April 20, 2007
Why does the Hugh Tracey Kalimba have an Overtone at One Octave Above the Fundamental?

Remember last week? We looked at the spectrum from a single tine (we deadened all the other tines when we took this spectrum), and found that an overtone existed at exactly one octave above the fundamental. Now, a string has an overtone at one octave, but the theoretical overtones of a kalimba tine are anharmonic, which means they are not pretty, and nothing is supposed to be present at one octave.

OK, I'm disappointed - nobody got the answer. Of course, I forgot to post the figure which gave the big hint until a few days ago. [Wow, is that cool of Mark to say this. It was really MY fault that diagram didn't get posted right away--I'm Mark's webmaster.]

So here is how it works:

When the tip of the tine goes down, the "in between part" - in between the bridge and the "Z-bracket" - goes up. When it goes up, the tine actually slides just a bit towards the "Z-bracket" because the distance along the curved part of the tine in the "in between part" is longer than when it is not vibrating. In the diagram, I show a zoom-in on the part of the tine that touches the bridge, and I've drawn a little marker just over the bridge.

If you drew such a marker, when the tine tip goes down at instant 1, that marker moves towards the "z-bracket", or left. When the tine is in the relaxed position at instant 2, that marker moves back to the right At instant 3, the "in between part" is bent again and the marker moves back to the left. At instant 4 as the tine again relaxes, the marker moves back to the right.

This "sawing motion" - going left and right on top of the bridge - makes a vibration which we can hear. But at what frequency? The fundamental vibration takes all four of these instants to complete a single cycle, but this back and forth vibration completes two complete cycles (i.e., left - right) in the same time, so its vibrational frequency is exactly twice the frequency of the fundamental. Cool, no? This also explains why the amplitude of the octave overtone is linked to the amplitude of the fundamental.

Another hint: this sawing back and forth is also related to the buzzing that we sometimes hear on the kalimba. Hypothesis: a simple clamping bridge mechanism would not produce such buzzing, but neither would it produce the octave overtone, which actually sounds rather nice.

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