The Mathematics of Musical Instruments

This article highlights several applications of mathematics to the design of musical instruments. In particular, we consider the physical properties of a Norwegian folk instrument called the willow flute. The willow flute relies on harmonics, rather than finger holes, to produce a scale which is related to a major scale. The pitches correspond to fundamental solutions of the one-dimensional wave equation. This “natural” scale is the jumping-off point for a discussion of several systems of scale construction—just, Pythagorean, and equal temperament—which have connections to number theory and dynamical systems and are crucial in the design of keyboard instruments. The willow flute example also provides a nice introduction to the spectral theory of partial differential equations, which explains the differences between the sounds of wind or stringed instruments and drums.