The Math of Music
By Evelyn Lamb on February 19, 2018
If someone plays a sine wave with a frequency of 440 Hz, you will most likely perceive an A. (In fact, 440 is probably the most famous musical frequency. It is the tuning standard for most modern orchestras and instruments.) However, that A will not sound like an A played by any instrument in the orchestra. The flute probably comes closest to producing a pure sine wave, but even its sound is much more complex. If a harmonic analyzer took the sound of an instrument or human voice as its input, it would break the input into sine waves of many different frequencies, generally all integral multiples of one lowest frequency. The process works in reverse as well: By adding sine waves of various frequencies, computers and electric keyboards can create decent imitations of the sounds of these instruments. But if you start playing with sine waves a little, you will hear some surprises. Frequency is not destiny. For example, if you play a sine wave with a frequency of 440 Hz by itself and one with a frequency of 660 Hz a few seconds later, you will hear two distinct pitches, one a perfect fifth higher than the other. (A perfect fifth is seven half-steps, the interval between an A and the E above it.) If you play the sine waves with frequencies 440 and 660 together, you will not hear a perfect fifth. Instead, you will hear a pitch an octave below the A 440. We hear pitch logarithmically, and an octave corresponds to a frequency ratio of 2 : 1. So, you will perceive the same pitch as a sine wave with a frequency of 220 Hz. Even if you know that nothing is actually playing a pitch with frequency 220 Hz, that pitch is what you will hear. The combination of the frequencies 440 and 660 creates a perceived pitch of 220. The fact that pitch is a perception, rather than an objective, measurable aspect of sound waves, is one of the challenges in trying to use mathematics to describe music.
Gareth E. Roberts’s textbook From Music to Mathematics: Exploring the Connections does a good job of separating the objective from the subjective in its discussion of pitch and frequency. There is a mathematical explanation for the auditory illusion in my example, sometimes called the missing fundamental: The perceived pitch is the greatest common divisor of the frequencies of the sine waves present. [Read more about the missing fundamental on this blog here and here.] The real explanation, however, belongs to cognitive science, not mathematics. Our pattern-recognizing brains, which must often make snap judgments with incomplete information, notice that 440 and 660 are among the expected frequencies that an instrument or voice would create when producing a note with fundamental frequency 220. The brain assumes it just missed picking up on sine waves with frequency 220 and fills in the gap, perceiving 220 where there is none. On a practical level, this effect is exploited by telephones, which do not pick up frequencies as low as most humans’ speaking voices, but nonetheless manage to transmit normal-sounding messages. The effect is also found in some pipe organs, which do not have room to make pipes large enough for the lowest notes on the organ, and instead cleverly trick the listener with precisely-calibrated smaller pipes. Roberts does not oversell mathematics as the explanation for the missing fundamental, but shows how mathematics can be used as a tool to predict the perceived pitch based on the spectrum of pitches produced.
Roberts, a mathematics professor at the College of the Holy Cross, developed the book for an undergraduate course in mathematics and music. Unlike some textbooks in this area, such as David J. Benson’s excellent but challenging Music: A Mathematical Offering, the mathematics required to read the text and understand the exercises is generally quite basic. The few times calculus is necessary, Roberts makes a note of it, and computations are generally explained clearly and thoroughly. The tradeoff is that some explanations are black boxes. We have to take his word for the ODE and PDE solutions that yield useful formulas for computing various aspects of pitch. The book would be appropriate even for a class in which some students need a refresher on fractions. In later chapters, particularly Chapters 5 and 6, on musical symmetry and change ringing, respectively, Roberts presents some abstract group theory. The pacing of the explanation, however, is such that students without a heavy mathematics background should be able to pick it up.
As Roberts writes in his introduction, we err when we assume students who are not math or science majors are not capable of or interested in learning advanced topics in math. One of the problems with the current high school mathematics curriculum is the way mathematics courses are often presented as a straight line to calculus. Most students who do poorly in math classes at that level never get an opportunity to learn about other mathematical topics that may be more natural or interesting to them. Roberts says that in his mathematics and music course, students who did not think of themselves as mathematically inclined have unearthed hidden talents and interests in the area. He writes,
For example, some students respond very well to learning group theory. In secondary school they struggled through limits, algebra, and precalculus, but completing the group table for the symmetries of the square and seeing its connection to an extent [a technical term in change ringing that refers to ringing all possible permutations of the bells in sequence] on four bells kindles a newfound interest in abstract algebra.
He says he has even had some students switch to being math majors after taking this class.
The book also does not assume a student is fluent in musical notation before taking the class. Chapter 2 contains the basics of reading music; in later chapters, understanding this notation is important. The accompanying website1 includes links to some listening examples and resources, but it would be great to have more such examples available to support students who are first learning musical notation. No book can be all things to all students, and Roberts seems to have created a gentle introduction to both the music and the mathematics in this course. A student with a very strong background in both may feel the book is too easy, but an instructor could modify the pace of the course and the content covered to provide an appropriate challenge, or just decide that such a student would be better served by a reading course using a different text.
The first four chapters cover standard material one would expect in a book about music and mathematics. Chapter 1 is on rhythm, Chapter 2 on the basics of musical notation and theory: reading a staff, naming musical intervals, and understanding key signatures. Chapter 3 discusses pitch perception and frequency, and Chapter 4 is on tuning and temperament. There is no one perfect way to organize a book on mathematics and music. From a strictly logical point of view, Roberts’s organization seems backwards: Chapter 2 relies on the assumption that an octave, the span of pitches between a frequency and its double, is split into twelve equal half steps the way a modern piano is, Chapter 3 explains why the octave would be a particularly important interval, and Chapter 4 justifies using a twelve-tone system and explains how we arrived at equal temperament to boot. The organization Roberts chose has the advantage of allowing students, particularly those with a musical background, to start in familiar territory, the piano, and probably prevents some confusion. A person who has played an instrument or sung in a choir will feel right at home with notation they understand, and a person without much musical experience has probably at least seen a piano keyboard. They may even have access to a real or virtual keyboard, where they can get a feel for intervals and other musical basics as they’re introduced in the text. Starting with pitch and interval perception instead and then moving to temperament could have the effect of pulling the rug out from under people. Twelve half-steps in an octave is practically an axiom in Western music, and it could be confusing to have to justify it at the outset.
The next four chapters feel more like a collection of case studies. Since each topic is largely self-contained, instructors would have ample opportunity to pick and choose the sections that will work the best for their particular classes. Chapter 5, on musical symmetries, explores Bach fugues and the more literally symmetric music of modern composers such as Bartok and Hindemith. Bach is par for the course with regard to the material covered in a basic mathematics and music class, but the part on Bartok in particular was an interesting exploration of whether Bartok’s famous use of the golden ratio was really as clear-cut as it is sometimes said to be. This chapter also contains a gentle introduction to group theory.
Chapter 6, on change ringing, delves more deeply into group theory. Change ringing is an unusual musical practice that originated in 17th century England. It takes place in bell towers with large bells that are mounted so they can swing in a full circle. One person stands under each bell holding a rope, and the bells are rung in sequence. Experienced change ringers learn to carefully control the timing of the bells, so bells can change positions in the sequence. Change ringing is not about artistic or emotional expression. Instead, melodies are basically combinatorial. Ringers will start by playing a descending scale, and in subsequent rounds, neighboring bells will swap positions. With a few rules to guide them, change ringers create sets of changes such as “extents” that perform all permutations of the bells using only allowable changes. My illustrious career as a change ringer lasted about three weeks at the beginning of graduate school, but in that brief time I was floored by the fact that we were actually performing permutation groups. Change ringing is often acknowledged as a musical form with very direct parallels in mathematics, and I found Roberts’s explanation thorough and clear.
Throughout the book, in another contrast to other textbooks about music and mathematics, Roberts often uses music as a motivation for introducing a mathematical topic and takes a detour into the mathematics itself. In the chapter about rhythm, he ventures into infinite geometric series using dotted rhythms for motivation. Frequency ratios lead to a detour into proving that √2 is irrational. Although I have a lot of experience in mathematics, music, and their intersection, I learned a few things myself. I had not known much about the way Indian classical music led to the discovery of the Hemachandra–Fibonacci sequence 1, 1, 2, 3, 5, 8,… in the twelfth century C.E., or about Peter Maxwell Davies’s use of magic squares in some of his music.
In the introduction, Roberts makes some suggestions about how to use the text for either a year-long or one-semester class on math and music. I have not taught such a course, but it seems like the text would be appropriate for a freshman seminar or liberal arts math class with a group of students that is diverse with regard to their mathematical and musical backgrounds. The text has plenty of worked-out examples, exercises, and two interesting projects for instructors to choose from. The project in the final chapter is particularly compelling: write music that is influenced by mathematics in some way. I only regret that I did not get to listen to any of the pieces his students wrote!
