Fourier analysis is a technique to analyze music whereby the sound is decomposed into sine-wave components (also called partials). It is comprised by a mathematical addition of the more basic waves into a more complicated combined wave that represents the sound.
The reversal of Fourier analysis – the construction of complex sounds from a set of individual sine waves – is termed Fourier synthesis. Fourier synthesis involves the creation of functions from sine waves. An example of Fourier synthesis is shown in chapter 3 of the Thompson reading (p. 49), in which a square wave is created from addition of the odd harmonics of a sine wave.
It’s important to note that, in spite of Ohm’s acoustic law, we usually don’t hear the differences in the constituent waves, thus making the concept of Fourier analysis a bit difficult to understand at first. In our minds, these different sound waves fuse together and contribute to the quality known as timbre.
Fourier analysis is extremely useful in many subject areas, such as math and physics. Some applications include noise filtering and x-ray crystallography. Fourier analysis is named for the French mathematician Jean Baptiste Joseph Fourier (1768-1830).