I’ve made a little demonstration of Fourier series that shows you how a string vibrates when plucked.
Red is the actual string: L is its length and b and h control where it’s plucked. The solid green curve gives a Fourier series approximation to it, whose precision you can adjust with M. The solid blue curve shows how that Fourier series evolves over time. The dotted green and blue curves do the same for a single term in the series -- that is, a single harmonic, which you can choose with q. k controls the wave speed.
For the uninitiated, here’s how this works. A string that’s fixed at both ends can vibrate in certain special ways called harmonics or normal modes (which are displayed on the graph as the dotted curves). In the simplest of these, q=1, the string takes the shape of a half-sine wave, which then oscillates up and down with all points on the string moving in phase. In the next simplest, q=2, the string is shaped like a whole sine wave; the midpoint of the string is perfectly stationary, and the two halves of the string oscillate in opposite directions. In the q=3 mode, three half-sine waves fit on the string, and so on.
Each of these modes oscillates with a certain characteristic frequency, and these frequencies have an elegant relationship: the frequency of mode q is q times the frequency of mode 1 (called the fundamental frequency). So if the fundamental frequency is 440 Hz, the A above middle C, then the second harmonic is 880 Hz, the third is 1320 Hz, and so on.
Now, when I pluck a string, I’m not making any of these smooth sine wave shapes, but a crude, pedestrian angle. But Fourier analysis says that we can see how the wave evolves over time -- in fact, how any initial waveform will evolve over time, no matter how complicated -- by following a simple recipe:
Write the initial waveform as a sum of normal modes.
Each normal mode in this sum then evolves independently, by oscillating at its own characteristic frequency.
At time t, you can figure out the shape of the string by calculating where each normal mode is at time t, and adding them back together.
This is exactly what you see on the graph.
So that’s the math. How about the consequences for music? First, though you can’t see it in real time, this graph shows that a plucked string actually always has a sharp corner, which moves up the string to the mirror image of the plucking point, and then shuttles back. Hermann van Helmholtz described first described this phenomenon in the more complicated case of a bowed string, where the corner traces out a lens shape as it moves.
Second, the string as a whole vibrates with the same frequency as its lowest normal mode, i.e. the fundamental frequency. But this graph makes it clear that the real vibration has some mixture of the higher harmonics. Exactly how they’re mixed determines the unique timbre of the instrument. So in a violin (with vibrating, continuously bowed strings), a trumpet (with a vibrating air column), or a clarinet (with a vibrating reed), the same principles are at play, but the actual mixture of harmonics is different, producing different sounds. (To complicate things further, the bodies of these instruments vibrate in special ways too, thus amplifying or suppressing certain modes!)
Third, by stopping the midpoint of the string from vibrating, one can silence all the odd normal modes while allowing the even ones to continue. This produces a pitch whose lowest mode is the second harmonic of the previous pitch, making it an octave higher. Stopping the string at one of its one-third points makes it sound at the frequency of the third harmonic, and so on. Guitarists call this “playing harmonics” -- you touch the string at one of these points without fretting it all the way, and produce an unusually high and clear tone. The higher of a harmonic you want to isolate, the harder this gets to do: the higher harmonics occur in the plucked-string vibration with weaker amplitudes, and the margin of error for where to touch is smaller.
Fourth, one can also do the opposite, which you can see on the graph by setting b to be an integer fraction of L. For example, if you pluck the string at its midpoint, then the second, fourth, and so on harmonics don’t sound at all. I’ve heard this is important in piano design. The hammers are supposed to hit the strings exactly 1/7 of the way along their length in order to silence the seventh harmonic, which is two octaves and a very flat minor seventh above the fundamental, so sounds dissonant to us.
Theoretically, this sort of thing is applicable to guitar or violin technique, too, though. I haven’t seen this actually used, but I have seen classical techniques where one plays close to the bridge or end of the string (sul ponticello). As you see if you make b close to L, this accentuates the higher harmonics relative to the fundamental, producing what I’d call a sharp, metallic sound. (You might also notice that the approximation gets worse as b approaches L, which is another instance of the same phenomenon: the higher terms in the series which are cut off by the approximation become more relevant.
I got into this from the happy coincidence of playing guitar and teaching Fourier analysis. Do you know things about the acoustics of other instruments, or other aspects of music production that can be mathematically modelled? I’d love to hear from you!
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