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 emi@windworld.com.    

A few years ago Phil Dadson and I put out a book on plosive aerophones and how to make them. Plosive aerophones is the generic term for things like slap tubes, udu drums, and other instruments in which a body of air is excited by percussion or other sudden  agitation. Sometime after the book was printed, I had a sudden realization: we forgot to include POP GUNS!  Too bad, because pop guns do qualify as plosives, and their sound is one of my favorites. 

As it happens, I did include a section on pop guns, in an earlier book, Funny Noises for the Connoisseur. So if you want the full story of popguns, that’s where to look.  But for a quick start, a short primer on the topic follows here.

Here’s a sketch showing the essentials of a pop gun.

The idea is to have a tube with a cork lodged in one end. From the other end the player pushes a snug-fitting plunger forward. This compresses the air and forces the cork to pop out, producing the popping sound.

Notice that two corks are needed, the popping cork and the plunger cork. Both have the truncated cone shape, with the large end a little larger than the inside tube diameter, and the small end a little smaller. The plunger cork serves as the piston-plug on the end of the plunger, fitting the inside of the tube closely so as to push the air through. To get that fit, stick the cork in one end of the tube as far as it will go. Cut off the portion that is still sticking out and discard. Sand the cork remaining in the tube flush with the tube end, and then remove it from the tube (push it out from the other side by poking through with a narrow stick or dowel).

At the end where the popping cork is to be inserted, use a rat tail file or similar tool to bevel the tube edges inward. Even if you do a somewhat sloppy job of this, it seems to help the popping effect.

The string that you can see in the drawing prevents the popping cork from flying away and getting lost. It also lets you pull the cork back into popping position simply by pulling the plunger back. The string length should be a little less than the tube length.

The drawing should tell you everything else you need to know. And to give you an idea of the sound you’re after, here’s recording of pop music.

Audio clip: Adobe Flash Player (version 9 or above) is required to play this audio clip. Download the latest version here. You also need to have JavaScript enabled in your browser.

A pop gun can produce a variety of tones, depending on how firmly the popping cork is seated, how rapidly you move the plunger, and how far you draw the plunger back before starting the forward thrust.

Another sound I get a kick out of, closely related to pop gun pops, is suction pops (think of the sound of a cork being pulled out of a bottle). Maybe in another post I’ll talk more about them.  Suction pop devices are less variable and more predictable in their sound, and it’s possible to tune suction poppers to definite pitches.