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

Mathematical Analysis of Melodies: Slope and Discrete Frechet Distance