(The following post is Part 4 in a four-part series on studying math through the lens of other disciplines. We believe that students thrive when they can form meaningful connections across different areas of study. Previous installments in the series have included History, Art, and Philosophy.)
A pizza is divided into half, and then half again, and then half again. Sally eats one slice of pizza. What fraction of the pizza is left?
For many of us, word problems like this feel like torture. But did you know that your brain is doing similar calculations when it listens to music? The mathematician Gottfried Leibniz famously observed that “music is the pleasure the human mind experiences from counting without being aware that it is counting.”
If your musically-inclined student is struggling to enjoy math, exploring the connections between the two disciplines can help to reinspire your student in both studies. Music theory is the formal study of music, with a focus on the interplay of number and sound. And this post will feature some light introductions to music theory. But music is also intuited and felt through the body, and this post will also try to provide opportunities for you and your student to feel the math in music.
Simple Addition, Ratios, and Fractions in Music
Let’s begin with musical notation, the sheet music that tells musicians what and how to play. In notation, we have whole notes, half notes, quarter notes, etc., which all designate how long a note should be held in proportion to other notes. We also have time signatures, which specifies how many beats are contained in each measure, which are expressed as ratios such as 4/4. The top number indicating there are four beats per measure, and the bottom number indicating that one quarter note equals one beat. So for example, if my time signature is 4/4, I know I can fit four quarter notes (♩) into that same measure. To see this in action, write 1234, 1234, 1234, 1234 (that’s the four beats of the measure, in four measures); then have your student count those numbers while clapping for each number. Next, try out the half note which in 4/4 signature, takes up two beats. Write out 1234 (four times) again, but this time underline the 1s and 3s and tell your student to only clap when they say those numbers. (You can also circle 1 and 2, showing that half note takes up two parts.) Now try whole notes which are notes that last for a full measure. Underline the 1s and only clap when you say them.
Now it’s time to experiment with how ratios can overlap. Try this exercise with your student: have your student clap the quarter notes while you clap the half notes. If there are other people available, have them clap the whole notes. Try to keep tempo together as best you can. As a bonus, see if your student can demonstrate how to continue the pattern by halving the quarter notes while keeping the time signature the same. (Hint: there are two eighth notes in every quarter note.) In solving this problem, your student is doing the exact same work as in the pizza word problem. A whole pizza is like a whole note, half a pizza like a half note, and then quarter, eighth, and etc..
Okay, now let’s practice counting out a famous excerpt of music from Beethoven’s Ninth Symphony. While watching the video below (it’s 50 seconds, so watch it several times for these exercises), have your student start by counting aloud 1234 over and over, practicing the 4/4 time signature and quarter notes. Your student can then experiment with clapping the whole notes (1, 1, 1), the quarter notes (1234, 1234), or half-notes (1, 3, 1, 3, 1,3).
Your student can also experiment with adding in the other notes. Identify each type of note and add it to your clapping. You will quickly notice there are a few extra notes types (dotted quarter notes are ⅜ of a beat). Or you can follow along to the bottom line of music which is only half notes and whole notes. If your student is looking for an extra challenge, see if they can figure out how to count out the eighth notes.
I studied piano for several years. And I also took tap dance lessons and participated in musical theater. Dance is another discipline that lets us feel the math in music, and also the math in our movement. Tap dance most obviously corresponds to counting because the tapping sound actually helps us to keep the count. But all dancers have to keep count in their head to be successful. In this famous tap dance routine from the beloved classic film White Christmas, notice how the dancers use the tapping of their feet along with clapping and even snapping to help keep track of the intricate rhythms. What you’re seeing and hearing in this routine is ratios brought to life. As you and your student watch, try clapping or tapping along with the beat and don’t feel bad if you lose track of it from time to time, just listen and try to regain it again. See if you and your student can count it out at various points in the song.
You probably found this a much harder exercise than Beethoven’s Ode to Joy. The main reason for this is that the dance number uses a musical technique called syncopation which is all about switching which beats get stressed. When you and your student count and clap 1,2,3,4 in Ode to Joy, notice that 1 and 3 sound naturally stronger as beats, while 2 and 4 are softer. But if you reverse this, so that 2 and 4 are the strong beats, and 1 and 3 are the soft beats, it changes the sound and feel of the count dramatically. Oh, and in case you were wondering, I definitely can’t tap dance this routine; in fact, I can barely keep track of the count!
Properties of Multiplication in Math
In thinking about this relationship between math and music, one of my inspirations is Dr. Eugenia Cheng, a professional mathematician who is also a concert pianist. Dr. Cheng is passionate about the intersection of music and math, and she has created videos that blend instruction in both music and math.
Most of us do not have the level of knowledge in abstract mathematics needed to understand the kind of advanced research that Dr. Cheng is involved with. But the mathematics in music are often far more accessible to us on the level of intuition. Recall Leibniz’s insight quoted at the beginning of this article about how we make sense of music through counting though we typically don’t realize that’s what we are doing.
Here’s a fun example that Dr. Cheng highlights. The principle of commutativity in multiplication tells us that we can multiply numbers in any order without changing the product. So for example, if we know that 2 x 3 = 6, then we also know that 3 x 2 = 6. As it turns out, it’s this principle that allows us to have varied meter in music in a way that still feels meaningfully patterned. By varied meter, I mean when a song starts out with one time signature (say 4/4 rhythm) but slides into another one (for example 3/4 meter which is often heard in waltzes.) In the short video below, Dr. Cheng uses several pieces of music including a familiar Broadway song from West Side Story to illustrate how this principle sounds in music. As Dr. Cheng states: “the commutativity of multiplication sounds like an obscure rule in mathematics but it turns out to be something that we can feel in music. Feeling things in music is a kind of obvious thing to do; feeling things in math is less obvious but we can feel things in math as well.”
If you and your student enjoyed this video from Dr. Eugenia Cheng, be sure to check out this website that features ten more videos where she explains cool aspects of math in music.
Hopefully, these exercises have helped demonstrate that disciplines like math and music are deeply interconnected. Once we realize this, we are empowered to approach these disciplines in less conventional ways. Whether your student is currently taking piano lessons, inventing their own songs on a guitar, or listening to their favorite album on repeat, they are actually engaging with mathematical ideas and processes, though it is mostly subconscious. And of course, the more they engage with music theory, the more they can see those mathematical ideas consciously as well. The goal, in the end, is for your student to interact with all kinds of music from various vantage points – listening, studying, making, and playing. All of these modes of engagement provide your student with opportunities to feel the math in music, and perhaps later, to hear the music in math.