Mathematical Monday 11/15/10: Overtones

Did you know that there’s math in music? Lots of really interesting math actually about what frequency the various tones are and why some sound good together and some don’t. So, for today’s quick Math Monday, I want to talk about my favorite mathematical musical phenomenon: overtones.

This is a reunion of the Allegheny College Choirs led by the man who taught me overtones. I’m in there somewhere!

An overtune is a higher frequency than the lowest frequency of the sound being produced. Every time I hit a key on the piano, you near a main sound called the fundamental frequency, and then above that there are numerous other sounds at higher frequencies and lower volumes. Different instruments make different overtones based on physical aspects of how they are made. Overtones are the reason that a Steinway and a Yamaha piano can play the same note and sound completely different. I tend to hate Yamaha pianos, by the way.

Now that you know what an overtone is, you can manipulate it. When I sing, I can make minute changes in the shape of my throat to change the overtones, but this is not easy. My favorite overtone game was some I used to do in choir, and it was adding notes (see, more math!). When two different fundamental frequencies are produced at the same time, they can produce the effect of an audible higher note somewhere else in the room. Ghost notes! This is very common in barbershop quartets.

The one I always used to do was to have an octave: the same note being sung high by the women and low by the men. The notes add together to make a 5th above the women’s note ring in the room. If both parts were singing F’s, then a C could be heard.

Here’s a youtube video where the choirs do some of the same things I used to do!

There you go: Math in music! Two things I love. Well, I was a music theory minor back in the day, so I would hope I loved it. There is more math in music, so expect other posts like this in the future.

3 Comments (+add yours?)

1. Whoa, I never knew about “ghost notes” that you could hear when two notes are sung together. That’s really cool!

I love overtones, though. Flute players rely on overtones to play many notes The same fingerings are used in some cases to play the same note in two octaves, we just blow harder/more directed to force it to play the higher overtone (I’m not sure exactly how that works, though–how blowing differently can eliminate the fundamental harmonic). Some fingerings are only used for their first overtone, and sound bad if we play it “down the octave”. Blowing even harder can force out even higher overtones, but they never sound very good so they aren’t used…except for fun experimentation.

Most people know how to blow on a bottle to make a tone, but I like to use my flute skills to get overtones with the bottle. Of course, because the bottle, unlike the flute, has only one open end, the first overtone isn’t an octave above, but a whole octave + a fifth above. With a nice large wine bottle, I can sometimes get a second overtone–two octaves + a fifth above. Fun (and nerdy) times at a dinner party đź™‚

• Oops, I meant two octaves + a third above for that last one. Brain got crossed with the previous sentence.

2. Love it! I sang in a concert choir and show choir and I love stuff like this.