Mathematical Analysis of Melodies: Slope and Discrete Frechet Distance

TL;DR

This paper introduces a mathematical framework using M-graphs and transposed discrete Frechet distance to analyze melody properties, including singability, symmetry, and similarity detection.

Contribution

It presents a novel approach linking M-graph slope positivity to singability and introduces transposed discrete Frechet distance for melody similarity analysis.

Findings

Positivity of M-graph slope relates to melody singability

Symmetry detection in M-graphs reveals structural properties

Transposed discrete Frechet distance effectively measures melody similarity

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.