By James S. Walker

Since 2009, I have taught an interdisciplinary course on mathematics and music at UW-Eau Claire. The course is a wonderful blend of science, technology, engineering, mathematics, and the art of music. Music in the course ranges from jazz, rhythm and blues, African drumbeats, Afro-Latin rhythmic techniques, Indian raga, and Chinese classical folk, in addition to Western European music. This music from around the world can be discussed using mathematical techniques in the course, such as spectrograms, greatest common divisors, and modular arithmetic, giving us a way to connect mathematics to a diverse range of cultures and peoples.

One particular activity, based on a PBS Frontline video featuring Yo-Yo Ma, demonstrates the wide-ranging opportunities for discussion in such a class. I ask the students that when they watch this video they should take special note of the following five musical passages:

1. Excerpt from a Bach cello piece. Yo-Yo Ma has played this piece, Prelude to Cello Suite #1 in G-major, to varied audiences around the world. It is part of his project to play each of Bach’s many preludes in as many countries as he is able, in order to foster appreciation of Bach’s music and engage with other musical cultures.

2. Excerpt from a duet with Rhiannon Giddens. Rhiannon Giddens plays banjo, an instrument which descended from the African banjo, while singing Build A House. Yo-Yo Ma accompanies her on cello. An interesting math and music aspect of this duet is the use of two unfretted string instruments, cello and unfretted banjo, which are not tied specifically to a discrete tuning, such as the 12-tone equal tempered tuning used for fretted banjos, guitars, and pianos. This allows for more pitch flexibility. Gidden’s voice is also capable of beautiful pitch flexibility. We examine a spectrogram displaying the fundamentals and overtones that make up the musical tones, both instrumental and vocal, within this duet. The course discusses the windowed cosine series that underlie these spectrograms.

3. Excerpt from a piece by Saint-Saëns. A cello transcription of Carnival of the Animals is performed simultaneously over Zoom by 24 young cello players from all over the world, including China, Korea, and the US. Yo-Yo Ma points out that the Zoom performance is actually a melding of 24 separate recorded performances. This highlights the importance of accurate rhythm—in terms of simultaneous note onsets and matching note durations—which is a basic mathematical aspect of music that is taught in the course. (Although it could be just a coincidence, it is also amusing to note that there are 24 major and minor chords—something else my students study!)

4. Duet of Bach piece. Yo-Yo Ma plays a duet with a 1960s video of William F. Buckley playing the harpsichord. This is a charming use of technology, allowing us to interact musically with past performances. It reminds me of Natalie Cole’s duet with her father, Nat King Cole, on Unforgettable. The performance is part of a discussion of the predominance of recorded music today and its effect on the general public’s participation in music. In the course, we discuss many of the techniques of modern music processing, including auto-tuning, phase vocoding, and adaptive synthesis. All these techniques are based on dynamic processing of windowed cosine series.

5. Excerpt from a Dvořák piece. Yo-Yo Ma plays a cello transcription from Dvořák’s New World Symphony. This passage was inspired by African-American music, as were several other passages. In addition to symphonic music, the course deals with various types of folk music from around the world, emphasizing the algebra and geometry of chord changes and rhythmic transformations.

More sensitive topics such as cultural appropriation in music and who has access to arts and music programs can be discussed in a classroom where students and faculty are ready for thoughtful conversation on these more nuanced issues.

Music is certainly not the only way of bringing the world into the mathematics classroom. Connections of mathematics and various art forms (such as the visual arts) can create opportunities for embracing diversity and further exploration.

James S. Walker is a professor emeritus at University of Wisconsin-Eau Claire. Together with music professor Gary Don, he wrote the textbook for his course, Mathematics and Music.