The Fourier Transform: Our Key to Digital Music

Written by Aryan Sholapure

This is the Fourier Transform. It and the associated Fourier series can be considered one of the most important mathematical tools in physics. Take it from the Physicist Lord Kelvin himself:

“Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics.”

The Fourier Transform deals with time-based waves – and these are one of the fundamental building blocks of the natural world. Over the years, the Fourier Transform has come to manifest itself in multiple fields. You can thank it for providing the music you stream every day, squeezing down the images you see on the Internet into tiny little JPG files, and even powering your noise-canceling headphones.

Now let’s try to understand how it works. The Fourier Transform was developed by Jean-Baptiste-Joseph Fourier in 1822 and published in his book, The Analytical Theory of Heat. The remarkable thing Fourier noticed was that any wave - no matter how complicated - could be represented as the sum of sinusoidal waves. Now, the real genius of this realization came about when it was reverse-engineered. If you can work out which sinusoids need to be added together to create the final waveform, you would know exactly which frequencies of waves need to be added together—and in which quantities—to represent the original signal.

I will demonstrate this through the use of a musical chord:-

The graph above is the sinusoidal form of a musical chord, composed of individual notes which have their own frequencies. After applying the Fourier Transform to this particular function we can identify the sound as the composition of these individual sine functions.

That’s what the equation at the top of the page does in one fell swoop. It's no wonder why the Fourier Transform and its applications are so versatile. As a musician, however, I feel inclined to highlight how it has managed to transform the field of Music into what it is today.

Fourier Transforms are used in many types of software applications to recognize the different frequency components of a sound. They can also be used to distinguish between the dominant frequencies and the ones that barely register. This information is very important to the processing of sound in today’s world. Identifying these properties of a sound are extremely helpful in compressing the size of audio files.

For example, when transmitting an audio file in the MP3 format, the Fourier Transform recognizes the barely perceptible frequency components as well as those out of our hearing range. These can then be tossed away to save space thus reducing the size of the file. Thus, this creates a way smaller sized audio file while also creating an accurate version of the original track.

The industry has also benefited hugely from the Fourier Transform, as it can be used to isolate the different instruments during a music recording so that they can be tuned individually. Many recording and music processing software such as Audacity uses the same technology.

Even devices such as Noise-Cancelling headphones use Fourier transforms: microphones record the ambient noises around you and measure the frequency content across the entire spectrum. Fourier Transforms are used to identify the individual frequency components of that ambient noise. The headphones then produce frequencies that help cancel out those frequencies so that the ambient noise is canceled out. There are countless other applications of the Fourier Transform in Music, like the Fast Fourier Transform (FFT) has in recent years revolutionized the entire field.

I guess all I can say is, Math is closer to Music than we think and perhaps it can even be considered its own art form. If you want to delve deeper into the mathematics behind it, definitely check out my paper on the same topic in the link below.

Mathematics Investigation.pdf