Distance and Similarity of Time-span Trees

Time-span tree in Lerdahl and Jackendoff’s theory [12] has been regarded as one of the most dependable representations of musical structure. We first show how to formalize the time-span tree in feature structure, introducing head and span features. Then, we introduce join and meet operations among them. The span feature represents the temporal length during which the head pitch event is most salient. Here, we regard this temporal length as the amount of information which the pitch event carries; i.e., when the pitch event is reduced, the information comparable to the length is lost. This allows us to define the notion of distance as the sum of lost time-spans. Then, we employ the distance as a promising candidate of stable and consistent metric of similarity. We show the distance possesses proper mathematical properties, including the uniqueness of the distance among the shortest paths. After we show examples with concrete music pieces, we discuss how our notion of distance is positioned among other notions of distance/similarity. Finally, we summarize our contributions and discuss open problems.