Mathematical Analysis of Melodies: Slope and Discrete Frechet Distance

Abstract

A directed graph, called an M-graph, is attached to every melody. Our chief concern in this paper is to investigate (1) how the positivity of the slope of the M-graph is related to singability of the melody, (2) when the M-graph has a symmetry, and (3) how we can detect a similarity between two melodies. For the third theme, we introduce the notion of transposed discrete Frechet distance, and show its relevance in the study of similarity detection among an arbitrary set of melodies.