I just returned from a speaking engagement in Australia and learned quite a few things about down-under. Koalas are not bears; they are marsupials, so it’s wrong to call them Koala bears even though everyone I know, does. In certain, less populated areas it’s a smart idea to shake out your shoes before putting them on in the morning in case some eight-legged Australian citizen has crawled in there during the night. And I learned some things about the acoustics of the didgeridoo (also spelled didjeridu, and I wouldn’t be surprised to see it spelled didgeridoo as well).
The didgeridoo is an Australian aborigine musical instrument dating back to between 1000 to 1500 year ago. We’ve all heard it even though we didn’t know it was a didgeridoo- it’s that constant buzzing sound instrument that sounds a bit like a cicada heard on a warm summer evening.
Didgeridoos are made from either large branches of trees or the main trunk of smaller trees where the dead inner pulp has been eaten away by termites and other small insects. Interestingly enough, the tree can remain alive despite being hollowed out since the resin from the healthy part of the tree acts as a deterrent for termites- only the dead inner core is hollow. The result is a tube of roughly circular diameter or even a conical shape and numerous “side branch” resonators- small tunnels into the wood carved out by the termites. The rough interior, coupled with the numerous side-branch tunnels creates a tube that when blown, gives the didgeridoo its unique sound.
Having said all that- I still can’t get a single sound out of my didgeridoo. Maybe the battery needs changing. And yes, I bought a didgeridoo for my son- I sent it Fedex since I had no intention of bringing it with me on the plane over 14 time zones and an international date line.
The didgeridoo, in its simplest form is a quarter wavelength resonator. It is “open” at the bottom end and “closed” at the other end where the mouth and vibrating lips blow in to the tube. In this condition, its resonant frequency is given by the formula F = (2k-1)v/4L or more simply as v/4L where v is the speed of sound (340 meters/second) and L is the length of the tube in meters. The one that I bought is about 1.25 meters long so its resonant frequency can be calculated as F= 340/(4 x 1.25) = 68 Hz. This roughly is the key of C (well, actually about half way between C and C#). There were other didgeridoos for sale that had keys of E# and some other non-Western frequencies. This is both a function of the aboriginal peoples not being restricted to the western European 12-note system, as well as the reality that hollow branches and trunks were used and they had the variable lengths that nature had ascribed to them.
With current digital recording techniques it would not have mattered in any event- an E# tuned didgeridoo can easily be transposed into any key using a couple of key strokes on a laptop, but it would be nice to have a ”conventional” key for live music so that it would not clash with other western musical instruments such as a guitar or piano. The advantage of having an instrument play in the key of C is that we don’t need to worry about sharps and flats. (The same would be true of A minor, but that would be another blog).
In addition to my didgeridoo being a quarter wavelength musical instrument with the fundamental being 68 Hz, there are other odd numbered multiples at 3 x 68, 5 x 68, 7 x 68, and so on. This harmonic structure is common to all quarter wavelength musical instruments such as the trumpet, and the clarinet (in the lower register), but interestingly not the tuba.
The tuba looks like a very large trumpet- open at the bell end and closed at the mouth piece end, but that’s where the similarity ends. Unlike the trumpet, the tuba has a uniformly increasing cross sectional area; one can even place a ruler on the outer side of the tuba to verify this. The tuba is “conical” whereas the trumpet has an “exponential” horn. Although this may seem to be a minor bit of trivia, the acoustics change quite a bit. The saxophone and the oboe look like a clarinet, with a reed on the “closed” end and “open” on the other end but that’s where the similarities end. The saxophone and the oboe, like the tuba, are conical with a uniformly flaring sound bore. Despite the fact that they are both “closed” at one end and “open” at the other end, conical instruments function as one half wavelength resonators.
Some didgeridoos are conical and have a uniform flare from the mouth end to the open end. And these instruments function as half wavelength resonators with the relevant formula being F = kv/2L or more simply just F = v/2L. Since the bottom of the formula only has a “2” rather than a “4” (as in the quarter wavelength model), the fundamental resonance is double that of a uniform diameter cylinder such as a clarinet. That same 1.25 meter long didgeridoo now has a resonant frequency of 2 x 68 Hz or 136 Hz. And because it is a one half wavelength resonator, it would have integer (as opposed to odd numbered) harmonics at 2 x 136 Hz, 3 x 136 Hz, 4 x 136 Hz, and so on.
To easily demonstrate the difference between a half wavelength and a quarter wavelength resonator, get a drinking straw. With both ends open blow across the top- then do the same thing with the bottom pinched off. In this second pinched case, the straw is a quarter wavelength resonator and its resonant frequency will be one half, or an octave lower, than the straw when blown with both ends open (a one half wavelength resonator). It’s more difficult to demonstrate the reasons why a conical tube can function as a one half wavelength resonator but it’s worthwhile thinking about our unoccluded ear canal which is conical and closed at the ear drum and open at the lateral meatus- the literature states that this is a quarter wavelength resonator, but there is evidence that it also resonates as a one half wavelength resonance with a natural frequency, not of just 2700 Hz, but of 2 x 2700 Hz or 5400 Hz. Perhaps what we have ascribed to the “concha resonance” at 5400 Hz is merely the first resonance of the ear canal functioning in its conical mode?
In part 2 of this blog series, the effects of the termite induced side branch resonators will be discussed along with other “side branch resonators” we see in everyday life, ranging from the articulation of nasal sounds in speech to how Dyson fans are made quieter. Also, does it matter if there is a curve in the Didgeridoo?