# How do harmonics on stringed instruments work?

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How do harmonics on stringed instruments work?

In: Physics

There are a lot of really good discussions online already, but what it boils down to is that the string doesn’t only vibrate (oscillate) along its full length (the fundamental)—it also vibrates as if it was three strings each having a third the length, and also as if it was five strings each having a fifth the length, and so on and so on.

The frequency of an oscillating string is inversely related to its length (other things being equal, namely density and tension). So a shorter string will oscillate faster and produce a higher frequency than the fundamental.

When you pluck a string, it vibrates. That vibrating string knocks into the air around it, creating sound.

Different strings vibrate at different rates, called “frequencies,” and we hear those different frequencies as different “notes” or “pitches.” For example, a guitar has 6 different strings, each named after the different sounds produced: E A D G B E

But wait, how can two *different* strings both sound like “E” on a guitar? Well, even though they both sound like “E,” the strings are made of different materials/thicknesses so that you get one “high-pitched E” and one “low-pitched E” string.

(Remember, pitch is the same thing as frequency!)

So now I’m saying two *different frequencies* are both called “E?” Yes! If you have two strings and one is vibrating at twice the frequency of the other, then they produce notes called “octaves,” that sound similar.

The “low-pitched E” string on a guitar vibrates at a frequency of 82.41 Hz (times per second), and the “high-pitched E” string vibrates at a frequency of 329.63 Hz, which is almost exactly four times 82.41 Hz (that’s no accident!)

The important idea here is that: all multiples of a “fundamental frequency” will “resonate” and “sound good.” These are called “resonant frequencies” in physics.

Here’s the super cool part: when you pluck a string, it actually starts vibrating at *more than one frequency*. The sound you hear is really made up of many different frequencies, and all these frequencies except the fundamental are called “overtones.” (These overtones depend on a lot of factors, but this is why different instruments/people’s voices have different “tones” or “timbres” even if they’re the same fundamental frequency. Like how the “E-string” on a violin sounds very different than a same-frequency-note “E-string” on a guitar.

And you know how “resonant frequencies” were just multiples of the fundamental? That’s so they fit nicely together, and that’s the same reason overtones are created when you pluck any string: they fit very nicely! They’re friends with the fundamental! (google the “overtone series” if this explanation is unsatisfying.)

So now we can finally talk about harmonics on stringed instruments: those soft, light, high-pitched sounds you get by first plucking a string, and then gently resting the tip of your finger on top of the string, as if to mute the string, but in a very particular place that somehow lets some sound through? That must mean the string is still vibrating somehow!

Indeed, the string is still vibrating, but no where near as much as before your fingertip created the harmonic. Remember how strings don’t just vibrate with one frequency? When you place your finger on the string to create harmonics, what you are doing is forcing the string to “not move” and “be still” at that exact point. Your finger is preventing the string from moving, but *only at that point*. The thing is, all frequencies naturally have some of these points “built-in,” but each frequency has different locations of those “not-moving” points.

Here is a picture of what a vibrating string looks like for various frequencies, just remember strings don’t only vibrate with one frequency at a time: https://upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Harmonic_partials_on_strings.svg/1200px-Harmonic_partials_on_strings.svg.png

So where do you put your finger? Those special points are called “nodes” but they just mean points where the string hasn’t moved up or down, so the height = 0. In that picture, the first node for each frequency is labelled with a gray dot, but you can see more and more nodes appear as frequency increases.

So by placing your finger on a known node of one frequency, you are eliminating any other frequency that *doesn’t* have a node at that point. Why? Because your finger is making sure the string can’t move at that point. Only frequencies with nodes at that point are allowed to vibrate.

And lastly, depending on which “known node” you select (try saying “known node” ten times fast), you can isolate different “musical intervals” aka “multiples of fundamental frequencies!”