November 24, 2006
Vibrational Modes of a String

Most anything physical can vibrate at certain predictable frequencies. The most visible example I can think of is a kid on a swing. If you make the swing longer, the time it takes you to complete a swinging cycle increases while the vibrational frequency DEcreases.

If we understand the way something vibrates, we can predict the pitch it will make. Before we look at the vibrational modes of a kalimba tine, let's look at the vibrational modes of a string, which is fixed at both ends (i.e., a guitar string, or any other type of string instrument).

Most sorts of vibrations will die out quickly, or will not be sustained by the string. However, certain types of vibrations will bounce off one fixed end of the string and travel to the other end, and the vibration will support itself in a way that makes it continue for a long time. It should make a pleasing tone. These sustainable vibrations are called vibrational modes. If you pluck a guitar string and look at what it does, the vibrations are much faster than what the eye can pick out as separate images of the string in motion (the eye refreshes at something like 1/30 of a second, or 30 Hz, while vibrations on a guitar string are 100's of a Hz), so we just see the blurred range of motion of the string.

In the figure below, we show the shape the string makes in its first vibrational mode, or the fundamental. The top figure shows the string pulled up; the tension of the string will pull the string down again, and then up, and down, and so on at a particular frequency, which we will hear as the pitch of the note. The bottom of the three "First Node" figures shows the extent of motion we see with our eyes in the blur of time.

The second mode makes an "S" shape. You can see this on a guitar string if you make the "first harmonic" by touching your finger tip to the string exactly at the halfway point (i.e., the 12th fret). In fact, the first overtone of the guitar string is the pitch at which the second mode vibrates. It turns out the wavelength of this vibration is half as long as the wavelength of the first mode, which means the frequency is twice as high. An INTERVAL in which the higher note has twice the frequency of the lower note is called an octave.

This is highly significant. When you play any note on a guitar, you don't have a pure tone, but you have a combination of the vibrational modes (at least if you wait 1/10 of a second for the transients to shift their energy into the vibration modes). You will have most of the energy in the fundamental or first mode. Then you'll have a bit of energy in the second vibrational mode, and some in the third, fourth, fifth, etc. BUT, as the second vibrational mode has a frequency which is one octave above the first vibrational mode's frequency, this means that they sound good together, and harmonize. In fact, the second mode, which is an overtone, is usually called a "harmonic", because it harmonizes with the fundamental.

The bottom three figures in the above diagram are the third vibrational mode, which has a vibrational frequency which is three times as high as the fundamental. So, for specifics, let's say we are plucking the A string on the guitar. The fundamental is 220 Hz (a Hz is one vibration per second). The second vibrational mode has a frequency of 440 Hz (also an A). The third vibrational mode has a frequency of 660 Hz. Now, without going into the WHY of it, I can tell you that the frequency of E, the 5th of A is 659.255 Hz. (If you don't know what the 5th is, see the last two weeks of Tues/Wed tips.)

In fact, the note which is 660 Hz is called the PERFECT FIFTH with respect to A 440 Hz, and the note which is 659.255 Hz is the "tempered fifth" (which will remain a mystery for now). In other words, the 5th which the guitar string's harmonic makes is more perfect than the 5th interval a piano would make, or a guitar can make with frets. Needless to say, the 5th an octave and a half above the fundamental harmonizes great with the other vibrational modes' frequencies. When you keep going up higher and higher vibrational modes, the modes' frequencies don't all harmonize so well, but it doesn't matter, as they have less and less energy in them.

So, the basic lesson here is that the overtones of a guitar string harmonize with the fundamental, which makes string instruments sound great. The same sort of thing happens with vibrating columns of air, as you have in organ pipes, marimba resonators, and flute-type instruments. Next week, we'll see that kalimba tines don't work quite the same way.