Measures of Temporal Pattern Complexity

In this study, three measures of temporal pattern complexity were compared with regard to their perceptual validity. The first measure, based on the work of Tanguiane (1993), uses the idea that a temporal pattern can be described in terms of (elaborations of) more simple patterns, which occur simultaneously at different levels. The second measure is based on the complexity measure for finite sequences proposed by Lempel and Ziv (1976), which is related to the number of steps in a selfdelimiting production process by which such a sequence is presumed to be generated. The third measure, newly developed here, is rooted in the theoretical framework of rhythm perception of Povel and Essens (1985). It takes into account the ease of coding a temporal pattern and the complexity of the segments resulting from this coding. The perceptual validity of the three measures was evaluated in an experiment in which subjects judged the complexity of 35 temporal patterns. Correlations between the three measures and the collected complexity judgments indicated that the third measure is a much better predictor of temporal pattern complexity than the other two measures. This is probably due to the fact that this measure, unlike the other two, is based on an empirically tested model of rhythm perception that takes into account the isochronous frame against which the rhythm is perceived. Reasons for the differences in performance between the three measures are discussed. INTRODUCTION The notion of complexity has generally been studied in the context of information theory and is closely connected with concepts such as randomness, information, regularity, and coding (Calude, 1994). Classical information theory, as well as notions of randomness, based on Shannon's concept of entropy (Shannon, 1948), relies on a priori knowledge of a probability distribution. In that respect, it does not allow one to speak of a particular object or outcome as being random or complex. In general, an object's complexity reflects the amount of information embedded in it. The representation of the object's information is achieved via coding. When a human being enters the equation, however, care must be taken in interpreting the notion of complexity, which necessarily becomes subjective. Moreover, depending on the context, only certain types of codes may be perceptually significant and hence coding efficiency or complexity must be considered within such constraints (Chater, 1996). This is well known, for example, in the field of visual perception (Leeuwenberg, 1971). In order to obtain complexity measurements from subjects, Pressing (n.d.) suggests equating complexity with 0929-8215/00/2901-061$15.00 #Swets & Zeitlinger Journal of New Music Research, 29 (2000), No. 1, pp. 61^69 Measures of Temporal Pattern Complexity Ilya Shmulevich and Dirk-Jan Povel Tampere International Center for Signal Processing,Tampere University of Technology, Tampere, Finland Nijmegen Institute for Cognition and Information, University of Nijmegen, Nijmegen,The Netherlands Correspondence: Ilya Shmulevich, Signal Processing Laboratory, Tampere University of Technology, P.O. Box 553, 33101 Tampere, Finland. Tel.: +358-3-365-3869. Fax: +358-3-365-3817. E-mail: [email protected] difficulty of learning, which in turn could be expressed by recognition or production. In this work, we consider the complexity of temporal patterns. Our aim is to construct a measure of complexity that corresponds to a high degree with a human's subjective notion of complexity. Pressing (n.d.) discusses three notions of complexity. The first is termed hierarchical complexity, which refers to structure on several levels simultaneously. The receiver is then able to perceive structure on one or more levels, inducing an appropriate complexity judgement. A general approach to hierarchical structure in perception has been proposed by Leyton (1986). The divisible nature of Western rhythms lends itself to hierarchical subdivision and reveals regularity of time organization on several levels simultaneously (Lerdahl & Jackendoff, 1983). The second type of complexity is referred to as dynamic complexity. This notion refers to the degree of stationarity or change with respect to time, in the sensory input. A highly nonstationary stimulus would tend to be perceived as being complex. In music, rhythms tend to be stationary or periodic in that events or groups of events are repeated in time. Finally, the third type of complexity, called generative complexity, refers to a tendency towards the most economical description (Hochberg & McAlister, 1953; Chater, 1996). In this paper, we examine three new measures of complexity of temporal patterns. The first measure is based on the work of Tanguiane (1993), and uses the idea that a rhythmic pattern can be described in terms of (elaborations of) more simple patterns, which occur simultaneously at different levels. The second measure is based on the complexity measure for finite sequences proposed by Lempel and Ziv (1976), which is related to the number of steps in a self-delimiting production process by which such a sequence is presumed to be generated. Finally, the third measure proposed is rooted in the theoretical framework of rhythm perception discussed in Povel and Essens (1985). This measure takes into account the ease of coding a temporal pattern and the (combined) complexity of the segments resulting from this coding. The measure presupposes the existence of a `̀ temporal grid'' or time scale consisting of isochronic intervals, which is selected among a set of possible grids according to the `̀ economy principle'' (Povel, 1984). All three measures, which will be discussed in detail below, capture one or more of the types of complexity discussed above. SELECTION ANDNOTATION OF RHYTHMS Before we proceed to explain the three measures, we must define the domain of rhythms studied. First, we restrict ourselves to quantized rhythms, i.e., rhythms as notated in a score, without timing deviations due to performance. Therefore, these rhythms can be described fully in musical notation.Without loss of generality and for the ensuing discussion, we use the sixteenth note as the smallest note duration (unit of length). Furthermore, the rhythms studied are supposed to repeat or loop infinitely and thus form an infinite sequence of events. Finally, we notate a rhythmic pattern as a string of ones and zeros, in which the symbol `1' represents a note onset and `0' represents no note onset. Of course, the smallest quantization level must be used for the encoding. For example, the pattern would be represented by 1011100010011000. We should emphasize that in this notation, only internote intervals are relevant and so the pattern is represented by exactly the same string as above. Now, we are ready to give the definitions of the complexity measures. T-Measure (Tanguiane measure) The first measure we consider is based on the work of Tanguiane (1993). A basic notion in the theory of Tanguiane is that of elaboration (Mont-Reynaud & Goldstein, 1985). Figure 1 gives an example of the elaboration of a quarter note. As can be seen, the quarter note is elaborated into patterns consisting of two notes on the second row, which in turn are further elaborated into patterns of three notes on the third row, which are all finally elaborated into a pattern of four notes on the last row. Thus, all patterns that are linked by a line, either directly 62 I. SHMULEVICH. ANDD.-J. POVEL.