Fourier discovered the Fourier series as a solution to a boundary value problem [33, 303, 512, 620] related to the heat wave equation. Fourier’s work on heat is still in print [455]. In this section, we derive the wave equation for the vibrating string and show how the Fourier series is used in its solution. The solution, in turn, gives rise to the physics of harmonics used as the foundation of music harmony. We contrast the natural harmony of the overtones to that available from the tempered scale of western music. The tempered scale is able to accurately approximate the beauty of natural harmony using a uniformly calibrated frequency scale. The wave equation is manifest in analysis of physical phenomena that display wave like properties. This includes electromagnetic waves, heat waves, and acoustic waves. We consider the case of the simple vibrating string. A string under horizontal tension T is subjected to a small vertical displacement, y = y(x, t), that is a function of time, t, and location, x. As illustrated in Figure 13.1, attention is focused on an incremental piece of the string from x to x +Dx. Under the small displacement assumption, there is no movement of the string horizontally (i.e., in the x direction), and the horizontal forces must sum to zero. T = T1 cos(θ1) = T2 cos(θ2). Let the linear mass density (i.e., mass per unit length) of the string be ρ. The mass of the incremental piece of string is then ρ. The total vertical force acting on the string is T2 cos(θ2) − T1 cos(θ1).