# Finding Harmony

Winter 2017

Why music could have the power to entice more young people into STEM fields.

As a longtime saxophonist and bass guitar player, I’ve long been a big fan of Herbie Hancock. So when I first heard the jazz great talking about music as one solution to the crisis in STEM education, it really struck a chord (pardon the pun). Hancock was concerned about students in the United States falling behind in STEM fields—and also about diminishing arts education. Why not, he astutely asked, combine the two?

Why not, indeed?

So last summer, as part of a national educational effort launched by Hancock, I stepped into the breach to teach The Mathematics of Music, a three-credit, monthlong course I developed for the Johns Hopkins summer program. The course attracted mostly high school students, all of whom, not surprisingly, had some interest in music. My challenge as an instructor was to introduce ideas in music to steer the students toward topics that I could use to help enhance their mathematical maturity.

It’s been understood for a long time that mathematics and music are intimately intertwined. The ancient Greeks devoted a great deal of attention to understanding harmony and scales, and wrote at length about the relationships between pitches, in terms of ratios of their frequencies. Ratios and proportions are also the key to understanding rhythm.

I focused initially on the problem of tuning a piano. Harmony between pitches is something that is explained by the fact that the ratio of their frequencies can be reduced to fractions involving small integers in the numerator and the denominator, like 3/2. This particular ratio corresponds to the interval referred to as a perfect fifth in music. It has been recognized for centuries that we cannot tune a piano and ensure that all of the fifths have the proper harmonic ratios, and the standard workaround for tuning that has been settled on is referred to as “equal temperament.” Here, the ratio of the frequencies associated with adjacent piano keys is always the same value: namely 2 1/12 ≈ 1.06.

Armed with this understanding of how piano notes work, we were able to ask simple questions, such as, “If a hummingbird flaps its wings 50 times per second, what note does the sound make?” In answering such a question, my students were motivated to think and work with one of the most important functions in all of mathematics—namely, the exponential function and its inverse, the logarithm function.

As the course progressed, we began to record the sounds of musical instruments and voices; one particularly popular session included opera singer and Whiting School Assistant Dean Christine Kavanagh. It quickly became quite natural to get students to think of digitized sounds as nothing more than series of numbers. Linear algebra is an indispensable tool, both within mathematics and in areas in which mathematics is applied, and in studying sound, linear algebra becomes extraordinarily powerful.

Another reason I introduced linear algebra was to help students understand one of the most profound contributions to applied mathematics: the discrete Fourier transform. Students learned in a very real sense that any series of numerical sound measurements taken over a sequence of N discrete time steps can be viewed as a superposition of the sounds of N instruments, each producing its own unique pitch. Along the way to them appreciating this extraordinary fact, I was able to expose the class to a host of mathematical subjects: complex numbers, the complex exponential function, Euler’s formula, vector spaces, their bases, and linear transformations.

As a teacher of mathematical ideas, in this course as well as others, I am convinced that establishing links to real-world phenomena in the classroom gets students excited about the possibilities for mathematics and motivates them to engage in further investigation.

Daniel Q. Naiman is a professor of applied mathematics and statistics at the Whiting School.