The banjo is a stringed instrument that produces a distinctive metallic sound often associated with country, folk and bluegrass music. One of the most famous examples is the duelling banjos scene in the film Deliverance which clearly shows the difference in the sound produced by a banjo and a guitar.

But here’s the thing: while the instrument is straightforward in design and the metallic timbre easy to reproduce, acoustics experts have long puzzled over exactly how the instrument produces its characteristic tones. What gives the banjo it’s a bell-like ring and why is it so different from other stringed instruments such as the guitar?

Today, we get an answer thanks to the work of David Politzer at the California Institute of Technology in Pasadena, who in his spare time, is a Nobel prize-winning theoretical physicist. He says the distinctive notes are the result of the complex way banjo strings stretch when they are plucked. And he shows how a similar effect can be easily reproduced using the free sound editing software, Audacity.

First, some background. The banjo is essentially a drum with a long neck attached for securing a set of strings. The strings are fixed at the end of the neck, stretched across the drum and fixed on the other side. However, they are supported by a bridge that sits on the drum membrane.

A key point is that each end of a string is secured at a fixed point. However, the bridge vibrates up and down as the drum resonates, usually at the frequency of the lowest note being played.

Politzer says this vibration is crucial. But to understand how requires a little more background knowledge about frequency modulation.

Back in 1973, John Chowning at the Stanford Artificial Intelligence Laboratory in Palo Alto showed that modulating the frequency of notes in a certain way leads to a surprising audio phenomenon.

One way to modulate the frequency is to change the tension in the string. A guitar player can do this by pushing the string sideways as it is plucked at a rate of a few times per second. This leads to the familiar warble or tremolo sound.

But Chowning found that when the frequency of this modulation increases, the warble eventually disappears and the timbre of the note changes instead. He found that this occurs when the frequency modulation reaches audio frequencies of a few tens of hertz and upwards.

That is hugely significant for the sound of the banjo, says Politzer. Whenever a stringed instrument is plucked, the tension of the string changes as it vibrates. However, this effect can usually be ignored when calculating the sound the string produces.

In a banjo, this change in tension is crucial. Politzer says that when the string is plucked, its tension changes as it vibrates. But at the same time, the drum resonates at a frequency of a few tens of hertz or higher. The movement of the drum pushes the bridge up and down and this also changes the tension in the string.

So there are effectively two different periodic changes to the tension of the string at the same time. There is the change caused by the string’s natural frequency and this is modulated by the change caused by the drum’s frequency of vibration. “The important picture to take from this discussion is that each string’s tension is modulated by the motion of the bridge,” says Politzer.

It is this modulating frequency that produces the banjo’s bright metallic timbre, just as Chowning showed in the 1970s.

Politzer say it’s relatively easy to reproduce this effect using the free, open source sound software, Audacity. Record a low note on a banjo and then push on the membrane near the bridge at a rate of a few times per second. This produces an audible warble in the note caused by the string stretching.

Next, use Audacity to speed up the recording and so increase the frequency of the note. Of course, this also increases the frequency of the modulation.

When this modulation reaches the audio level — Politzer says 36 Hz should do — the quality of the note changes. “The sound will have acquired a definite metallic plink, akin to banging on sheet metal,” he says.

This also explains why other stringed instruments do not share this effect. In a guitar, for example, both the bridge and the string is fixed to the soundboard and so move together as the soundboard vibrates.

But what of other stringed instruments where the bridge moves relative to the fixed ends of the string like the banjo? These include the violin and the mandolin but they do not produce a metallic ring.

Politzer says this is because the soundboard on these instruments is made of wood and so does not move nearly as much as the membrane on a banjo. Indeed, he points out that if the membrane on a banjo is replaced with the wood, the quintessential banjo features disappear.

A fascinating insight into a remarkable instrument—and the musical interests of a Nobel prize-winning physicist.

Ref: arxiv.org/abs/1406.4907 : Banjo Timbre From String Stretching And Frequency Modulation