State of the Stomp: Harmonic Convergence, Pt. 1

We hear the words fundamental, harmonics, distortion and total harmonic distortion (THD) quite often when discussing music and sound. Let’s consider these from a couple of angles and see how they relate to guitar tone. We’ll start with some basic terminology and take a look at how guitars and guitar strings produce sound. Once we grasp those things, it’s easier to understand the ways that stompboxes and amps generate distortion—our topic in Pt. 2.

One of the first things guitar players learn (after “Smoke on the Water”) is how to play a harmonic at the 12th fret. (The beginning to “Roundabout” is a classic example of the musical use of harmonics.) The fundamental is the lowest frequency that a fretted or open string produces. The second harmonic is twice the frequency of the fundamental—an octave up. The third harmonic is three times the frequency of the fundamental—an octave plus a fifth.

Let’s Get Physical
Understanding this requires a detour into the physics of vibrating strings. When you pluck a string, its vibration creates a wide range of frequencies. But soon after the note is struck, all that remain are the fundamental and harmonic frequencies. When you pluck the string while touching the 12th-fret harmonic, you are effectively “cutting the string in half.” This eliminates the fundamental and any odd-numbered harmonics.

A node is a point on a string at which the string doesn’t move. The guitar has nodes at the nut and the bridge (see image), much like a playground jump rope has nodes where the kids hold the rope (ignoring the fact that the kids have to move a bit as they add energy to the rope). A 12th-fret harmonic forces a node in the middle of the string—like stepping on the middle of the jump rope.

Octaves are a way of describing factors of two. The fundamental frequency of the low E on a guitar is about 82 Hz. The low E on a bass guitar is an octave lower, or around 41 Hz. Notes on the 12th fret are an octave above the open string—twice the frequency. The harmonic that is twice the frequency of the fundamental is called the second harmonic. (That makes the fundamental the first harmonic, though you rarely hear it referred to that way—we usually just say fundamental.)

Higher and Higher
What about harmonics on other frets? The 7th fret is one-third the distance from the nut to the bridge. This makes a harmonic struck above this fret three times the frequency of the fundamental. (Extra credit for finding the other fret that creates the same harmonic as the 7th fret.) The 5th fret is one-quarter the distance from the nut to the bridge, and a harmonic struck there is two octaves above the open string.

You’ve probably noticed that it gets quite a bit harder to produce harmonics below the 5th fret. That’s because it becomes increasingly difficult for the string to vibrate in smaller sections. (This is where a real string and an idealized physical model start to diverge. A theoretical “ideal string” has an unlimited number of harmonics.)

To understand why a piano string sounds different from a nylon guitar string, or why your tone changes when you switch between 1st strings gauged .009 and .011, we need a few more facts about string physics.

The Tension Mounts
When we tighten a string with a tuner, pull up on the whammy bar, or bend a note, we increase string tension. Pitch increases as the square root of the tension. Another factor for an ideal string is the linear density, or mass per unit of length. A heavy string at a given tension vibrates at a lower frequency than a lighter string at the same tension. Conversely, a heavy string tuned to a particular pitch is under greater tension than a thinner string tuned to the same pitch. (You may have noticed that your truss rod and vibrato springs needed tightening when the ’80s ended, you ditched the spandex, and you switched from .009s to .011s.)

The higher the tension, the brighter the string sounds. If we want high tension and low frequency, then we need a long length, which is why a grand piano has such long strings. The difference in the tone of strings is in the balance of the fundamental and harmonics. Brighter-sounding strings have louder harmonics than mellow-sounding strings.

The way the fundamental and harmonics decay over time also determines the character of a string. Energy gets put in the string when we pluck it, and that energy decays as it vibrates the air and the instrument body. The materials of the string and instrument body determine the rate at which varying frequencies decay.

A guitar’s body has various resonances, depending on its shape, wood type, and finish. When a string frequency matches one of the body’s resonant frequencies, the body vibrates and sucks energy out of the string. This makes the note decay more quickly as it absorbs the string’s energy. (The body’s resonance actually decreases the string’s sustain. But if we’re playing through an amplifier, the guitar’s body absorbs energy from the sound waves produced by the amp, and sustain increases as energy is transferred back to the string.)