Trigonometry, Beats, and Instrument Tuning

Sometimes mathematical formulas that only seem to be interesting from a theoretical point of view can reveal unexpected connections with some applied problem.

This is the case of the “sum to product” trigonometric formula

$$\displaystyle \sin(x)+\sin(y) = 2\sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right)$$

This formula has a direct application in explaining an acoustic phenomenon called beats. Let’s see what this is all about.

Beats

When you play two notes with slightly different pitch, the resulting sound seems to appear and disappear as someone raises and lowers the volume with a certain frequency. This phenomenon is known as a beat. Trigonometry will help us explain it.

The two notes played can be represented by the trigonometric functions

$$\begin{aligned} f_1(t) & = \sin(\omega_1 t)\\f_2(t) & = \sin(\omega_2 t) \end{aligned}$$

that oscillate in time with the two frequencies $\omega_1$, $\omega_2$ (to be precise those are the angular frequencies but in this post I will call them simply frequencies).

Acoustic phenomena are (to a good approximation) linear, and the sound of two notes played together is equal to the sum of the two single notes

$$S(t) = \sin(\omega_1 t) + \sin(\omega_2 t)$$

With the aid of the sum to product formula, we can write the “sound” function as

$$\displaystyle S(t) = 2\sin\left(\frac{\omega_1+\omega_2}{2} t \right) \cos\left(\frac{\omega_1-\omega_2}{2} t \right) = 2\sin\left(\omega t\right) \cos\left(\delta t\right)$$

where in the last term we have defined

$$\begin{aligned} \omega & = \frac{\omega_1+\omega_2}{2} \\ \delta & = \frac{\omega_1-\omega_2}{2}\end{aligned}$$

If the two frequencies are very close to each other $\omega_1\sim\omega_2$ we have that $\omega$ is also similar (the mean between them) and $\delta$ is a very small value compared to $\omega$ (the difference between two similar values).

With $\delta$ being much smaller than $\omega$, we can interpret it as a periodic change in the amplitude $A(t) = 2 \cos(\delta t)$ applied to the note $\sin(\omega t)$

$$S(t) = A(t)\sin(\omega t)$$

so we can see that $A(t)$ acts as a periodic volume change of the sound $\sin(\omega t)$. The smaller the difference is between the two original frequencies, the slower is the beat frequency $\delta$.

In the following image you can see an example of the two functions $f_1$, $f_2$ (top graph) and the resulting function $S$ (bottom graph).

And the following is the corresponding sounds. At the beginning you can hear the two notes separately and then the effect created by the two notes played together.

Instrument Tuning

Beats are used to tune instruments. Let’s see how this works.

Assume you have to tune a guitar string using a diapason as a reference. If the guitar string is tuned somewhere near the note of the diapason, playing them together will create a beat.

Usually it’s difficult to understand if you have to raise or lower the string tension to perfectly tune the string. It’s easier to proceed by trial and error.

Let’s say you try raising the string tension (and so the note frequency), you play the string again and the diapason, and hear that the beat frequency has increased. That means raising the tension is going in the wrong direction. You slowly lower the string tension until the beat frequency is so small that it’s no longer noticeable. Congratulations! Now your guitar string reproduces (for every practical purpose) the same note as the diapason.

And the other 5 strings? The process is the same but instead of the diapason you can take other notes played on the string you already tuned as reference.

Of course, there are apps you can use to tune instruments, but if you are playing your guitar on a beach surrounded by a group of appreciative listeners, you’d better know how to tune your guitar the old-fashioned way or you’ll instantly lose their trust of your playing skills!

Ciao,

Enrico