Chapter 5 Strings, mainly plucked

5.1 Stringed instrument overview

Figure 1: a selection of acoustic stringed instruments

Stringed instruments come in many forms: a few are illustrated in Fig. 1. This chapter will largely be concerned with plucked-string instruments (because we are sticking as long as we can with linear theory: remember that idea from section 2.1?). But we will give a broad description of how an acoustic stringed instrument works, and for that purpose all stringed instruments have some things in common, whether plucked, hammered or bowed. The process of bowing a string is definitely nonlinear (we will come back to it in Chapter ?), but the body vibration and sound radiation characteristics of a violin or cello are essentially linear.

The musician devotes virtually all their effort and skill towards making the strings of their instrument vibrate in particular ways: at the right pitch, at the right time, and with subtle control over the initial transient and the tonal balance. But, as we can understand from section ?, a vibrating string cannot radiate much in the way of sound waves into the surrounding air. It is simply far too thin compared to the wavelengths of sound in air at audio frequencies. Most strings are only around 1 mm in diameter, whereas the wavelength of sound in the mid-audio range is of the order of hundreds of millimetres, and even at the very highest frequency audible to humans, around 20 kHz, it is about 17 mm. Air simply flows around a vibrating string, getting out of the way rather than being compressed to produce sound waves.

In order to make sound that is loud enough to be useful, some of the energy in the vibrating string must be used in one way or another to cause vibration of something bigger, comparable in size to the wavelengths of interest. In an electric instrument, this larger vibrating object is the loudspeaker cone, set into vibration electrically by an amplifier. But in an acoustic instrument, it is usually woodwork of some kind.

As the string vibrates, the end in contact with the instrument body at the bridge will have a time-varying angle of contact. The string is, of course, under tension, and this varying angle means that a component of the tension is exerted on the bridge, perpendicular to the string in its plane of vibration. This force drives the body into vibration, and the vibrating structure will radiate sound waves into the surrounding air. Figure 2 gives a sketch to illustrate what is going on, for the situation where the string vibrates in a plane perpendicular to the soundboard, so that the time-varying angle lies in that plane.

Figure 2. Sketch of a vibrating string on an instrument. The changing direction of the string tension at the bridge produces an oscillating force on the bridge, which excites vibration of the body.

There is a complication. For definiteness, it will be described in the context of the guitar. A string can indeed vibrate in the plane perpendicular to the soundboard as sketched in Fig. 2, but it could also vibrate in the plane parallel to the soundboard, or anywhere in between. If the string vibrates in the plane parallel to the soundboard, the force on the bridge is also parallel to the soundboard and this is not an efficient way to make the soundboard vibrate. Vibration perpendicular to the soundboard is much better: the force is now oriented in the best way to make the soundboard vibrate, and the sound is likely to be louder. This difference lies behind the guitarist’s techniques known as apoyando and tirando: these different ways to pluck a string with the fingers make a difference to the plane of vibration of the string. An apoyando stroke (sometimes called a rest stroke) produces more string vibration in the plane perpendicular to the soundboard, and gives a louder and fuller sound.

Now most of the bowed instruments have a problem. As will be seen in section ?, a bowed string vibrates, mainly, in the plane of the bow-hair. In order to be able to access the strings without the bow hitting the body, in most bowed instruments (like the violin) this means bowing approximately parallel to the soundboard. But we have just seen that this is not a recipe for making a loud sound. This is the reason that a violin or cello needs the familiar high bridge, quite different from the squat bridge of a guitar. The string exerts a force at the top of the bridge, approximately parallel to the soundboard. But this force is translated by a rocking action of the bridge into forces at the bridge feet that are perpendicular to the soundboard.

Some bowed instruments escape this problem. In the Chinese erhu, for example, the bow is threaded between the two strings and they are bowed perpendicular to the ‘soundboard’ (which is actually a stretched snakeskin membrane in this particular instrument). There is no need for a high bridge: the erhu has a low bridge more like a guitar bridge, and it is quite loud.

Figure 3. PICTURE OF ERHU

There is an important consequence of the sequence of events just described, and illustrated in Fig. 2. To a good first approximation, all interaction between the string and the body occurs at a single contact point: the string notch on the bridge. So to characterise the vibration behaviour of the body in so far as it affects the string motion, what is needed is a measurement of the frequency response function at this point.

Such a measurement was illustrated in Fig. 4 of section 2.2, and Fig. 5 of that section showed some results for the toy drum. A (very) small hammer can be used to tap the bridge by the string notch, and the response can be measured with a laser vibrometer or an attached sensor with very low mass. The waveform of force is measured by a built-in sensor in the hammer. The two signals, force and response, are recorded into a computer, and the FFT is used to convert them into frequency spectra. The output spectrum (FFT of the response) is divided by the input spectrum (FFT of the force waveform), and the result is the frequency response function we are seeking. The most common version is based on an output signal proportional to the velocity at the measurement point, and is called the bridge admittance or bridge point mobility.

Figure 4. A banjo set up for measurement of the bridge admittance. The hammer taps the bridge near the notch for the top string and the red spot from the laser vibrometer is nearby. Vibration of the strings, both in the playing lengths and in the afterlengths between the bridge and the tailpiece, has been suppressed by weaving a piece of paper through them.

Figure 4 shows a typical setup, on a banjo. Figure 5 shows a slightly modified procedure being used on a violin. As mentioned above, violin strings vibrate predominantly in the plane of the bow-hair. To measure the most relevant bridge admittance the force has to be applied in the bowing direction, rather than perpendicular to the soundbox, and the response needs to be similarly measured. It is easier in practice to measure the response from the corner of the bridge on the other side, on the right in this picture. This means that the result is not strictly a point admittance, but provided the top portion of the bridge moves without deforming, it is a good approximation to it [1].

Figure 5. Bridge admittance being measured on a violin. The force needs to be applied in the direction of bowing, because the string vibration occurs predominantly in that direction.

As with all measurements, there are some tricks of the trade for getting best results. It is prudent to collect several sets of data, and use an averaging procedure to combine them into a best estimate of the bridge admittance. Some details of the process are given in the next link. A side effect of the procedure is something called the coherence function, which gives a very helpful measure of data quality: an example is shown in the link. The hammer is held in a pendulum fixture to guarantee that it hits the same spot every time. We want to characterise the body behaviour separately from the strings, in order to couple them together in a controlled way later (see section 5.4). So for the admittance measurement the strings are damped to suppress their vibration: in this case using a piece of paper woven through the strings. But we don’t want to remove the strings entirely: on instruments like the banjo or the violin, the tension of the strings is needed to hold the bridge in place. A more subtle effect of the strings is that the body vibration can be affected significantly by the stiffness of the strings along their length: especially for metal strings, this stiffness can be quite high.

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There is a final step: in order to obtain a bridge admittance that is quantitatively correct, some kind of calibration procedure is needed. The measurement must be repeated on some system for which the answer is already known, so that a suitable scaling factor can be determined. The most common approach is to weigh a mass, then hang it as a pendulum. Applying the hammer/laser measurement method to this, we should obtain a result that agrees with Newton’s law: force=mass$\times$acceleration.

It is time to see some examples. A typical measured and calibrated admittance is shown in Fig. 6, for a violin. We will say something about what it shows in a moment, but first it is helpful to convert it to a different form. In the course of this chapter we will use three contrasting instruments to illustrate the range of possibilities: a classical guitar, a violin and a banjo. How can we make a fair comparison between instruments of different types, employing different strings and tuned to different pitches? To give a good answer to that question we need one more piece of background information. For reasons explained in the next link, the strength of coupling between a string and the instrument body is determined by the product of the admittance we have measured and a property of the string called its wave impedance or characteristic impedance. This impedance is defined as

$$Z_{string}=\sqrt{T m}$$

where $T$ is the string tension and $m$ is its mass per unit length.

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Figure 6. The magnitude of the bridge admittance for a violin made by David J Rubio.

To show the most useful comparison between different instruments, we can scale the bridge admittance by the impedance of a typical string: we will use the top string. We can also adapt the frequency scale to take account of the different tunings: we will show a scale in semitones, starting from the fundamental frequency of the lowest string. The result, for the three chosen instruments, is shown in Fig. 7. The frequency scale covers 5 octaves, a frequency ratio of $2^5=32$. The scaled admittance needs no units: it is a dimensionless quantity. It directly describes the rate at which energy from the vibrating string can ‘leak’ through the bridge into the instrument body. An informal description would be that the higher the level in the plot, the louder the played note is likely to be. Some relevant parameters for these three instruments are listed in Table 1.

Figure 7. Bridge admittance magnitude multiplied by the wave impedance of the top string, for three different stringed instruments. Red curve: the violin as in Fig. 5; purple curve: a classical guitar made by Martin Woodhouse; blue curve: the banjo seen in Fig. 1. The frequency scale is expressed in semitones, starting from the lowest tuned note of each instrument. Successive octaves are indicated by ticks.
ViolinGuitarBanjo
Lowest frequency196 Hz82.4 Hz147 Hz
$Z_{string}$ for top string0.18 Ns/m0.16 Ns/m0.15 Ns/m
$Z_{string}$ for lowest string0.35 Ns/m0.67 Ns/m0.29 Ns/m
Table 1. Key parameters for the three instruments whose bridge admittances were shown in Fig. 7. String impedances are based on typical string sets: different string choices give slightly different values.

The three curves in Fig. 7 all show a pattern of peaks and dips. At low frequencies, there is a very simple interpretation of the peaks: they correspond to individual vibration modes of the respective instruments. However, things get more complicated at higher frequencies. Recall from section 2.2.7 that each mode contributes a peak with a characteristic bandwidth determined by its level of damping. It is usually characterised by the half-power bandwidth, the frequency range between the two points where the amplitude falls to $1/\sqrt{2}$ of the peak value. Energy is proportional to the square of amplitude, so these correspond to the points where the energy involved in the vibration has halved. On a decibel plot like the ones shown here, these points occur where the amplitude has fallen by approximately 3 dB (because $20 \log_{10} (1/\sqrt{2}) = -3.01$). In terms of the modal Q-factor which we have used to characterise damping, this half-power bandwidth is equal to the peak frequency divided by the Q-factor.

Now we can learn something interesting by combining two facts that have been mentioned earlier. For any plate-based structure, like a guitar or a violin, the spacing between adjacent modal frequencies is, on average, constant (see section 3.2.4). On the other hand, it is an empirical observation that the modal damping of such structures usually shows, approximately, a constant Q-factor for all modes. So, roughly, the peaks are uniformly spaced along the frequency axis, while the half-power bandwidth of each peak increases approximately proportional to its centre frequency. The result is that the first few resonant peaks may be well separated, because the spacing is large compared to the bandwidth, but as we look higher in frequency, the spacing stays the same while the bandwidth increases, so the peaks begin to overlap. Eventually, there may be several modes with resonance frequencies within the half-power bandwidth of each separate mode: this number is called the modal overlap factor.

Once modal overlap becomes significant like this, the admittance function at any particular frequency is influenced by several modal contributions. But these will all be in different phases, so that interference will occur. We can no longer rely on the fact that a peak corresponds to a resonance: peaks will occur where several overlapping modes have responses that combine constructively because of their phase relations.

Returning to Fig. 7, all three of the curves show this pattern. (The argument given above doesn’t apply directly to the banjo, which is not plate-based, but we will see in section ? that the modal density and hence the modal overlap increases more rapidly with frequency for a membrane-based system.) There is a great deal that can be learned from the details in Fig. 7: so much so that we will need to take it in easy stages. A detailed look at the similarities and differences of the three instruments will be deferred until section 3.3. First, we will look at a deceptively simple question, often asked of stringed instrument makers: “I like this instrument, but can you make it louder?”

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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).