7.3 Multiple strings and double decays

One obvious difference among the instruments presented in section 7.1 is that some instruments use a single string for each note (guitar, harp, violin, banjo) while others used pairs (lute, mandolin) or even triples (piano). In this section we tease out some of the consequences of using multiple strings. These may go some way to explain what gives each instrument its characteristic and recognisable sound. This account is inspired by a classic study of coupled piano strings by Gabriel Weinreich [1].

Suppose we have a pair of nominally identical strings, attached to the soundboard of an instrument at essentially the same point. Our first guess about what will happen is very simple. We imagine that the two strings are identical in every way, including being tuned to perfect unison. We also suppose that vibration of the soundboard is in one direction only, perpendicular to its surface. Under those circumstances, we can predict the behaviour without needing any detailed calculations.

This idealised version of the system has a symmetry: imagine a mirror between the two strings, oriented normal to the soundboard. As we saw right back in Chapter 2, if a system is symmetrical like this, every mode must be either symmetric or antisymmetric in that mirror. So every string overtone splits into a pair: one is symmetric, with the two strings vibrating in unison; the other is antisymmetric, with one string going up when the other goes down. Figure 1 shows a sketch, based on the lowest string mode. The symmetric mode exerts force on the soundboard, which will vibrate in response. But the antisymmetric mode exerts precisely zero net force, because the forces from the two strings always cancel out.

Figure 1. Idealised model of two coupled strings, tuned identically. The upper plot shows symmetrical motion, the lower plot shows antisymmetric motion with the two strings vibrating in opposite directions.

There are three results of this. First, the antisymmetric mode will make virtually no sound, because it does not excite vibration in the soundboard. Second, the symmetric and antisymmetric modes will have slightly different frequencies: the antisymmetric mode will be at the frequency of a string with a rigid termination at the bridge, but the symmetric mode will have a slightly different “effective length” because the bridge is moving a little. Third, the two modes will have different damping factors, and thus different decay rates. The symmetric mode is losing some energy to the soundboard, while the antisymmetric mode is not.

The net result of all this is that the system behaves for all practical purposes as if it had a single string, with twice the mass, twice the tension and twice the impedance of the two separate strings. If that was all we could achieve, it might have been simpler to use a single, thicker string. But, in reality, the behaviour is more complicated and the story is more interesting. Something always interferes with the exact symmetry, and this can have profound consequences. The most obvious possibility from the perspective of a player or a piano tuner is that the two strings might have very slightly different tensions, and that is the case we will examine first. But the soundboard vibration may also violate the symmetry condition: the point on the bridge where the strings are attached may move in a three-dimensional way rather than purely in the normal direction. We will return to that case a little later.

For a first view of what can happen, we will look at a simple model inspired by the sketch in Fig. 1. We will allow each string to have only a single “mode”, with a half-wave sinusoidal shape like the sketch. Both strings are rigidly anchored at one end, but at the other end they share an attachment to the “soundboard”, which for this purpose we will model as a single mass-spring-dashpot system, representing a single body mode. So our model has three degrees of freedom, describing the “body” motion at the bridge, and the amplitudes of motion of the two strings. We will calculate the modes of this coupled system; but the calculation will be a little more complicated than the modal problems we thought about back in chapter 2, because this time we will be particularly interested in the damping behaviour. The details of the model and the method used to compute the result are described in the next link.

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We can illustrate the range of behaviour through a case study, based approximately on the second course of a 12-string guitar. We will take a pair of steel strings with diameter 0.45 mm and length 650 mm, identical except for having slightly different tensions. They will both be tuned to frequencies near 250 Hz, but the two strings will be mistuned by a small amount, which we can characterise in cents — remember from section 2.3 that one cent is a hundredth of a semitone. Each string resonance, in isolation, is given a Q-factor of 3000.

The single body mode is chosen to match, approximately, one of the strong low-frequency modes of a guitar body. It will be given an effective mass of 200 g, and the stiffness will be adjusted to move the resonance frequency through a range around the string frequencies. Figure 2 shows the “bridge admittance” of this single-mode guitar body, for the 6 cases we will study. The resonance has a Q-factor of 50 in each case. The string tuning is indicated by the vertical dashed line. So you can see that the first curve, labelled (a), has the body resonance placed well below the string tuning. In the second curve (b) it is closer, and in the third case (c) the body resonance is right on top of the string tuning. Cases (d,e,f) have the body resonance progressively higher than the string tuning.

Figure 2. The bridge admittance of the single-mode “guitar body” used in the study of string mistuning. In all cases the effective mass is 200 g and the Q-factor is 50. The stiffness has been adjusted to 6 different values, to produce the 6 cases plotted. The labels ‘a’–‘f’ match the labelling in Figs. 3–5. The dashed vertical line marks the nominal tuning of the string investigated in those figures.

For each of the 6 cases, the degree of mistuning between the two strings will be varied through a range of $\pm8$ cents. The system has three degrees of freedom, so it has three modes. They will all involve motion of the strings and the body resonance, but the modes can generally be divided into two “string modes” and one “body mode” based on how the energy is shared between the degrees of freedom: we did something similar for the banjo model back in section 5.5. The frequencies, Q-factors and mode shapes have been computed at each stage, and the results for the two “string modes” are plotted in Figs. 3-5. First, Fig. 3 shows how the frequencies vary with the degree of mistuning. The 6 sub-plots (a–f) correspond to the 6 bridge admittances from Fig. 2, in order of increasing resonance frequency and labelled with matching letters.

Figure 3. Frequencies of the two “string modes” as mistuning is varied over a small range, corresponding to the 6 different placements of the single body mode shown in Fig. 2.

Concentrate in the first instance on sub-plot (a). This shows behaviour often called “curve veering” or “avoided crossing”, for reasons that are obvious in the plot. If you looked at the plot without your spectacles on, so that it was blurred, you might think that it shows two straight lines which cross in the middle of the plot — pretty much what you might have expected, since we have varied the tuning of one string while leaving the other one fixed. But look more carefully: the lines do not cross, but instead they veer apart at the last moment.

However hard you try, it is not possible to adjust the tuning of this pair of strings to produce exact unison, or in other words a state with no beats present in the sound. The coupling of the two strings through the bridge means that there is a minimum separation of the two frequencies. Really, we should have been expecting that from the earlier discussion. The minimum separation occurs when the strings are tuned with exactly the same tension, and the result is the symmetrical case we looked at earlier. The modes are as sketched in Fig. 1, and we already noted that the two frequencies will be different because one involves motion of the bridge, while the other does not.

Now look at the other cases in Fig. 3. They all show a similar “veering” shape, but as you go through the cases the closest approach of the two curves changes. In cases (b) and (c) the two curves get progressively further apart, and then they come closer together again once you get to the second row of plots. We already know from Fig. 2 what is special about case (c): this is the case where the body resonance falls very close to the string tuning. Figure 3(c) shows the effect of this nearby body resonance: the two curves stay well apart throughout the range of mistuning. Again, this conclusion is not a surprise: at the point of identical tensions in the two strings the body motion is particularly large because we are near a resonance frequency, so the frequency of the symmetric mode is shifted a long way away from that of the antisymmetric mode. In cases (d–f) the body resonance is above the string tuning, and progressively further away from it. What we see in in Fig. 3 is that the closest approach of the two curves gets nearer and nearer.

Figure 4 shows the corresponding behaviour of the modal Q-factors for the 6 cases, and Fig. 5 shows the ratio of amplitudes of motion in the two strings. In both cases the colours of the curves match the ones in Fig. 3. Looking at the patterns in Fig. 4, it can be seen that the Q-factors vary sensitively with mistuning, especially in the range where the frequencies were actively veering in Fig. 3. In the middle of each plot, where the mistuning is zero, the two Q-factors reach their maximum and minimum values. The minimum value varies significantly across the 6 cases, with the most extreme behaviour in case (c), where the body resonance falls close to the string tuning. This is the only case where the Q-factor of a “string mode” comes close to the Q-factor of the “body mode”, indicated by the dotted line. In all cases the red and blue curves come close together once the mistuning is relatively large, whether it is positive or negative.

Figure 4. Modal Q-factors plotted against mistuning, for the same 6 cases as in Figs. 2 and 3. Colours correspond to Fig. 3. The black dotted line indicates the Q-factor of the “body mode”. (Recall that Q-factor is the inverse of the loss factor we have talked about in some other sections: high Q means low damping, and slow decay; low Q conversely means high damping and rapid decay.)

Notice that in every panel of Fig. 4, the blue curve reaches a maximum value of 3000 at the point where the mistuning is zero. This is the case we talked about earlier: the two strings are identical, and the symmetry argument means that we know what the mode shapes must be. One is symmetric, the other antisymmetric. The antisymmetric case must have a Q-factor of 3000, because the strings cannot lose energy to the body mode so the mode exhibits the value we have assigned to the strings in isolation. On the other hand, the symmetric mode can lose a lot of energy to the body, leading to a low Q-value. Looking at the corresponding points in Fig. 5, this identification of the modes is confirmed. At that tuned point, the blue curve has the value $-1$ in every case, while the red curve has value $+1$: equal amplitudes, either with the same sign (red curves) or opposite signs (blue curves).

Figure 5. Modal amplitude ratios between the two strings, plotted against mistuning, for the same 6 cases as in Figs. 2 and 3. Colours correspond to Figs. 3 and 4.

But when the mistuning is non-zero, the mode shapes are no longer exactly “symmetric” or “antisymmetric”. What we can see in Fig. 5 is that when the mistuning is large, the ratio of amplitudes of the two strings gets either much bigger than unity, or else much less than unity, as can be seen in Fig. 5. This has a simple interpretation. If the strings are sufficiently out of tune, each mode simply has one string vibrating strongly while the other hardly moves: exactly what you would expect without taking into account coupling between the strings through the body.

However, one feature persists throughout the plots: the blue curves always describe a case in which the two amplitudes have opposite signs, while the red curves always correspond to the same signs. You can see this directly in Fig. 5: the blue lines are always below the x-axis, while the red curves are always above it.

Looking back at Fig. 3, we can see that the red and blue curves swap places once the body resonance is above the string tuning. What is happening is that the body influence switches from “mass-like” to “stiffness-like” depending whether the string frequency is above or below the body resonance (the reason for that was explained back in section 2.2.7). This determines which of the two string modes has the lower frequency.

Having seen the behaviour of the individual modes, we can put these together to make a kind of “pluck response” of our one-mode coupled strings. This will consist of a combination of the three modal responses, in a mixture governed by the assumed initial conditions: technical details are described in the previous link. There are two extreme cases that are illuminating to look at: either the two strings are initially excited identically, or just one string is driven. This is an idealisation of the two choices for playing a note on the piano: ordinarily the piano hammer strikes the pair or triplet of strings simultaneously, but if the una corda pedal is pressed the hammer is shifted sideways so that it only strikes one or two strings.

The output variable we are interested in is the motion of the “soundboard”, represented here by the single body mode. This motion corresponds, in our crude model, to the source of sound from the vibrating soundboard. To bring out the most interesting aspect of the behaviour, the results will be plotted in the form of an envelope. We smooth over the individual cycles of the vibration to show how the amplitude varies in time, then plot it on a decibel scale. We can start with the most extreme case, where the body mode nearly matches the string tuning: this is case (c) in Figs. 2–5, the case that gave the biggest effects.

Sub-plot (a) of Fig. 6 shows the envelope curves for the two excitation cases, when the strings are perfectly tuned in unison. The red curve shows the result from driving both strings, while the blue curve shows the una corda response. The main thing we see here is a pair of lines, sloping steeply downwards, parallel to one another. This is the case of perfect symmetry, in which one string mode is antisymmetric so that it does not excite the soundboard at all. We see the rapid decay associated with the symmetric string mode, which has a very low Q-factor as seen in the red curve of Fig. 4(c). At the start of the note, we also see some wiggles in the envelope: this is the result of transient excitation of the body mode. The only difference between the red and blue curves in this case is a vertical shift by 6 dB, arising simply because when both strings are initially driven the amplitude of the symmetric mode is twice as high as when only one string is driven.

Figure 6. Envelope plots for the transient response of the string model, for the case corresponding to sub-plot (c) in Figs. 3, 4 and 5. Red curves show the result of driving the two strings simultaneously, blue curves show the response to driving only one of the strings. The sub-plots correspond to different levels of string mistuning: (a) 0; (b) 1 cent; (c) 3 cents; (d) 5 cents.

Moving to Fig. 6(b) we see something more complicated. The two strings are now mistuned by just 1 cent, breaking the symmetry so that both string modes are now able to excite the soundboard. Note that 1 cent is a very small difference of tuning — surely near the limit of what guitarists or piano tuners are likely to be able to achieve. Both the curves in this plot show a characteristic double decay: a fast initial decay, giving way to a much slower decay in the tail of the note. We are seeing the effect of the very different Q-factors of the two string modes, shown in Fig. 4(c). Provided both modes are driven to some degree, a decay pattern like this is inevitable: the fast-decaying mode, however loud it may be initially, is bound to decay below the level of the other mode after a time, leaving that slower-decaying mode to dominate the sound.

In piano terms, the second, slower-decaying, phase is sometimes called the “aftersound”. The balance between the early sound and the aftersound is quite different in the red and blue curves of Fig. 6(b). Although the string modes are no longer exactly symmetric and antisymmetric, with this small degree of mistuning the two mode shapes have not changed a great deal. The result is that striking both strings together excites mainly the mode with the strings in the same phase, which is the one with the fast decay. It takes some time for this initially louder mode to decay enough to reveal its slower-decaying companion.

But in the una corda case both modes are excited to similar amplitudes. This means that the aftersound takes over sooner, and is a bigger component of the overall sound. As Weinreich pointed out [1], this is probably the main function of the una corda pedal on a piano. Although it is often called the “soft pedal”, the difference of loudness with and without this pedal is not very great. More important, probably, is the fact that the sound is less percussive and dominated by the aftersound: more “lyrical”, perhaps.

As we increase the mistuning, in Fig. 6(c,d), the pattern remains recognisably similar but the details change. The decay rate of the aftersound component gets faster, and the difference in level between the blue and red curves is somewhat reduced. At least for this special case, in this very simplified model, we get a strong hint that a skilled piano tuner might be able to exercise considerable influence over the sound by controlling very small levels of mistuning between the strings.

Figure 7 shows a similar set of envelope curves to Fig. 6, for the case corresponding to case (d) of Figs. 2-5. Case (a), with the strings in perfect unison, looks quite similar to the previous case, except that the slope of the parallel lines is less steep because the Q-factor of the symmetric mode does not fall as low this time. The other sub-plots show rather more complicated patterns. Case (b) shows a trace of the double decay profile in the blue curve (it starts below the red curve and ends above it), but all the curves show strong wiggles. These correspond to beats between the frequencies of the two “string modes”. These frequencies are not exactly the same as the nominal tuned frequencies of the two strings in isolation, but they are less affected by the coupling in this case because the bridge admittance at the string frequency is lower by almost 20 dB: compare the heights of the yellow curve and the purple curve where they cross the vertical dashed line.

Figure 7. Envelope plots for the transient response of the string model, for the case corresponding to sub-plot (d) in Figs. 3, 4 and 5. Red curves show the result of driving the two strings simultaneously, blue curves show the response to driving only one of the strings. The sub-plots correspond to different levels of string mistuning: (a) 0; (b) 1 cent; (c) 3 cents; (d) 5 cents.

Of course, we would like to know what this all sounds like. It is not very interesting to listen to the sounds from the simple model with just a single mode in each string: all you hear is a variety of modulated sine waves, not sounding at all like a plucked string of any kind, and therefore not suitable for making interesting musical judgements. But we can take the model of two strings coupled via a soundboard, and use the synthesis approach described back in section 5.4 to create versions of the sound of the open second course of a 12-string guitar, which motivated the model we have been looking at.  Sound 1 illustrates the “una corda” case, where only one of the strings is plucked.  Sound 2 is the “regular piano” case (or perhaps it is better thought of as a “regular harpsichord” case since we are modelling plucked strings rather than hammered strings) where both strings are plucked equally and simultaneously. In both cases you will hear three notes, corresponding to perfect unison, 1 cent mistuning, and 5 cents mistuning. The sounds are encouragingly familiar from the sounds made by a 12-string guitar: realistically, 1 cent is as accurate as any guitarist is likely to tune their strings, but 5 cents represents a serious degree of mistuning.

Sound 1. Synthesised pluck sounds, for two steel strings coupled through a guitar body (using a measured bridge admittance). The first note has the strings tuned in perfect unison, the second has 1 cent of mistuning, the third has 5 cents mistuning. These examples illustrate the “una corda” case, in which only one of the two strings is plucked.
Sound 2. Synthesised guitar-like sounds as in Sound 1, except that now both strings are plucked equally and simultaneously.

These two cases may be natural for discussing the piano, but they are rather artificial in the context of a guitar. But we still learn something interesting from them, because they give an indication of the extent to which a player may be able to vary the sound by subtle details of performance technique. It is hard to guess exactly how a player will pluck a course of two strings. They probably excite both strings to some extent, but not exactly equally and not exactly simultaneously. The cases demonstrated here, which sound strikingly different from each other, give extremes from the range of possibilities. Most normal notes probably lie somewhere in between the two.

There is one circumstance under which a note on a guitar, banjo or other stringed instrument can show the behaviour we have been looking at, even when it has only single strings. This is the case where sympathetic resonance occurs with one of the other strings of the instrument. A simple example would be to finger a note on a lower string at the same pitch as one of the open higher strings. You then have two strings, coupled at the bridge, tuned to the same nominal pitch: exactly the model we have been studying. But they are not a course, where both strings would be played together: you play one string, and the other will respond just like the una corda case, as just discussed.

However, this is not the end of the story. Multiple strings are not the only way to get sound envelopes showing a double decay. There is another aspect of string vibration, which we have been playing down: each string mode can occur in two polarisations. In other words, the string could vibrate in two different planes. If both ends of the string were rigidly anchored in a symmetrical way, these two modes would occur at identical frequencies. What would happen then is similar to something we saw earlier, when we looked at drums. As was shown in section 2.2.4, linear combinations of this pair of string modes can be found, which would correspond to vibration in any chosen plane.

But when the string is attached to a real instrument, things get more complicated. Motion at the bridge, and to a more limited extent at the other end of the string at the nut or the fret, leads us to expect that vibration in the plane perpendicular to the soundboard will be different from vibration parallel to the soundboard of the instrument. Instead of the plane of vibration being arbitrary, the combined system of string and instrument body will choose two particular orientations, resulting in a pair of modes with slightly different frequencies.

At first sight it seems obvious that these orientations will be parallel and perpendicular to the soundboard. Well, that is sometimes true, especially at low frequencies, but it is not always true. The details depend on the particular instrument. Think about the guitar, as a typical example. The string positions on the bridge saddle of a guitar stand a few millimetres above the soundboard. As the soundboard vibrates, the bridge participates in that motion. But that does not necessarily mean that the top of the bridge moves purely normal to the soundboard surface. Bending modes of the soundboard can also involve rotation of the bridge, like the sketch in Fig. 8 (showing a cross-section through the guitar body in the plane of the bridge). The strings experience more complicated three-dimensional motion, slightly different for each one: motion at the string notches associated with this mode is not purely normal to the surface. The sketch also tells us something else: through the rotational component of the bridge motion, string vibration in the plane parallel to the soundboard can excite soundboard vibration, and thus create sound radiation.

Figure 8. Sketch of a guitar body and bridge, undergoing a vibration mode of the soundboard. The six string positions on the bridge are indicated by black dots. The combination of displacement and rotation means that each string moves in a different way during this vibration

Pulling these features together, we find something interesting. Each “string mode” is in fact a pair of modes, with slightly different frequencies. They will also have different damping factors, because the rate of energy loss to the soundboard will be different. Both modes can create at least some sound radiation. When the player plucks the string, they will excite some mixture of the two modes. The result will be a “double decay”, for reasons that are very similar to the earlier discussion about the modes resulting from a coupled pair of strings. The mode that couples better to the soundboard will tend to make more sound, but it will decay faster. Sooner or later, it will have decayed below the level of the other mode, with its slower decay.

To complete the story of double decays, we can ask whether all instruments show the effect, or whether some are more prone to it than others. The physics we have just described is always present, of course. Every stringed instrument can exhibit effects of the two polarisations of motion, and any instrument with double or triple strings (or sympathetic strings) will have coupled mode behaviour similar to what has been seen here. But that does not necessarily mean that a double decay will occur at a level to be perceptible in the sound of the instrument.

We can use information we have already developed to give a very simple criterion for the effect to be significant. The crucial factor, for both physical mechanisms of generating double decays, is that the decay rates associated with the “early sound” and the “aftersound” must be significantly different. If they are not, the envelope will not be able to show the characteristic transition that we saw in Fig. 6(b,c,d). To get significantly different loss factors, either for the in-phase mode of multiple strings or the favoured polarisation of a single string, the energy loss into the body must be significant in comparison with the internal loss factor of the string.

A double-decay envelope is mainly dominated by the fundamental, or the very low overtones of the string. This is the frequency range where the string’s intrinsic damping is dominated by air viscosity effects: we gave an expression for the associated loss factor $\eta_{air}$ in section 5.4.5. (Remember $\eta$ is Greek letter “eta”.) This is to be compared to the loss factor $\eta_{body}$ associated with energy loss to the body: in section 5.1.2 we derived an expression for that in terms of the bridge admittance. So we can simply plot $\eta_{air}$ and $\eta_{body}$ on top of each other, as functions of frequency, for some representative instruments. That will tell us immediately whether the double decay effect is likely to be important for that instrument.

The first example, shown in Fig. 9, is for a steel-string guitar. The green line shows $\eta_{air}$, while the red lines show $\eta_{body}$. The solid curve is for a single string, the dashed curve is for a double string like the example discussed above: the loss factor is doubled (i.e. raised by 6 dB in this plot), because the combined impedance is doubled with a pair of strings as in a 12-string guitar. The figure reveals that the red curves only rise convincingly above the green curve around strong resonance peaks. That is exactly what we found in the discussion of Figs. 2–7: there was a strong effect in the case where the string tuning was close to the single body resonance, but only much weaker effects otherwise.

Figure 9. Comparison of loss factors for a steel-string guitar, based on the second string. The solid green line shows the loss factor due to air damping, $\eta_{air}$. The solid red line shows the loss factor due to energy flow into the body at the bridge, $\eta_{body}$. The dashed red line shows the effect of turning the guitar into a 12-string instrument, to match the example studied earlier.

This is to be contrasted with the situation revealed in Fig. 10, for a piano. This time the solid curve shows the effect of the actual triple string group, while the dashed line shows what would have happened if the piano had only single strings. This time the loss factor is raised by a factor 3 with three strings. The solid red line lies above the green line over the entire frequency range, often by a substantial factor of about 20 dB. That is precisely the condition for the double decay phenomenon to occur strongly, for all notes rather than just for ones that happen to fall near a soundboard resonance.

Figure 10. Loss factor comparison in the same format as Fig. 9, for a piano (the baby grand shown in Fig. 1 of section 7.1), based on plain wire string triples in the mid-range of the instrument. The solid red line shows the result with three strings; the dashed red line shows how it would change with only one of the same strings.

In the piano, the excitation direction of the strings is controlled in a very standardised way by the piano mechanism. This means that the multiple-string effect can operate strongly, including the effect of the una corda pedal, but the effect of polarisations is probably rather minor because the strings are excited predominantly in the vertical plane. Also, there is no equivalent of the discretion a guitarist has over the exact orientation of the initial string motion.

The final example, in Fig. 11, shows the corresponding plot for a banjo. The banjo has only single strings, but we have already seen in section 5.5 that they are coupled to the “soundboard” (which is of course a membrane in the banjo) far more strongly than occurs in a guitar. The result in Fig. 11 is that the red curve lies well above the green curve over most of the frequency range. So we can expect banjo notes to exhibit a strong double decay envelope: not in this case because of multiple strings, but arising from the two string polarisations. This has indeed been observed experimentally, by Stephey and Moore [2].

Figure 11. Loss factor comparison in the same format as Fig. 9, for a banjo, based on the thickest plain steel string (the 3rd string).

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[1] Gabriel Weinreich “Coupled piano strings”; Journal of the Acoustical Society of America 62, 1474–1484 (1977).

[2] L. A. Stephey and T. R. Moore, “Experimental investigation of an American five-string banjo,” Journal of the Acoustical Society of America 124, 3276–3283 (2008).