Cosine Contours: a Multipurpose Representation for Melodies

Melodic contour is central to our ability to perceive and produce music. We propose to represent melodic contours as a combination of cosine functions, using the discrete cosine transform. The motivation for this approach is twofold: (1) it approximates a maximally informative contour representation (capturing most of the variation in as few dimensions as possible), but (2) it is nevertheless independent of the specifics of the data sets for which it is used. We consider the relation with principal component analysis, which only meets the first of these requirements. Theoretically, the principal components of a repertoire of random walks are known to be cosines. We find, empirically, that the principal components of melodies also closely approximate cosines in multiple musical traditions. We demonstrate the usefulness of the proposed representation by analyzing contours at three levels (complete songs, melodic phrases and melodic motifs) across multiple traditions in three small case studies