5. The Frequency Domain

So far, we have seen several scenarios where combinations of frequencies lead to interesting musical results. In additive synthesis (Chapter 3), mixing basic sinusoids at integer multiples (harmonics) of a fundamental frequency gave rise to different timbres. In our study of scores (Chapter 4), combining notes at different fundamental frequencies (pitches) turned out to be a primary axis of musical composition. Both observations point to the same conclusion: variation in frequency is a fundamental property of musical expression.

But how should we reason about these variations? Are there mathematical tools to formalize them? And if we are handed a new sound that we know nothing about, how can we determine what frequencies it contains? This chapter develops the frequency domain and the Fourier transform, the essential tools for answering these questions.