Polynomial functions as a representation of melodic phrase contour

The present paper explores a novel way of characterising the contour of melodic phrases. Melodic contour is represented by a curve that can be derived from fitting a polynomial equation to the pitch values given the note onsets. To represent contour numerically, we consider the coefficients of the polynomial equation as well as the time limits of the melodic phrase. After a brief review of different theoretical, didactic, analytic, and computational approaches to melodic contour, a detailed step-by-step description is provided of how polynomials can be fitted to melodic phrases. In two application examples, it is demonstrated how polynomial contour can be used as a basis for the computational processing of melodic information. The first application estimates the frequency of occurrence or prevalence of a phrase contour: a probabilistic model is constructed based on a large sample of phrases from popular melodies using a standard density estimation procedure to obtain an indicator of contour occurrence frequency. The second application is a similarity measure that exploits polynomial curves graphically and is based on Tversky’s (1977) ratio model of similarity. Further applications of the approach as well as quantitative tests of the existing ones are discussed as options for future work.