Physics Of a Guitar String

when maths gets musical…

By considering the guitar string as a one dimensional object we can explain its motion with some basic wave physics.

Our string rests on the nut and the bridge, setting up the ‘boundary conditions’ or physical constraints of our system. The string cannot move at the nut or bridge, we call these ‘fixed node points’.

By drawing all the possible vibration patterns than can occur on a string physically constrained in this way, we arrive at all the harmonic modes our string can exhibit.

We can give each of these a number, n, to make it easy to refer to them.

The first and most simple harmonic (n=1) is called the ‘Fundamental’. This is the dominant frequency and dictates the pitch of the note we hear.

Other higher order harmonics will also be present in the vibrating string, creating a complex waveform made up of many frequencies which our ear hears as a single note with a particular timbre or tonality.

By examining the wavelengths of each harmonic in turn we can arrive at an equation for the frequency of each harmonic which is as follows:

Here f is the frequency of any given harmonic, v is the speed of wave propagation in the string, and L is the length of the string.

v is dependent upon the tension and mass per unit length of the string (you can think of each as how high the string is tuned and the gauge of string respectively) and L is dependent upon the scale length of the instrument and if any notes are being fretted.
For open strings, L would be the scale length.

Let’s assume that we aren’t bending any strings, or changing the string length; we are just letting an open string ring out.

This renders the v/2L portion of our equation as unchanging, a constant value. This has important implications because it shows clearly that, under such conditions, the frequency of the any harmonic is simply an integer multiple of the Fundamental frequency.

This creates a mathematically beautiful and simple set of limitations on what frequencies can share a vibrating string at any given time:

If we had a Fundamental frequency (n=1) of 440Hz (an A note), then n=2 would be 880Hz which is the octave above our original note.
n=3 would give 1320Hz, an E note which is the major 5th of our original note.
And so on…

Integer multiples of the Fundamental frequency are incredibly musical and it’s all because of maths!