Melodies as Curves

Abstract

"The curvature function of a 2D curve describes the shape of the curve regardless of how it is moved or rotated. In music, a motif is a string of notes that appears throughout a piece, although sometimes moved up or down. The measure of curvature is a number, and we can assign each note in a motif a numerical value that describes it’s distance from some fundamental frequency. Because each curve has a unique curvature function, we are able to create a unique song based on a curve, and vice versa. However, this does not easily allow for rhythm. A 3D curve has curvature and torsion, which measure how much it “bends” and how much it “twists”, respectively. We can make curvature rhythm, and torsion pitch, and get a unique song based on any space curve, and vice versa."