10.4 Measuring frequency response functions

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Frequency response functions are the “workhorse” measurement for a lot of musical acoustics research, whether they are used to characterise the vibration and sound radiation from a violin body, or to identify the acoustic signature of a clarinet or trumpet tube. We have already seen quite a few frequency response functions, and made use of them for various purposes. In this section some of this earlier material will be re-examined in the light of the specific question of how to measure a frequency response that is reliable and accurate.

When we first introduced the idea of frequency response, back in Chapter 2, we thought about a “black box” which represented the behaviour of a linear system with a single input and a single output. Figure 1 shows a repeat of Fig. 2 from section 2.1 (the box is labelled “Drum” because we used a toy drum as our example system in that first discussion). There were then two key ideas. The first is the “sine wave in, sine wave out” property of any linear system. The second is Fourier analysis, which allows us to express any input or output signal as a combination of sine waves. Put together, these mean that if we understand how a system responds to sine waves at every possible frequency, we know everything there is to know about it (at least in principle).

Figure 1: Schematic block diagram of an input-output “black box”, reproduced from Fig. 2 of section 2.1.

In earlier times, before the routine availability of computers, frequency response was measured in exactly that way. Some kind of actuator was fixed to the structure, and it was driven with a sine wave. The desired response was measured, and the frequency of the sine wave was slowly swept through the range of interest — for any musical problem, that range is usually limited to the range of human hearing, roughly 20 Hz—20 kHz. At each frequency the ratio of amplitudes of output/input was measured, and for a complete measurement the phase shift between the two was also measured. The equipment was usually large and expensive, and measurements were confined to laboratory settings. The frequency response plot was usually drawn by a chart recorder, often mechanically coupled by sprockets and a chain to the sine-wave oscillator.

The world has changed a lot since those days! Measurements are done with the aid of a computer, and this changes every aspect of the procedure. We need not use sine waves as the input signal: we can use any convenient input signal which contains the range of frequencies we are interested in, and use Fourier analysis (the FFT) in the computer to separate the different frequency components. This opens many possibilities, which we will discuss shortly.

The ubiquity of small but powerful computers means that measurements like this have moved out of the laboratory. Some instrument makers are making routine use of frequency response measurements in their workshops, to inform their working processes and decisions. This fact opens another area for discussion, because instrument makers are not necessarily highly trained in science. They sometimes need a little guidance to perform and interpret the measurements. I will try to provide something useful in the course of this section.

A. Actuators and sensors

Any measurement requires the instrument to be set into vibration somehow. For a measurement done in the traditional way, with a swept sine wave, this means attaching some kind of actuator capable of producing a continuous excitation — such things are usually called “shakers”. The most common type works on the same principle as a traditional loudspeaker, shown in Fig. 2 in a cutaway view taken from this Wikipedia page. The electrical signal is fed through a coil (2), which is close to a permanent magnet (1). Just as in an electric motor, the interaction of the electric current and the magnetic field produces a force that vibrates the cone (4), which then creates sound waves in the surrounding air. The cone and coil are supported by bellows-like structures (3) which keep things nicely lined up without interfering with vibration in the desired direction.

Figure 2. A cutaway view of a loudspeaker. 1: magnet; 2: coil; 3: support structures; 4: cone. Image: Svjo, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons

A commercial vibration shaker, like the one that can be seen in Fig. 3, has the same features, except that it does not have a cone. Instead, there is a threaded boss attached to the coil, which can be used to connect the shaker to the object you want to vibrate. Shakers like this come in many different sizes: Fig. 4 shows a selection. This kind of shaker is fine for driving a relatively heavy structure, like the undergraduate experiment seen in Fig. 3, but it is not immediately suitable for a lightweight object like a violin. If something like this was attached to a violin bridge it would act like a mute, changing the behaviour significantly by adding mass, and probably adding damping as well.

Figure 3. A vibration shaker (the blue cylinder) in use for an undergraduate experiment involving vibration of two coupled cantilever beams.
Figure 4. Vibration shakers in different sizes

However, moving-coil methodology was successfully used to measure frequency response functions of violins in earlier times. To solve the problem of added mass, some ingenuity was needed. One approach was to attach to the instrument just the bare coil extracted from a loudspeaker. A magnet could then be brought near, and by driving the coil electrically, a swept-sine test could be made. This was the favoured test method of Carleen Hutchins, the doyenne of scientific violin making in the United States in the second half of the 20th century. The converse strategy has also been used: a very light ceramic magnet can be glued to the violin bridge, and a coil brought close to it to provide the force.

However, neither strategy is commonly used these days in the musical instrument world. Moving-coil shakers have been almost entirely replaced by the second major type of vibration actuator, the impulse hammer. The great virtue of this kind of actuator is that once the hammer has bounced off the structure, there is nothing remaining in contact. No added mass, no added damping.

We have already seen an impulse hammer in action, back in section 5.1. As we just saw with shakers, impulse hammers come in a variety of sizes: two are shown in Fig. 5. Both these hammers can be seen to have a cable leading out of them: they both have a force-measuring sensor close to the hammer tip, allowing the precise waveform of force to be measured when a structure like a violin is tapped.

Figure 5. Two sizes of impact hammer, both incorporating force-measuring sensors. The larger one is arranged in a pendulum fixture, to ensure positional accuracy in repeated taps.

Why might we want different sizes of hammer? We already know the answer to that, from section 2.2.6 where we found the frequency spectrum of an idealised hammer pulse. That calculation showed that the frequency bandwidth of a hammer tap is inversely proportional to the time of contact during the bounce. If we want a measurement with a wide bandwidth, we need a tap with very short duration. That requires two things: the surfaces in contact during the tap need to be sufficiently rigid, and the mass of the hammer needs to be not too high. That is the reason that instrument measurements are usually done with a very light hammer like the one seen at the bottom of Fig. 5. But if you wanted to measure vibration of something like a railway bridge, you would need a heavier hammer in order to put enough energy into the structure to get clean results.

Of course, you could excite a violin body into vibration by tapping with something more familiar (and cheaper), like a pencil. You can learn quite a lot from such an informal hammer test: modal frequencies and damping factors, for example. But what you can’t obtain that way is a calibrated frequency response function that can be directly compared with measurements made by someone else with a different experimental setup. For that, you need a measurement of the input force as well as the vibration response. We will come back to this question of what you can expect to measure using more and less sophisticated equipment in the next subsection.

We have described the two main types of vibration actuator, but we haven’t exhausted all the possibilities. There are at least three other methods that have been used for measurements on violins and other stringed instruments, and they deserve a brief mention. First is the wire-break pluck, which we met briefly back in section 7.4. You take a length of very thin copper wire, loop it round some part of the structure, and pull gently until it snaps. The method is particularly useful for stringed instruments: it can be used for a controlled string pluck, as we saw in section 7.4, and it can also be used to excite body motion by looping the wire round a string right up against the string notch in the bridge.

This method allows the position and direction of the pluck to be very well controlled, and also gives a reliably repeatable amplitude between plucks because the wire breaks at more or less the same force every time. It can even be used, with care, to give a calibrated frequency response. The force waveform when the wire breaks will be a very sharp step. The magnitude of the step can be calibrated by hanging small weights from the wire to determine the breaking load, and the frequency spectrum of the step is then known mathematically so that even without measuring the input force, you can tell your computer a good estimate of the input spectrum. Figure 6 shows an example of this method being used to measure the frequency response of a violin, compared with the impulse hammer method. For more details, and examples for a cello, see Zhang and Woodhouse [1].

Figure 6. The bridge admittance of a violin, measured with three different excitation methods. Purple: impulse hammer; blue: wire-break; red: bowed glissando.

The second approach is perhaps one you already thought of — why not use real playing on a stringed instrument to excite the body vibration? This has the obvious advantage that it is the “real thing”, with no complications or doubts to do with artificial actuators. The disadvantage is that you cannot ordinarily measure the input force at the instrument bridge. But if you should happen to have a bridge equipped with the kind of force sensors we have seen before (see section 9.1.1), then you can indeed use regular playing to give a calibrated frequency response function.

An example is included in Fig. 6, using a one-octave glissando played on the lowest string of a violin. For more detail, and examples for a cello, see Zhang and Woodhouse [1]. Figure 6 demonstrates quite convincingly that the hammer, wire-break and bowing methods are all capable of giving very similar results: the differences between the three curves are hardly bigger than the differences you find if you repeat the same measurement with the same instrument, but on a different day.

However, the sharp-eyed may notice that the curves in Fig. 6 look a little different from the violin bridge admittances we have seen previously: there isn’t much evidence for a “bridge hill” here. The reason is an important one: the additional mass of the bridge-force sensors and their associated cables have a muting effect on the violin. So the comparison of the three excitation methods is entirely valid, but none of them gives a really good representation of how this violin would behave with its normal bridge, without the added sensors. This is the reason that the published version of this work [1] concentrated on the cello: the heavier bridge of a cello makes the added mass of the sensors much less significant. The moral is: always play the instrument with all your instrumentation in place, to check how much you might be changing the behaviour.

Without an instrumented bridge, you can still learn something useful about a violin by playing it. One of the very earliest steps in quantifying violin response was the “Saunders loudness test”, in which every semitone was played on the instrument, trying in each case to get the loudest sound possible. A sound level meter was used to record the level. Peaks in the loudness flag up resonances. The early literature talked about the “air resonance” and the “main wood resonance” based on such tests, but this terminology highlights an interesting issue. What was then called the “main wood resonance” is now understood to be a combination of the two modes called B1- and B1+ (see section 5.3, figures 5c and 5d). The loudness test, moving in semitone steps, was too crude to resolve the two peaks.

More recently, Oliver Rodgers investigated a large number of violins using a technique based on regular playing. As in the cello measurement described above, a glissando was played on each string. This was analysed with a variant of the spectrogram, which we have seen earlier. Although the input force was not measured, and the glissando playing was never precisely repeatable, he was able to obtain clear results which revealed strong low-frequency resonances — much more clear than the Saunders loudness curves. For some examples of his results by this approach, see [2].

The final approach to excitation is to use sound waves from a loudspeaker, in a similar way to making Chladni patterns (see section 10.3). An interesting application of this approach to the frequency response of a violin was developed by Gabriel Weinreich [3]. He was interested in measuring the monopole and dipole components of sound radiation from a violin as a function of frequency (see section 4.3.1).

He measured these by taking advantage of a general reciprocal theorem (which we mentioned back in section 5.4). Instead of driving the violin with a force at the bridge and measuring the sound at many points, then averaging over directions to give the monopole component, he did the converse. A set of loudspeakers surrounding the violin all played an identical signal, giving (approximately) an omnidirectional sound field. He then measured the mechanical response at the bridge of the violin, using a phonograph pickup. The same set of loudspeakers could also be driven in ways that produced dipole sound fields in all the possible orientations, and the measurement repeated with each of those.

There is a much more obvious application for a loudspeaker as an actuator. I have been concentrating up to now on stringed instruments, but if the frequency response you would like to measure is of a wind instrument, then some kind of loudspeaker is a natural choice. Figure 7 shows the input impedance of a saxophone being measured: the figure is reproduced from Fig. 5 of section 8.5. The mouthpiece end of the tube is being driven by a loudspeaker inside the canister.

Figure 7. Measuring the input impedance of a saxophone, reproduced from Fig. 5 of section 8.5. The measurement is taking place in an anechoic chamber: the wedges on the walls absorb the sound waves so that there is virtually no reflection. Image copyright Jean-Pierre Dalmont, reproduced by permission.

Now we have surveyed the main options for vibration actuators, we turn to the corresponding range of sensors to register the result of excitation. Again, the possibilities range from hi-tech and expensive to simple and home-made. If we want to measure sound pressure, as in the wind instrument example we have just seen, we are bound to use a microphone of some kind as our sensor (although the actual details of the measurement seen in Fig. 7 need ingenuity: see the next link). For a stringed instrument or a percussion instrument, things are more complicated. We might be interested in the sound radiated, and thus use a microphone as our sensor, but we might be more directly interested in the vibration of the structure. For that, there is a wide choice of sensors.

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Many of the options measure relative motion between two points, and so they require some kind of fixed reference position near the vibrating object. But there are exceptions. The most widespread of these is the accelerometer: some examples of accelerometers are shown in Fig. 8. You simply stick one of these to your structure, and it gives an electrical voltage signal proportional to the acceleration at the attachment point in a particular direction, usually perpendicular to the surface. Some accelerometer units actually include three separate sensors, aligned in perpendicular directions. Called “triaxial accelerometers”, they give three output signals so that all components of the acceleration are measured simultaneously.

Figure 8. Three different sizes of accelerometer: only the smallest one is light enough for use with stringed instruments.

Some accelerometers are very small indeed: they are made using silicon chip fabrication technology, and they can be integrated on the chip with their supporting electronics. Such devices are called MEMS accelerometers, standing for “Micro-electro-mechanical systems”. They are all around you, although you may not be aware of it. Many mobile phones include triaxial MEMS accelerometers so that your phone can respond to motion (and can also sense gravity, so that the phone knows which way up it is). All modern cars are fitted with airbag safety systems, and these use MEMS accelerometers to detect the sharp acceleration (or, usually, deceleration) which signals a collision and so fires the airbag. There are many other types of MEMS device, for example gas sensors used to detect pollution, but we need not go any further into that here. See this Wikipedia page for more information.

All accelerometers work in a similar way, taking advantage of Newton’s law. They contain a mass, attached to a force-measuring sensor. When the accelerometer is subjected to an acceleration, the mass inside it must be pushed by a force proportional to the acceleration, and this force is measured and gives the output signal. Because they have a mass supported on some kind of spring to provide the force sensor, all accelerometers have an internal resonance frequency. This governs their useful frequency bandwidth: once the frequency approaches the resonance frequency, the output is no longer simply proportional to acceleration.

There are two main types of force sensor. Accelerometers like the ones seen in Fig. 8, intended for high-frequency measurement, use a piece of piezo-electric crystal. Accelerometers that are intended for very low frequencies use some kind of mechanical spring with a strain gauge attached. The key difference is that a strain gauge works down to the very lowest frequencies, and even for static deformation. This means that such accelerometers respond directly to the force of gravity. This provides a very simple way to check the calibration of the device: turn it upside down, and gravity reverses. The jump in output level will be twice the value of $g$, the acceleration due to gravity: about $9.81\mathrm{ m/s}^2$ at sea level. The down-side is that the strain-gauged sensing element will be much less stiff than a piezo-electric crystal, so such accelerometers have a rather low resonance frequency.

We commented earlier that if you use some kind of shaker as your vibration actuator, you need to be careful about the mass this adds to your structure — especially if the structure is very light in weight, like most musical instrument bodies. Well, exactly the same consideration applies to sensors. Convenient though accelerometers are, they always add some mass. For musical instrument measurements, we usually need to use the very lightest of accelerometers. The smallest one shown in Fig. 8 weighs a bit under 2 g, and it is just about light enough. The others in that picture are far too heavy: they are designed for testing large engineering structures. Figure 9 shows one of these small accelerometers attached to the soundboard of a harp.

Figure 9. A small accelerometer being used to measure vibration of a harp soundboard

The best cure for the added-mass problem is, of course, to use a sensor that adds no mass. This is the big virtue of a laser-Doppler vibrometer. The only thing impinging on the structure is a laser beam, which adds no mass. You can see one in use in Fig. 10 — in a domestic room, because this measurement took place during Covid lockdown! The laser unit on its tripod is in the lower left of the image. The red spot from the laser beam can be seen on the bridge of a banjo. A miniature impulse hammer on a pendulum fixture is striking the banjo bridge near the laser spot. Both signals are then collected by the laptop visible on the right, to be processed into the frequency response function.

Figure 10. A laser-Doppler vibrometer (foreground) being used to measure vibration of a banjo. Note the red spot on the banjo bridge, where the laser beam strikes.

Laser vibrometers have two disadvantages. First, they are very expensive so they are usually only found in laboratory settings. Second, they are surprisingly fiddly to use — much more difficult than an accelerometer. In order to measure the Doppler shift from the vibrating structure, a sufficiently strong reflection of the laser beam must be detected by the vibrometer. This is usually helped by a small piece of retro-reflective tape, of the kind used for safety stripes on a cyclist’s clothing. But even so, it can be tricky to obtain a really high-quality signal.

These types of “laboratory-grade” sensors do not exhaust the possibilities. One obvious thing for a stringed instrument is to use one of the many types of commercial pickup, designed to allow a performer to add some amplification to the sound of an acoustic instrument. For example, these may be incorporated within or under the bridge, or they may be stuck to the soundboard. They will all give an electrical output that responds in some way to the vibration. The problem is that you usually don’t know exactly what the output means: pickups are designed to sound good, not to give a scientifically “clean” output signal. We will see what this means for the purposes of acoustic measurement in the next subsection.

Another class of sensors is related to the standard pickup of an electric guitar. The principle of these is essentially like a moving-coil shaker, but run in reverse. If you vibrate a magnet close to a coil, Faraday induction will cause a voltage to appear in the coil. This can be picked up and amplified. Alternatively, you can vibrate the coil while holding the magnet still. Or, as in the guitar pickup, you keep both the magnet and the coil still, while you vibrate a metal object like a string close to them both.

There are two other possibilities that are occasionally used, which both work on the same principle. The first is to sense string vibration by placing a concentrated magnet near the point you are interested in, then use the string itself rather than a coil: any string with a metal component will develop a voltage across its ends, proportional to the velocity at the magnet position. The second option is a standard phonograph pickup, which also works by Faraday induction. As we noted earlier, Weinreich used a phonograph pickup to sense the violin body vibration in connection with his reciprocal sound radiation measurement.

There is an important consequence of using different types of sensor for a measurement. All the sensors based on Faraday induction give a signal proportional to velocity, as does a laser-Doppler vibrometer. However, an accelerometer obvious gives a signal proportional to acceleration. The two are related by the fact that acceleration is the rate of change of velocity, or in mathematical terms it is the time derivative of it. Once we move to the frequency domain to plot our frequency response function, this translates into the fact that acceleration emphasises high frequencies and de-emphasises low frequencies, relative to velocity.

Figure 11 demonstrates the effect. This simply shows the same measurement, plotted as velocity per unit force in red, and as acceleration per unit force in blue. The red curve is a bridge admittance, of a kind we have seen in many earlier plots. The blue curve is called bridge accelerance. In this plot it has been scaled so that the two curves agree in the middle range. It is obvious that the accelerance falls to very low values at low frequency, but rises to higher levels than the admittance at high frequency.

Figure 11. Admittance (red) and accelerance (blue) measured at the bridge of a violin. The admittance shows calibrated values, but the accelerance has been scaled down by a factor of 10,000 so that it fits on the same vertical scale.

Indeed, we might have used a sensor that responded to displacement rather than velocity or acceleration. Such things are sometimes used: an example would be a capacitance pickup (working a bit like the touch screen of your phone). In that case, the resulting frequency response would be called receptance, and it would show the opposite trend, rising above the admittance at low frequency and falling below it at high frequency.

Every measurement has a limited range of reliable output, governed by the background noise which is always present at some level. So the result of the distinction between receptance, admittance and accelerance is to alter the effective bandwidth of the measurement. Choosing a sensor to emphasise the frequency range you are most interested in is important when designing any measurement like this. If you use an accelerometer you must resign yourself to getting rather noisy and poor data at low frequency, but good results at high frequency. If you chose a displacement sensor you would have the opposite effect: excellent at low frequency, but very poor at high frequency.

B. What do you want to get from your measurement?

Now we know a bit about the main options for actuators and sensors, we can think about the consequences for a measurement of frequency response. The key question, and one you should always ask yourself before plunging in to a measurement project, is “What exactly do I want to know, and how accurately do I want to know it?” To focus this question, it is useful to remind ourselves of the anatomy of a typical frequency response, as we discussed in section 5.3.

Figure 12 shows the bridge admittance from Fig. 11, annotated with some key details. At low frequency, we expect to see well-separated modal peaks. In the case of this violin the four peaks indicated by arrows correspond to the “signature modes” A0, CBR, B1- and B1+ which we showed earlier (see the animations in Figs. 5(a)—(d) in section 5.3). As we go higher in frequency, we expect the spacing of modal peaks to remain roughly constant, while the bandwidth of the individual peaks gets wider (roughly in proportion to the centre frequency). The result is that the peaks start to overlap, and by the time we reach high frequencies there may be several modes with significant amplitude at any given frequency.

Figure 12. The violin admittance from Fig. 11, labelled with key features.

It then ceases to be productive to ask questions about individual modes, but there may be important response trends that we can still say something about. In the case of a violin, the most important trend is probably the feature indicated by the green dashed line in Fig. 12, the so-called “bridge hill”. Underneath the small wiggles in the admittance function, there is a broad “formant” feature. Another hill-like feature seen in many violins is indicated by the grey dashed line: it is sometimes called the “transition hill”.

So what might you want to know, if you are trying to measure the frequency response of this violin? There is a kind of hierarchy of questions of increasing difficulty, and how you choose to go about the measurement depends critically on where you are in that hierarchy. Perhaps you just want to know the frequencies of the signature mode peaks? Perhaps you would also like to know the heights of those peaks? Perhaps you want this information purely for your own use, to compare with other instruments you have made or tested; but perhaps you want the information in a standardised form so that you can compare with measurements made by a friend with a different test setup. Perhaps you want to capture higher-frequency features like the bridge hill? The most challenging option of all is if you want the frequency response function in order to do some kind of clever processing with it, like modal analysis (see section 10.5) or the synthesised guitar and banjo sounds we heard earlier.

If you just want to know signature mode frequencies, the simplest possible measurement is probably the best. You can tap the corner of the bridge with a pencil, and capture the resulting clonk sound with a commercial pickup or a microphone, perhaps the one in your phone. An FFT of that sound will show the peaks you want. With a pencil tap, you do not of course know the exact waveform of the force, but you do know that it is a short pulse of some kind. That is enough: the frequency spectrum of any short pulse is bound to be rather smooth and featureless at low frequency, so any peaks you see in the response must come from the violin, not from the input tap. The bandwidth of those peaks should also be reliably captured, if you are interested in the damping of your signature modes.

If you want to know something about peak heights, things are immediately more tricky. Heights are influenced by the strength of your tap, and if you are using a microphone to capture the response they are also influenced by the microphone position and distance, and also possibly by the acoustics of the room you are in. If you only want the information for your own comparison between instruments you could still get away with something quite simple, though. Instead of tapping with a pencil you might swing a small pendulum of some kind. You still aren’t measuring the input force, but with a pendulum it is quite easy to make successive swings of the same height so that the force pulse should be quite repeatable. You could still measure with a microphone, provided you always use the same one and you place it somewhere reproducible, preferably quite close to the instrument to minimise any influence of room acoustics.

However, if you want peak heights that you can compare with someone else’s measurement, there is no avoiding some system that measures the input force, or tells you what that force must be in some other way. If you do not have access to a force-measuring hammer, this is where a wire-break pluck can be useful. If you have calibrated the breaking load of your wire, then the frequency spectrum of the input force is predictable, and with a small amount of effort with your computer you can use it to convert your output into a true frequency response measurement. But then you still have the issue of calibrating your microphone, or whatever sensor you are using. I will come back to microphones and also to calibration in general, a little later in this section.

If you want to know about features at higher frequency, such as the bridge hill, then you need to ensure that your input force contains enough energy at all the frequencies you are interested in. If it is a hammer tap of some kind, the hammer needs to be light and its face needs to be quite hard so that the bounce happens quickly, giving a short pulse and a high bandwidth. Alternatively, the wire-break pluck is still a good option because thin copper wire breaks extremely fast so that the force jump contains a lot of high-frequency information. You also have to worry about the added mass of any kind of sensor you may fix to the instrument: the influence of mass rises inexorably with frequency (which is why a violin mute has its main effect at high frequency).

Finally, if you want to measure something like a bridge admittance in a form that can be used for further processing then it is hard to avoid the full professional-grade procedure, with a force-measuring impulse hammer and either a very small accelerometer or a laser-Doppler vibrometer. For this final purpose, the wire-break pluck is not really good enough. The reason is that processing for modal analysis, for example, needs an accurate value of the phase of the frequency response, as well as the amplitude. It is not so easy to get accurate phase with a wire pluck, and (for a related reason) it is also not easy to do multiple takes and use averaging to improve the quality of your frequency response measurement. Similarly, a commercial pickup is not usually acceptable as a sensor, because it is hard to know exactly what the signal means. Such pickups have often been designed with features to improve the sound, by modifying the frequency-dependent amplitude and/or phase in a way that the manufacturer doesn’t tell you.

C. The challenge of microphone measurements

If you want to measure a frequency response function to characterise a stringed instrument, it seems very natural to measure the sound radiated by the instrument, using a microphone. After all, it is surely the sound of the instrument that is of most interest? However, there are interesting subtleties and snags associated with microphone measurements, and it is important to be aware of them. These have to do with physics, and even more to do with psychoacoustics and perception. “Sound”, which seems so simple, is actually a very slippery thing.

When you listen to an instrument being played in a normal room, you probably have the impression that there is a characteristic “sound” of that particular instrument which does not change in its essential character if you move your head, or if you walk around in the room. However, as we will see shortly, this is something of an illusion, created inside your head by the formidable processing power of your hearing system.

Your visual system does something comparable. You probably have an impression of being surrounded by a three-dimensional world full of visual details. Actually, your eyes flit around the scene you are viewing, and only a small region in the centre of your visual field has really high resolution. Your brain somehow puts together a composite image from the rapidly varying signals your eyes are supplying.

It is not even an “image” in the sense that you might stitch together camera images in your computer to form a bigger panorama. Your visual system is doing something far more clever: it is constantly interpreting the stream of visual input in terms of recognisable components (people, clouds, trees and so on). What your brain assembles is a kind of simulation model of the world around you, made up from these components. This simulation model is the “scene” that you are consciously aware of. Sometimes your brain’s interpretation system is fooled, and that produces what you would describe as an optical illusion.

To start thinking about the corresponding auditory system, we can do a measurement that seems intuitively sensible. If you are trying to choose a new violin, you would probably play a few candidate instruments in a suitable room — not too small, not too dead, but probably not a concert hall or a church. So let us put a violin in a room like that, and measure the frequency response from tapping the bridge with a small impulse hammer, and measuring the sound with a microphone at a typical listener’s position. Figure 13 shows two examples of such a measurement, with two different microphone positions about 2 m away from the violin, and about 1 m apart.

Figure 13. Measured frequency responses of a violin. Black: bridge admittance; red and blue: microphone responses to a bridge tap, at two different positions in a medium-sized room.

Figure 13 also shows the bridge admittance of the same violin, with the three “signature modes” A0, B1- and B1+ indicated: these are the three low-frequency modes of a violin body that have the strongest sound radiation. Well, all three do show up in the microphone measurements, but they do not look like the simple peaks in the admittance plot. Instead, they look more like the “formant” features we talked about before: broad peaks “dotted out” by a lot of narrower peaks. Furthermore, the details of those narrower peaks are different for the two microphone positions.

What is happening, as you have probably guessed, is that we are seeing evidence of the acoustics of the room, as well as the acoustics of the violin. The room acoustics is producing many, many extra peaks. Indeed, at higher frequencies the extra peaks are so dense that they cannot be resolved in this plot. Figure 14 shows a zoomed view of the same two curves, in a frequency range around 2.5 kHz.

Figure 14. Zoomed view of part of Fig. 13, showing some of the characteristic features of room acoustics.

You might imagine that the peaks seen in Fig. 14 correspond to acoustic modes of the room, but you would be quite wrong. The medium-sized domestic room where this measurement was done is a bit bigger than the one that was modelled in order to generate Fig. 17 of section 4.2, so we can deduce that in this frequency range the room has more than 10,000 modes in every 100Hz band! But if you count the small wiggles in the red or blue curves, you see no more than about 15 in each 100 Hz band.

Each of these wiggles is a combined result of perhaps 1000 modes of the room, which happen to conspire to produce a bigger sum total at the wiggle peak. Between the peaks, we sometimes see very sharp dips: these are frequencies where the 1000 room modes conspire to cancel at that particular microphone position. The combination of rounded peaks and sharp dips is characteristic of any system with high modal overlap, like our room. Notice that both peaks and dips fall at entirely different frequencies in the red and blue curves: another characteristic of high modal overlap.

The conclusion seems to be that a measurement like the ones seen in Fig. 13 is not a really good way to see the characteristics of the violin, independent of the room. There are various things we might do to improve things — but we will see that those come with their own issues, and even a hint of paradox. Two things we might try are to move the microphone closer to the violin, and to choose a room which is more “dead” in order to do the measurement. Both of those seem likely to reduce the relative influence of the room acoustics.

Figure 14 illustrates the result of those two things. The red curve shows a measurement of the same violin in the same room, but with the microphone moved to be directly in front of the violin, at a distance corresponding approximately to the length of the top plate of the instrument. The blue curve shows a measurement with the same setup and microphone placement, but in a deader room. This is still a domestic room, but one with carpets, curtains and a lot of soft furnishing.

Figure 15. Frequency responses of the same violin as in Fig. 13. Black: bridge admittance as before; red: microphone response in the same room as Fig. 13, but with the microphone closer to the violin; blue: the same microphone position relative to the violin, but measured in a more “dead” room.

It is clear from the figure that both changes have had an effect. The low-frequency peaks in both curves are much more recognisably related to the bridge admittance plot, and the blue curve is significantly smoother than the red curve. But already there is a snag we should consider. The microphone is now sufficiently close to the violin that it is definitely not in the acoustic “far field”. The sound measured in this position will not reflect accurately the way this violin will send sound energy out into a large concert hall. Of course, both ears of a violinist are at least this close to the vibrating surface of the violin body, so the player also cannot judge how the instrument will sound from a distant seat in the auditorium.

The ultimate extension of the idea of a “dead” room is an anechoic chamber, like the one seen in the background of Fig. 7 above. This is a laboratory facility specifically designed to have virtually no sound reflections from floor, walls or ceiling so that a sound source will behave as if it is in empty space, radiating sound which never comes back. It achieves this effect by lining all the walls with deep wedges of foam, as you can see in the picture. So the “scientific” answer to measuring sound radiated by a violin is to place the microphone far enough away from the instrument in such an anechoic chamber.

But this does not eliminate difficulties. As we saw back in section 4.3, a violin (or any other vibrating object) will not send sound equally in all directions except at the very lowest frequencies. As frequency goes up, the instrument becomes increasingly directional — look back at the animation in Fig. 9 of section 4.3, which showed the directivity pattern of a violin as a function of frequency. So it is not good enough to have a single microphone position in the anechoic chamber. Ideally, we need to surround the instrument with microphones, in order to sample the sound field in all directions. This indeed is how the animation from section 4.3 was generated: data was collected by George Bissinger using 264 microphone positions in a latitude-longitude distribution around the violin, as shown in Fig. 15 [4].

Figure 16. The distribution of microphone positions around a violin body, used in the measurements by George Bissinger. Red stars are positions on the front side of the violin, blue ones on the back.

But surely we have been led to a bit of a paradox? An anechoic chamber is the worst possible place to perform music, or to listen to it. Players can’t hear themselves clearly, nor the others in their ensemble. A listener is acutely aware of the directional nature of sound from an instrument: if you walk around a violinist performing in an anechoic chamber, the sound is very different from different angles — the opposite of the common experience in a normal performance space. So to obtain a “scientific” measurement of the sound radiation behaviour of a violin, we need to test it in this least musical of environments, the exact opposite of where you would choose to judge the quality of the instrument.

This brings us back to the nature of human perception. When we listen to a violin being played in a normal room, our brain apparently builds some kind of stable “sound image” of that instrument, in a way that is not fully understood. This allows you to walk around, quite unaware of the huge sound changes from point to point suggested by Figs. 13 and 14. We already know about one ingredient of the process, from section 6.2. We talked there about the “precedence effect” and the idea of “echo fusion”. Our brains are capable, at least to an extent, of recognising early echoes as being extra copies of the direct sound. The information in these echoes is combined with the information in the direct sound to build up a composite perception. Far from confusing the sound, these early echoes can fill in missing information.

Of course, the early reflections in a typical room come from a range of directions: from the walls, floor, ceiling, and perhaps furniture. This means that these early echoes give the brain an opportunity to sample the directional sound field of the violin from several directions, and then combine these into a composite perception. This must be at least part of the underlying explanation for the “stable sound image”, compared to the situation in an anechoic chamber where the brain only has the direct sound from a single direction to work with.

Our computers and measurement systems are not as clever as our brains. We do not yet know how to process measurements like the ones in Fig. 13 to put together a corresponding “sound image” of that violin in that room. But, naturally enough, people have tried to devise ways of doing measurements that give at least some of the benefit of a reverberant listening space. Arthur Benade advocated measuring a “room-averaged response” by deliberately moving around in the room during the measurement sequence [5]. A somewhat similar effect is sometimes achieved in noise measurements by using a microphone suspended on a swinging pendulum.

Another variant was advocated by Erik Jansson. He made single-microphone recordings of violin music being played in a reverberation chamber, which is the opposite of an anechoic chamber: a kind of super shower cubicle, with very hard walls and minimal sound absorption. He then calculated a long-time averaged frequency spectrum from the recording [6]. The reverberation chamber allowed the microphone to sample the directional radiation of the violin very thoroughly, then the averaged spectrum gave a representation of the frequency distribution of total sound energy of that particular violin, during normal playing.

Finally, there is another way to use a small microphone to get some information about an instrument body: put the microphone in through a soundhole, and measure the internal sound in response to tapping on the bridge. The justification for this apparently odd procedure relies on some things we learned back in Chapter 4. At very low frequency the most important source of radiated sound comes from the monopole component of the motion, associated with volume change.

That volume change is a combination of the motion of the wooden parts of the body and the motion of the invisible “Helmholtz piston(s)” in the soundhole(s): one of them for an instrument like the guitar, two of them for a violin. But we saw in section 4.2.2 that the same net volume change also governs the pressure inside the body cavity, which will be approximately uniform throughout the space when frequency is low enough that the wavelength of sound is much longer than the body dimensions. So measuring the internal pressure is a surrogate for the far-field monopole sound radiation.

This approach has been developed into a useful measurement procedure by Colin Gough [7]. One big advantage of the approach is that it minimises the issue of room acoustics, because the inside of the body cavity is reasonably well insulated from external sound. Another advantage is for large instruments, like the double bass. It is very hard to get far enough away to be in the far field of an instrument that large, with a playing range extending to very low frequency. But the internal sound measurement is actually easier on a bass than on a violin, because the f-holes are bigger so it is easier to manipulate the microphone into the correct position inside.

However, there are limitations to the usefulness of the method. The justification just explained relies on very long wavelength, or very low frequency. How low? From section 4.1 we know that the key condition is that the “Helmholtz number” should be small. In section 4.3 we expressed that approximately in terms of the circumference of the radiating body: the wavelength of sound needs to be significantly longer than this circumference. For a violin body, taking the circumference around the widest part of the lower bout of the body leads to the condition that frequency must be lower than about 680 Hz. That just about encompasses the three “signature modes” A0, B1- and B1+, but the argument breaks down after that.

Something else happens at a frequency in the vicinity of B1- and B1+: the first resonance of the internal air in the cavity, which is a standing wave having approximately a half-wavelength in the length of the cavity. Gough’s procedure limits the influence of that mode by placing the microphone on a nodal line of this mode, and also on a nodal line of the first cross-wise standing wave mode.

But higher in frequency, internal air modes come thick and fast. The inside of a violin is just like a very small room, and the density of modal frequencies will show the same rising trend that we saw in Fig. 17 of section 4.2. As we showed in section 4.2.4, the typical spacing between adjacent modes decreases in inverse proportion to the square of the frequency. Somewhere in the low kHz range, the number of cavity modes inside a violin will overtake the number of “wood” modes from the violin body vibration. All these modes will create a trend in the internal sound spectrum, which has very little to do with the external sound radiation of the violin, and so is potentially misleading. So the conclusion is that the Gough approach is useful to study low-frequency signature modes, but it is likely to become increasingly less useful at higher frequencies.

D. The housekeeping of frequency response measurement

We have discussed a range of general issues to do with measuring frequency response functions. But if you actually want to do such measurements yourself, there are some details that you need to be careful about. These are probably not of much interest to the general reader, so I have tucked them away in the next link.

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[1] A. Zhang and J. Woodhouse, “Reliability of the input admittance of bowed-string instruments measured by the hammer method”   Journal of the Acoustical Society of America 136, 3371-3381, (2014).

[2] Oliver E. Rodgers and Pamela J. Anderson, “An engineering approach to the violin making problem”, Catgut Acoustical Society Journal 4, 2, 13—19 (2000).

[3] Gabriel Weinreich “Sound hole sum rule and the dipole moment of the violin”, Journal of the Acoustical Society of America 77, 710—718 (1985).

[4] George Bissinger and John Keiffer: “Radiation damping, efficiency and directivity for violin normal modes below 4 kHz”, Acoustics Research Letters Online 4 (2003), DOI 10.1121/1.1524623.

[5] Arthur H. Benade; “Fundamentals of Musical Acoustics”, Oxford University Press (1976), reprinted by Dover (1990).

[6] E. V. Jansson “Long-time-average spectra applied to analysis of music, Part III: A simple method for surveyable analysis of complex sound sources by means of a reverberation chamber”, Acustica 34, 275—280 (1976).

[7] Colin Gough, “Acoustic characterisation of string instruments by internal cavity measurements”, Journal of the Acoustical Society of America 150, 1922—1933 (2021), DOI: 10.1121/10.0006205.