Telephones lie about sounds because odd numbers aren’t even. Once again with those integers and sound perception!

Telephones can only pick up frequencies above 300 or 400 Hertz (cycles per second, also called Hz), but most adults' speaking voices are lower than 300 Hz (approximately the D above middle C). And yet every day, people manage to have telephone conversations. (Don’t believe me? Try it!) Thank the missing fundamental. It’s a trick you (or your phone) can play on your brain with sine waves and ratios, and I was inspired to include it in my chat with the Girl Scouts about math and music after watching Vi Hart’s video What is up with Noises?

Hart demonstrates the effect starting around the 11:13 mark of the video, but you really should watch the whole thing.

A pure sine wave produces a recognizable tone, but the human voice and other instruments do not produce pure sine waves. In addition to the fundamental frequency dictated by, say, string length and thickness on a viola, instruments and voices produce harmonics whose frequencies are integer multiples of the fundamental frequency. So when a viola plays a C, the sound can be thought of as the addition of a sine wave with the fundamental frequency of the C and sine waves with frequencies of the C an octave above (2:1 ratio), the G a fifth above that (3:1 ratio), the C a fourth above that (4:1 ratio), and so on.

What does that have to do with the missing fundamental? Let’s start with two notes an octave apart, so their frequencies have a 2:1 ratio. The harmonics of the higher note are also harmonics of the lower note, but not all harmonics of the lower note are harmonics of the higher note. If we start with 100 and 200 Hz, then 400, 600, and 800 Hz, the even multiples of 100, are harmonics of both pitches, but 300, 500, and 700 Hz, the odd multiples of 100, are only harmonics of the lower pitch.

Let’s say you cook up some sine waves with frequencies of 200, 400, 600, 700, and 800 Hz. When you listen to that mix, your brain recognizes that this set of frequencies does not match the harmonic series of 200 Hz, the lowest frequency that is present, because of that interloping 700. But all of the frequencies are in the harmonic series of 100 Hz. So you'll perceive a pitch of 100 Hz, even though that is lower than any frequencies actually playing.

And that's how telephones lie to us. And it's a good thing they do—it's the reason our voices don’t sound substantially higher on the phone than they do in real life. When we speak, we produce a fundamental frequency and a bunch of overtones. The phone can’t pick up the fundamental frequency, but it can pick up some of the overtones, and the ear and brain of the person on the other end of the line supply the fundamental. Efficiency!

Strictly speaking, this isn't just an even/odd issue. It's about the greatest common factor of the frequencies playing. If I played sine waves at 300, 500, and 700 Hz, you would probably hear 100 Hz because that is the highest pitch that has 300, 500, and 700 in its overtone series. In mathematical language, 100 is the greatest common factor of 300, 500, and 700. So the factorization properties of integers, while inconvenient when it comes to piano tuning, are helpful for telecommunication. (Incidentally, it's helpful for a different aspect of piano tuning; the lowest notes on a piano sound as low as they do due to the missing fundamental effect.)

An illustration of the experiment Peter Schneider et al. used to separate people into "fundamental listeners" and "spectral listeners." The lowest frequency actually playing goes up from the first pitch to the second, but the fundamental frequency goes down. Image from Peter Schneider et al., 2005. Click picture for link to article.

When I visited the Girl Scouts, I played various mixes of frequencies for them, and it was shocking to all of us when adding a higher note caused the perceived pitch to drop an octave, which is what happens if you start with the frequencies 200, 400, and 600 and then throw in 700 or another odd multiple of 100. Even when you know it's coming, you can't will yourself to hear the higher notes instead of the missing fundamental, or at least I can't.

Almost everyone hears the missing fundamental when there are a lot of overtones to reinforce it, but a 2005 paper in Nature Neuroscience (paywall) explores the limits of the missing fundamental effect. Researchers Peter Schneider et al. found that both musicians and nonmusicians could be separated into "fundamental listeners" and "spectral listeners" based on whether or not they perceived fundamentals when only a few high harmonics were present. The test subjects would hear two tones back to back. The fundamental would go down from one note to the next while the lowest frequency went up, or vice versa. I found an mp3 of some examples of the effect on an overtone singing message board, which also has an "answer key" so you can find out whether you're a fundamental listener or not.

If you'd like to play with the missing fundamental effect for yourself, check out Audacity, a free audio editing program, and make yourself some sine waves.