Complexity measures of musical rhythms

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 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 to a high degree corresponds 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. Leyton (1986) proposed a general approach to hierarchical structure in perception. The divisible nature of Western rhythms lends itself to hierarchical subdivision and reveals regularity of time organization on several levels simultaneously (Lerdahl and Jackendoff, 1983). Another 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 tendency towards the most economical description (Hochberg and McAlister, 1953; Chater, 1996). We examine three new measures of rhythm complexity. The first measure is based on the work of Tanguiane (1993), and …