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A Mathematical Model for Lyrical Singing


Starting my studies in telecommunications engineering almost came as a relief to me. The new academic year meant I would no longer be obliged to take subjects such as philosophy, languages, and history, which, frankly, I didn't like. From then on, I could concentrate on my true interests: physics, mathematics, and computer science.

But going to university also meant that, after 7 years of combining my studies with music training--I play the accordion--I had to choose between an engineering and a musical career. This may sound like a tougher choice than it really was; I knew that I probably wasn't persistent and hard-working enough to make it as a musician. Little was I to know that my interest in physics and mathematics would eventually bring me back to the world of music.

From September 1993, I spent 5 years at the Public University of Navarra in Pamplona, Spain, taking part in theoretical and practical lessons in telecommunications engineering. Among other areas of study, these encompassed electronics, photonics, signal reception, and signal processing, as well as more general subjects such as algebra and calculus. Like most of my classmates, I thought I would go on to get a good job with a telecommunications company after graduation.

But toward the end of my fifth year, I decided to apply for a grant from the Spanish Agency for International Cooperation ( Agencia Española de Cooperación Internacional), which organises the exchange of students, researchers, and teachers between Spanish and South American universities. This gave me the opportunity to spend two-and-a-half months at the Corporación Tecnológica de Bolivar in Cartagena de Indias in Colombia, where I got to take more courses in telecommunications engineering, but where I also met Ph.D. students working in various research areas. This introduction to the world of research made me want to try some research of my own .

I returned to the Electrical and Electronic Engineering Department of my university in Pamplona to finish my final-year project on signal processing. Whereas some scientists working in this area focus on the mathematical tools used to process information, my project involved using some of these tools to find out what is happening with a given natural process. During that year, I also decided to take on another research project in the same department.

I finished my final-year project in November 1999, and a few months later I decided to stay on to do a Ph.D. in the Electrical and Electronic Engineering Department. Working on the two signal-processing projects had really given me a taste of what research life is like, and I enjoyed it. I also saw it as a good way to stay here in Pamplona and be close to my family.

Building on my knowledge in applied signal processing, my Ph.D. research would also intersect with a seemingly distant field, but one that had also been a passion of mine: music! This all came about because a lecturer in the physics department was studying the singing voice from an acoustic point of view, but he also saw real potential in analysing singing techniques from the signal-processing point of view. He helped me kick-start the project, and we collaborated for the first few months.

So from spring 2000 to summer 2004, I looked into the generation of the vocal vibrato, using signal processing to relate mathematically what is perceived acoustically and what is happening in the singer's throat. The vibrato is a very important tool for classical singers because it makes it easier for them to match the pitch of a note with the music; it is also important for the listener, as it prevents them from losing their sensitivity after having listened for the same sound for too long. From an acoustic point of view, the perceived variations in the pitch, timbre, and loudness of the voice that characterise the vibrato are due to variations in the amplitude and frequency of its harmonics. And of course, all of this starts within the human voice production system, which is itself an acoustic signal generator relying on the lungs and vocal cord movements.

The main goal of my work has been to try to determine how the voice production system gives rise to the particular features of the vocal vibrato. To this end, I have collected singing voice recordings in collaboration with the local conservatory and amateur singers. I have also received advice from relevant experts to help me understand the physiological implications of voice production.

I have found this topic very interesting for many reasons. First, it gave me the unexpected opportunity to use my years of musical training, because this project has relied on not only mathematical, but also musical, concepts. Even though a musical background was not critical to this project, it gave me the ability to look at the issue from a different angle. Looking at the bigger picture, this topic has allowed me to combine my interest in science and music, and it shows that mathematics, which may be perceived as one of the most abstract scientific fields, can be applied in the world of the arts.

Indeed, we are now starting to work with a singing school in order to apply the mathematical tools to the singers' learning process. The idea is to try to detect with our mathematical model imperfections in the singing voice that traditionally are detected subjectively by the teacher based on their experience. These imperfections can be related to the musical technique or voice pathologies. The applicability of my research (in the context of a wider research project) has been recognised by the European Centre for Companies and Innovation of Navarra ( Centro Europeo de Empresas e Innovación de Navarra), which gave us their Prize for the Most Transferable Research Results ( Premio a la Mejor Transferencia de Resultados de Investigación).

Now that I've finished my Ph.D., and with funding ending at the end of this year, I am trying to decide what career step I would like to take next. Although the option is always there for an engineer to work in industry, I would like to stay in research based in Spain. This will be tricky because of the practical difficulties. The Spanish university system is trying to prevent Ph.D. students from staying on at the same university, at least for 2 years after graduation, and national and international funding organisations seem driven to promote the mobility of postdocs to foreign countries. In theory, these measures are good; I agree that if one could forget one's private life and for women, the ticking of their biological clocks (I'm now almost 29), this strategy would be a good bet to become a competent and well-qualified scientist. However, it is more difficult for individuals to overlook private life than it is for governments.