Science and Mathematics Professor Luis Nasser Publishes Research on Musical Harmony

Luis Nasser, PhD, professor of Science and Mathematics at Columbia College Chicago, has always stood at the crossroads of art and science. A professional bassist as well as a physicist, Nasser recently published a research paper in the peer-reviewed journal "PLOS One" that explores the mathematical and physical roots of musical harmony.

Nasser's work seeks to explain why certain combinations of notes are pleasing to the human ear-using thermodynamics, the branch of physics that deals with energy and heat.

The paper, titled "A Thermodynamic Foundation for Musical Harmony," proposes that our perception of harmony is not just a product of cultural conditioning or music theory, but may actually be grounded in natural physical laws. The research was a collaboration with colleagues from NYU Abu Dhabi and The National Autonomous University of Mexico.

We sat down with Nasser to unpack his findings-and how science might help explain the timeless magic of music.

Q&A With Columbia College Chicago Associate Professor Luis Nasser

Can you explain, in simple terms, how the thermodynamic principle helps explain the way musical scales are formed across different cultures?

The idea is as follows: in matter, there is a quantity we call the Helmholtz free energy, F. It is the difference between the total thermal energy of the system, U, and the proportion of the energy that is disordered, which is the product of the temperature T and the entropy S:

F = U - T·S

Ordered phases in matter will arise as this free energy is minimized. So, liquids arise from gases and solids from liquids, and whatever internal structures are present depend on the details of this free energy. By analogy, it is possible to write a "musical" free energy, where the internal energy U is replaced by a dissonance function D. This function D results from a mixture of biological and cultural factors. As a result, when we begin with every possible frequency of sound and minimize this free energy, ordered phases of sound will appear. How many notes are selected will depend on the details of the dissonance function, and it is in this way that different cultures give rise to different scales/divisions of the octave.

What are some of the key ways you and your collaborators expanded on Berezovsky's original idea in this new paper?

Berezovsky's paper was very ingenious, but it made use of several assertions that we proved are not necessary for his method to work. For example, in music, timbre refers to the specific mixture of harmonics present when you play a note on an instrument-also called the overtone series. In his paper, Berezovsky used a very specific timbre-the so-called square wave timbre-which is rarely (if ever) observed in acoustic musical instruments. We show that his method works for many timbres, including those found in instruments and the human voice. We also generalized it to systems of intonation that do not have periodic octaves, as is the case with gamelan music, made primarily with non-harmonic bells and other percussion instruments.

Why is it important to view musical harmony through a universal, physics-based lens rather than through a Western classical framework?

One of the great benefits of being funded by the National Science Foundation was that it allowed me to travel abroad and give many invited talks. Among them was a lecture in Chile where part of the audience was Mapuche-natives dating back to 600 BC. They have their own musical idiom, which is looked down upon by traditional music academia, where harmony is totally centered on 17th-century harmonic style. Other forms of harmony are not viewed as equal and have traditionally endured an air of cultural inferiority. This method reveals that all forms of harmony encountered in human culture correspond to different perceptions of dissonance but are fundamentally just as valid as any other-which is important from the perspective of diversity and inclusion. To ignore these forms of harmony is a bit like teaching physics but only teaching mechanics, while ignoring thermodynamics, electrodynamics, etc.

How might your findings open new possibilities for research in physics, music theory, or other fields?

There is a geometric interpretation of harmony due to Euler called the Tonnetz. This is a triangular tiling of the plane wherein each of the 12 notes of the traditional scale of Western music is represented by an equilateral triangle, each one placed in a specific order such that all the chords are explained geometrically. The minimization of musical free energy appears to recreate the Tonnetz. We hypothesize that by exploring different forms of the dissonance function, the ordered phases of sound that emerge can be compared in a meaningful way to the patterns of order observed in simulations of the early universe.

However, in the case of cosmology, we have no real way of interpreting many of these early stages of simulations because we don't really know enough about the physics then. However, if we observe similar forms emerging with sound, these can be understood in the context of music, which could give some meaningful insight into cosmology. It is an example of how in physics everything is done visually. But physics is the art of finding the patterns of nature, and there is no reason we can't make use of our other senses.

What do you hope students and fellow researchers will take away from this study?

That we tend to falsely believe people are either "right- or left-brained." That in science, we need to be just as creative and imaginative as in the arts-but this creativity is limited by what the experiments can verify. It is also important to see that the artistic perspective can sometimes be informed in a meaningful way by science and vice versa. We talk about the importance of interdisciplinary work but never really do it-here's a chance to put our money where our mouths are.

Read the full study: https://arxiv.org/pdf/2501.05467

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