Understanding Harmonics

Several meanings of 'harmonic' are used. One is a method of playing stringed instruments - playing the instrument while holding a finger against the string, but very lightly so that it doesn't touch the fretboard. Another meaning used has to do with the relationship between frequencies. Although these meanings are related, try to be sensitive to these different definitions.

Physics of the Plucked String

Pitch

The pitch of a plucked string is dependent on three factors: the length of the string, the tension in the string, and the mass density of the string (i.e. the mass divided by the string length). If you increase the tension (like when bending a string) or make the string shorter, the pitch increases. The frequency also goes up as the mass density decreases. This last property is why guitar strings get thinner as you move towards the higher strings, and why lighter gauge strings are 'slinky' and easier to bend.

Wave Theory

There are some basic concepts about waves that will make it easier to understand what follows. If you have experience with waves and vibrations already, you can probably skip this section.

First of all, there is an inverse relationship between frequency and wavelength. As frequencies increase, the wavelength becomes shorter. But if you double the wavelength, the frequency would be halved.

The simplest wave is a pure tone (a sine wave, or sinusoid). Any waveform can be broken up in to a series of sinusoids of different frequencies (although there might be an infinite number of frequencies). If we view the shape of a string just before it is released as a waveform, we can then consider a single sinusoidal component at a time. In a vibrating string, each sinusoidal waveform defines the boundaries that the string moves between. Places where the string does not vibrate are called nodes or zeros.

Below are several figures showing the pattern of a vibrating string. Think of these images as a series of snapshots taken of the vibrating string. Figure 1 shows a string vibrating where the length of the string is equal to a half-wavelength. The only nodes are at the ends of the string. Figure 2 shows the string vibrating with twice the frequency in Figure 1. The wavelength is now the same as the length of the string, and there is an additional node at the center of the string.

Figure 1: A vibrating string with nodes at the ends.

Figure 2: A string vibrating at a higher frequency, with a third node at the center.

If you could pluck the string in the shape of a sine wave, then you would only hear a pure tone and you would see a pattern like one of those above. But when you use a pick, you are creating a shape more like a triangle, where the three points are the nut (or a fret), the bridge, and the point where you are plucking the string. This shape adds other frequencies to the string which is why you normally don't see those patterns above. Those patterns are mixed in with many other patterns.

Natural Frequencies

A frequency is called a natural frequency if it can exist without any driving source. With our string, there is a physical interpretation of natural frequencies. A frequency is a natural frequency if its waveform has nodes that match up with the ends of the string. When you pluck a string, you are causing it vibrate at these natural frequencies. The lowest frequency at which this occurs is the fundamental - the pitch you hear.

Of course, the waves in the string can have other nodes, in addition to the nodes at the ends of the string. These waves come from the other natural frequencies, which are the upper harmonics (also referred to as the higher partials). When you pluck a string, you are hearing a combination of these natural frequencies, and not a single tone. Looking back, we can see that figures 1 and 2 show a string vibrating at two of its natural frequencies - the fundamental and the second harmonic respectively.

The Harmonic Series

The harmonic series is a mathematical definition, generally used when talking about frequencies. The harmonic series is important in musical applications because most instruments (including guitar) produce sounds that contain harmonic frequencies. The natural frequencies of the string mentioned above form a harmonic series.

A frequency is harmonic if it is an integer multiple of the fundamental frequency. The fundamental is the first harmonic (although it's generally referred to as the fundamental). The second harmonic is two times the frequency of the fundamental, the third harmonics is three times the fundamental, and so on. So with a fundamental of 100 Hz, the second harmonic is 200 Hz, the third is 300 Hz, the fourth is 400 Hz, etc.

Rather than working with numbers, you can think of the harmonic series in terms of musical intervals. The first harmonic is the fundamental. The second harmonic is an octave above. The third harmonic is an octave and a fifth above the fundamental. The fourth harmonic is two octaves above the fundamental. The fifth harmonic is a another major third higher, the sixth is another minor third higher, and the series continues.

Playing Harmonics

The natural frequencies of the string are determined by the ends of the string, which can't move. When you play a harmonic, you very gently hold you finger against the string at some other point. You have added a third point along the string that can't move. Now, the only frequencies that can exist must have nodes at this point, as well as at the ends of the string. That's really all there is to it. Let's look at some examples.

First of all, play a fretted string and notice that the portion of the string between your finger and the nut is not vibrating. You can touch it and it won't affect the sound. Now play a harmonic, say at the twelfth fret. Now the entire string is vibrating, and touching it will dampen the sound. When you fret a string, you are changing the amount of string being used, but when you play a harmonic, you are using the entire string, just imposing an additional constraint.

Now let's see how playing a harmonic at the twelfth fret is different than playing the open string (The twelfth fret is located at the midpoint of the string). The left side of Figure 3 shows the vibration patterns of a string for the lowest natural frequencies. The right side shows what remains when you play the harmonic. You are forcing the center of the string to remain stationary, which eliminates some of the original natural frequencies. But the frequencies where there is a node at the center of the string can still exist. Try playing an open string, and then gently move your finger to the string at the twelfth fret while the note is ringing. You stop some of the vibration, but not all of it. What remains is what you would hear by playing the harmonic. Try playing a harmonic, but lift your finger off the string while you pluck. You can control the balance of the harmonics this way.

Figure 3: Some of the frequencies in an open string (shown at the left) can be eliminated by playing a harmonic at the twelfth fret. The frequencies that remain (shown on the right) all have a node at the center of the string. A vibrating string will contain more frequencies than shown however.

You can view harmonics as a method of selecting the lowest natural frequency that can exist in the string. You can select the second harmonic by adding a node at the twelfth fret that eliminates the fundamental, but still allows the second harmonic to exist. You can get the third harmonic by moving the node to a different location - one that permits the third harmonic to sound, but that eliminates the fundamental and the second harmonic. This point is also known as the seventh fret, or one-third of the string length. To get the fourth harmonic, place the node at one-quarter of the string length (the fifth fret). You can continue by placing your finger at one-fifth of the string length, one-sixth, one-seventh, and so on, but they get weak rather quickly. In each successive step, the lowest frequency that can exist has an additional half-wavelength in the length of the string. Also, the number of the harmonic is equal to the reciprocal of the distance of the third node to the end of the string (i.e. if you put your finger at one-third the string length, the pitch you hear is the third harmonic).

If your guitar had a fretboard over the entire length of the string, harmonics wouldn't give you any extra notes. Since the fretboard can't continue the entire length of the string, harmonics gives you a way to get some high pitches that you don't have frets for. Harmonics also allow you have notes besides the fundamental of the open string to ringing.

Calculating Pitches

Perhaps the way to figure out what pitches the harmonics are is to use a tuner, but if you want to be able to do without a tuner, you can use the harmonic series. Forget the numbers though, and think in terms of intervals. Here's how I do it.

Begin with the pitch of the open string, and consider it as the base of the harmonic series (the fundamental). Playing a harmonic at the twelfth fret generates a pitch equivalent to the second harmonic in the open string, which is an octave above. The next harmonic, at the seventh fret, produces a pitch equal to the third harmonic in the open string - an octave and a fifth above the fundamental in the open string. A harmonic at the 5th fret is the fourth harmonic, two octaves above the open string. Using this process to select one of the harmonics in the open string may be why the playing technique is called a harmonic.

Other Directions

If you understand the physics of the plucked string, you can also make sense of some other aspects about playing guitar that shape your tone.

You've probably noticed that the sound of your guitar changes depending on where you pluck the string. Plucking close to the middle of the string, there's a nice round tone, but as you near the bridge, the sound becomes brighter. I believe this has to do with the shape of the string as you're plucking it. At the center of the string, you are creating a shape similar to the waveform of the fundamental (such as in Figure 1). But near the bridge, the shape of the string when you pluck is quite different, and this emphasizes the higher frequencies. In effect, you are changing the harmonic content in the string by selecting where to pluck it.

In an electric guitar, the pickups are looking at a small point along the string, and they create an electrical signal based on how the string vibrates at that point. So even if you are using identical pickups in two locations, it should be clear that they will produce different outputs since different parts of the string can have different behavior.