If you read about science at all, you’ve heard of Heisenberg’s uncertainty principle. It’s the canonical example of quantum weirdness, the strange idea that you can’t simultaneously know the position and momentum of a particle. Pack a particle into a small enough box and your accurate knowledge of position will necessarily cause that particle to have a very uncertain momentum, “bouncing” around crazily inside that box.

What you may not have read is that this isn’t just quantum weirdness, it happens just as often in the classical world of waves. In fact, the very fact that quantum particles have a wave nature creates the bridge between the weird quantum uncertainty principle and the classical uncertainty principle that you literally *hear* all the time.

Let’s start off with a graph of a sine wave with a frequency of 1 Hz:

There’s two equivalent ways I can specify this wave. I can say, as I did, that it’s a sine wave of frequency 1 Hz. Mathematically you could say I’ve given the spectrum of this wave, describing it in what we in the business call the *frequency domain*. (And phase, but that’s beyond our scope for this post.) The other way I could specify this wave is in the *time domain*, where I just write down the function of t that describes the time behavior of the wave.

One representation is not more fundamental than the other. Given the frequency domain, it’s possible to write down the one unique waveform of that frequency spectrum – which, for the record, is itself a function of ω, the frequency. Given the description in the time domain, I can calculate the frequency spectrum.

So what if I have a wave that looks like this?:

It’s almost the same 1 Hz wave as above, but it’s not the same. It’s contained within a envelope, and only maintains that 1 Hz for a few cycles before dying away. Clearly in the frequency domain, “1 Hz” isn’t exactly the right description because it has already been taken by that pure sine wave. So what’s the right frequency representation?

We can calculate it by a method called the Fourier transform, named after the French mathematician Joseph Fourier. By doing this, we can see what the right frequency representation is. It’s this:

Here the units are in Hz. We see that it’s sharply peaked right at 1 Hz, which makes sense because it’s mostly a 1 Hz signal. But because it’s not exactly a pure sine wave, the sine wave in its envelope is actually a mixture of different frequencies spread near the 1 Hz mark.

What if we plot another 1 Hz wave, but force it into an even briefer envelope, like this?

We can do the Fourier transform and see how it looks in the frequency domain:

Ah. It’s even more spread out. Despite being in some sense a “1 Hz” pulse, it actually involves frequencies as low as 0.5 Hz and as high as 1.5 Hz. The general principle continues to hold as we compress the pulse more and more. Shorter pulses require a broader range of frequencies. In general, short in one domain means broad in the other. Very short pulses of sound contain such a broad range of frequencies that they can scarcely be said to constitute anything resembling a musical note at all.

Now that we’ve seen it, we ought to listen to it. Our 1 Hz example is far below the range of human hearing – or most speakers, for that matter – so we’ll use a base frequency of 440 Hz. Here’s a 440 Hz tone in an envelope that’s 1/2 of a second long, measured from half-maximum on the upslope to half-maximum on the down:

It is a pulse of sound, but it’s obviously a nearly pure note without a lot of frequency spread. What if we pack the same wave into a smaller envelope, one that’s just 1/10 of a second long?

Shorter, but still clearly identifiable. Quite a few cycles – well, 44 of them – happen in that time, so it’s not so shocking we can still pick out what the note is. But what if we make the envelope just 1/100 of a second long?

Hmm. At this point it’s possible to calculate that the frequency spectrum is spread almost from 380 to 510 Hz. Compared with a piano keyboard, that’s starting to spread into the frequency territories of the neighboring keys. When you listen to the note, it’s possible to tell that the pitch is not very easily identifiable as one definite note. It’s more of a “pop” with a hint of its original tone.

Finally at 1/500 of a second even the hint of the original tone is gone. As a musician to duplicate this note on a piano and they’ll just shrug their shoulders:

The frequency content of that little pop spans several octaves. It no longer resembles a pure note at all. The large degree of certainty with respect to the time of the note means a large degree of uncertainty with respect to the frequency of the note.

It’s the uncertainty principle and it’s not even quantum!