Question Time! Tonedeaf Overtones

"just wondered do you think its mathematically/physically possible to make a single note thats in disharmony with itself? like the close overtones in it are not in harmony?" - Philip Chuah

Answer: YES.

Brief intro to overtones: Everything you hear, you hear because something is moving, causing the air around it to move, which pushes a wave of air into your ear where it is heard. Musical notes come from a specific type of movement - vibrating at a constant speed. When you pluck a string, blow into a tube, hit a metal bar, or anything like that, something vibrates at a regular speed, producing a note. The faster it vibrates, the higher the note you get.

Here's a weird thing though - Only the simplest waves - sine waves (like in http://www.youtube.com/watch?v=R7D1f6U6TpU) - come from just a smooth vibration back and forth. Every real-world instrument actually vibrates in a more complicated way. But the cool thing is that these complicated vibrations can actually be split up into a bunch of simple vibrations - called overtones - that add up to the original!

You can see overtones at work quite easily if you have a stringed instrument like a guitar, or violin, or piano. Touch a string lightly at exactly the halfway point and play that string - you will see that the string still vibrates, but at a higher pitch. This is one of the overtones of the string. The same thing happens if you hold your finger one third of the way down the string, or one fourth, or one fifth. In fact, all the overtones of all instruments that vibrate along a straight line (strings, pipes, etc) can be found by dividing up the length evenly. And whenever you play the string, all of those overtones actually are played at once. And in fact, it's precisely the relative strengths of the overtones that give each sound a different quality. Why does flute have such a "pure" sound? Because only the first one or two overtones are very strong. What gives brass instruments their piercing texture? Because of the shape of the instrument (more cone-like than tube-like), it brings out the odd harmonics, while the even harmonics are really weak. In instruments with a "rich" sound like violin or human voice, you can go up a long ways in the overtone series and they'll still be pretty strong. But while the strengths of each overtone change from instrument to instrument, the pitch of each overtone is always the same in most instruments - you can get each overtone by evenly dividing the length.

But, as briefly mentioned in the video, some instruments don't have quite as nicely placed overtones. In particular, 2 or 3-dimensional vibrators have stranger overtones. If not controlled right, usually the overtones will just be messy and result in noise - like most drums. But sometimes the overtones can be controlled to sound not so bad. This 3-d aluminum bar shows this well - the 1st overtone is an octave above the fundamental (so it sounds ok), but the 2nd overtone is just a half step above the 1st! With 1-d vibrators, the overtones almost always sound nice with the original note, because they're so nicely divided - you'd have to go past the 10th overtone before you'd get two overtones a half step away from each other. But 3-d objects aren't limited in that way.

So in conclusion - yes, a note can be out of tune with itself. Case closed.