7.4 Add a touch of nonlinearity

Some stringed instruments give the impression that they should be approximated well using linear methods, but in fact crucial details for the sound involve nonlinear effects. In this section, we dip into two examples. Both of them were included in the set of plucked-string example sounds in section 7.1: the harp and the lute. In both cases the nonlinear effects are fairly weak, and we will be able to see roughly what is going on by simple approaches. But a full discussion of nonlinear effects needs more care, and after this introductory “toe in the water”, we will give a more systematic description in the next chapter.

If you were able to pick out the harp from the set of instruments in Section 7.1, you will know that there must be something characteristic about the sound, even of a single note. Probably, though, you find it hard to put the distinctive quality into words. We can start by looking for physical features that distinguish a harp from other plucked-string instruments. First, the physical configuration is unusual. In the vast majority of plucked or struck stringed instruments, the strings lie approximately parallel to the soundboard. However, the strings of a harp meet the soundboard at an angle: Fig. 1 shows the small harp used in this study, where the string-soundboard angle is 35$^\circ$.

The other conspicuous physical feature that distinguishes the harp from instruments like the guitar is that it has a separate string for each note. This means that there are usually a large number of sympathetic strings contributing to the sound of a given note, and the layout of strings also influences many aspects of performance style. Two of these will be particularly important here. Harpists normally pluck their strings closer to the centre than is usual in other instruments. They can also impose a much higher initial displacement than a guitarist is able to use, because there is no equivalent of the danger of a “fret buzz”. The amplitude is only limited by the spacing between strings, about 15 mm for the harp shown in Fig. 1. As we will see shortly, string motion with large amplitude is likely to bring nonlinear effects into play.

Figure 1. The small harp used in this study, and also for the sound demonstration in Section 7.1.

A harp string, like all other instrument strings, can vibrate in two polarisations. One polarisation, in the plane of the strings, generates a component of force that can excite soundboard vibration efficiently. String vibration in the perpendicular plane, however, will drive the soundboard less strongly. The player will normally excite string vibration with some mixture of both polarisations, which means that harp notes might exhibit the double decay phenomenon discussed in section 7.3. However, the effect does not in fact seem likely to be a very strong one: Fig. 2 shows a plot similar to Figs. 9–11 of section 7.3, using the string properties and measured bridge admittance of the small harp. It can be seen that the red curve rarely rises above the green curve. This means that the decay rates for vibration in the two polarisations will not be significantly different. We must look elsewhere for an explanation of “harpish” sound.

Figure 2. Loss factor comparison for the harp in the same format as Figs. 9–11 of Section 7.1, based on the string to be used for tests later in this section. This is a nylon string with length 390 mm and diameter 1.045 mm, tuned to E$_4$ (329.7 Hz). It is the 20th string from the top of the instrument.

One candidate might involve longitudinal string vibration, something that is often ignored in discussions of plucked instruments. The geometric arrangement of strings and soundboard in a harp means that there is a direct coupling between transverse and longitudinal motion in the string. The harp soundboard will prefer to vibrate normal to its surface, moving the string’s termination in a direction that must involve both transverse and longitudinal motion.

This coupling can be modelled using linear methods, similar to the synthesis models described earlier (see section 5.4). The conclusion is that this effect, too, seems to be rather minor: unimportant enough, in fact, that we needn’t go into details here: see [1] for the full story. Our quest for the distinctive source of “harpish” sound seems to be thwarted at every turn. However, we must not be too hasty. What has been mentioned and then dismissed here is longitudinal string motion driven via linear vibration. But this is where nonlinear effects start to appear: longitudinal string vibration, able to drive the soundboard efficiently because of the string-soundboard geometry of the harp, is also generated by another mechanism.

There are two aspects of nonlinear excitation of longitudinal string vibration, and both have been shown to give audible effects in some musical instruments. They both have their origin in the fact that when a string vibrates transversely, the length of any small element of the string must increase a little as a simple consequence of Pythagoras’ theorem: this is called geometric nonlinearity. The first effect is that the small increase of the total string length causes an increase in mean tension. Higher tension raises the pitch of the transverse string vibration, of course, so during the early part of a vigorously-plucked note a “pitch glide” can occur, starting a little higher than the expected note. The pitch moves rapidly downwards as the string motion decays, allowing the tension to fall back towards its nominal level.

We can illustrate with a spectrogram. Figure 3 shows a portion of the spectrogram of a strongly-plucked harp note: you can listen to it in Sound 1. The spectrogram contains some stripes that are approximately vertical, showing the frequency components of the note decaying away with time. But if you look carefully, you can see that the stripes all bend a bit towards the right in the bottom half of the plot. This is the pitch glide effect: in the early part of the note the tension is a little higher so that all the component frequencies are raised a bit.

Figure 3. Spectrogram of a strongly-pluck note on a harp, showing the pitch glide effect
Sound 1. Soundboard acceleration from a regular fingertip pluck of a harp string. Level in dB is indicated by colours, according to the scale on the right-hand side.

The second effect of nonlinearity is more subtle. Dynamic axial forces are generated in the string, leading to unexpected frequency components in the sound, sometimes described as “phantom partials”. These forces can also directly excite the longitudinal string resonances. Figure 3 shows examples of both effects. Notice that the pattern of stripes is not regular. Starting with the stripe around 2.7 kHz, instead of a single line there are two lines close together. The next group near 3 kHz shows a pair slightly more widely separated, and this progression of wider and wider spacing continues as you move across towards the right-hand side of the plot. We will find out in a bit how this pattern arises.

Figure 3 also shows a bright patch at early times (bottom of the plot) around 3.4 kHz. This is the frequency of the first longitudinal resonance of this particular harp string. Conklin [2] has demonstrated the importance of longitudinal string resonances for the tone of the piano, to the extent that a piano designer needs to take the effect into account to achieve the best tonal balance: some convincing sound demonstrations from Conklin’s work can be found in this web link. All these effects of nonlinearity in string vibration are usually discounted for non-metallic strings like nylon or gut harp strings, but Figure 3 demonstrates that they can in fact appear strongly in plucked harp notes. They may well play a significant role in the distinctive character of harp sound.

To investigate the pattern of phantom partials, and related nonlinear effects, we will look at some more careful measured responses. To pluck a string in the most well-controlled way, the best approach is to take a length of very thin copper wire, loop it round the string, and pull gently until it snaps. 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. This kind of wire-break pluck will not sound the same as a normal harpist’s finger pluck, of course, but it can reveal the underlying physics in the clearest form.

To bring out the first piece of key behaviour we will look at a carefully-chosen special case. The string was plucked at its mid-point, in the direction in the plane of the strings and also in the perpendicular direction. The responses were measured by a small accelerometer fixed to the soundboard, very close to the attachment point of this string. The resulting frequency spectra are shown in Figs. 4 and 5, for two different frequency ranges. The pluck in the plane of the strings is shown in blue, the pluck in the perpendicular plane in red. Because the two curves are sometimes on top of each other, the positions of the main “string mode” peaks are indicated by markers in corresponding colours.

Figure 4. Frequency spectra of soundboard acceleration following wire-break plucks at the mid-point of the string, in two perpendicular directions. The blue curve shows the direction in the plane of the strings, the red curve shows the perpendicular direction. Short horizontal markers indicate the heights of the main peaks, in the corresponding colours. This plot shows the response at low frequency: it continues in Fig. 5.

Look first at Fig. 4, and concentrate initially on the blue curve. At first glance, this shows what we expect to see: a sequence of tall and sharp peaks corresponding to “string modes”, with some wiggles at lower level resulting from the resonances of the harp body. But in fact the plot reveals a surprise. Remember that the plucks were at the mid-point of the string. As we found in section 5.4 (with details in section 5.4.1), linear theory would predict that a mid-point pluck can only excite odd-numbered modes of the string, because all the even-numbered ones have a nodal point at the centre. But Fig. 4 shows peaks 1, 2 and 3 with roughly similar heights. After that, mode 4 is indeed very weak, then mode 6 is moderately strong.

How does the high peak around mode 2 arise? To understand what has happened, we need to know a bit more about the mechanics of nonlinear excitation of longitudinal vibration. An approximate theoretical model is derived in the next link. This model shows that the excitation of longitudinal motion is governed by a function involving the square of the transverse motion: this is the essential source of nonlinearity. A direct consequence of this square law, also explained in the next link, is that the longitudinal motion may contain extra frequencies compared to the transverse motion: it can involve components at double the frequency of transverse components, and also the sum and difference of pairs of transverse frequencies.

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This gives an immediate clue about the unexpected peak number 2 in Fig. 4. There are several ways that this frequency could be generated through the square law: or, strictly, there are several ways that frequencies close to this can be generated. If the transverse string motion consisted of perfect harmonic frequencies, we could generate the frequency “2” by doubling the fundamental frequency “1”, but we could also obtain it as the difference between pairs like 3 and 1, 5 and 3, 7 and 5, and so on. But things are a little more complicated, because of the inharmonicity associated with bending stiffness. These various combination frequencies all fall close to the frequency “2”, but they will all be slightly different from each other, leading to a cluster of frequencies. This effect is hardly visible in Fig. 3, but a similar effect will become important shortly when we look at phantom partials at higher frequency. The previous link explains the effect in more detail.

Figure 4 contains direct evidence that this strong peak at approximately double the fundamental frequency is caused by nonlinearly-generated axial force. The key is to compare the peak heights of the blue and red curves. For the odd-numbered peaks, the red curve is typically 10 dB lower than the blue curve. But for the even numbers, the red and blue curves have very similar heights. This is the expected pattern for nonlinear excitation. Figure 2 has already shown that coupling to body vibration makes little difference to the string damping, so the transverse string motion in the two plucks should be very similar. This means that the nonlinear excitation of longitudinal motion will also be very similar. The measurement is of soundboard acceleration. The transverse string motion in the plane of the strings will excite the soundboard more efficiently than the pluck in the perpendicular plane, giving the observed pattern of heights of the odd-numbered peaks. However, longitudinal force from both plucks will excite the soundboard identically, so that approximately matching heights for the even-numbered peaks are expected. That is exactly what the plot shows, both for the strong peak 2 and for the smaller peaks 4 and 6.

The wire-break plucks used here were relatively weak compared to regular harp playing. Even so, the nonlinearly-generated octave peak in Fig. 4 is of similar height to peaks 1 and 3 which will be, at least in large part, linearly excited. The conclusion is surely that nonlinear generation like this is not a small perturbation to the sound of a harp. The special nature of the frequency content from a mid-point pluck has been obliterated by the nonlinear effect. Harp strings are indeed often plucked quite near the mid-point, and they are also commonly plucked with far larger amplitude than was the case in Figs. 4 and 5. Because of the square law nonlinearity, the relative importance of nonlinear excitation in the sound is likely to be even greater with more vigorous plucks. This mixture of linear and nonlinear effects is a strong candidate for imparting a recognisable “harpish” sound quality, and it may also give the harpist a way to vary the tone quality depending on the strength of the pluck, even when the plucking point remains the same.

Figure 5. Frequency spectra as in Fig. 4, showing the next section of the frequency range. The rounded peak labelled “L” is the first longitudinal mode of the string.

Figure 5 shows the next frequency range for the same two spectra as in Fig. 4. This brings out some extra features: we have already seen a hint of them in the spectrogram of Fig. 3. Instead of a single peak 8, we see a pair labelled 8a and 8: these correspond to the close pair of lines near 2.7 kHz noted in Fig. 3. After that we have pairs 9a and 9, 10a and 10, and so on. The peaks labelled with an “a” are the phantom partials. They are generated from the transverse modal frequencies of the string by frequency doubling or by sum and difference combinations, and as we already noted these extra frequencies don’t quite match the linear modal frequencies of the string because of the effect of inharmonicity from bending stiffness.

Recall from section 5.4 that the effect of bending stiffness is to make the overtone frequencies of a stiff string progressively sharp, relative to an ideal harmonic pattern. As a result, a frequency-doubled component based on, for example, mode 5 will have a lower frequency than the actual mode 10, because mode 10 has been sharpened by the stiffness effect. These are the two frequencies we have labelled 10a and 10 in Fig. 5. Similarly, the sum of frequencies 4 and 5 is a little lower than the actual frequency of mode 9: these are the two frequencies labelled 9a and 9. But for our string plucked at the mid-point, mode 4 was only driven very weakly, and that is the reason that the peak 9a is very weak in Fig. 5: it relies on the strengths of both peak 4 and peak 5. It is shown in reference [1] that the observed pattern of peaks, including the phantom partials, can be very well explained across the whole frequency range by the effects of bending stiffness.

Notice that all the “a” peaks in Fig. 5 show identical heights in the blue and red curves, whereas the “non-a” peaks are higher in the blue curve. This is the same pattern discussed earlier, for the peaks 2, 4 and 6 compared to peaks 1, 3 and 5. Peaks with equal heights in the two spectra are a hallmark of nonlinearly-generated longitudinal string motion. This suggests that to be strictly consistent we should have labelled the peaks as 2a, 4a and 6a in Fig. 4: for this special case of the mid-point plucks, these were all “phantom partials”. But we didn’t recognise them as such because they didn’t obviously fall at unexpected frequencies. Another interesting detail: the rounded peak labelled “L” in Fig. 5 is the first longitudinal resonance of the string. It is not very prominent because it has high damping, but notice that, again, it has virtually identical levels in the blue and red curves.

The next step is to synthesise some sounds, so we can listen and compare them with the measurements. There are several possibilities here. First, we can use a version of the linear model developed in Chapter 5, extended to allow for the string-soundboard angle of the harp, and the consequent coupling between transverse and longitudinal motion. But then we want to give at least an impression of the effect of the nonlinear coupling, generating phantom partials. But nonlinear synthesis is a complicated business, still very much an active research topic. So for the moment we will be content with the simplest approximate model, as described in the previous link.

Even this is somewhat complicated. The procedure is first to find the transverse string motion, then to use this to calculate the nonlinear driving force for longitudinal motion. This approach involves several assumptions and simplifications. First, the method is intrinsically approximate: the governing equation derived in the previous link is based on discarding everything except the leading-order effect. Second, it only works in one direction: the transverse motion influences the longitudinal motion, but not vice versa. Among other things, this means that the method is not capable of giving the pitch glide effect.

Third, to compute the nonlinear driving force requires knowledge of the transverse vibration at all positions on the string, not just near the bridge. But the linear synthesis method we have been using does not provide this: it only gives the motion at the bridge resulting from the combined effects of string and soundboard. So we will use a crude approximation. We saw in Fig. 2 that the transverse string motion in the harp is not very strongly influenced by coupling to the soundboard. That being the case, we can try using the simple closed-form response of a plucked string developed in section 5.4.1, for the case of a rigidly terminated string. We can easily extend that treatment to allow for the string’s damping and for the effect of bending stiffness.

This approximate version of the transverse motion can now be used to calculate the nonlinear driving force, then solve the equation from the previous link for the longitudinal motion. Forces will then be exerted on the harp soundboard by both transverse and longitudinal motion, and these can be used in conjunction with the measured bridge admittance to give an estimate of the soundboard acceleration. This, finally, can be compared directly to the measurements from the wire-break plucks.

But there is still one more twist to the story. As shown in detail in the previous link, it turns out that there are two contributions to the longitudinal force exerted on the bridge by the vibrating string. The balance between these two depends on the details of exactly how the string makes contact with the soundboard. There are two extreme cases. One has both effects fully active: this corresponds to assuming what is known as a pinned boundary condition at the end of the string. The other case has only one of the two effects, and corresponds to what is called a clamped boundary condition. We will show results for both cases: in a real harp, the answer is likely to be somewhere in between the two.

Figure 6 shows a selection of frequency spectra for these various synthesised and measured responses, all for a pluck near the mid-point of the string. The top curve, in green, shows the result of the simplified linear synthesis. This curve shows why I chose to pluck near the mid-point, rather than exactly at that point. In the computer it is easy to be accurate, but in the measurements there is something of the order of 1 mm uncertainty about the exact plucking point. For a special case like the mid-point this makes a difference, especially when we plot results on a decibel scale showing a very large range of amplitudes. If we had plucked at the exact mid-point, in this simplified model of transverse motion, the even-numbered string modes would have been completely absent. Instead, the assumed plucking point is 1 mm away from the centre, to reflect the accuracy of the measurements and give a more representative comparison. You can still see a strong even-odd pattern of peak heights, but the even-numbered peaks are all visible to some extent in the green curve.

The second and third curves in Fig. 6, in magenta and black, show the result of the approximate nonlinear model with the two different boundary conditions: pinned in magenta, clamped in black. All these synthesised curves are to be compared with the measurements in the lowest two curves. These are the same results as shown in Figs. 4 and 5, and the blue/red colour scheme is the same as in those previous plots.

Figure 6. Comparison of synthesised and measured spectra for a harp string plucked near the mid-point. The top (green) curve shows the result of linear synthesis. The magenta and black curves show synthesised results using an approximate model to incorporate nonlinear effects: the magenta curve assumes a pinned boundary condition at the bridge, the black curve assumes a clamped boundary there. All three synthesised results assume a plucking point 1 mm from the geometric mid-point of the string. The two curves at the bottom show the same measured responses as Fig. 4 and 5, using the same colour codes: the blue line for a pluck in the plane of the strings, the red line for a pluck in the perpendicular plane.

We have already talked about some the main features of the measured results, especially the pattern of phantom partials. The three synthesised cases give very different results for this pattern. The linear synthesis, of course, does not show them at all. The magenta curve, for the nonlinear model with a pinned boundary condition, shows a very high peak near the position of the second string mode, as we expect from the earlier discussion. However, at higher frequency it shows far too many phantom peaks, compared to either of the measured plots. The black curve, for the clamped boundary condition, comes far closer to matching the pattern of measurements. It is by no means a perfect match — look for example at the height around peak 4 — but in the main it shows encouraging agreement.

So, finally, what do these all sound like? Sounds 2–7 allow you to hear all the cases we have been talking about. Sounds 2 and 3 are the two measured wire-pluck waveforms. Don’t forget that these are all giving the “sound” as measured by an accelerometer on the harp soundboard, so we don’t expect them to sound the same as a microphone recording. The sharpness of the wire-break plucks also gives them a lot of high-frequency content, in comparison with the finger pluck in Sound 1.

Sound 4 is the result of a linear synthesis using the method of Chapter 5, while Sound 5 gives the result of the simplified linear synthesis described above. The simplified method does not take proper account of coupling to the soundboard, so it lacks the initial “thump” coming from transient response of the modes of the harp body. But it does contain part of the effect of the body, because the string force has been converted into a version of the soundboard acceleration using the bridge admittance: you can see in the green curve of Fig. 6 some evidence of body resonances, in the low-level wiggles at low frequency in between the narrow string peaks.

Sounds 6 and 7 give the two versions of the approximate nonlinear synthesis, with pinned and clamped boundary conditions respectively. They sound, to my ears at least, different from each other, and also clearly different from the linear syntheses on the one hand, and from the measurements on the other. What do we conclude? Perhaps that all the effects discussed in this section have some influence on sound, but that we don’t yet have a fully convincing synthesis model. There is plenty of scope for further work here.

Sound 2. Measured soundboard acceleration from a harp string, plucked with a breaking wire in the plane of the strings
Sound 3. Measured soundboard acceleration from a harp string, plucked with a breaking wire perpendicular to the plane of the strings
Sound 4. Synthesised soundboard acceleration from harp string pluck, using a linear model similar to the one developed in section 5.4
Sound 5. Simplified linear synthesis of the plucked harp string, ignoring the influence on the string of coupling to the body.
Sound 6. Approximate nonlinear synthesis of the plucked harp string, assuming that the string has a pinned boundary at the soundboard
Sound 7. Approximate nonlinear synthesis of the plucked harp string, assuming that the string has a clamped boundary at the soundboard

That is as much as we will say about the harp. But we now turn to another plucked-string instrument with a distinctive sound, the lute. We will discover that a different type of nonlinearity is significant here. Figure 7 shows the strings of a typical lute. This particular one has 8 courses: during the long history of the lute the number of courses gradually increased from 5 or 6 in the earliest years, up to 10 or more by the time of Bach. The top string is single, the rest are in 2-string courses. The first 4 pairs are tuned in unison, the lowest 3 in octaves. All the strings are plain monofilaments; in this case they are all synthetic polymers, but originally gut would have been used.

Figure 7. The strings and bridge of an 8-course lute

An intriguing puzzle is revealed by comparing the sounds of the lute and a classical guitar. The top three strings of a classical guitar are nylon monofilaments, apparently similar to the strings of the lute. A guitarist normally uses fingernails to pluck the strings, while a lutenist uses the flesh of the fingertips. When played in the usual way, both instruments make a satisfyingly bright sound, but if a guitar is played using lute technique, with finger flesh rather than nails, the result is very dull and unsatisfying. The most important physical difference between the two instruments is that the strings of the lute are thinner than the corresponding strings of the guitar.

This poses a puzzle because of something we can deduce by putting together a few things covered in earlier chapters. If we combine information from sections 5.4.1 and 7.2.1, we can deduce that for an ideal textbook string, of the kind undergraduates are usually taught about, the diameter of the string should not make any difference whatsoever to the sound of a plucked note! Specifically, we think about a string of a given length and material, tuned to a given frequency, and plucked with a perfect step function of force like a wire-break. We then calculate the response at the bridge of the instrument, and it turns out that the string diameter simply cancels out.

We need some data. First, let’s listen to some sounds. An experiment was done by fitting a guitar with four plain nylon strings of different diameters, all tuned to the same frequency. Three of the strings were the top three from a standard set of classical guitar strings. But all three were tuned to the usual frequency of the top string (E$_4$, 329.7 Hz). A fourth string was added, with a typical diameter for the top string of a lute, and this was also tuned to the same note. This experiment was not done using an expensive guitar! Tuning the second and, particularly, the third strings up to the top E put more stress on the guitar and its tuners than it was really designed to withstand. Fortunately, no disaster ensued.

We will concentrate on three of these strings: the light-gauge lute string (“string 1” in what follows), the regular guitar top string (“string 2”) and the tuned-up guitar second string (“string 3”). The diameters of these three strings were respectively 0.50 mm, 0.69 mm and 0.78 mm. Sounds 8, 9 and 10 give single notes on these three strings, plucked with a thumb using lute technique. As with the harp results earlier in this section, these “sounds” are actually the result of recording the output of a small accelerometer fixed to the guitar bridge. I think you will agree that there is a big difference of sound between these examples: brightest for string 1, but quite dull for string 3 even with the high-frequency boost provided by the accelerometer recording.

Sound 8. String 1, plucked with a thumb.
Sound 9. String 2, plucked with a thumb.
Sound 10. String 3, plucked with a thumb.

Sounds 11, 12 and 13 are the result of plucking the same three strings with a breaking wire. All three of them sound brighter than the corresponding thumb plucks. Comparing Sounds 9 and 12 illustrates the lute/guitar contrast commented on at the start of this discussion. These two sounds are both based on the normal top E string of a guitar, and the difference of brightness in the sound between the two is very marked. Even with the artificial “treble boost” provided by the accelerometer recording, the thumb pluck gives a rather unsatisfactory dull sound. The fingernail of a classical guitarist will give an effect somewhere between a thumb pluck and a wire-break pluck, giving the guitar a sufficiently bright sound.

The contrasting brightness between the thumb plucks and wire-break plucks is no surprise: this is exactly the difference predicted by linear theory. We saw in section 5.4.1 that the contribution of each mode to the pluck response of a string is found by expressing the initial shape as a mixture of the mode shapes, and for a string this amounts to expanding that initial shape as a Fourier series. The sharp corner given to the string by the wire break requires a lot of short-wavelength terms in that Fourier series. This in turn creates a lot of high frequency content in the sound. The more rounded shape from a thumb pluck only involves relatively long wavelengths, and so it excites the higher modes of the strings rather little, and produces less high frequency content in the sound. A fingernail pluck will fall somewhere between the two in terms of sharpness.

Sound 11. String 1, plucked with a wire break.
Sound 12. String 2, plucked with a wire break.
Sound 13. String 3, plucked with a wire break.

So can we track down what is responsible for the difference of sound between strings with different diameters? Several effects we have already encountered might play a role. First, we know that strings of different diameters will have different inharmonicity, and also different damping behaviour. Both effects are associated with a difference of bending stiffness: a thicker string will have higher stiffness, leading to greater inharmonicity and also to a reduced cut-off frequency associated with damping. We have already seen some data about this damping effect in Fig. 2 in section 7.2, reproduced here as Fig. 8 to remind you. The strings shown in black and red here are in fact string 1 and string 2 of the present study. (The string shown in green is not relevant here, though. That string had diameter 1.68 mm, far thicker than any of the four strings used in the guitar experiment.)

Figure 8. A repeat of Fig. 2 from section 7.2, showing the damping behaviour of three different nylon strings. The black and red points correspond to strings 1 and 2, respectively, in this section.

Both these effects associated with bending stiffness can be captured accurately by linear synthesis: Fig. 8 reminds us that we have a good theoretical model of the damping of these strings (shown in the lines which follow the experimental points quite closely). Sounds 14, 15 and 16 give synthesised results for the three strings, corresponding to wire-break plucks at the same position as the measured responses in Sounds 11, 12 and 13. To my ears, these all sound remarkably similar. This suggests that differences in bending stiffness, inharmonicity and damping are not a very big contribution to the contrast between the three strings that we heard in Sounds. 8, 9 and 10.

Sound 14. String 1, synthesised by linear theory.
Sound 15. String 2, synthesised by linear theory.
Sound 16. String 3, synthesised by linear theory.

Perhaps the nonlinear effects we investigated in the harp might be important in the guitar and lute as well? It is certainly true that “phantom partials” can be seen in detailed spectra of the wire-break plucks of the strings of a guitar or lute: see [3] for some examples. Sounds 17, 18 and 19 give synthesised plucks on the three strings using the approximate nonlinear approach used for the harp in Sounds 6 and 7. The three sounds are clearly different from the linear versions in Sounds 14, 15 and 16. They are also slightly different from each other, but nowhere near enough to match the differences heard in the three wire-pluck measurements, let alone in the three thumb plucks.

Sound 17. String 1, synthesised by approximate nonlinear theory.
Sound 18. String 2, synthesised by approximate nonlinear theory.
Sound 19. String 3, synthesised by approximate nonlinear theory.

So what remains? The full detective story of tracking down the evidence is explained in [3], but you may have already guessed the most likely answer after listening to Sound 8 and Sound 11. The very thin string, under rather low tension, is almost certainly hitting some other part of the structure when it vibrates. There are various possibilities, but a plausible culprit is the fret closest to the nut. The early part of the transverse string motion following a pluck will be similar to the pattern for an ideal textbook string: see the animation in Fig. 3 of section 5.4. After half a cycle, the string shape becomes an approximate mirror image of the initial displacement. The plucking point was 30 mm from the bridge, so the maximum displacement will occur 30 mm from the nut, fairly close to the first fret at about 37 mm. If the string hits a fret, a short force pulse is likely to occur. This would excite additional vibration, with a very important extra feature. If the original pluck was band-limited, like the thumb plucks, a fret impact could excite vibration over a far wider bandwidth, just as was heard in the sound files (and also seen in measurements: see [3]).

The original observation, that lighter-gauge strings sound brighter, would then be explained by the simple fact that a player will naturally tend to excite a lower-tension string to larger amplitude, and thus make impacts or other nonlinear interactions more likely. In the case of wire plucks, the pluck force is explicitly kept constant, so that different amplitudes of string motion are excited. Musicians will probably not be thinking about plucking force as such, but they will be aware of the loudness of sound they wish to create. To achieve comparable loudness on strings of different tensions will require a larger vibration amplitude in a lighter string.

A musician or instrument maker would probably describe a fret impact as a “buzz”, but that term is usually applied to cases of multiple impacts with a fret so that a sustained effect is clearly audible. Guitar makers normally go to some lengths to avoid such buzzes. But we are suggesting here that a similar effect at a low level is probably essential to the tone quality of the lute, and other early plucked-string instruments in which the flesh of the fingers is used, rather than fingernails or any kind of plectrum. The brightness and crispness of tone associated with the lute and related instruments relies on their light-gauge stringing, probably via some manifestation of the effect discussed here.

This idea has some interesting consequences. First, certain design details of the lute are perhaps clarified. The frets on a lute (or a viol, or other stringed instruments of this early period) are not made of metal like the frets of a guitar. Instead, they are made of lengths of gut tied around the neck of the instrument. The result is softer than a modern metal fret, and this may reduce the abruptness of a string-fret contact so that the contact force is more band-limited and the sound is less obtrusive than a buzz on a guitar. In early lutes, the frets were conventionally tied with a doubled length of gut. This slightly curious feature is, perhaps among other things, a “buzz promoter”. The vibrating length of the string is probably terminated at the rear strand of the fret, but the vibrating string might buzz against the front strand to give some increased brightness to the sound.

Another consequence is that the desired effect of a slight buzzing contact might be extremely sensitive to the set-up of the instrument. The maker is probably able to shape the sound by adjusting various small details, but it is also likely that the sound is quite sensitive to changes in temperature and humidity which may cause movement in the wood structure of the instrument body and change the set-up details. To a musician, the instrument may seem rather “twitchy” to changes in the weather, sounding harsh on some days but more mellow on others.

There is another effect on the sound of a lute connected with humidity sensitivity, to do with the way the instrument is played. The recommended technique is to press down with the flesh of a finger pad on a string (or a pair of strings in a course), then pull sideways. The “pluck” is thus produced by frictional slipping against the skin. Now, it is well known that the coefficient of friction of human flesh is sensitive to the state of hydration: dry skin has lower friction. That means that when a player has dry skin, they will have to press down harder to increase the normal force in order to achieve the required frictional force for a pluck of a given strength. This will change the orientation of the string vibration, and the extra normal displacement may make buzzing more likely. The result is that the instrument may sound harsh.

This fits well with the experience of players, who know that soaking their hands to hydrate the skin can have a big effect on tone quality. Once this mechanism is understood, it can be seen that players may have other choices to achieve the same effect: there are various substances available to enhance friction. Powdered rosin is one, but this makes the fingers feel undesirably sticky. A better option may the “chalk” (magnesium carbonate powder) that rock climbers use to improve friction in their finger-tips. The effect may be longer-lasting than that of hydration, if playing in a dry environment.

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[1] J. Woodhouse, “The acoustics of a plucked harp string”, Journal of Sound and Vibration ?? (2021)

[2] H. A. Conklin, “Generation of partials due to nonlinear mixing in a stringed instrument” Journal of the Acoustical Society of America 105, 536–545, (1999).

[3] J. Woodhouse, “Influence of damping and nonlinearity in plucked strings: Why do light-gauge strings sound brighter?”, Acta Acustica united with Acustica 103, 1064–1079 (2017).