the new information in observations made sequentially. The PIR captures a dimension of temporal structure that is not accounted for by previously proposed measures.…”