December 8, 2006
Vibrational Modes of a Kalimba Tine: Data!

Last week, I tried to explain about the vibrational modes of a kalimba tine which has been fixed at one end, and stated without proof that the theoretical frequency of the first overtone should be about 6.3 times the fundamental frequency.

A few months ago, I linked to some work that David Chapman did on the kalimba's overtones. The upshot of David's work was that the first overtone was coming in between 5.3 and 5.86 times the fundamental frequency. David's kalimba is a simple 5-note model that you might see on eBay, and it got me wondering: what about the Hugh Tracey kalimba? What do its overtones look like?

So as to study the tine's vibrations and not the box's vibrations, I used a TM Alto. I was careful to dampen the other tines I wasn't interested in. I recorded using SONAR and then exported the WAV file to AUDACITY where I formed the power spectrum every 0.2 s, so I could see how the strength of the different modes changed with time.

Below, you can see the power spectrum for the first 0.2 s of the low D note, and then from 0.2 to 0.4 s of the same note. The fundamental is the biggest peak. The ratty stuff to the left of that peak is just noise that looks funny due to the log frequency scale.

In the first graph, the "first overtone" is the next highest peak, and it has a frequency of 1958 Hz, or 6.68 times higher than the fundamental D at 293 Hz. But there is another note that just ain't supposed to be there: that slightly smaller peak at 586 Hz. This is exactly an octave above the fundamental. Kalimba tines aren't supposed to vibrate in octaves.

Also note there is no sign of the theoretical "second overtone" which should have a frequency 17.5 times higher than the fundamental.

Power spectrum of t = 0.0 s to 0.2 s of the low D tine's vibration.

 

Power spectrum of t = 0.2 s to 0.4 s of the low D tine's vibration.

Now, the big thrill comes when we look at how the power spectrum changes from the first 0.2 s to the next 0.2 s (i.e., the lower graph). The overtone at 1958 Hz has completely faded away, but the vibration at the octave is still hanging in there. Well, what in blazes is going on?