Overtones – Fourier transform

Let us consider a superposition of spinning wheels, which are arranged in a very specific way. We have a fundamental frequency f, along with double, triple, quadruple, quintuple, and sextuple of that frequency. The amplitudes of the spinning wheels decrease to one-half, one-third, one-fourth, one-fifth, and one-sixth of the amplitude of the fundamental frequency f. What is the time course of this superposition? The spinning wheels, with their respective frequencies and amplitudes, form a kind of a sawtooth wave.

This is the time course of the sawtooth wave. It is possible to present the same information in a different way. Let us consider the specific spinning wheels with their rotation frequencies. Let’s start with the fundamental tone having the frequency f and the amplitude r, that is, with the radius of this wheel. Let us now add the amplitudes of the multiples of the fundamental frequency. We get the so-called spectrum of the wave, which covers all frequencies with their respective amplitudes that are present in the wave. The spectrum helps us to reconstruct the time course of the wave.

The mathematical operation that associates the frequencies and their respective amplitudes with the time course is the so-called Fourier transform, named after French mathematician Fourier. Nowadays, it is very easy to investigate frequencies of sound waves using a computer. We can thus describe the wave either over time, or in the so-called frequency domain. The Fourier analysis is one of the key methods of analysis used in the natural sciences.

Let us return to the issue of unambiguity. Is it possible to infer the sound source from the sound wave? Is the frequency spectrum a unique fingerprint for the musical instrument used? We shall answer this question in the next station.