harmonics pt. 1: where are they?

We’ve looked at the physics behind harmonics, but before I go into nitty-gritty technical matters, let’s just see how you go about finding those harmonics.

At the 12th fret position, place your finger lightly on the string. Pluck the string. Excellent: You’ve got the harmonic at twice the frequency of (one octave higher than) the fundamental.
Same again at the point above the 7th fret. You’ve trebled the frequency (gone one octave and a fifth higher).
Now, take position over the 5th fret. Quadrupled the frequency (two octaves).
Position over the 4th fret. Quintupled (two octaves plus a major third).

Fine, fine, fine, you say. We’ve all done this before. In fact, those are the harmonics that many guitarists use to tune their instruments. Ho-hum. Tell me something new.

Let’s take a few steps back and look again at the 12th fret harmonic. What we’re doing here is dividing the string’s length in half and effectively doubling the frequency.
The 12th fret position corresponds to the node (stationary points) in the modes of vibration we’re interested in (think back to the discussion of the physics behind all this).
Now look again at dividing the string’s length into thirds. The 7th fret position corresponds to one of two nodes. There should be another node since we’re dividing the string into three equal parts.
The other node is at the 19th fret position. Try placing your finger there to get the same harmonic.
Similarly, when dividing the string into quarters, there should be three nodes. They are at the 5th, 12th and 24th fret positions.
Note that the 12th fret position will not sound at two octaves since it is shared with the one octave harmonic. (These ‘unusable’ nodes will be present whenever a non-prime number division of the string length is made.)
Dividing the string into fifths gives you four nodes. They are at the 4th, 9th, 16th, 28th fret positions.
I leave it as an exercise to find the other harmonics (divide string by 6th, 7th, 8th...), but at some point you will stop getting useful results.

Okay, why all this information? Since some nodes will be technically more accessible or comfortable under certain circumstances, as you begin to explore playing harmonics, knowledge of these alternative positions will become valuable. As I will describe in a future post, knowledge of node positions will also inform your choice of where to pick the string.
Additionally, each different way of playing a harmonic has a different tonal quality. (Compare, for example, the 7th harmonic you get at the 2 2/3 and the 10th fret positions.)

Some final comments: The above recipe for finding the nodes (by dividing the string up into equal parts) assumes a perfectly flexible, zero diameter string with mass evenly distributed along it. When dealing with real-world guitar strings, you will find that the nodes are not exactly in the expected positions, especially with the upper harmonics, and when dealing with heavier gauge strings (I leave it as an interesting exercise to figure out why wound and unwound strings behave, in some cases, contrary to this). In addition to the above factors, the guitar’s action and neck relief will make the correspondence of fret and node position inexact: Use you’re ears and don’t stubbornly expect fret-positions to correspond.