Musical strings come in a great variety. Some are simple single-strand monofilaments: made of steel for the piano or steel-string guitar, or of various polymers for classical guitars, harps and so on. Others have a more complicated construction, with a core of some kind over-wound with one or more layers of wire or tape, as in the lower strings of a piano or guitar, or any of the strings of a cello. Figure 1 shows some examples. The piano has plain steel strings for the higher notes, three strings per note. Lower down, the strings switch to having a steel core, over-wrapped with copper wire. The lowest notes have just one string per note, slightly higher notes have two strings per note. The classical guitar, on the other hand, has one string per note throughout. The top three strings are plain nylon (which you can see is transparent), while the three bass strings have a core of stranded nylon (not visible in the picture), over-wrapped with copper wire which has been plated to appear silver. Other instruments might have strings using natural gut, either plain (as in many harps), or over-wrapped (as in some traditional violin strings). Why is there so much variety, and how does a player or instrument maker set about choosing the “right” strings for an instrument?
Back in Chapter 5, we already met some of the key ideas. Now we will approach the question more systematically, and support the discussion with sound examples so that the perceptual effect of some of the key factors can be experienced. The first stage is to put ourselves into the player’s position, and try to understand what they want to achieve or to avoid when they choose a string. As often with these musical questions, the answer is surprisingly complicated. There are some aspects that are common to all stringed instruments, plus some extras that are specific to instruments of particular types.
The starting point is to be clear about what we already know, before we begin the selection process. We have the instrument in front of us, so we know the length of the string (by which we mean the vibrating length, between the bridge and the nut). We also know what note the string is to be tuned to.
One thing we have already seen is that the impedance of a string is important, because it determines how strongly the string vibration couples to the instrument body. This in turn governs the loudness of the note when that string is played. To remind you of two key facts, the impedance is governed by the product of the tension and the mass per unit length of the string. But, for a string of a given length, the fundamental frequency is governed by the ratio of those two quantities. So that ratio is fixed by the fact that we know the tuning. This seems to leave us with just one choice. We can choose a heavier or a lighter string, and we will then adjust the tension in proportion to the mass per unit length in order to tune correctly. A heavier string needs a higher tension, and so has a higher impedance; a lighter string vice versa.
But impedance is not the only thing that changes when you choose a heavier, tighter string. We can start by thinking about monofilament strings. Once you have chosen a material, with a particular density, the only choice you have is the string diameter. This determines the mass per unit length, in proportion to the cross-sectional area of the string. But, as we saw back in section 5.4, other things change when you replace a thin string with a thicker one. First, the bending stiffness increases. That makes the natural frequencies deviate more markedly from ideal harmonic relations.
It is easy to use computer-synthesised sounds to illustrate the effect. We can use the synthesis method described in section 5.4, and in the computer we can try combinations that would be difficult to achieve on a real physical instrument. Sound 1 gives three guitar-like notes, and the only thing that has changed between them is the diameter of the string. The chosen note represents the third string of a guitar, played at the 12th fret: in other words, the note G (392 Hz) on a string of length 325 mm. To avoid complications associated with damping, which we will discuss in a moment, all modes of the string are assumed to have the same Q-factor with value 3000. This is a number typical of a steel guitar string. The string is coupled to a guitar body, but the admittance has been artificially reduced by a factor of 1000, to create an essentially rigid bridge (again to avoid confusion from extra effects).
These “strings” are made of steel, with the appropriate density and Young’s modulus. The first note assumes a string of 0.3 mm diameter: a typical value for the top string of a guitar, much thinner than you would expect for the third string. The second note assumes a string of 0.6 mm diameter. It sounds perceptibly different from the first note, with a slightly “jangly” quality. This diameter is about what you might expect if you were to fit a solid steel third string to a guitar, and follow the trend from the first and second strings. But sets of steel guitar strings do not usually use a solid third string, probably to avoid the jangly sound just illustrated. Instead, a thinner core is used, with over-wrapped wire to increase the mass without increasing the bending stiffness so much.
The third note in Sound 1 assumes a steel string of diameter 1 mm, comparable to a piano string. Now the sound is drastically different. You probably hear the “pitch” going up compared to the earlier notes, although if you listen out for it you can hear that the fundamental frequency hums on at exactly the same pitch as the other two notes. This strange, percussion-like sound is an extreme example of the effect of bending stiffness: no-one would want to choose a guitar string making a sound like this, unless the player is looking for special timbres along the lines of the “prepared piano”.
This example brings out another factor that influences string choice. The three “strings” used here require very different tensions to tune them to the assumed note: the tensions would be 36 N, 143 N and 398 N respectively. The middle value is typical for actual steel guitar strings. The first tension would probably strike a player as far too low: such a string would be very floppy, and in danger of giving fret buzzes all the time. The third one, on the other hand, has such a high tension that it is probably more than the structure of a guitar body is designed to take. It might also feel uncomfortably tight to the player, but that issue is probably less significant than the danger of the bridge coming unstuck from the soundboard! This highlights an important thing for the process of string selection: for a given string on a given instrument, there will be a minimum and a maximum acceptable tension. We will return to that issue in a bit, when we build up a design chart for string selection.
But first we need to be aware of a second thing that happens when the string diameter is changed: the damping of the string vibration changes. This doesn’t matter very much for metal strings, but for nylon or gut strings it can make a very big difference to the sound, to the extent that it is a critical factor to take into account in the process of selecting strings.
Back in section 5.4, we discussed two sources of damping in a vibrating string. One is associated with the air surrounding the string: energy is lost to viscosity of the air. It was shown in section 5.4.5 that this effect is strongest at low frequencies. The second source of damping arises from the fact that string materials like nylon or gut have very high intrinsic damping. Think of the sound you would hear if you tapped a nylon wine beaker: it would be a dull thud, quite unlike the ringing sound of a glass one. This high material damping can influence our vibrating string through the effect of bending stiffness: it depends on exactly the same parameter that determines the inharmonicity we just heard in Sound 1 (see section 5.4.4 for the details). Energy loss by this mechanism increases rapidly at high frequency.
The combined effect of the two things is that the damping as a function of frequency follows a U-shaped curve: from highish values at low frequency due to the air viscosity, it falls with increasing frequency until the bending effect comes in, making the curve turn round and grow. Somewhere in between it will hit a minimum value where the string can ring the best. Strings of different diameters will follow different versions of this curve. Fat strings, with high bending stiffness, will show the rising trend sooner and more strongly.
This expectation is borne out by measurements. Figure 2 shows some results for three different nylon strings, all attached to the same guitar body. The discrete symbols show measured values of frequency and loss factor, in three colours for the different strings. The black points are for a very thin string, the red ones for a regular guitar top string: both are acceptably “musical-sounding”. The green points, though, show the results for a very thick string, which gave a very unsatisfactory dull thump when plucked. The results show why: except for the very lowest frequency, where the three strings show comparable values of loss factor, the green points indicate very high damping, increasing rapidly with frequency.
Figure 2 also shows, as solid lines, the predictions of the theory from section 5.4. The U-shaped curves are evident, and the curves follow the measured points satisfyingly well. The dash-dot lines show the contribution to the loss factor from the bending effect on its own, rising rapidly with frequency. This rising trend creates a high-frequency roll-off in the sound of each plucked string. This can be used to formulate a criterion for a “musically acceptable” string, which distinguishes the red and black strings from the unacceptable green one. We will come to the details shortly.
I said earlier that this damping effect is not important for metal strings. To see why, Fig. 3 shows results corresponding to Fig. 2, but for a steel guitar string. The scales are the same as in Fig. 2. This time the loss factor is still falling when we reach 10 kHz. The internal damping of the kind of steel used for musical strings is very low, so we simply don’t see a dramatic rising trend of damping associated with string bending stiffness. Comparing Figs. 2 and 3, it is no surprise that a steel string can sound quite different from a nylon string, even a very thin one. The steel string can deliver long-ringing sound up to very high frequencies, most evident if it is played with a hard plectrum in order to excite frequencies that high.
Building a string selection chart
We are now ready to pull together the things we have seen about string selection in graphical form to make a design chart. We will develop this in stages, for the particular case of nylon strings. For reasons explained in the next link, it will turn out that we can express all the things we need in terms of a single parameter, which is the product of the string length (in metres) and the tuned frequency (in Hz). The value of this parameter, remember, is something we already know for any given string. Also, the value does not change if you play different notes on the same string: the fundamental frequency of a string is proportional to the length so the length cancels out. We will put this product along the horizontal axis of a plot, and on the vertical we can indicate the string diameter. The result of the first stage of construction is shown in Fig. 4. Two things have been added to the axes just described. First, the set of curves in shades of copper colours are lines of constant tension: the colour code for values of tension is shown in the sidebar.
Two vertical lines have also been added to the plot. These are connected with the ultimate strength of a nylon string, before it breaks. Somewhat surprisingly, if you take a nylon string of any diameter and stretch it until it breaks, the highest frequency you will reach is the same, regardless of the string diameter. You need higher tension with a thicker string, of course, but breaking is determined by the stress in the string (force per unit area), and it turns out that stress depends only on the position on the horizontal axis. The algebra lying behind this claim is given in the next link. There are two versions of the vertical line. To the left of the solid line, you can expect a nylon string to last more or less for ever without breaking. But once you are to the right of the line the string will continue to stretch, more or less rapidly, and eventually it is likely to break. The further right you move, the shorter the lifetime. The dashed line marks the most extreme string I could find in actual use on a professionally-played instrument.
The next stage of constructing our selection chart is shown in Fig. 5. A new set of curves has been added, in magenta dashes. These are lines of constant impedance. They have a somewhat similar falling trend to the constant-tension curves, but quite different in detail. Again, the algebra behind these lines is explained in the previous link.
The final step is to add a limit on string choice based on the damping effect described above and to be illustrated shortly in Sound 2. The details are described in the next link. This time, the limiting condition for a musically-acceptable string does not give a single curve in the chart, because it also depends on the length of the string. Figure 6 shows some examples of the result. The upward-trending black curve is the limit for the open string of a guitar, with length 650 mm. The sound of that open string should be acceptable if it lies below this black curve. The dashed curve is the corresponding limit for playing at the 12th fret of the guitar, so that the vibrating length has halved to 325 mm. We see immediately that a given string will be in increasing danger of crossing the limit line as you play it in higher and higher positions: the string’s point on the chart stays fixed, but the limit curve moves progressively downwards.
Figure 6 also includes some discrete points. The blue stars mark the position in the chart of the top three strings of a typical classical guitar set. It can be seen that this set of strings has been chosen as some kind of compromise between constant tension and constant impedance. The third string, the leftmost of the three points, comes the closest to the limit lines associated with damping: indeed, the chart suggests that if that third string is played up to the 12th fret the sound quality will become muffled to an extent that may be considered marginal. This is indeed an effect familiar to guitarists.
But, of course, you want to hear the effect for yourself. Sound 2 gives three synthesised notes, corresponding to the three red stars in the plot. The note played corresponds to the 12th fret, so the important comparison is with the dashed limit line. The first sound goes with the lowest star. It is well below the line, and sounds fine. The second corresponds quite closely to the actual third string played at this fret, and it is close to the line. The sound is clearly less “twangy” than the first note. The third note is well above the line, and illustrates what I mean by “not musically acceptable”.
To see what the chart in Fig. 6 is telling us, a schematic version is shown in Fig. 7. The string needs to be chosen to be not too tight and not too slack. Obviously, the string should not break, or at least not too quickly. Finally, it should lie below the limit line associated with damping, for the highest note to be played on the string (with the shortest vibrating length). The player’s choice of string diameter allows them to move along a vertical line in the chart. Choosing a thicker string allows a louder sound (because the impedance is increased), but if the string lies in the left half of the plot, you run the risk of getting too close to the damping limit. So a thicker string tends to be louder but duller, and conversely a thinner string is quieter but can be brighter-sounding.
It should be emphasised that the damping limit plotted in Fig. 6 is not a precise one. The lines plotted there give an indication of when the sound quality may start to deteriorate, in the manner illustrated by Sound 2. But the criterion, explained in the previous link, is based on a very crude assumption: an “acceptable” string should have at least 10 overtones with damping lower than a certain threshold value. For the strings of a guitar, this empirical criterion works quite well. But in a moment we will look the harp, where the range of string lengths and tuned frequencies is far larger. For that case it doesn’t entirely make sense to have a criterion based on a fixed number of overtones: 10 overtones of a low string on a concert harp might only take you to 1 kHz, whereas 10 overtones on the top string of the same harp would take you beyond 30 kHz, well beyond the range of human hearing. Under those circumstances, you would guess that for the low strings you would like to be well below the threshold as plotted here, but by the time you reach the highest strings you could afford to be significantly above the threshold line.
All of this has been about solid, monofilament strings. As the chart makes clear, for an instrument with fixed-length strings like a guitar there is an intrinsic problem associated with choosing bass strings. The length is fixed, so as the tuned note gets lower you automatically move to the left in the chart. We have already seen that the third string of a classical guitar is close to the limit for a solid nylon string, if you want to be able to play it in high positions. For the lower strings, if you want a bright and twangy sound, you have to do something different. One choice is to use steel strings rather than nylon strings, but that takes you into the different sound world of the steel-string guitar, and we have also heard (in Sound 1) that thick steel strings don’t really work, for a different reason. Although the damping problem goes away, the inharmonicity due to bending stiffness produces an undesirable sound.
You are left with two choices. Most modern instruments, like the classical guitar, opt to use over-wound strings for the lower notes. This construction allows the string to be heavier without too much penalty in terms of bending stiffness. The result is the typical appearance of a set of classical guitar strings, as we saw in Fig. 1.
Things are more challenging if you play a period instrument like a lute, and you want to aim for an “authentic” sound and feel. At least in the earlier part of the lute’s heyday, the technology of making wire-wrapped strings may not have existed, or at least such strings would have been expensive and not widely available. Players used plain gut strings, and the result in terms of damping was very similar to the nylon strings we have been looking at. To help with this intractable problem, lute-players used a trick. The strings of a lute are in pairs called courses. For the lower courses, instead of using a pair of fat strings tuned in unison they could use a thick-thin pair, with the thinner one tuned an octave higher. When you pluck the two strings together, you get the necessary low fundamental frequency from the thick string, but the thinner string can supply higher overtones to add brightness to the sound, because it lies further to the right in the chart and so is further away from the damping limit line.
Case study: the harp
The harp makes an interesting case study in the use of the selection charts. Unlike the guitar, a harp has strings of different lengths for each note. This raises a new issue, to be taken into account when choosing strings: the “feel” of the strings to the player. A simple way to characterise feel is to ask “how hard do you have to pull the string, in order to create a desired displacement before you let go and start the vibration?” A simple static force-balance calculation reveals that this measure of feel is governed by the ratio of tension to length. So if a harpist wants the feel to stay roughly the same across the range of the instrument, they need to choose higher tensions for the longer strings.
This is indeed what they do. Figure 8 shows the same selection chart for nylon strings, annotated with a set of recommended string diameters for a concert harp, from a string manufacturer. The pattern is quite different from what we saw for the strings of a classical guitar. The tensions get very high indeed for the lower, longer strings. The harp body has to be designed to withstand this amount of internal force; and even so, a common mechanical failure in harps involves the soundboard pulling away from the rest of the body.
Because all the strings have different lengths, it is hard to show the associated damping limit curves. In this plot, changing colours have been used to indicate the pattern. The discrete points for the strings are coloured red and blue, in alternating batches. A limit curve is plotted for the length associated with the last string of each colour batch, in the corresponding colour. It can be seen that these lines always fall rather close to the change of colour in the batches of points. What this means is that all these strings fall close to the suggested damping limit.
But many harpists do not choose nylon strings at all: they prefer the sound of natural gut strings. Figure 9 shows the corresponding selection chart for gut strings: it is similar in some ways to the nylon chart, but details matter here and when you look in detail, every string material requires a different version of the chart. The chart in Fig. 9 is again annotated with discrete points based on the string manufacturer’s recommended choice of gauges for a concert harp. The same scheme of changing colours is used to indicate the pattern of the damping limit curves.
This time, the colour changes between the batches of discrete points all occur before the corresponding limit line, even though the actual values of tension are higher than in the nylon example. Comparing with Fig. 8, each colour change for the strings falls consistently further below the plotted limit line for gut strings compared to nylon strings: gut strings allow the harp to have a brighter sound than is possible with nylon strings. Probably this is at least part of the reason why players continue to prefer gut strings, despite their higher cost and higher sensitivity to changes in humidity and temperature.
The most obvious difference between the charts for nylon and gut is in the shapes of the damping limit curves. For nylon the curves bend down, whereas for gut they are essentially straight. The reason for this difference lies in the way the two materials respond to changes in tension. Man-made polymers like nylon consist of long-chain molecules, tangled together a bit like a bowl of spaghetti. As a string stretches under tension, these chains get straightened out; more and more so as the tension is increased. This has a consequence for an important mechanical property of the string, the Young’s modulus, which determines the stiffness along the length of the string. As the molecules straighten, the string gets stiffer. This is ultimately what causes the damping limit lines to bend over (for details, see [1]). But gut does not behave like this: its Young’s modulus stays essentially the same whatever the tension. That results in the straight lines seen in Fig. 9.
Now look again at the points representing the harp strings in the two materials. Because of the desire to use high tensions in order to improve the feel, the string choices push into a region of the charts where they are significantly different. The straight lines in the harp chart gave a bit more “headroom” in the string choice, compared to the nylon chart. It looks as if harpists exploit this to the full.
Just to complete the story, we should note that harpists actually use a more complicated combination of strings. They do indeed prefer gut strings in the middle range of the instrument. But a comparison of Figs. 8 and 9 reveals something we haven’t mentioned: the black lines indicating the breaking strength are further to the left for gut than for nylon. So for the very highest strings, nylon is generally used in order to improve the life expectancy. For the very lowest strings, a different issue arises. The charts indicate the use of very fat strings, which influences both the damping effect we have discussed, and the inharmonicity we met earlier. For both reasons, harpists typically choose over-wrapped strings for the lowest notes, usually with a steel core.
What about bowed instruments?
The strings with the most complicated construction are not found on plucked-string instruments, but on instruments like the violin or cello. The vibrating string length of a cello is similar to that of a guitar, but you can pay as much for a single cello string as for several full sets of high-quality guitar strings. What is going on? The same range of physical behaviour involving impedance, inharmonicity and damping is still important. We will see in a bit that there are some extra details that also matter in a bowed string, but in themselves these are probably not the main issue that has driven string design towards such complicated and expensive constructions.
A more important aspect of the explanation lies, probably, in the different ways that a violinist and a guitarist interact with their strings during performance. Think for a moment about the sounds made by a beginner on these two instruments. A complete novice can pluck a single note on a guitar, and obtain a satisfyingly “musical” sound. But, notoriously, a beginner on the violin is likely to produce some very unpleasant noises before they learn how to create an acceptable sound.
With a plucked string, the performer’s input is mainly limited to setting the initial shape of the string. Once they let go, the string “does its own thing”, producing a mixture of almost-harmonic overtones, each with its characteristic decay profile. In a bowed string, by contrast, the player is interacting with the string throughout the motion, all the time the bow is in contact. The player is trying to make the string do particular tricks (with a range of bowing techniques with names like sautillé and martelé). The string may have other ideas, and respond to a wrong bow gesture with an undesirable squeak or groan rather than the crisp note the player had in mind. Once a player has learned to control these different ways to bow a string, they can be rewarded by an instrument that offers enormous flexibility of musical expression. But first they need to put in many hours, weeks and years of practice: as the saying goes, there is no free lunch.
This is a first glimpse into the wild and complicated world of seriously nonlinear instruments: we will engage with it properly in the next few chapters. For the moment, all we need take away is that a violinist or cellist cares more than a guitarist about some subtleties of string behaviour. They are constantly fighting against a string that doesn’t really want to do what they have in mind. Any small improvement in the string’s mechanical behaviour might make their life easier, and allow them to play better and more reliably. A similar thing happens with sports equipment: for a tennis player or a cellist, any small enhancement that improves their performance is valued, and worth paying serious money for.
I mentioned that there are some properties of a string that we have not considered up to now, which can matter for a violinist or cellist. To satisfy your curiosity, I will give a brief account now. But things will become clearer when we look at bowed strings in detail, in Chapter 9. A bowed string is driven into vibration by the friction force between the bow-hair and the string, mediated by the rosin with which a player coats the bow-hair. This introduces two new things to consider in string design and selection. First, the surface of the string has to be suitable to interact with the rosin. It helps if the surface is smooth, so wrapped strings need to be made with ribbons of tape rather than with round wire. The physics and chemistry of the string’s surface may also play a role: bowed instruments do not generally use nylon strings, for example, because rosin does not stick effectively to them.
The second consequence of driving a string by friction from a bow is that a new type of string vibration can come into play. The friction force is applied tangentially across the surface of the cylindrical string. So as well as exciting transverse string vibration, it can excite torsional vibration in which “waves of twisting” travel along the string. Torsional string vibration does not (usually) contribute directly to much of the sound from an instrument, but it does influence the details of the string motion. It can provide an additional source of damping, and it can also contribute more subtle effects: more on this in section 9.5.3.
[1] J. Woodhouse and N. Lynch-Aird “Choosing strings for plucked musical instruments”. Acta Acustica united with Acustica 105, 516-529, (2019).