Guitars are harmonic oscillators

The dots on the fretboard correspond to different harmonic positions.

If you play guitar, you’re probably somewhat familiar with natural harmonics. By lightly placing a finger on specific positions of a string (oftentimes directly above or near certain frets), clear bell-like tones can be produced when the string is plucked. Most guitars even have special markers on the fretboard to indicate these harmonic positions. From Niccolò Paganini to Billy Gibbons of ZZ Top, artists have been using harmonics to spice up their guitar playing for generations. How are these sounds created? What is the science behind guitar harmonics? Keep reading to find out!

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The science behind guitar harmonics

First is the concept of a harmonic oscillator in physics. This term simply describes something that undergoes periodic motion, in which the restoring force is proportional to the displacement. A basic example is a ball hanging from a spring. The further the ball is pulled down, the further the spring is stretched, and the greater the spring tension will be. If the ball is let go, it will spring back to its original position; but not before bouncing up and down several times (oscillating) until eventually coming to a stop at its equilibrium position (thus technically a ‘damped’ harmonic oscillator because the amplitude of vibration is decreasing over time).

Harmonic oscillators are common in nature, from vibrating molecules to dynamics of star systems. In this case we have a vibrating guitar string. Imagine a string fixed at both ends (the bridge on one end, and the nut on the other) under tension. If I pull on the string, it moves from its equilibrium position. When I release the string, it experiences a restoring force due to the tension in the string, and oscillates quickly back and forth at a certain frequency. The frequency at which the string vibrates corresponds to a musical tone. For example, plucking the open ‘A’ string on a guitar will cause it to vibrate at a frequency of 110 Hz (cycles per second) which corresponds to an A2 note. In the case of an acoustic guitar, the string vibrations are transferred to the wooden guitar body, which in turn amplifies the sound as the flat surfaces push air molecules around. This causes pressure waves in the air which are detected by our ears and interpreted by our brains as an A2 note.

Next up is understanding the different patterns in which a string can oscillate. When a string vibrates back and forth in a single large arc, this is called the fundamental mode. However, it can also vibrate in more complex ways called higher-order modes of oscillation. As illustrated in the figure below, the string can oscillate in such a way that standing waves are created with a number of nodes (points on the string where displacement is zero) and anti-nodes (points on the string where displacement is maximum). When a guitar player places their finger on a node, they dampen other potential frequencies and force the string to vibrate in a specific pattern. Thus depending on where a finger is placed, the string will divide up into different ‘sections’ and be coerced into higher-order modes:

user:Mjchael, Flageolette, CC BY-SA 2.5

The order of the mode corresponds to the number of ‘sections’ the vibrating string divides into, i.e. the length fraction. The table below gives these values for various modes, along with the guitar frets where the modes occur:

Mode Length fraction Fret Interval
1 1 open unison
2 12 12 1st octave
3 13 7 1st octave + perfect fifth
4 14 5 2nd octave
5 15 3.9 2nd octave + major third
6 16 3.2 2nd octave + perfect fifth
7 17 2.7 2nd octave + minor seventh
8 18 2.3 3rd octave

Linking science and music

Now you’ve added a bit of the science behind guitar harmonics to your musical knowledge, so lets put that information to good use! For example, the next time you’re rocking Top Jimmy by Van Halen and reaching for those 7th fret harmonics, you’ll know that your strings are vibrating with two nodes and three anti-nodes. You should also know that vibrational modes correspond to musical intervals of a given string, relative to the fundamental frequency. For example, the 7th fret harmonic on a guitar ‘A’ string corresponds to an E note, which is a perfect fifth. Taking this idea further: if the first seven harmonics on the A string are played in succession, a dominant 7th chord is implied. This is true for any string on the guitar. Playing the first seven harmonics outlines a D7 chord on the D string, an E7 chord on the E string, and so forth. Putting these together in the correct order, we end up with the building blocks of blues music. Below is a video showing what I mean, played on an Epiphone ES-339 guitar with Ernie Ball strings:

Once again we’ve found another deep connection between physics and music. Seems as if pioneering blues musicians were tapping directly into fundamental principles of nature to create their unique sound. Amazing!

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