Maths And Music: The Unlikely Relationship

I discovered the unlikely relationship between mathematics and music during my 2nd year of university, where I was presented with the opportunity to write an essay about any mathematical topic of my choosing. After much debate, discarded plans and useless ideas, I eventually settled on a mathematical concept known as Fourier Analysis. I have always loved maths, but have never had much of an interest in music aside from some fairly novice violin playing back in Primary school! During my research, I stumbled across the math-music rabbit hole and spent a fair amount of time learning some extremely fascinating concepts. I have decided to share some of the ideas which I found most surprising and interesting.

First, a brief introduction to Fourier Analysis. Way back in 1768, a rather famous (and super intelligent) mathematician was born: Joseph Fourier. (I think you can guess that he was the brains behind Fourier Analysis!) This particular mathematician was having a great deal of problems trying to solve a rather tricky equation involving a complicated function. One day whilst pondering his problem, he had an epiphany and thought to himself "I wonder what would happen if I considered my function as a combination of bunch of waves all with different heights and lengths?" Amazingly, this was exactly the idea that led Joseph Fourier to a solution to his mind bending problem. You might be thinking having multiple waves is much more complicated than one weird function, but in fact that is not the case. Sine and Cosine waves are very symmetric and are always repeating at regular intervals, so at every point on the wave, you can predict what it will look like in the future! Being able to predict the future is pretty handy if you ask me. So, in summary, Fourier analysis is the splitting up of complicated functions into a series of waves (in particular, sine and cosine waves) so that when you add all of these waves up, you get the original function back!

Okay now for a bit of music theory. I know you probably clicked on this article and expected a discussion on counting beats, or keeping in time with music. Sensible assumptions for a maths article about music, but we`re here to talk about the more interesting and abstract relationship known as harmonics. As simply as possible, a harmonic is a sound wave that has a frequency which is a whole-number multiple of a fundamental tone, where the fundamental tone is the lowest possible frequency that a sound wave could have. Confusing right? Let`s consider a simple example to understand:

Imagine you had a sound wave with fundamental frequency 2 Hertz (Hz), so the wave completes 2 cycles per second. . A harmonic would be a wave with frequency equal to 2 x "any whole number". So a sound wave of frequency 6 Hz would be a harmonic since 6 = 2 x 3. A sound wave of frequency 5 Hz would not be a harmonic of this wave since 5 = 2 x 2.5, and 2.5 is clearly not a whole number. There are infinite possibilities of harmonics, essentially any wave with even frequency would be a harmonic of the wave with fundamental frequency 2 Hz.

Back to maths, when an instrument plays a note, the fundamental frequency is heard as well as its harmonics. Interestingly, only certain whole numbers will produce harmonics that sound "nice" when played simultaneously with the fundamental note. Examples of frequency ratios that produce "nice" sounds are:

2 : 1, 3 : 2, 4 : 3.

This series of ratios is known as the harmonic series on the Fundamental note C.

Other notes have harmonic series but I think that is enough abstract thinking for this article! Thanks for reading, I hope you found it insightful and interesting.