Resonant frequencies and spectrum, a comparison between different profiles

In my previous post I mentioned the difference in timbre between a cylindrical, conical and generically shaped didgeridoo. To quantify this there is a simple and accurate mathematical construction that enables one to calculate the resonant frequencies of a didgeridoo of a given shape. It is not too difficult to implement this as a computer program and then plot the resonant frequencies to obtain a graphical representation of my qualitative statements about form and sound. Below we can see the internal profiles of four different didgeridoos. The four shapes have been chosen such that the generic didgeridoo lies between the two extremes of conical and cylindrical, while the fourth “radical” profile has been included to show the effects of more creative changes to the internal profile. 

Figure 1. The profile of four different possible didgeridoos all tuned to have a fundamental tone at 60 Hz.

The acoustic impedance, plotted in the diagram below, provides information on the resonant frequencies of an instrument (corresponding to the peaks of the spectrum) and also on the backpressure (related to the relative magnitude of impedance at a given frequency).  Below we see the superimposed impedance spectra of a conical, generic, cylindrical and radical didgeridoo, all tuned to have a fundamental frequency of 60 Hz.

Figure 2. The impedance spectrum showing the resonance peaks for the four different instruments of Figure 1. The vertical lines correspond to the harmonics over the common fundamental frequency of 60 Hz (at 60, 120, 180… Hz). The vertical dashed line corresponds to a note that would be a musical interval of a 10th above the fundamental frequency, in this case it is at 150 Hz. 

Observing this figure we see clearly that the greatest spacing between resonances occurs for the cylindrical form and closely follows the odd harmonics of 60 Hz as one expects. The smallest spacing is for the conical profile with the second resonance being close to a musical interval of a 10th above the fundamental.  The generic shape has a spectrum that lies between these two extremes while the  radical profile has a less regular behaviour. This enables one to have a general feeling for the spectrum of an instrument given its internal profile. Clearly one could make strange expanding and contracting internal forms that can have greatly varied spectrums although for a completely generic form the quality of the resonances can be seriously degraded. For example one can see that for the radical profile didgeridoo the third resonance (second overtone or “toot”) at around 270 Hz has a lower impedance when compared to nearby resonances for the other three profiles. As a consequence the second toot will be more difficult to play on this instrument in comparison to the second toot on more conventional instruments.

The final figure shows a zoom on the impedance close to the fundamental frequency and here we can see that in general a conical instrument has lower backpressure than instruments with more cylindrical profiles. The radical instrument probably has a higher backpressure also as a consequence of the constriction in the first part of the profile. From figure 2. on the other hand, one sees that the conical instruments have a slightly higher back pressure on overtones than the other instruments (apart from the anomalous second toot of the radical instrument already discussed above).

Figure 3. A zoom in on the impedance spectrum of the four instruments around the fundamental frequency (at approx. 60 Hz). 

 One can also learn more about the actual timbre of the notes played from the impedance spectrum. The peaks also correspond to frequencies that are easier to accentuate while playing the drone and modifying the shape of the vocal cavity. The general timbre of the instrument, when playing the fundamental tone, is determined by the amount in which the various resonances of the instrument are excited by the harmonic overtone series above the drone. For example, when playing the fundamental tone on the radical instrument one should hear a strong accentuation at about 480 Hz as a consequence of the alignment between the instrument spectrum and the harmonic series at that frequency as is clear from figure 2.

An additional interesting observation is the first toot of the generic, conical and radical instruments, which is between a musical 10th and a musical 12th above the fundamental – a fact which any didgeridoo player with some experience on different instruments has surely noticed while playing.  

To obtain the actual audio spectrum of the instruments, meaning the frequency components of the actual played sound, one needs to combine the above results with the vibrations of the players lips. This will be the issue of an upcoming blog post.   

 

The sound of a Didgeridoo

The didgeridoo is a musical instrument that, for thousands of years, was known and played only by the Australian aboriginal people. I have already written much about this instrument on this blog, but since I have not written anything here for a long time I thought I should reintroduce the instrument and explain why I find it so fascinating.

The didgeridoo, which actually has various names coming from the different languages of the many aboriginal groups that share it, is a simple tube of wood or sometimes bamboo or hollowed pandanus palm trunk. In the case of wooden instruments they originate from termite hollowed eucalyptus trunks. They are instruments that are found in nature. The work of the artisan is in the discovery of appropriately hollowed trees, the cutting down of the tree, cleaning of the interior and sculpting of the exterior to follow the internal shape. Clearly the sound of every instrument is different although each artisan generally searches for instruments that have a characteristic timbre.

To play the didgeridoo one produces a basic, or fundamental, tone by sealing the mouthpiece with pursed lips that then vibrate as air flows through them thus producing standing waves that resonate with certain frequencies depending on the geometry of the tube.  This basic tone is predominantly of a low frequency between about 65 and 90 Hz, between a C and an F. Traditionally slightly higher and lower fundamental frequencies are also used though they are not as common.

My initial and continuing fascination with this instrument is related to the richness of its sound and the personal expressiveness available to every didgeridoo player. The supreme expressive musical instrument is arguably the human voice. The didgeridoo displays many elements of this expressiveness that interact with the intrinsic musical characteristics of the instrument being played. 

To understand better this claim it is useful to consider the acoustics of the instrument. Of course at a basic level the didgeridoo is simply a tube with resonant frequencies and playing it with vibrating lips will cause this air-column to vibrate with these frequencies which are determined by the physical characteristics of the instrument. Obviously the most important resonance is the the fundamental one, the basic note that the instrument plays and this is roughly dependent on the length. The actual timbre of the instrument is then determined by the series of higher resonances and they can vary dramatically. To understand this point in a little more detail it is useful to have a basic understanding of the harmonic series.. 

The harmonic series is most intuitively understood as the frequencies of vibration of a taut string of fixed length. The lowest frequency harmonic has frequency f. The subsequent harmonics then all have frequencies that are simply multiples of f, i.e. 2f, 3f, 4f, …. In musical terms, if the lowest frequency corresponds to the note A, then the higher frequencies in the harmonic series correspond to another A (an octave above the first), then E then another A (two octaves above the first), then in order C sharp, E, G, A,…

Getting back to the didgeridoo, a perfectly cylindrical didgeridoo has as resonances only the odd-harmonics, i.e. f (A), 3f(E), 5f(C sharp), 7f(G) ..etc.. whilst a conical didgeridoo has resonances corresponding to flattened harmonics of the cylinder (depending on mouthpiece radius and how conical it is). In a typical  traditional instrument one does not have any of the frequencies of the harmonic series above the fundamental frequency f. 

Up to this point there is nothing in particular that distinguishes the didgeridoo from other musical instruments. Actually it seems to have a very big disadvantage as it can produce a series of notes that have no particularly nice musical relationship to each other. Like a badly out of tune alphorn. 

Physically the crucial difference is easily observed. It is the mouthpiece! Of course in size it is just like the mouthpiece of say a trombone or a bass tuba, but there is a crucial difference. Brass instruments have mouthpieces that have the form of a small cup with a hole in the bottom, didgeridoo mouthpieces have almost parallel walls and although in some cases there is a slight narrowing in the first 10-20cm (typically yidaki have mouthpieces of this form) it is negligible compared to the extreme narrowing of brass instrument mouthpieces. The cup shape of the brass mouthpiece is necessary to align all of the harmonics of the instrument into a full harmonic series, without the mouthpiece the instrument does not have a nice harmonic series, and this is one of the almost magical properties of the cup (effectively a Helmholtz resonator). This cup with small hole also quite effectively isolates the player’s vocal cavity from the instrument while it is precisely the coupling of the player’s vocal cavity to the didgeridoo sound column that allows the player to produce incredible variations in the sound of the instrument. I like to think of the didgeridoo as a resonant extension of the vocal/oral cavity and the sounds produced are a consequence of the acoustic properties of the instrument plus vocal cavity of the player.

At a very simple level this coupling between vocal cavity and didgeridoo means that when you use your voice in the instrument it can easily be heard together with the notes of the didgeridoo. The voice thus is a second sound generator and when the sung note is in is not the same as the didgeridoo note one begins to create interesting effects akin to ring modulation used in in synthesisers. However the more interesting effects are related to the beats and vibrations that one can hear when making certain tongue movements. The simplest tongue movements produce simple sounds, tongue movements like ta, da, ki, ka together with the simplest tongue and cheeks movements used during the inbreath phase of playing. The more unique didgeridoo sounds are those produced by the turbulent and pulsing movement of air inside the mouth, a little like waves that bounce around inside a breakwater compared to waves that travel freely in the sea. These produce all the additional pulsations and embellishments of the basic sound and the true masters of control of these effects are the aboriginal Australian custodians of this instrument. For them the basic tongue movements are more complex than those mentioned above, coming from their spoken language, often incorporating also the tongue movement for breathing. These movements create often drastic changes of pressure inside the players mouth which in turn modulate the sound of the didgeridoo producing the complex characteristic sound of the instrument.  

In future blog posts I would like to return to this discussion and carefully analyse a couple of the simplest traditional sounds, and also the wobble – a contemporary technique nevertheless originating from aboriginal players Mark Atkins and Alan Dargin – as examples of the above described effects.  

Gravitational Waves

The announced detection of a gravitational wave signal arriving from the inspiral of two black holes resulting in their inglobation into a final more massive black hole has now travelled around the world.

We all have an enormous practical experience in the detection of waves. Actually, almost all of the information that we receive about the surrounding world arrives to us in the form of waves: Sight – our eyes are extremely sensitive detectors of electromagnetic waves; Sound – our ears can detect a large range of sound waves; Touch – our body detects certain frequencies of electromagnetic radiation coming from the sun (it warms our skin).

Each of these waves is detected as a function of the way that it travels through an elastic medium. The medium of electromagnetic waves and also of gravitational waves is the vacuum. To observe the universe more completely science has developed detectors to extend the range of seeing and hearing well beyond the range accessible by our body, and this has enabled us to observe a huge variety of events in the universe.  These detectors have extended our range of vision in electromagnetic waves well beyond the visual ight of our everyday experience – and this has led our investigation of the properties of matter and spacetime deep into the microscopic and cosmological realms. As a consequence of this extended vision the theory of quantum mechanics was developed and refined.

Gravitational waves are a simple prediction of Einstein’s general theory of relativity, a theory that celebrated its 100th anniversary in November of 2015, and which continues to be confirmed as a spectacularly successful theory of gravity with no close contenders. The construction of gravitational wave detectors began in the early 1970’s but up until last year they were never sensitive enough to detect the gravitational waves that we expected should arrive from cataclysmic events in remote regions of the universe. The construction of gravitational wave detectors is extremely demanding due to the incredible weakness of the gravitational field and it is only with the dedication of experimental physicists continually refining the detectors that we have finally arrived at the actual observation of at least one, and probably various other, gravitational waves.

The magnitude of this discovery is completely out of reach or our everyday experience. The difference in strength between the gravitational force and the electromagnetic force is on the order of forty zeroes. Forty zeroes. This number is really beyond imagination. If we take a huge number, for instance the distance to the big bang is on the order of twenty zeroes in seconds, then it is still excruciatingly small compared to a number with forty zeroes.

What will the future bring now that we have opened a new window of perception on our universe? The most evident lessons are related to black hole physics. The observation of this gravitational wave is the most direct evidence of the existence of black hole like objects almost all the way to their horizon – the famous point of no return and the source of all the subtleties of black hole physics.

Black holes, like gravitational waves, we’re first discovered as solutions to Einstein’s equations almost 100 years ago. It is beautiful to ponder that these two predictions are finally coming into view at the level of observation and are amongst the more profound confirmations of Einstein’s theory and at the same time the most likely to lead to the further evolution and extension of this theory into the quantum world.