A couple of times people have asked me what the mathematical relationship is between length and pitch in kalimba tines and other sorts of plucked rods held at one end. My answer has always been “I don’t know.” I’ve never run across this information myself, and I’ve often wondered.
I wasn’t able to intuit my way to a physical analysis of the situation which would point the way to a formula, so I recently tested the question empirically. That’s a fancy way of saying that I spent a few minutes in my shop setting up rods of different lengths and noting what pitches they produced, looking for a consistent relationship between length and pitch.
Here, in brief, is what I found:
Given a rod or tine of length L, to make it produce a tone an octave higher, it should be shortened to approx .685L. For an octave below, lengthen to approx 1.460L.
From this, a bit of math tells us that the semitone-up would be at 0.9699L, and the semitone down would be 1.0320L. You can arrive at other intervals in 12-tone equal temperament by repeatedly applying the semitone-up or semitone-down factor.
BUT WAIT! Please don’t take these observations as gospel, or at least please do read the following notes before working with them.
First keep in mind that these numbers only work for uniform rods or tines. They won’t work if the tines are irregularly shaped – say, thicker at the far end — or if you’re comparing two tines of different thickness or diameter. Also, the tines shouldn’t be significantly bent over at the end.
Second, I should mention that my testing set-up and observations weren’t all that precise, and my results weren’t perfectly consistent. The octave-up factor ranged from about .67 to .69. (I notice that these results are fairly close to the the square root of 1/2 at .7071, which sugggests the possibility of an inverse-squared relationship between length and pitch, but I’m just speculating here.)
I’m wondering if anyone reading this has performed similar experiments. Or, whether have any readers come across or come up with a theoretical formula for this relationship. This news page isn’t currently set up for reader comments are present, so if you can add to the discussion, please email me at email@example.com.