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{reprinted with permission}
{reprinted with permission}
The following is a reprint of several articles by Albert Prak of the Netherlands that were posted to the Djembe-L List during July 1997.
TABLE OF CONTENTS
Starting with the tone and slap analysis
An interesting experiment: visualising the membrane resonances
Still searching the origin of the vibrations
The master drummers' slaps and tones
How to hit that djembe (a physicist's view)
INTRODUCTION
The `slap' thread of last week (July 1997) focused on the physics of the djembe. I posted some ideas to the list, ideas based on common physical sense. I felt that this topic deserved a more serious approach, and got a little captivated.
Friday I printed a copy of fellow djembe-lister Ben Sibson's web page The Relationship of Design to Sound Production in the Djembe Drums of Log Drums Inc. and took it home. I read Ben's page with great interest. I was stunned by how far he got with the analysis. It gave me new insights in the physics of the sound produced by the magic drum. I was impressed by the relations that Ben gathered in literature, by experience of craftsman and by own experiment.
The motivation of this kind of analysis is pointed out perfectly on Ben's webpage. I can imagine that many listers prefer to play the djembe instead of analysing it. I doubted if I should send this mail to the list since many people will not be interested, or are principally against this physics stuff in relation to the magic drum. But I am sure there are some listers who ARE interested.
A friend of mine has a lot of experience in building drums. He made a few hundreds of them. Once he said that he has an eye for good sounding djembes. The djembes he sees don't need a skin, they don't need to be played. He just looks at them. He knows what skin to put on it to get the sounds he wants. And once the drum is finished he appears to be right most of the time. But for him its very difficult to explain what he looks at when he observes a djembe and decides what skin to put on it. It's a feeling, he says. Hocus pocus? Can his intuition be grasped by physical laws?
To answer this question (if possible at all) we first have to know the origin of the resonances. Do they come from the air in the bowl, the skin, the wood of the shell, the room you play in, the fingers, the toes, the mind ......?
ORIGIN OF THE BASS
Ben's approach of the bass being a result of the Helmholtz / tube cavity resonator (air vibrating in the shell) was new to me. The Helmholtz resonator never crossed the path of my research. My first reaction was "but what about the skin, doesn't the skin have an effect on the sound?" Troke's formula for the frequency, given on Ben's web-page, only shows parameters of the shell. Doesn't the skin play a role? What happens to the frequency if we change the rope tension?? No change is predicted by Troke's formula. Let's elaborate somewhat on the physics. As Ben said, there is little known about the skin / air cavity interaction.
The Helmholtz resonator is, as the word says, a resonator. It needs to be brought in vibration. It needs an exciter. And the exciter is the skin. But the skin in turn is a resonator as well, though much less powerful than the Helmholtz resonator at low frequencies. We hardly hear the skin at these low frequencies, the sound comes from down the pipe, not from the top of the skin. The two resonators interact (coupled oscillators is the keyword). The loudest and purest bass is obtained if the resonance frequency of the skin matches the resonance frequency of Helmholtz resonator (matched impedance in physical terms). If the resonance frequencies of the two resonators are far away from each other, the weaker one (weaker in terms of energy involved) will obey the stronger one. The weak one is put into a forced vibration at the resonance frequency of the strong one. From the wave visualisation experiment (see below) it appeared that the frequencies are indeed far away from each other, the Helmholtz resonance being much lower.
As Ben already pointed out, the Helmholtz resonator produces the bass, not the membrane. I still had difficulties to believe that the rope tension doesn't have an effect on the bass frequency. So I made up for the decisive experiment. I measured the bass frequency (mic at the end of the pipe of course): 77.6 Hz. I untied 5 knots. Slaps sounded awful. Same experiment. Guess what? 76.5 Hz. The rope tension did indeed have little effect on the the bass frequency. (PS I just read that Kent Multer has the same experience).
STARTING WITH THE TONE AND SLAP ANALYSIS
I found values for the harmonics of a tensioned membrane in a physics textbook. The frequencies are approximately given by the series: 1, 2.33, 3.66, 5.00, 6.33, 7.66, 9.00 ..... These are the harmonics whose vibrations are characterised by nodal circles, ie concentric circles around the centre of the skin with zero amplitude. The skin only vibrates in between the nodal circles, the vibration being the strongest just halfway. Every harmonic adds another nodal circle. There are also harmonics with nodal diameters (straight node lines, from one edge via the centre to the opposite edge) In fact the majority of resonances have nodal diameters, but they seem to be of less importance as the textbook said. Note that the situation for a string (guitar) is different. In that case the harmonics are given by the series 1, 2, 3, 4, 5, .....
I couldn't wait to put these numbers against Ben's measured frequencies as posted to the list lately: 60, 280, 380, 570, 720, 900, 1200, 1350. Well, the 60 Hz is the Helmholtz frequency, clear. I didn't succeed to fit the others to the membrane harmonics 1, 2.33, 3.66, 5.00 etc. I wondered what is going on here. I decided to set up an experiment. I wanted to make the membrane harmonics visible, face them with my own eyes.
AN INTERESTING EXPERIMENT: VISUALISING THE MEMBRANE RESONANCES
During my PhD I did a lot of measurements on vibrating structures. So I know how to ask questions to the tensioned skin in such a way that it will betray its secrets. A good experiment is to excite the membrane with a sine wave, instead of with the hands as usual. When using a sine wave, the membrane always responds with exactly the frequency of the sine wave (linear physics, for the insiders). No other frequencies are present. The closer the sine wave is to a resonance, the larger the amplitude of the skin. When tuning exactly to a resonance, the membrane gets crazy: it vibrates very strong.
The experiment is not very difficult, you don't need advanced scientific equipment. I bet a lot of listers have the equipment available. You need a djembe, an audio set, a computer with basic sound facilities, a level and some dry sand (you're lucky if you live in the desert). Put the loudspeaker with it's back on the floor. Put the djembe right on top of the loudspeaker so that the sound goes directly through the pipe to the skin. Use the computer to generate the sine wave. Put a thin layer of fine, dry sand on top of the skin (sieve the larger grains out with a tea strainer). The layer should be so thin that you still can see the skin underneath. Now play around with the mouse (or use a real keyboard if you have one) to generate sine waves of different frequencies. If you hit a resonance the sand starts hopping, jumping. Shift one single note and the sand will freeze immediately: you've tuned out of resonance. Experiment with the loudspeaker volume. I was fascinated by the sand grains dancing on the skin. Look at that black one, where does it go?
The most fascinating thing of the experiment has yet to come. Where do the sand grains go? Keep the keyboard pressed at resonance for a minute or so, and you'll see a miracle happen. The sand arranges in patterns! If the skin is positioned exactly horizontal, so that the sand grains have no gravitational preference (use a level) the sand goes to places of relative small amplitude. It arranges in circles, not perfect circles because this is a goat skin, a product of nature, not a mathematical ideal membrane. You'll find no circles for the lowest membrane resonance, one circle for the first harmonic, two for the second etc. I was surprised by the success of the experiment!! The third started to be irregular, but there was a pattern, definitely. The circles are the node lines, lines where the skin is perfectly silent. You need little imagination to see how the skin vibrates once you have seen these node lines. I took some pictures of the sand patterns.
I measured the following:
resonance nodal diameter of frequency
number circles circles in cm in Hz
1 0 - 60 .. 90 peak at 75
2 0 - 230 .. 280 peak at 255
3 0 28 (close to rim) 355
4 1 11 .. 12 844
5 2 7 .. 9 and 19 1335
6 3 irregular 1785
In the low range (up to 355 Hz), the membrane responded to most frequencies, though the response was stronger on some broad-band resonances at 75, 255 and 355 Hz. Above 355 the skin is just frozen, apart from very narrow frequency bands: the higher harmonics are very well defined.
I put these values against the theoretical ones (1, 2.33, 3.66, 5.00) It was not difficult to identify the membrane resonances. The match is within a few percent, almost perfect: fundamental at 355, first harmonic 844, second harmonic 1335, third at 1785. I was surprised because I expected the goat skin to behave far from an ideal membrane. The 75 Hz peak clearly was the Helmholtz resonance, which got excited by the loudspeaker as well. But what about the 255 Hz peak ? Could this be a harmonic of the Helmholtz resonance? And which resonances are produced by slaps and tones of my own djembe / hands. Do they produce any other resonances than the Helmholtz and the Membrane resonances discussed so far (as Ben's measurement data suggest)? And if so, what is their origin. Questions, questions, questions.
Now I wanted to analyse tones and slaps and look if I could find the membrane resonances (which I now knew very accurately) in their spectra. What I needed now was either detailed info of Ben's spectra (which I didn't have available) or even better slap/tone analysis of my own djembe. At this point I got stuck because I don't have a good mic available to record good slaps and tones. They sound terrible when played back.
STILL SEARCHING THE ORIGIN OF THE VIBRATIONS
Well, the first post 'physics of djembe sound' (Monday 14 july 1997) seems read by quite a number of listers. I was hapilly surprised since it had quite a physical viewpoint. Among the reactions were numerous suggestions for measurements and inquiries for details of the experiment and the theory etc. They have encouraged me to do a next step in my search for the sectrets of tones and slaps.
In the mentioned post, an experiment was described to make the djembe's skin vibrations visible by means of sand that arranges in the form of nodal circles. The measured resonance frequencies of the djembe skin were 355, 844, 1335 and 1785 Hz. The frequencies corresponded very well to theoretical values. Were these the frequencies that produced the tones and slaps? Measurement data from other sources suggested that there was more....
In order to find out the origin of the vibrations, I did a few experiments. I recorded some tones and slaps with that borrowed B&K mic of all mics (thanks Micheal Pedersen). I spent an hour or so to make the room sound dead, and to throw every rattling thing out. I also recorded some very soft touches, since they suffer less from non-linearities (I didn't know if any non-linearities show up during 'normal' playing). I digitised the samples using an SB AWE32 soundcard and calculated the Fourier spectra in Mathcad. The strongest frequencies were at 350, 502, 705, 888, 1060, 1240, 1410, 1580 and 1745. Where I earlier found only 4 frequencies, I now found a whole bunch of them.
An experiment that might help me identify the frequencies was to release the skin tension a bit. The frequencies that originate from the skin should all decrease. I untied two knots. And they all shifted by 2,5% ... 4%, the 350 Hz peak showing the largest shift, and the high frequencies showing the lowest shift.
The conclusion that the skin resonances caused practically all sounds produced by the djembe (except for the bas of course, which comes from a resonance of the air in the shell, the Helmhotz resonator) was unavoidable.
In some way, the wave-visualisation-experiment had only exposed the vibration modes related to the nodal circles. Fortunately, the theory also predicts resonances with nodal diameters. I didn't pay attention to them because a textbook said they were not important. However, for the djembe they are !!! (this book must have been written before the djembe was invented, because it also stated that the Helmholtz resonator is not very suitable as a source of sound !!!??) Below is a list of the most important resonances of a tensioned membrane, sorted according to frequency.
resonance #nodal #nodal frequency
number circles diameters (lowest = 1)
M1 1 (rim) 0 1
M2 1 1 1.59
M3 1 2 2.14
M4 2 0 2.30
M5 1 3 2.65
M6 2 1 2.92
M7 1 4 3.16
M8 2 2 3.50
M9 3 0 3.60
M10 1 5 3.65
M17 4 0 4.91
The series continues ad infinitum. The spectrum becomes more and more crowded at higher frequencies. Another textbook (Rayleigh / Strutt, the theory of sound) noted that resonances with the numbers M2, M3, M5 and M7 (all from the family with 1 nodal circle) form a consonant chord in very close approximation. Could this be a secret of the magic drum?
TYPICAL DJEMBE SPECTRA ?
I have tried to plot the above frequencies in a graph (as far as plain text allows, use proportional font) and compare them with the measured data of my own djembe. The lowest theoretical membrane frequency was scaled to fit with the experimental data (350 Hz) in order to make the comparison possible.
The resemblance is striking. The resonances were most pronounced in the very soft touches that I had recorded.
In the past week, I have sampled and analysed a few djembes. The above pattern keeps coming back. I saw the pattern in the spectra of my own djembe, in a sample from Mamady Keita (Nankama, track 2, Yankadi-Makru the slaps/tones in the call) and Famoudou Konate (Rhythmen der Malinke, track 17, Sofa, the slaps and tones in the first 10 seconds). But I also found spectra that didn't show the pattern. Hey listers, how about all your djembes ????? A challenging question is the following: does an instrument that is regarded as a 'good djembe' show a spectrum like above ???
The spectrum is explained as follows:
Going from low to high, we first encounter the bas, (not shown above). The bas has little to do with membrane vibrations. It is typically at 70 or 80 Hz.
M1: At 350 or 400 Hz (depending on the instrument and its tension) we find the fundamental membrane resonance. The highest M1 that I've found so far is in Mamady Keita's Yankadi Makru (415 Hz).
M2: The first overtone, characterised by 1 nodal diameter, is typically found in the 500 Hz region. The weird thing about nodal diameters is that they can be oriented in every possible way (e.g. from the left rim to the right rim, or from the front rim to the back rim, and everything in between). I tried to find it's orientation by pressing a ruler against the skin while playing. If the ruler coincides with the nodal diameter we should still be able to hear the M2. I didn't find the orientation.
M3 and M4: Next on the list is a 'twin peak' around 700 or 800 Hz. It is formed by a vibration mode with 2 nodal diameters (perpendicular to each other) and a mode with one extra nodal circle in addition to the rim. The radius of this circle is predicted by theory as 43% of the skin's radius (internal radius of the rim of course).
M5: The next one, the first one that might exceed 1 kHz, is the last one of the prominent peaks. It is a vibration mode with 3 nodal diameters, forming 6 pies of 60 degrees each, and only the rim as a nodal circle.
M6 and up: Higher resonances tend to have lower intensity, however they might contribute to the timbre significantly (attack, ping). They are hard to identify since the spectrum is so crowded. The triple peak at approx 1200 Hz is sometimes clearly recognized. The spectrum continues with audible peaks up to 4 kHz or even more.
The main sound intensity, however is concentrated in the frequencies up to 2 kHz. In this frequency region the difference beteen a tone and a slap is found. Also I found very different peak shapes and peak behaviour of me playing on my djembe compared to the master drummers playing on their djembes .......
THE MASTER DRUMMERS' SLAPS AND TONES
Here are the results of the master drummer samples that I analysed. I thought this would give us an impression of the 'ideal slap and tone'. The M resonance numbers refer to the vibration modes of the membrane (skin) as posted in 'physics of djembe sound, part II' (Monday 21 July). The B1 number refers to the bas, whose origin is not found in the skin. See annex for how the numbers were calculated.
Mamady Keita, Nankama, Yankadi Makru intro
-------------------------------------------------------
resonance frequency tone slap difference
number in Hz intensity intensity slap-tone
in dB in dB in dB
-------------------------------------------------------
B1 74 84 90 +6
M1 415 104 97 -7
M2 633 97 93 -4
M3 810 84 97 +13
M4 872 90 101 +11
M5 980 77 85 +8
M? 1120 87 95 +8
M? 1582 73 81 +8
M? 1770 75 83 +8
M? 2000 82 83 +1
M? 2950 72 82 +10
-------------------------------------------------------
Famoudou Konate, Rhythmen der Malinke, Sofa intro
-------------------------------------------------------
resonance frequency tone slap difference
number in Hz intensity intensity in dB
in dB in dB
-------------------------------------------------------
B1 81 88 77 -11
M1 370 102 88 -14
M2 550 79 90 +11
M3 770 81 96 +15
M4 810 88 96 +8
M5 1030 84 96 +12
M? 1220 78 78 0
M? 1250 75 78 +3
M? 1440 75 75 0
M? 1650 72 82 +10
M? 1700 72 78 +6
M? 1850 71 80 +9
-------------------------------------------------------
HOW TO HIT THAT DJEMBE (A PHYSICIST'S VIEW)
When playing tones, both players put most energy in the M1 resonance. In other words: the tone is mainly formed by the fundamental membrane resonance. The players suppres higher modes as much as possible, though they cannot be completely suppressed. Physics gives us some rules how to strike the fundamental mode (= tone) optimally, meanwhile suppressing higher modes. It goes beyond the scope of this post to explain the theoretical details, but you have to think of 'pushing' the skin in the fundamental mode as you touch it. The rules can be summarised as follows:
1) The contact area of the hand to the skin should be as large as possible (fingers held together, parallel to the skin). You kind of press away the higher resonances with their numerous nodal circles and diameters. If you hit a vibration mode symmetrically around a nodal line, you don't excite the resonance at all !!
2) The contact period should be half a period of the tone sound (= 1.5 milliseconds). If you touch the skin longer, you dampen the tone that you just created. If you touch it shorter, you excite more higher resonances than necessary (you have not dampened the higher resonances as much as possible).
When playing slaps, both players put their energy in higher vibration modes (with possibly M4 being the major contributor). Other modes follow very close, so may be a conclusion is that the slap is formed by striking as many vibration modes as possible. This is done (for the same reasons as above) by:
1) hitting the skin as locally as possible, so preferably with the fingertips. How many fingertips do use when playing slaps?
2) hitting the skin as short as possible. May be the hard skin on the fingers might help you in doing so. This would be an extra handicap for beginners to get the tone/slap difference under control!
THE MYSTERY OF SHORT TONES
I have also looked at the decay of the resonances (how long does the sound last). Physically seen, this depends very much on the thickness of the skin: the thicker the skin, the slower the decay, the longer the sound lasts (this was confirmed in the posts about solo and accompaniment djembes). In Famoudou's djembe I found a fast decay, but the decay was not constant over the frequency range up to 2000 Hz. I found a huge absorbtion activity from 200 .. 600 Hz, so the M1 (tone) included.
Mamady's djembe had a much smaller decay, so I guess he used a thicker skin. Mamady's djembe also had a much more constant absorbtion activity over the entire range up to 2000 Hz (bas excluded). Also his tone decays faster than surrounding frequencies, but by far not as fast as Famoudou's.
Indeed, the tone sounds are generally very short. It's a kind of dead 'plop', isn't it? Famoudou's tone lasts only for 12 cycles or so (0.03 seconds). It collapses like a pudding.
The rapid decay of the tone can be partly explained from the huge acoustic radiation of the M1 mode shape. But this would not explain the difference between Famoudou's and Mamady's djembes.
It is possible that the resonance looses a lot of energy in the shell, which might function as a kind of absorber in this frequency range. In this case a heavy, rigid shell would make the tone longer.
I found another clue a few days back on the list: the slap is formed by the curvature of the bowl just above the stem (or something like that). I was thinking of soundwaves reflecting from the bottom of the bowl, the bowl being a parabola with te center of the skin at it's focal point. The pressure shock would help us to push the membrane in the M4 (slap) mode: the center is pressed up by the shock wave, while we press down the region near the rim, a splendid nodal circle in between. Sounds like it makes sense, doesn't it? In perfect agreement with the analysis of the master drummer samples. The reflecting sondwaves would also have a kind of self-dampening effect on the M1 (tone) resonance! Does the shape of the djembe enhance the M4 (slap) resonance, and attenuate the M1 (tone) resonance ??? May be another mystery of the magic drum.
WHAT'S NEXT
Looks like we have arrived at the point that the shell needs to be included in the analysis. Up to now, it was only skin skin skin (at least in my analysis). The shell was only necessary to explain the bas, but I bet the slaps and tones of a plain membrane sound different than those played on a skin stretched over a djembe. But how do they sound different?
Who has a clue?
ANNEX
The sound intensity in dB (decibels) was calculated as follows:
dB = 10 * 10log|(FFT(s(t))^2|
were s(t) is the measured signal as a function of the time, expressed in a 16 bit (-32535 ... 32535) number. The sampling frequency of s(t) was 44.1 kHz; 8192 samples. FFT stands for fast fourier transform. The resulting dB value depends of course on the recording level, so the reference level of the used dB scale is quite arbitrary. At least it has little to do with the reference level that is used in acoustics. But nevertheless, every 10 dB difference corresponds to a factor 10 in sound intensity, 20 dB a factor 100 etc.
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