Fractals in music

von Ismael Sträuli
 

While the internal assessments of the IB syllabus are a lot of work to do, they give students opportunities to engage in subjects more freely than the ordinary school curriculum, as you can write about almost anything you like. I knew I wanted to make my math IA about music, as I play piano in my free time and music is one of my passions. The choice to investigate one of Bach’s fugues was made, as his musical pieces, as well as fugues in general, tend to be rather calculated and mathematical in their composition. Lastly, fractals were familiar to me and they seemed like a topic that was different from anything else we had learned in our math classes up to that point. But what exactly are fractals? And how does one go about using them to analyse music?

Fractals are patterns or shapes which look the same no matter how much you zoom in. They are «self-similar» as smaller parts of themselves have similar properties to bigger parts and vice-versa. An intuitive example would be a head of broccoli, a small stem of it looks like the bigger whole; a green stalk with darker bulbs at the end. In fact, one of the most well-known fractals occurring in nature is the romanesco broccoli, which has self-similar properties down to micrometer-small scales.

Broccoli aside, this characteristic of «self-similarity» can be, and has been, abstracted from pure geometry. It can be found in any and all forms of structures outside of mathematics, like in biology, technology and notably in the arts. There has been extensive research about how the golden ratio (which has fractal properties!) has been used in renaissance paintings. Yet, it is also possible to find fractal patterns in music. This is what Charles Madden attempts in his book Fractals in Music; Introductory Mathematics for Musical Analysis. He uses many complex tools to try and uncover hidden fractal patterns in all kinds of music, some of which I adapted to analyse Bach’s Die Kunst der Fuge: Contrapunctus I for my mathematical investigation.

Firstly, I graphed the piece’s subject (motif), using the relationships between individual notes and afterwards their individual pitches. The result were two graphs which I was able to compare with each other in hopes of finding some semblance of self-similarity. I was able to find slight traces of self-similarity by rearranging and contorting the smaller parts of the graphs to fit the whole.

Afterwards, I applied the golden section (the inverse of the golden ratio) to the bar-count of the piece, trying to find important structural elements at places corresponding to the golden section. This is the musical equivalent of placing the golden spiral over a painting and finding interesting things on the curve.
Lastly, I tried to figure out how random/ chaotic this piece is by comparing it to different kinds of «coloured noise». Randomness and noise are important parts of fractal theory as they themselves are fractal. They stay equally random/noisy at all scales, making them self-similar. Fractals are even one of the best ways to describe them, as it is possible to state just how «random» a particular case of «randomness» is through the use of fractal dimension. Different types of coloured noise are distributions of frequencies with varying randomness dependent on the «colour». The most famous example is white noise, which is perfectly random, meaning none of the frequencies depend on the frequencies coming before them. The frequencies of pink noise are somewhat more influenced by what frequencies came before them, thus the- re is noticeable correlation between the individual frequencies. This pattern follows into red noise, which has strong correlation between frequencies, meaning one frequency is almost predetermined by the frequency that came before it.

By graphing all notes in the soprano voice of the fugue, so that the x-axis displays the pitch of the note and the y-axis displays the pitch of the note that came before it, I was able to compare it to coloured noise and show that the fugue follows a pattern similar to pink noise. I then used a box-grid method to find the soprano’s fractal dimension, as mentioned above. This let me see how «random» it was in numerical terms. However, the fractal dimension of the soprano did not coincide with the fractal dimension of pink noise, probably due to the fact the pink noise relies on continuous values of frequencies, while music depends on discreet pitches of notes. The following graphs show the pitch correlation in the soprano voice of Bach’s fugue and the frequency correlation of pink noise respectively.

To conclude, applying math that was unfamiliar to me was rather finicky. Fractal theory is a very complex topic, one that I don‘t fully understand nor ever expect to fully understand. Using Madden’s book as a guide was helpful, but other methods might have proven more suitable. Despite that, I do believe the Investigation led to some interesting results. On the one hand it serves as evidence that certain composers might have been conscious and active users of fractals in their music, for example through the utilization of the golden ratio. On the other hand, it also gives an insight into how fractals work and how they affect us. In truth, fractals are oftentimes better than traditional mathematics at describing real things, because real things are messy and random. While art might seem like it doesn‘t hold to this characteristic, art stems from humans, who are famously messy creatures. Ultimately, based on this investigation, one could even make the argument that humans think in terms of fractals more than they may realize.
 

Ismael Sträuli ist Schüler der Klasse 6i.
Illustration: Victoria Schaller