Kick Sound Drum

[Frauke Vilandre]

We'll start by casting our mind back to the kettle drum. If you remember my analysis of two months ago, you'll know that the environment in which a membrane finds itself influences its modes of vibration. I covered some of this ground last month, but let's now go into a little more detail.

Figure 1(a) shows the idealised case of a circular membrane suspended in a vacuum. As we already know, it is this that produces the enharmonic set of frequencies determined by the Bessel Functions. Of course, it's highly unlikely that either you or I will ever experience this sound. Firstly, no sound is carried in a vacuum. Secondly, much as I would like to eject most of my band's former drummers from the airlock of a passing Vogon Constructor Vessel, the opportunity to do so has not yet arisen. Therefore, we must consider the vibration of the membrane when suspended in the atmosphere, as shown in Figure 1(b).

The closed shell of the kettle drum discussed last month is a particular example of this, with the vibrations of the membrane strongly influenced by the modes of vibration of the air inside the shell itself. Nonetheless, as shown in Figure 1(c), the membrane still has to shift the air on its outer surface against atmospheric pressure.

The physics of this example is, as we have already discussed, rather complex, but we know that the consequences of mounting the membrane in this way are twofold. Firstly, the pitches of the modes rise even further. Secondly, the frequencies of the important radial modes become almost harmonic in their distribution.

OK, so far we've discussed nothing new. But let's now take a step backwards, and ask what happens when we remove the body of the kettle drum and replace it with a tubular shell capped by a second movable membrane (see Figure 2). Clearly, we have designed a conventional drum of one sort or another. Welcome to the world of the unpitched membranophones.

Table 1: [top-left] The modal frequencies of a dual-membrane bass drum. Table 2: [top-right] The modal series shown in Figure 4. Table 3: [bottom-left] The modified series is almost precisely harmonic. Table 4: [bottom-right] The modal frequencies of a dual-membrane bass drum with the carry head detuned with respect to the batter head.

The physics of this is too advanced for Synth Secrets, but we can understand the measurements of its behaviour observed by academics. Let's start by considering Table 1, which shows a set of bass drum modes when both membranes are stretched to the same tension.

First, I had better explain why there are two frequencies shown for the 0,1 and 1,1 modes. Again, without describing the physics involved, we observe that the membranes' vibrations are affected by the air between them in such a way that, for these modes, not one, but two, frequencies are produced. Weird, eh?

Now, the frequencies in Table 1 may not seem related in any way, but when we plot them on a chart, something unexpected happens. If you look at Figure 3, the frequencies may look enharmonic, but if we remove the doubled 0,1 mode at 118Hz and the doubled 1,1 mode at 86Hz, we obtain Figure 4.

Yikes! I don't know how it looks to you, but to me Figure 4 looks very close to a harmonic oscillator. This is, however, an illusion (damn... just when you thought you were getting the hang of things!). Although the spacing of the modes is very regular (see Table 2), the frequencies of the upper partials do not lie at integer multiples of the lowest component frequency. In fact, they appear to be in the form of a harmonic series, frequency-shifted upwards by about 7Hz. I can demonstrate this by subtracting 7Hz from each of the frequencies in Table 2, and writing the results in Table 3... which gives us an idea about how to synthesize the spectrum in Table 2.

Figure 3: [left] Frequencies of the modes of a bass drum. Figure 4: [top-right] A simplified plot of the bass drum modes. Figure 5: [bottom-right] Generating the spectrum of an orchestral bass drum using a frequency-shifter.

Let's take a complex harmonic waveform with a fundamental frequency of 43Hz, and pass its output through a device called a frequency-shifter (see box). If we use this to increase by 7Hz the frequencies of each component in the signal (ie. at 43Hz, 86Hz, 129Hz, 172Hz... and so on), we obtain a signal with partials at 50Hz, 93Hz, 136Hz and 179Hz... which is almost exactly the spectrum we require.

But before we get carried away, analysing and developing this model still further, let's take a step backwards for a moment. We started by assuming that the bass drum had two membranes of equal tension, and was completely sealed at both ends. This is seldom true, because players of orchestral drums rarely tune their instruments like this, tending to tune the carry head at a much lower tension.

Table 4 shows the measurements for a drum tuned in this fashion. As you can see, the dual frequencies previously exhibited by the 0,1 and 1,1 modes have disappeared. But, more important than this, the result is close to a true harmonic series without any offset. This means that, when we synthesize the sound of a drum with a detuned membrane, we can dispose of the frequency-shifter.

Of course, the bass drums we encounter in most modern music do not have two complete membranes. We call them kick drums and they often have a hole cut in the carry head, into which we stuff pillows, microphones, the guitarist's head, and the occasional empty lager can (see Figure 7, below).

Figure 7: [left] A bass drum with a hole in the carry membrane. Figure 8: [right] Coupling of the membrane and the internal modes of the kettle drum.It might seem that we now know everything we need to know in order to create a bass drum patch, but there are two important attributes left to investigate: first, the pitch shift that occurs every time you excite the batter head; and second, the sound of the beater 'click'. Let's look at the first of these.

If you again cast your mind back a couple of months, you may recall that the cavity modes within a kettle-drum shell act as frequency regulators on the equivalent vibrations of the membrane. Therefore, no matter how hard or softly you hit the membrane, the frequencies of its modes remain tightly locked to those permitted by the air in the cavity. Figure 8 depicts the 0,1 mode of a kettle drum, showing how the vibration of the air is coupled to the vibration of the membrane.

Now let's look again at the bass drum in Figure 7. This has a flexible interface with the outside world (ie. the air section at the aperture in the carry head), so the cavity modes are very much less constraining that those of the kettle drum. This means that, if the batter head wants to vibrate at a different pitch, it is relatively free to do so. But what would make it change pitch?

Figure 9(a): [top] A membrane at rest. Figure 9(b): [bottom] A struck membrane.Consider Figure 9(a). This shows a stretched membrane seen from its edge. I have arbitrarily made it 30 inches in diameter.

Now let's beat the living daylights out of this, smacking it with a beater and displacing its centre by an inch or so. This is an unrealistically large displacement for any tightly stretched membrane, yet it only increases the distance across the surface by about one sixteenth of an inch, as shown in Figure 9(b).

What it does do, however, is increase the tension of the membrane by an amount that is proportional to the square of the displacement. And, since pitch is determined by tension, this increases the pitches of the modes by a considerable amount. It's a small leap of understanding to realise, therefore, that the pitch of every mode will be higher at the start of the sound (when the maximum instantaneous displacement is large) and will drop as the amplitude of the vibration decays. Indeed, the pitch of a typical kick drum can shift by a couple of semitones from start to finish, and we must build this into our patch if it is to sound realistic.

Fortunately, we do not need a second contour generator to implement this. After all, the loudness and the pitch of the sound are both determined by the maximum instantaneous displacement of the membrane, so a single AR Generator should do the trick. However, whereas the VCA Gain will change by 100 percent from the start to the end of the sound, the pitch should only shift by around 10 percent, so the patch requires some form of attenuator at the oscillator's pitch CV input (see Figure 10).

There are two ways we can model this. One would be to use a short noise burst (which you would normally call a click); the other uses a contour generator to shape the spectrum of the partials generated by the FM components in Figure 10. If we choose the latter, we split the band-pass filter into its low-pass and high-pass components, and apply a rapid AR contour to the cutoff frequency of the LPF. This allows many high-frequency components (almost a noise spectrum) to pass for a very brief time, before the patch settles down to the sound generated in Figure 10 (see Figure 11, below).

So there we have it: a simple bass-drum patch. Simple? Well, we've skirted over the true nature of the enharmonic partials, approximated the decay rates, disregarded the (albeit reduced) effects of the cavity modes, and totally ignored the presence of any shell resonances. Consequently, I think that it's fair to say that this is a simplified patch. Nonetheless, it will produce extremely usable results. If you have access to a patchable analogue synth with three oscillators, cross-modulation, three suitable filters, a couple of contour generators, a mixer and a couple of VCAs, you're in business.

Unlike a Harmoniser or pitch-shifter, which alters the frequency of each component in the sound spectrum by a fixed ratio, a frequency-shifter moves each of the components by a fixed amount in Hertz. To understand the difference, we'll start by considering the first of the diagrams. This shows the spectrum of a 100Hz signal with four harmonics.

If we pass this signal through a pitch-shifter set up to increase the pitch of the fundamental by one octave, we obtain the spectrum shown in the second diagram. And, since we have doubled the frequency of the fundamental, all the harmonics have moved too: the second harmonic still lies at twice the fundamental frequency, the third harmonic still lies at three times the fundamental frequency... and so on.

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