On the combinatorial complexity of fuzzy pattern matching in music analysis

In music analysis it is a common requirement to search a musical score for occurrences of a particular musical motif and its variants. This tedious and time-consuming task can be carried out by computer, using one of several models to specify which variants are to be included in the search. The question arises: just how many variants must be explicitly considered? The answer has a profound effect on the computer time needed. In this paper, recurrence relations and closed form analytic expressions are derived for the run time complexity of two models of "fuzzy pattern matching" for use in music analysis; each model assumes the existence of an atomic exact pattern matching operation. The formulae so obtained are evaluated and tabulated as a function of their independent parameters. These results enable a priori estimates to be made of the relative run times of different music searches performed using either model. This is illustrated by applying the results to an actual musical example.