How did a theory about heat end up taking music apart?

Through a trick that works on any repeating wave. In 1807, the French mathematician Joseph Fourier was trying to solve the heat equation in a metal plate. His idea was to model a complicated heat source as a combination of simple sine and cosine waves, then build the solution out of the simple solutions matching each of those waves. Sound is a repeating wave too: Fourier analysis can split a sampled musical note into simple waves at different frequencies, and mixing those waves back together re-synthesizes the same sound.

That is how you can see why a violin and a trumpet playing the same note sound different. An orchestra's tuning note is a combination of 440 Hz, 880 Hz, 1,320 Hz, 1,760 Hz and so on, and each instrument produces a different combination of these frequencies; the balance between them is a major factor in each instrument's characteristic sound. Fourier analysis reads that recipe straight out of a recording. Fourier himself announced his discovery to the French Academy in 1807 and published his full theory of heat in 1822.

Fourier's own colleagues were not immediately convinced. When he submitted a later essay in 1811, the review committee, which included the mathematicians Lagrange and Laplace, judged that his method for deriving and solving these equations still left something to be desired in generality and rigor. It took later mathematicians, including Peter Gustav Lejeune Dirichlet and Bernhard Riemann, to put Fourier's results on firmer footing, since the nineteenth century had no precise definition yet of a function or an integral.

Once the method was trusted, it spread far beyond heat and sound. Because so many physical processes reduce, in simplified form, to equations whose basic solutions are sine waves, the same trick used to find the frequencies hidden inside a musical note also became, in modern computing, the fast Fourier transform: an efficient algorithm for pulling those frequencies out quickly.