April 13, 2007
How the Kalimba Tine Vibrates: Explaining the Octave Overtone Which Shouldn't Be There

We are awaiting some more African tunings to complete our 15-part series on the tempered scale and all that, so now is a good time to show something I figured out recently.

Go back to the Friday Tips from 11/24/06, 11/31/06, and 12/08/06. This work looked at the vibrational modes of a kalimba tine, but considered the tine to be fixed at the bridge. Back then we plotted this spectrum:

This shows the fundamental tone as the big peak at 293 Hz (D). The first predicted overtone is at 1958 Hz, and this is more or less what is predicted. What isn't predicted is the very skinny peak (the third highest), around 586 Hz, or exactly one octave above the fundamental. And long after the 1958 Hz overtone has died down, this one octave overtone is still ringing. Where does it come from? I know now.

Consider the following diagram:

This shows the kalimba tine during the different phases of the vibration of its fundamental mode. The main difference between this and what we did on 11/31/06 is the way the tine is supported in a more realistic manner in this figure: you can see the bridge near the center, the "Z" bracket holding down the tine, and the wooden backstop with a half-circle cross section. The numbers to the right refer to the different phases of the vibration.

• Phase 0: this is the relaxed shape of the tine when it isn't vibrating. The tine's shape from phase 0 is duplicated in all the other phases for reference.
• Phase 1: the tip of the tine has been depressed, but this actually makes the part of the tine between the "z" bracket and the bridge bulge up.
• Phase 2: after letting go of the tine, the tip comes back up and the in between part goes back down, and the tine actually has the same shape as in the relaxed "Phase 0" case - except that the tine is only passing through on the way to Phase 3 less than 0.001 s later.
• Phase 3: the springy tine now is the opposite of "Phase 1", with the tip up and the in between part down. When it reaches this point, it will spring back the other way.
• Phase 4: after going as far out as it can in "Phase 3", the tine passes through the relaxed shape again on the way back to "Phase 1".

For a tine vibrating at 250 Hz (this would be close to a C), that means that this entire 4-phase process is repeated 250 times a second. The whole process takes 0.004 seconds (i.e., 250 times 0.004 seconds makes 1 second), and there is 0.001 seconds between each of these pictured phases.

Now, here's the trick: the tine is not firmly attached to the bridge, but it can slip over the thin metal rod on the bridge - left and right in these diagrams. You now have the crucial information for understanding the source of the 1-octave overtone in the kalimba spectrum. If you figure it out before next week, drop me a line and let me know. I'll send you a CD or something!

I feel a little wimpy putting out data that I don't understand. But I feel blessed to have found the explanation for this. That's the way science works - you have to put up with some partial understanding, but if your data doesn't match what you believe it should, you always put the true data out anyway - what we believe should be the case is only part of the story, and the deviation between the true data and our belief will point us to the true understanding.

On the other hand, when Einstein's theory of general relativity was experimentally confirmed after World War I, someone asked Einstein what he would have done if his theory had been disproved. He said he would feel very sorry for God, because the theory was correct.

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