Maths and Music: The importance of interdisciplinary study

A reflection on the motivation for my research project and the benefits of interdisciplinary study.

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Knowing that I would have to completely immerse myself in a topic for 10 weeks, I wanted to choose a research project that I would truly enjoy pursuing. With none of the pre-defined project titles particularly catching my eye, and with pure maths and music being my two main academic interests, I came up with the title of a self-defined project, ‘Investigating Mathematical Patterns in Music’.

This is incredibly broad (since mathematical patterns can be found in almost all aspects of music e.g. melodic patterns, rhythmic patterns, timbral patterns, patterns in the sound waves themselves, patterns common to particular musical genres etc.), so we narrowed it down to looking at twelve tone technique, which involves manipulating a tone row, an ordered sequence of the twelve tones in a chromatic scale, by applying various transformations such as retrograde (reversing the order of the sequence’s pitches so that the first pitch becomes the last and the last pitch becomes the first) and inversion (reversing the intervals between each pitch), to develop melodic material.

A diagram (created by me) showing the effects of retrograde and inversion on a tone row.

The aim of this project is to find out whether composers bias towards using certain transformations and creating certain patterns, or whether they use the full range of cases available to them. To do this, we have used abstract algebra, an area in pure mathematics which looks at algebraic structures rather than numbers.

This summer I spent most of my time looking into the mathematical side of things, familiarising myself with the relevant knowledge I will need for analysing the music and thus collecting data. With the experimental work happening next summer, I thought it would make sense to save the details of my findings until later, and tell you more about the motivation for this particular project instead.

Abstract Algebra and Twelve Tone Technique

You may be familiar with the idea that there are connections between mathematics and music. However, you may not be aware that abstract algebra heavily ties in with music, or even that abstract algebra exists at all.

‘Non-mathematicians’ never usually get the chance to encounter abstract algebra which I think is a real shame, because for me it is my favourite branch of mathematics, yet it is perhaps one of the furthest from what you find in the school curriculum. However, this is not to say that it is hard to understand, as like with all mathematics, if explained well, it can be understood by anyone. So I guess the ultimate long-term goal of this project is to not only extend the connotations of ‘maths and music’ to abstract algebra on top of the more typical connotations of musical theory requiring a ‘mathematical mindset’, and the physics of sound waves, but to also popularise the branch of abstract algebra itself, and thus the area of pure mathematics in general – something that not many people get the pleasure of encountering, especially after obtaining the common perception that maths is all about using formulae and performing calculations from school.

Furthermore, this project is yet another example of how mathematical concepts can be applied to a completely different context. Often when people learn maths, a frequently asked question is always ‘what’s the point?’. Therefore it is incredibly important to make people aware of the applications of even the most abstract mathematics, and remind people that these applications can lead you to make fascinating discoveries that can tell you so much.

I found there was a lot of overlap between abstract algebra and modern 20th century music. Having grown up with a fondness of music in an academic setting, I have always kept an open mind to different types of music and enjoy listening to almost all genres. However, some of the more obscure 20th century music, including twelve-tone music (which is often not very listener-friendly at all!), is something which I have always struggled to fully appreciate. Therefore, I saw this project as an opportunity to learn more about it, as I truly believe that the more you understand something, the easier it is to see the underlying beauty. This may sound like an awful risk considering I would need to spend 10 weeks on it, but I knew that the possibility of me being able to appreciate something I have not been able to appreciate before, and being able to share this appreciation with others, would make it totally worth the risk. Consequently, it gave me the opportunity to demonstrate one of my overall favourite things about music: on the surface, it can seem meaningless, but the deeper you immerse yourself in it, the more meaningful it becomes.

Inevitably there will be genres of music that people are not the greatest fan of at first, but different perspectives (in this case, a mathematical perspective) can give people another way, and thus, another chance, of being able to appreciate them. After all, people learn and understand things in different ways, but just because a person does not understand something via one way, does not mean they should be deprived of understanding it completely.

Maths and Music

Various mathematicians have been popularising the connections between mathematics and music, so I urge you to read some of the articles they have written, or watch some of the talks they have given, as I found many were super engaging and highlighted some of the connections extremely well.

It is generally considered that mathematics is the most abstract of the sciences, and music is the most abstract of the arts, so it would seem intuitive that they would complement each other to some extent. The hot debate lies more around the idea that music contains too much emotion to be somewhat rationalised by mathematics, but I believe that Marcus du Sautoy constructs a perfect response as to why this is not the case.

That aside, it was really rewarding doing a project which combined maths and music – I found myself improving my knowledge in both subjects, and it was so satisfying to discover more and more connections between the two, whilst also being prompted to view things from different perspectives to either find out new things, or strengthen things I already knew in one subject but had not realised in the other, allowing for more new things to happen. This enabled me to nurture my affinity with both even further, as I got to witness how they interact with each other in both ways that I would and would not have intuitively thought. It was an alternating cycle of curiosity and discovery that fuelled my motivation to continue doing research.

Importance of Interdisciplinary Study

I do realise that mathematics and music is not absolutely everyone’s cup of tea, so here is a summary of the reasons why interdisciplinary study is so important no matter which subjects are involved, drawn from my experiences throughout this project and the points raised above:

  • It widens an appreciation for both subjects – it encourages people to gain new knowledge and new passions in different subject areas, and may even help people get over their academic fears in the process
  • It makes learning more meaningful – seeing the applications of subjects makes learning more enjoyable and memorable, and being more interested encourages people to dig deeper
  • It encourages you to realise more efficient ways to learn – having more than one perspective to look at things enables you to explore alternative learning pathways, enhancing the learning experience
  • It helps you to progress – viewing things from multiple perspectives helps you to see things in a new light and helps you spot things you would not have spotted otherwise
  • It helps you discover unexplored areas – you find gaps in research that have been missed due to lack of collaboration with the knowledgeable people
  • It helps you develop critical thinking skills and creativity – making mental comparisons between subject areas prompts you to think deeply and consider new possibilities

I may not have managed to persuade you to embark on a maths postdoctorate or to become a musicologist, but I hope you can still take away the fact that interdisciplinary study is hugely beneficial to both learning and research, and perhaps this may convince you to take some time to explore the connections between various disciplines you personally enjoy yourself.

I would like to say a huge thank you to my supervisor, Professor Peter Cameron, for his insightful advice and support, Lord Laidlaw for allowing me to undertake this research project, and the Laidlaw team for organising such a brilliant programme.

Molly Chung

Mathematics Student, University of St Andrews

I'm a Mathematics student at University of St Andrews. My research looks at twelve tone music from an abstract algebra perspective, with the hopes of gaining insight into the thoughts and compositional methods of various composers.

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