November 31, 2006
Vibrational Modes of a Kalimba Tine

Last Friday, we looked at the vibrational modes of a string. Before I move on to the vibrational modes of a kalimba tine, let's let the idea of a mode really sink in.

Most vibrations or ways of moving on a string just don't work. The string doesn't respond. If you try to force the string to vibrate that way, the vibration gets confused and sort of damps itself out, and the energy gets shifted into other vibrations that DO work. Those vibrations that do work end up having an integral number of half-wavelengths that fit onto the string, and these vibrations can live on for thousands of vibrational cycles before they finally lose their energy to the world (that's how we hear them). These vibrations that work are called vibrational modes. The vibrational mode with the lowest frequency is called the fundamental mode, and we call the frequency at which it vibrates the "fundamental tone," or just the "fundamental." The frequencies of the higher modes are called "overtones" or "harmonics." The fundamental tone determines which pitch we hear, and the details of the overtones determine the coloring of the note, and will make one guitar sound different from another, or will make a guitar sound different from an oboe.

For a vibrating string, we saw that the overtones harmonize with the fundamental, a very lucky accident. The kalimba's overtones follow a different logic. The diagram below shows three figures for the first and second vibrational modes of the kalimba tine. The first vibrational mode is more or less what you would expect - like a dog wagging its tail, but up and down instead of back and forth. You start by pushing the tine down, and then it springs up, and back down, etc.

The second mode has a part of the tine which starts out down and part which starts out up, and these trade places. In the case of the guitar string, the second mode has a vibrational frequency which is twice the fundamental frequency. However, in the case of the kalimba tine, the second mode (or the first overtone) has a vibrational frequency which is about 6.3 times higher than the fundamental tone. (On a string, the first overtone is 2 times higher in frequency, which is one octave - 4 times would be two octaves, and this is getting close to 8 times higher, or 3 octaves). Why is this? Well, the first mode vibrates in a way such that you are getting only about a quarter of a full wavelength of the vibration, and the second mode is getting close to a full wavelength of the vibration (that gives us a factor of 4, which is getting close to 6.3 - in physics or mathematics, we call that a "hand waving explanation", because it might give you the flavor of the answer, but it isn't the whole answer, and if the person doing the explaining waves their hands quickly while talking, it may be that nobody actually notices that the answer has not been adaquately explained).

By the way, the third vibrational mode of a kalimba tine will have a vibrational frequency that is 17.55 times higher, which is more than 4 octaves high.

So, numbers like 6.3 and 17.55 result in overtones which are not at all related to the western scale and which do not "harmonize" with the fundamental tone. I would argue that the western scale grew out of the harmonics one gets from strings and vibrating columns of air (flutes). Here's an idea: could African scales have grown out of the overtones one gets with the kalimba tines? Possibly, but I do not actually hear the overtones of the kalimba tines as distinct notes. Their anharmonic nature does not sound dissonant for three reasons: first, the harmonics are so high above the fundamental, and our ears can accept "dissonant" notes better if they are very far apart (ie, a major 7th sounds fine, and a flat 2nd sound very bad). Second, the overtones are so high that they might actually be out of our hearing range. Take the note A 440 Hz played on a kalimba tine; it's 2nd overtone is 17.55 times higher, or 7722 Hz. OK, it's not out of range for people with good hearing (20,000 Hz is the limit for the human ear), but older people won't hear it so well. And the upper kalimba notes will have 2nd overtones, which are out of hearing range. Third, these high frequency overtones die out much faster than the fundamental tone, so they flavor the "attack" of the note, and what is left later is a nice pure tone dominated by the fundamental.