In ancient Greece, Pythagoras of Samos was the first philosopher to identify a connection between music and mathematics. He discovered the mathematical relationships between sounds and, for the first time, considered the study of music as a branch of mathematics. The link between the two fields grew clearer as they advanced: understanding sound as a pressure wave travelling through air led to the study of frequencies and the discovery of overtones, which can be predicted and examined through a specific branch of mathematics called ‘harmonic analysis’. This offers insight into why certain intervals are more pleasing than others and why different cultures employ distinct musical scales.
Composers in the early 20th century explicitly used mathematical structures to create dodecaphonic works, while many Baroque paradigms are based on symmetry and other mathematical patterns. This relationship appears to extend beyond such examples, reaching into the realm of aesthetics and, more broadly, pattern recognition. Music seems to be the art form of choice for scientists and mathematicians, and an interest in mathematics and science is deeply connected with an interest in music.
We discussed this subject with Bertrand Maury – professor of mathematics at the Université Paris-Saclay in France, as well as an accomplished pianist. His mathematical career began with a degree from the École Polytechnique, continued with postgraduate research at the University of Houston, and was crowned with his recent election to the Académie des Sciences in France. His scientific interests, initially centred on fluid dynamics, expanded to the mathematical modelling of different areas of reality: from equations describing airflow in the human lungs and bronchial tree, to the movement of grains in fluid, to crowd movements and safety at large gatherings. Today, his research is increasingly involved with social sciences: the mathematical modelling of social behaviours. Alongside his successful mathematical achievements, Prof. Maury has had another muse accompanying him throughout his life: music.
His passion for the piano was born when his younger sister began her piano lessons, which encouraged him to do the same. Within a few weeks, he recalls becoming deeply passionate about playing and beginning to understand the importance of music for his well-being. He studied music at the conservatoire in Versailles while simultaneously completing his mathematical preparation. “The preparatory classes for admission to the École Polytechnique were very demanding and time-consuming, but I never stopped or slowed down my musical studies,” Maury recalls. “I never had any plans to turn it into a job and become a professional musician. I simply needed it.” However, in his youth he did gain some experience performing in live concerts, with orchestra or as a soloist.
With such a background, it is certainly unsurprising that he attracted the interest of the founders of Festum Pi, a conference and festival now in its fourth year, which seeks to explore the connection between music and mathematics. The festival was created by mathematician Cédric Villani (winner of the 2010 Fields Medal, roughly equivalent to the Nobel Prize in mathematics) and Dionysios Dervis-Bournias, conductor and artistic director of the festival. It consists of two different conferences – one concerning music, the other mathematics – offering deep, technical lectures in the respective areas, along with a concert series where mathematicians and musicians can gather, share ideas, and discuss their research. The first two editions of the festival were held in Samos in 2022 and 2023 to honour Pythagoras’ homeland, and the following two in Crete.
As well as giving lectures in maths at the first three editions of the conference, Prof. Maury has given several performances, including a programme based on Bach and Debussy. I asked about this perhaps unusual combination; Maury explains that Johann Sebastian Bach was his first love when he approached piano playing in his younger years. “It’s very common for a scientifically oriented mind to be fascinated by Bach,” he says. “The architecture of his compositions exhibits patterns very pleasing to a mathematician.” As a young musician, he was not particularly fond of French music, but then about ten years ago he became interested in Debussy and “discovered how sophisticated his music is; what sounded fuzzy and random, almost improvised to my young, immature taste, I now recognise as accurate and structured.” This led to the production of a recording, released by the Greek record label Minos-EMI.
Regarding the relationship between music and mathematics, Maury observes that while the connection is certainly present, it is far from obvious what it consists of. “For me, as a professional mathematician and a semi-professional musician, the two disciplines are quite separate. Music cannot be reduced to mathematics, and the opposite is even less true, naturally. The connections are there, but they seem somewhat random. Very hard to pin down.” This is what makes the effort at Festum Pi so interesting. At the festival, there are mathematicians who have never attended a concert, and they discover classical music. “The mathematical spirit makes one ready to appreciate music,” Maury says.
Often, musicians were not particularly successful in mathematics at school but are nevertheless fascinated by maths as an idealisation. Both domains share the same abstract quality: they don’t serve any practical purpose in themselves. Of course, the applications of mathematics to the real world are abundant, but the study of mathematics itself is the creation (or discovery?) of an abstract framework, and the same can be said of music. “Also, they share the same approach to learning – the endless grinding of details until things finally come together,” he adds.
“Of course, you can analyse a Bach fugue from a mathematical point of view,” he says as an example. “At Festum Pi, I have given talks aimed at both musicians and mathematicians where I describe how you can look at the score of a fugue as a geometric object, visualising the symmetries, the transposed or inverted subject, but that doesn’t explain why a Bach fugue is a masterpiece, why it touches our souls. Mathematicians tend to find this sort of thing aesthetically pleasing, though.”
On the other hand, many musicians do not find mathematics pleasant, but this, he muses, “could be because mathematics lacks the emotional component. There certainly are emotions involved in the study of mathematics” – “frustration,” I suggest, and we laugh – “but it’s a different kind of emotion; it’s not shared. The emotions tied to music are much easier to share, not only with fellow listeners but with fellow performers, and at times even with the composer, we feel. Mathematics is a much more solitary intellectual effort.”
But there is a very interesting problem at the border between mathematics and music that has caught the professional interest of Prof. Maury, and it has to do with the mechanics of the piano. “We all know that there are pianists who sound better than other pianists. Some of them you can sometimes recognise from a recording,” he tells me. However, when sound is produced in a piano, with the hammer hitting the string, the player has no control over it. “It’s not like a violin, where you have your finger on the string and can directly affect the sound production.” After you press the key, the hammer is lifted and then released; the release always happens in the same position, and only its velocity affects how the hammer hits the string.
“But then,” he asks, “how can two different pianists sound different from one another? How can a sound be more or less beautiful?” We look forward to a paper in the future explaining this mystery to us all.
See more information about Festum Pi:
www.festum-pi.eu
Bertrand Maury’s Résonances is on Minos-EMI and available to stream now.
This article was sponsored by Festum Pi.
