The Physics of Making Your Guitar Cry

The mechanics of turning the flick of a plectrum on an electric guitar into a sound—and, if you’re you’re lucky, something resembling music—isn’t rocket science. But less scientific scrutiny has been granted to the finer embellishments of making those strings moan and wail like Eric Clapton or Brian May.

One physicist who gave up his ambitions of being a professional guitarist for the (slightly) more reliable pay checks of the research world has stepped up to the plate with a set of formulae that explore the more idiosyncratic techniques beyond my three-chord knowledge; the flamboyant rockstar stuff like string-bending and whammy bars.

David Grimes at the University of Oxford published his paper on “string theory” (groan) today in PLOS ONE. He told me he used to play as a session guitarist and would often wonder what he could do to improve his technique. “It is an artistic pursuit, but there’s a lot of mechanics and beautiful physics underpinning it,” he told me.

“One of the things that’s real beautiful about the guitar is it shares some characteristics with the human voice,” he said. Unlike a piano, where you’re restricted by the limited keys to changing pitch in semitones, bending strings lets you modulate pitch much more freely. So, as Grimes puts it in his paper, “String-bending and vibrato add much to a guitarist’s palette, and as these techniques are heavily influenced by the physical constraints of the guitar and strings used, the underlying mechanics are worthy of analysis.”

That’s precisely what he goes on to do, complete with formulae for string-bending, fretting force, vibrato, and “whammy-bar dynamics.” Take the string-bending equation, where “l is the length of the vibrating element, T is the string tension and μ the linear density or mass per unit length of the string”:

Right. The basic take-away for players is that the type of strings you use have a big impact on the effect you’ll get—something he admits most guitarists figure out intuitively. But theoretically, you could use Grimes’s work to select the perfect string for your requirements.

One of the most important properties is known as the Young’s modulus, which essentially measures the stiffness of the string. It might not help you emulate your heroes with scientific precision, but if you want to cry like Clapton, one of the equations dictates that you’d go for strings with a lower Young’s modulus that are less stiff. If you want to shred metal, you’d probably go for stiffer strings. “If you know a bit about what’s affecting your tone and pitch, you could probably buy strings that are more suited to what you want to do,” Grimes said.

Not content with exploring the phenomenon theoretically, Grimes sacrificed one of his own guitars in the name of science (he said he has about 17, so it was no biggie). He stripped down an old Gibson Epiphone and hammered in two nails on the fretboard, which acted as fixed-place fingers to bend different strings at the same angle. He loaded different strings supplied by Ernie Ball onto the guitar and recorded their pitch unfretted, fretted at the 12th fret position, and then bent on each of the makeshift pegs. Sure enough, the effects of the string-bending agreed with his model.

But even with experimental backing, players will need to rely on intuition. “Even bending, I found that string-bending isn’t linear,” said Grimes. If you bend a string to two degrees, the effect on pitch isn’t twice that of one degree. “So a lot of this ends up being intuitive, even though you can quantify it and the paper does that.”

Where it could perhaps be more useful on a practical level is in digital instrument modelling. “If you’ve ever played a keyboard replication of a guitar, they’re pretty woeful,” said Grimes. Someone looking to really emulate the mechanical manipulation you get with a guitar might be able to use the mathematical models to get a closer match.