Choosing Strings for Plucked Musical Instruments

The various factors constraining the choice of synthetic polymer and natural gut strings for musical instruments are discussed, including a newly-formulated constraint based on the internal damping of the string material. It is shown that all these constraints can be summarised graphically in a design chart, and calibrated versions of the chart are presented for monofilament strings of nylon, fluorocarbon and natural gut. Based on these charts, detailed case studies are presented for the stringing of a lute and a harp. An explanation is suggested for why harpists continue to favour gut over nylon. © 2019 The Author(s). Published by S. Hirzel Verlag · EAA. This is an open access article under the terms of the Creative Commons Attribution (CCBY4.0) license (https://creativecommons.org/licenses/by/4.0/).


Introduction
Aperennial problem for players and designers of stringed musical instruments is howtochoose the best set of strings for ag iven instrument. This question has long been discussed, and the basic science underlying this question is not new: see for example [1,2,3]. However, recent experimental measurements highlight ap henomenon which may be familiar to some musicians, butwhich has not previously attracted scientificattention. If aplain nylon or gut string is fitted to aplucked-string instrument and tensioned just enough to begin to makeamusical note, the sound is muffled and unsatisfactory.Asthe string is tuned upwards, the sound first becomes musically acceptable butr ather mellow, and then becomes progressively brighter until the point where the string breaks. It will be demonstrated that this perception of increasing brightness is largely the result of am aterial damping effect: there is ar oll-off frequency above which the string overtones progressively become so highly damped that theyare no longer "musical", and this roll-off frequencyrises dramatically as the string is tightened.
In this paper the relevant results are reviewed, and a design chart is constructed that encapsulates the damping roll-off as well as other aspects of practical stringing choices. The main focus will be on plucked-string instruments liket he harp, guitar and lute. The issue of string choice can be particularly pressing for period instruments liket he lute or vihuela, because the musician is not pre- sented with ar eady-made choice in the form of sets of strings selected by instrument makers or string manufacturers, in contrast to the situation for am odern guitar or harp. The clearest results will be obtained for sets of monofilament strings, such as plain nylon or gut strings for ah arp or lute. The rationale for over-wrapped strings will become clear in the course of the discussion, butdetailed constructional choices for such strings will not be explored.
When choosing strings, amusician is likely to be interested in questions of "loudness", "tone quality" and "feel". All three concepts are decidedly slippery from the perspective of ascientist, butitiseasy to identify physical correlates that capture at least part of what each means. All will play ar ole in the design charts to be presented here, and in detailed case studies for the lute and the harp to be presented in Sections 4.2 and 4.3.

Frequency and impedance
The linear theory of free vibration of astretched string is quite familiar,b ut keyr esults needed in the later discussion will be summarised here. Consider astring of circular cross-section of diameter d and length L,under tension T and made of material with Yo ung'smodulus E and density ρ.The nth natural frequency ω n = 2πf n satisfies where m = πd 2 ρ/4i st he mass per unit length, and the second moment of area I = πd 4 /64. This result follows from the Rayleigh quotient (see for example [1]), and the approximation sign arises because it has been assumed that the corresponding mode shape is where 0 ≤ x ≤ L is the position variable along the length of the string. Areal string will deviate slightly from this assumption because of end effects: coupling to anonrigid structure at both ends, and other effects of evanescent fields arising from the detailed end boundary conditions. Nevertheless, it has been shown previously [4,5] that Equation (1) holds up very well for real strings, especially at higher values of n which will be important in this work.
The second term on the right-hand side of Equation (1) arises from the non-zero bending stiffness of the string. Forr ealistic musical strings, the bending stiffness effect is relatively weak so that the fundamental frequency( i.e. n = 1) is always well approximated by neglecting this term, Twodifferent musically-important effects are governed by the influence of the bending term: inharmonicity of natural frequencies, as shown by Equation (1),a nd the damping behaviour as afunction of frequency, to be discussed in the next subsection. It is useful to introduce anon-dimensional parameter to express the proportion of the potential energy associated with this effect, where the final expression makes use of Equation (3). Using Equations (3) and (4) in Equation (1),the natural frequencies can be expressed in the form Fors u ffi ciently high mode numbers n the bending term ceases to be as mall perturbation. However, it will be argued shortly that, for musically-relevant natural frequencies of strings of the kind to be discussed here, λ is always quite small. It follows that with λ ≈ Eπ 2 d 2 n 2 64ρL 4 f 2 1 .

(7)
It has been shown [6,7] that the mechanical properties of an ylon string depend significantly on the stress state and history.I np articular,t he Yo ung'sm odulus increases by roughly af actor of 3b etween the unstressed state before it is fitted to the instrument and the state just before it breaks. Similar results have recently been shown for fluorocarbon strings [8], which are becoming increasingly popular with musicians. Natural gut strings are also still popular,especially with harpists, and for these it has been found that no such "strain stiffening" effect occurs: within the limits of accuracyofthe measurements, asingle value of Yo ung'smodulus is consistent with the results overthe entire stress/strain range [8]. However, all these strings, including the gut strings, showsignificant frequencydependence of Yo ung'sm odulus, as is normal for viscoelastic materials. The value of E relevant to the equations stated above is the high-frequencyv alue called E B in the previous work [7,8]. Twoo ther parameters for vibrating strings should be mentioned. First is the wave impedance which is an important contributory factor to the strength of coupling between the string and the body of the instrument, and hence to the loudness of the played note. Asecond quantity has been proposed by Firth [9,10,11] as important to harpists: what he called "feel", although no doubt this simple quantity only captures part of what a musician may understand by that term. Firth'squantity is defined as the plucking force necessary to produce agiven initial displacement. Foram id-point pluck, to produce a small displacement δ the required force F p is givenb ya simple static equilibrium calculation as so that the "feel" γ is

Damping model
The other keyi ngredient needed for this study concerns damping. There are three main physical mechanisms of energy loss in avibrating string: loss by coupling to the instrument body,viscous dissipation due to the surrounding air,and viscoelastic loss within the material of the string. All three effects can be estimated, leading to amodel that has been shown previously to give ag ood fit to measurements [3,4,5]. That model will be fine-tuned in the light of more extensive measurements nowa vailable, and then an important conclusion will be drawn relating to the influence of damping on "brightness" of plucked strings and hence on string selection. Energy loss via the bridge to the instrument body will vary strongly with frequency, especially at lower frequencies, depending on the proximity of individual body resonances [12]. However, as imple approximation to the energy loss at higher frequencies can be obtained using Statistical Energy Analysis, giving aloss factor η body [4]. Substituting typical numerical values for the high-frequency behaviour of musical instrument bodies reveals that this loss mechanism is usually insignificant compared to the other twomechanisms in the frequencyrange that will be of interest here [4]. Examples of its effect will be shown in Figures 2and 3.
Energy loss due to viscosity in the surrounding air can be estimated using ac lassical analysis going back to Stokes. The associated loss factor is givenbyFletcher and Rossing [2] in the form where ρ a is the density of air,and where η a is the kinematic viscosity of air.T extbook values will be used: ρ a = 1.2kg/m 3 and η a = 1.5 × 10 −5 m 2 /s. In the light of tests on manymusical strings covering awide range of string gauges, it has been found that this formula does not quite capture the variation with diameter to best accuracy: it slightly underestimates the damping of thick strings and overestimates that of thin strings. It has been found that the measurements can all be approximated well enough for the present purpose by applying an ad hoc correction factor (d + 0.2) to η air ,w ith the string diameter d expressed in mm. Some examples will be shown in Section 3, buti ti sn ot possible to reproduce the entire set of results here: between them, the previous studies covered 20 strings of various materials, each tested at up to 8different tuned frequencies [4,5,7,8,13,14]. To see more clearly howt his damping contribution varies with frequency, it may be noted that for all the strings and tuning frequencies under consideration here, the value of M is fairly large so that The third damping mechanism will prove to be the most important for the purposes of this investigation. Energy loss from viscoelasticity in the string arises from the influence of bending stiffness. Yo ung'sm odulus becomes ac omplexv alue E(1 + iη E ). An argument based on Rayleigh'sp rinciple [15] can be used to yield an expression for the associated loss factor of the nth string mode, which takes avery simple form in terms of the parameter λ introduced earlier, The total modal damping factor can finally be estimated by η n ≈ η body + η air + η bend .
As described above,t he contribution from η body is small and can largely be ignored. From Equation (13), η air varies inversely with the square root of frequency, so it is dominant at lowfrequency. On the other hand, from Equation (7), λ ∝ n 2 so that η bend dominates at sufficiently high frequency. The result is aminimum of damping at an intermediate frequency, above which η n increases rapidly with frequency. It will be argued that this increase accounts for the change in the sound of strings as theyare tuned to different frequencies, as described earlier.The damping model will first be confirmed and illustrated with measurements.

Measured mechanical behaviour of strings
Data collected in the course of previous projects [4,7,8] can be combined to give af airly comprehensive picture of the mechanical behaviour of monofilament strings of synthetic polymer or natural gut. Those earlier papers give full details of the experimental methodology,i ncluding the confidence limits associated with measurement errors.
Most of the strings were tested using ap urpose-designed measurement rig that included provision for control of temperature and humidity,a nd an automated system for monitoring and maintaining the string'stuning. These tests were conducted overas u ffi ciently long period that the string'splastic response at anygiven tuning levelhad time to levelo ffto as tate that am usician would describe as "settled". The tests also took full account of the slight reduction in the string'sdiameter under tension, butfor the purposes of the present work this effect is small enough to be insignificant, and it will be ignored throughout this paper. It should be emphasised that all the tested strings were regular commercial strings marketed for musical instruments: the authors were not able to obtain detailed information about the chemical composition despite asking the manufacturers. So, for example, for strings described here as "nylon" we do not knowe xactly which members of the family of nylons were involved: indeed, evidence was found that strings marketed as ahomogeneous set for use on the harp were not all made of the same precise material. However, for the present purpose these uncertainties were sufficiently small that theyw ill turn out not to be significant in the context of the "broad-brush" design charts to be presented later.The behaviour of Yo ung'smodulus at high frequencies (E B in the earlier work)was found to be very similar for all the nylon strings tested, and similar uniformity wasa lso found for the gut and fluorocarbon strings tested.
Damping behaviour can be extracted from the string response to wire-pluck excitation [4]. The string data collected in connection with [4] is restricted to asmall number of strings, butithas high quality because the response wasc ollected under laboratory conditions, from an accelerometer on the bridge of the guitar to which the tested strings were attached. The accelerometer gave excellent high-frequencyr esponse data. The data associated with [7,8] covers afar wider range of strings. It wascollected using ap air of purpose-built test rigs, described in detail Hz. Levels are plotted in dB relative to the maximum in each case, to adepth of 40 dB. This string waslabelled "string 5" in the earlier study [7].
in [7]. The data quality is not as good because the pluck responses were only collected as secondary material in the course of tests primarily devoted to the mechanical and thermal response of strings overal ong timescale, buti t is sufficient for at horough test of the proposed damping model. The first use of these results is to illustrate the underlying phenomenon described earlier,inwhich the brightness of the sound of agiven string depends on its state of tuning. Figure 1shows spectrograms of the pluck response of an ylon string of diameter 1.20 mm and vibrating length 0.5 m, which wasl abelled "string 5" in the earlier study [7]. The plots showt his same nylon string tuned to two different frequencies, the lowest and highest used in the testing sequence. At the higher tuning frequency, it can clearly be seen that the decay rate at anygiven frequency is much slower,and that the bandwidth containing identifiable strong string overtone lines is much wider.I ti sno surprise that the sound of this string wasperceivedasgetting progressively brighter as it wastuned higher.
Next, it is useful to showresults to support the damping model presented in Section 2.2. Figure 2shows results for damping factors of three different nylon strings fitted to a guitar.T wo of them are the thinnest and thickest strings used in an earlier study (called "string 1" and "string 4" there) [4]. The third string shown here is asection of very thick harp string from the same original piece as "string 23a" in adifferent earlier study [7]. The twothinner strings were tuned to the same frequency, butt he thickest string wast uned to al ower frequencyt han the thinner strings to reinforce the main point to be learnt from these results (and because the structure of the guitar would not tolerate tuning such aheavy string to the same pitch).
The discrete symbols in the plot showt he measured points, while the various lines indicate the theoretical comparison. The separate contributions to the total damping η n are shown: values for the loss factor η body associated with energy transfer to the guitar body are included in this plot, calculated as explained in earlier work [4]. The plot confirms the earlier statement that this contribution is small enough that it can be ignored for the purposes of this study.
To calculate η bend ,av alue for the loss factor η E was needed. As has already been remarked, the real part of the Yo ung'smodulus E for nylon strings has been shown to vary significantly with both frequencya nd stress state. However, careful examination of the damping results suggested that ag ood fit wasg iven to all measurements by assuming that the imaginary part remained constant, independent of stress. Furthermore, the frequencydependence can be ignored at the high frequencies relevant here, in the kHz range: as can be seen in Figures 2a nd 3, as atisfactory fit to the measurements wasg iven by this approximation. In terms of the usual terminology of the theory of viscoelasticity (see for example [16]), E = E +iE where E varies with stress but E does not. As imple fit to the results for E B in Figure 10 of [7] gives where σ is the stress expressed in GPa, while the damping results are consistent with av alue E = 0.25 Gpa. The result is that η E = E /E varies with stress, falling from approximately 0.04 to 0.02 as stress increases over the range tested here. The solid curves in Figure 2s howt he combined loss factor η n for each string predicted according to the analysis of Section 2.2, and it is immediately clear that each curve follows the trend of the measured data points remarkably well. This confirms the damping model developed in Section 2.2, and also illustrates the pattern implied by that model. Damping is relatively high at lowf requency, falls to am inimum around the frequencyw here η air and η bend are equal, then rises rapidly at higher frequencies so that there is an effective roll-off frequencyabove which string modes have damping that is too high to sound "musical". The thinnest string has this roll-off around 13 kHz, the middle string has it around 7kHz, butfor the thick string all modes have ahigher loss factor than the plotted points for the thinner strings, and it waso nly possible to determine values up to about 1.5 kHz. It will come as no surprise that this thick string produced athoroughly unsatis-   factory sound on the guitar: more am u ffl ed thud than a ringing guitar-likenote.
Forthe twothinner strings, which produced acceptable musical sounds, it can be seen that the data points run out at roughly the same value of η n ≈ 10 −3 .I nt he light of Equation (14),t his suggests that the useful bandwidth of ag iven plucked string might be associated with at hreshold value of λ,s ince the material loss factor η E is fixed for ag iven string and tuning. Of course there is not a crisply-defined threshold for damping, butfor the purposes of ad esign criterion with the right order of magnitude, a threshold value λ ≈ 0.05 will be used in the subsequent development: the appropriateness of this choice will be confirmed later in the light of the twocase studies in Sections 4.2 and 4.3. The chosen value justifies as tatement made earlier in connection with Equations (6) and (7),t hat for musically-relevant string resonances the value of λ is always small.
The full set of results for other nylon strings and for fluorocarbon and gut strings cannot be reproduced here because of space constraints, butplots similar to Figure 2 have been examined for every tested case. Asuitably calibrated version of the proposed damping model wasfound to give as atisfactory fit for each material, overt he full range of string diameters and tensions. Four examples for fluorocarbon strings are shown in Figure 3: these strings were fitted to al ute and tuned to their normal playing pitches, and will form part of acase study of lute stringing in Section 4.2. Note that for the present purpose, the important aspect of these results is the high-frequencydamping trend, leading to the effective roll-off frequency. The three thinnest strings give an excellent fit in this region. The thickest string givesaless clear scatter of points with higher damping, somewhat similar to what wass een in Figure 2. This plot wasp roduced using fitted values for the Yo ung'smodulus of fluorocarbon strings as afunction of stress similar to the results for nylon shown in Equation (16), Forthe imaginary part of Yo ung'smodulus, as wasfound for nylon, ac onstant value givesas atisfactory fit: E = 0.18 Gpa. Forg ut strings the corresponding fitted model is simpler: fixed values E = 6G pa, η E = 0.04 are appropriate. It may be remarked that the gut strings did not generally give measured results as clean as those shown here for nylon and fluorocarbon strings. The trend is always clear,b ut the individual points usually showm ore scatter relative to the predicted curves. This scatter is probably adirect consequence of the construction of gut strings. The twisting and polishing processes are carried out by hand, on each length of string individually.S ome variation of detailed structure along the length of the string is bound to occur,a nd this may lead to spatial variation in the complexY oung'sm odulus. That variation will interact with the different mode shapes of the string overtones, and the twod i ff erent polarizations of vibration in each mode, to produce variation in the modal loss factors. Indeed, if the detailed distribution of Yo ung'sm odulus wasknown, the Rayleigh'sprinciple argument could be applied mode by mode to predict these variations in damping, in asimilar waytoearlier predictions of modal variations in damping factor: see for example the results for composite plates [17]. Table Is ummarises the values of E and E obtained for the three materials, together with the corresponding bulk density values used here. The earlier studies of harp strings found differences between the bulk densities of the thinner and thicker monofilament nylon strings tested [7,13], and between the thinner (monofilament) and thicker (wound)fluorocarbon strings [8,14]. The bulk density values shown in Table Ia re representative of the measured values for the thinner nylon and fluorocarbon harp strings, and were selected as being appropriate for the string diameters common to other instruments. Forthe construction of the string design charts to be shown here, however, the exact values of the material bulk densities are not particularly important; what matters is the relative difference between the densities of the three materials, with fluorocarbon having asignificantly higher density than natural gut or nylon.
Fora ll three materials, at lows tress levels where the damping effect will turn out to be most important, η E is around 0.04. This coincidence of values is very convenient, because it means that if the same threshold value of λ is used for all three materials, that will correspond to essentially the same values of modal damping factor for the string modes. This will allowd irectly comparable string design charts to be presented for the three materials, in the next section. Furthermore, it wasn oted earlier that λ also governs the degree of inharmonicity of astring. This means that design guidelines based on at hreshold value of λ will set al imit on inharmonicity as well as damping. Both damping and inharmonicity have been associated in earlier literature with the perceptual discrimination of "warmth" versus "brightness". The damping roll-off affects the spectral centroid, and there is awell-established correlation of perceivedbrightness with variation in spectral centroid [18]. Quite separately,t he perceptual consequences of inharmonicity have been investigated in literature extending back at least as farasthe classic work of Fletcher et al. [19]. The fact that both effects are governed by the same parameter may give anew perspective on relevant perceptual questions, as will be discussed further in Section 5.

Development of the chart
The criterion of athreshold value of λ can be expressed in graphical form. From Equation (4),itcan be seen that the value of λ for ag iven material depends on three parameters relevant to string choice: d, f 1 and L.H owever,t he expression for λ can be written in terms of twocombinations of them, The parameter α is anatural one to bring in, since if bending stiffness is ignored, α remains constant as ag iven string on ag uitar,s ay,i sfi ngered in different positions. Using Equation (3), α ≈ c/2where c is the wave speed on   the string. Formaterial of agiven density,its value determines the stress: It is straightforward to drawac ontour map of λ in the (α, nβ)p lane, noting that for nylon or fluorocarbon, E is afunction of α through Equations (16), (17)and (19).An example is shown in Figure 4, for nylon strings. Contours of λ have been plotted at intervals of 0.01 up to the value 0.1. Beyond that value the string overtones will surely be too highly damped to be of interest: recall that the suggested threshold value is 0.05, in the middle of the plotted range.
Points corresponding to particular strings can be calculated and added to the plot, butb ecause of the presence of n in the quantity plotted on the y-axis, ag iven string givesapoint for every relevant overtone. These overtones all have the same value of α,sotheymakearegular vertical column in the plot. Points are included here for the four nylon strings for which results have already been shown. The open symbols showt he three strings from Figure 2, while the stars showt he string from Figure 1, at all the tunings tested. The twocases plotted in Figure 1were Figure 4; stars: all strings and tested frequencies from [7,13].
leftmost and rightmost of the set. Foreach string, the lowest plotted symbol shows the fundamental n = 1, and then to indicate the pattern without cluttering the plot with too manypoints, symbols are plotted above it for n = 10, 20, 30. .. To interpret the plot, consider first the thinnest string of the set in Figure 2, indicated by square symbols towards the right-hand side of Figure 4. Locating the contour corresponding to λ = 0.05, it can be seen that the closest square symbol to that contour marks the value n = 50, so the prediction is that this string should have about 50 overtones with damping lower than the chosen threshold. The middle string from Figure 2i sindicated by circular symbols, and because this string had the same length and the same tuning as the thinnest string, theyappear at the same value of α.However,the circular symbols are wider apart, and the λ = 0.05 contour passes between the twos ymbols marking n = 20 and 30. So for this string, roughly 25 overtones should have damping belowt he threshold. Comparing the two, the prediction is that the bandwidth of lightly-damped string modes should be roughly twice as big for the thinner string. Looking at where the plotted points run out in Figure 2, this prediction matches the observations quite well.
The thickest string from Figure 2i si ndicated in Figure 4bydiamond symbols, towards the left-hand side. For this string, even the symbol corresponding to n = 10 lies above the λ = 0.05 contour,s ot he prediction is that this string should have very fewlightly-damped overtones. Recall that the criterion underlying this plot captures only the damping due to viscoelasticity: for avery thick string like this, the damping due to air viscosity takes overa tl ow frequencywhile the viscoelastic loss is still quite high, so that in fact the model predicts that this string should have no modes at all with lowdamping. That is exactly what the measurements in Figure 2revealed.
This criterion based on damping can nowb ei ncorporated into ad esign chart for string selection. Forag iven string on ag iven instrument, the desired values of L and f 1 will be known, and the task is to select astring material and gauge d.Itispossible to summarise all the constraints on string selection into as ingle chart with α = Lf 1 on the horizontal axis and d on the vertical axis. The string tension T = πd 2 σ/4 = πρd 2 α 2 .T his means that for a givenmaterial, contours of equal tension can be plotted in the chart, as illustrated for nylon in Figure 5. Note that the values of tension shown here will be somewhat inaccurate because theyare based on the unstretched linear density of the string, whereas the actual linear density of the stretched string is alittle lower [7,8]. However, this is asmall effect and it makes no significant difference to the broad-brush argument underlying the design chart presented here.
From Equation (8),t he string impedance can be expressed as Z 0 = πρd 2 α/2a nd so impedance can also be indicated on this chart. The set of magenta dashed curves, falling towards the right, shows some selected contours of equal impedance. If one wished to select as et of strings with constant impedance, in the interests of equal loudness, these lines indicate the trend that should be followed.
Discrete symbols have been added to Figure 5t oi ndicate previously-studied nylon strings. Stars showt he strings from [7], each appearing as an approximately horizontal rowo fs tars showing the different tested tunings. Open symbols correspond to those in Figure 4, for the three strings whose properties were shown in Figure 2. The vertical lines give an indication of the ultimate breaking stress of nylon strings. The solid line marks the highest value of α for which astring survivedthe sequence of testing described in [7] without breaking. But nylon does not break immediately if this threshold is exceeded: what happens instead is that the string nevers tops creeping (and thus requiring to be re-tuned), and eventually it will fail. But manymusical instruments are fitted with nylon strings requiring ah igher value of α:t he most extreme the authors have been able to findf or an instrument in regular professional use is the top string of the 8-string "Brahms guitar" developed by luthier David Rubio in collaboration with guitarist Paul Galbraith [20,21]: the original version of this guitar has at op string of length 630 mm, tuned to 440 Hz (A 4 ), giving the value of α shown by the dashed vertical line.
Nowt he damping criterion can be added. Because the vertical axis depends on d rather than on β as in Figure 4, the length L will makead i ff erence. Forag iven value of L,i ti se asy to takee ach value of α and use the expression for λ to calculate the threshold value of d for which as tring of that length would have as pecified number of overtones with damping lower than the chosen value. For the purpose of plotting something that givesag ood indication of practical limits, the lines shown here correspond to requiring 10 overtones with λ<0 . 05. This leads to the rising curving lines in Figure 5 To understand what this chart shows, it is helpful to look at the schematic version plotted in Figure 6. Forany given instrument, the mechanical structure will impose an upper limit on string tension. Practical considerations of playing anyplucked-string instrument enforce alower limit on tension. The string needs to be belowi ts yield stress, or at least to be not too farabove that limit so that it survives for long enough to be useful. Finally,the damping criterion must be satisfied. The result is that the string needs to be chosen from within aregion of the plot likethe one shown shaded here, bounded by these various limiting conditions. Foragiven value of α,varying the gauge results in moving along av ertical line within this region. Moving upwards will increase the tension and the impedance. However, especially if the value of α is fairly low, it also results in moving closer to the curvegiving the trend of the damping limit. So, in very broad terms, athicker string will tend to be louder butless bright. Conversely,athinner string will have lower tension and impedance, and so be quieter but brighter-sounding.
Corresponding charts can be plotted for the other string materials for which mechanical properties are available. Figure 7shows the pair of charts for fluorocarbon strings. These are qualitatively similar to the charts for nylon, but with subtle differences in the shapes and positions of the curves that will be shown to have musical consequences in Sections 4.2 and 4.3. Figure 8shows corresponding plots for plain gut strings, and nowt he curves are quite different. The lines in Figure 8a are straight rather than curved, and this results in the damping threshold lines in Figure 8b also being straight. The explanation lies in the "strain stiffening" effect discussed earlier.T he curvature in the lines for the twos ynthetic polymer materials is ad irect result of the variation of E with α.Gut has astrain-independent value of E,a nd for that case it can be deduced directly from Equation (4) or (7) that the contours of λ are straight radial lines as seen in Figure 8a. That in turn produces straight lines in Figure 8b.
Both sets of plots include points for individual strings similar to those shown for the case of nylon. Figure 7b includes points for all fluorocarbon strings and tunings tested in [8]. One of these strings wass elected to include in 7a, the one with closest match of d to the 1.20 mm string chosen for Figure 4. The vertical line in Figure 7b marks the highest value of α for which afluorocarbon string survived the test programme described in [8]    surprising since the plot reveals that fluorocarbon breaks at al ower value of α than nylon, so for instruments requiring extreme values of α,n ylon is likely to be chosen in preference to fluorocarbon. Similarly, Figure 8b shows stars for all the gut strings and tunings tested in [8], and  [8,14]. the string with the closest d value to 1.20 mm has been selected to showi nF igure 8a. The solid vertical line in Figure 8b again marks the highest value of α for which a string survivedthe test programme, and the dashed vertical line marks the value corresponding to the highest string of the harp which will be studied shortly.T ounpack the significance of all these charts, twoc ase studies of practical string selection will nowbepresented.

Case study: the lute
String choice can be ap articularly trickyi ssue for period instruments. There is some guidance from historical sources about the choice of stringing, butt his typically dates from before the availability of precise measuring equipment, and so the guidance is purely qualitative.T he lute is ac ase in point. Lutes begant oa ppear in European music and art around about the 15th century, and the instrument continued to be played until about the time of Bach, gradually evolving in form: see for example [22]. The instrument has undergone arevivalinthe last 50 years or so. Acase study to "reverse engineer" aparticular choice of stringing for an 8-course lute will be presented here. This style of lute is typical of the late 16th century, the time of Dowland. Alute of this kind would originally have been strung with plain gut strings, butthe set exam- ined here consists of synthetic polymer strings, intended to produce asimilar sound and feel to gut. The particular set studied here is offered commercially for this purpose [23], without anye xplanation of the rationale behind the choices. The details of string length, gauge, tuning and material are giveni nT able II. The strings are amixture of nylon and fluorocarbon, and at first glance the underlying logic is not obvious. However, by plotting them on the design charts for nylon and fluorocarbon it will emerge that the choice is entirely rational. Figure 9a shows the nylon design chart as in Figure 5, with various relevant points added. The three open symbols are the same as in the earlier plots, for orientation. The three star symbols showthe top three strings of atypical modern classical guitar,which are usually of plain nylon. The remaining symbols, plotted as filled squares of various kinds for nylon and filled circles of various kinds for fluorocarbon, correspond to the set of lute strings. The full set has been plotted here, including the ones that are in fact fluorocarbon. In as imilar way, Figure 9b shows the design chart for fluorocarbon as in Figure 7b, with the same full set of points for the lute strings. In both plots, the damping threshold curveh as been plotted for twov alues of L:625 mm, the open string length of the particular lute tested (solid lines), and 312.5 mm, the length at the octave (dashed lines). Lutes of this period usually have no more than 12 frets, so the dashed lines mark the shortest relevant length. Armed with all this information, the string choices will be discussed, starting from the highest string.
The top string of the lute lies to the right of the vertical line in Figure 9a, whereas the top string of the classical guitar lies to the left of it. It is indeed the case that guitar top strings rarely break, whereas the top string of the lute has afinite lifetime before it breaks through progressive creep and eventual necking to failure. This top string is traditionally asingle string, butall the remaining strings of the lute come in pairs called "courses". It can be seen in the plot that the 2nd, 3rd and 4th courses (all in nylon) have essentially the same tension, around 30 N. There is some historic evidence for choosing strings of equal tension in instruments of this period (for example it is recommended by Mersenne in Harmonie Universelle [24]), so this choice is obviously deliberate. Note that in an instru-  Figure 5; stars: top three strings of atypical set of classical guitar strings; star-in-square: nylon strings for top four courses of lute; star-in-circle: fluorocarbon strings for 5th course of lute; plus-in-circle: fluorocarbon strings for courses 6, 7, 8o fl ute; plus-in-square: nylon strings for octave strings of courses 6, 7, 8oflute. ment likethe lute where all strings have equal length, equal tension automatically means "equal feel" from Equation (10).The top string has somewhat higher tension, perhaps to raise its impedance to compensate somewhat for the reduction in loudness as aresult of being single, or perhaps to increase the feel for asimilar reason. The contour lines for impedance showthat the first and second courses have similar values of impedance. As an aside, the top three strings of the classical guitar (stars)s howarather similar pattern in this chart to the top three courses of the lute.
The 5th course, shown as as tar-in-circle symbol, presents ap roblem for nylon strings. If an ylon string at this value of α waschosen with agauge to give it the same tension as courses 2, 3and 4, it would lie close to the solid curvei ndicating the damping roll-off for the open string length. This would lead to av ery unsatisfactory sound. But the designer of the set of strings investigated here has taken the sensible decision to switch to fluorocarbon for this course. The corresponding symbol in Figure 9b falls on the dashed curve, butlies well short of the solid curve.
This has been made possible by the subtle difference of shape between the limit curves for nylon and fluorocarbon, having its origin in the higher density and slightly different Yo ung'smodulus behaviour of fluorocarbon (see Table I). The chosen gauge of the fluorocarbon strings for the 5th course has resulted in essentially the same tension as the nylon strings for the higher-frequencyc ourses, as can be seen by counting the contour lines in the twoplots. However,t he charts reveal that the impedance of this fluorocarbon string is rather greater than that of the higher nylon courses.
Forthe three bass courses of the lute, even fluorocarbon strings suffer from the problem of high damping. The three strings are plotted as plus-in-circle symbols: all three lie near or beyond the solid curvei nt he fluorocarbon diagram, although it can be seen that the gauges have been chosen to continue the constant-tension pattern. To deal with this problem, standard lute stringing uses atrick. Instead of having twostrings in unison, as wasthe case for the other courses, the second string of each pair is tuned an octave higher: these octave strings are indicated by the plus-in-square symbols, and the string set specifies nylon for these strings at approximately the same tension as the other strings. The player plucks the twos trings simultaneously,w ith the result that the bass string givest he desired fundamental frequencycomponent, while the octave string (which lies belowthe solid curve) is able to give acceptable brightness to the tone by contributing aspread of higher overtones.
In summary,the designer of this string set has done an excellent job by mixing the twodifferent string materials. The only detail one might question is whether the three octave bass strings might have been specified in fluorocarbon (with suitably modified gauges)rather than nylon, to give them al ittle extra brightness. But even here, the decision may have been deliberate: aplayer probably does not want the octave string in abass course to sound too prominently, because that might impair the illusion of acombined string sound with both lowf requencyc omponents and brightness at higher frequencies. As afinal comment on this case study,itshould be noted that the damping roll-off predictions based on the chosen threshold value λ = 0.05 are in very good general agreement with the subjective impressions of aplayer.

Case study: the harp
Asharply contrasting case study in string choice is given by the harp. Again, this discussion is an exercise in reverse engineering: the information to be shown here is based on the chosen set of gauges for either nylon or gut strings on aparticular pedal harp. There is an interesting issue to be explained. Manyharpists, especially in Europe and North America, prefer gut strings, on grounds of "sounding better". Theyt end to regard nylon strings as only suitable for beginners. But virtually all classical guitarists have switched from gut to nylon strings, the only exceptions being those specifically choosing to play period instruments in an authentic style. Ap ossible explanation for this difference of opinion will be suggested in this section.   Table III. The lower notes of this harp used metal overwound strings, outside the scope of these design charts. Adistinctive feature of the harp, of course, is that each string has ad ifferent length. This means that each string strictly needs it ownv ersion of the damping roll-off curve. In an attempt to showthe pattern sufficiently clearly,aset of curves has been plotted for string lengths 0.2, 0.4, 0.6, 0.8 and 1.0 m, covering the range of the actual string lengths. The curves are plotted in alternating colours, and the discrete symbols for the individual strings switch colour in the same pattern, as the string length passes the values corresponding to these plotted curves. (The twos trings longer than 1m are shown with black symbols.)W hat is revealed in both plots is that the string choice has stuck rather close to the damping threshold used in these calculations, overthe entire range: the symbol colours change in am anner that is approximately synchronised with the plotted curves.
The other conspicuous feature of these plots is that the string tension increases steadily from treble to bass over Table III. String lengths and diameters for the harp string sets. Strings are numbered according to the harp convention, and the corresponding notes on the piano scale are given. The gut string gauges are from Bowbrand's" Pedal Light" range [25]. The nylon string gauges are based on Bowbrand's"Pedal Nylon" range, with scaling adjustments to provide abetter comparison with the gut "Pedal Light" range, which uses slightly thinner strings for some notes than the "Pedal Standard" gut range. Fluorocarbon ("Carbon")gauges are suggested, based on the scaling approach described in the text. the entire range of the instrument, reaching tensions far higher than those seen earlier for the guitar,l et alone for the lute. The tensions approach the 300 Nlimit of the contours plotted here. It may have been noted in the earlier plots, Figures 5, 7b and 8b, that the test points for the earlier studies [7,8] ceased at about the same tension. This is not acoincidence: the test rig for those earlier studies was designed with harp strings in mind, with aload cell set up for aforce measurement limit of 300 N. As aresult of these twofeatures of the stringing choices, the harp explores the region of the design charts where the nylon and gut results are most different. The straight lines of the gut chart give al ittle more "headroom" than the curves of the nylon chart, and this may be the keyt o harpists' preference for gut strings. Figure 11 givesdirect comparisons for several important quantities between the nylon and gut stringings, shown as the twos olid lines in each plot. The first subplot shows the string impedance, as afunction of string number (harp strings are numbered from highest to lowest, starting with the highest strings being labelled 00 and 0, then 1, 2, 3e tc.). The values for nylon and gut followasimilar trend, butthe nylon strings have consistently lower values across the whole range. The string impedance rises dramatically towards the bass end of the instrument, since both mass per unit length and tension have been chosen to increase as the strings get longer. Harp strings are certainly not selected with an eyetoconstant impedance, to result in constant loudness: as is shown by the contour lines of impedance in Figure 10, that would require tension to decrease, not increase, for the longer strings.
The clue to this apparently perverse choice may come in the second subplot of Figure 11: the increasing tension for longer strings means that the "feel" only varies rather slowly across the range of the instrument. From Equation (10),longer strings require higher tension to maintain "feel" at similar values. As with the impedance, the gut and nylon strings follows imilar trends in this plot, with nylon having consistently lower values.
The remaining subplots showtwo different views of the predicted damping roll-off frequency, according to the criterion used in all calculations here. The first of these plots shows the number of string overtones passing the threshold test, while the second turns this number into af requencyb andwidth by using the relevant fundamental frequencies. Aclear difference is seen between nylon and gut: the gut strings have as ignificantly wider bandwidth than the nylon strings, across the entire range. The reason is the increased "headroom" mentioned above.T he attempt to limit the variation of "feel" has forced the use of very high tensions, and this inevitably runs the danger of approaching uncomfortably close to the damping roll-off threshold. Nylon suffers from this problem to ag reater extent than gut, and so these results strongly suggest that gut strings will sound brighter than nylon strings overthe entire range of the harp. This is true even though the gauges of the nylon strings have been chosen to give lower tension than the gut strings: ac ompromise has been struck between loudness and brightness. This observation seems as trong candidate for explaining players' continued preference for gut strings, despite the disadvantages of higher cost and higher sensitivity to changing environmental conditions [8]. Figure 9a shows whyg uitarists can afford to makead i ff erent choice. The plain nylon strings of ac lassical guitar do not fall in the region of the design chart where there is such abig difference between gut and nylon, and so theyo pt for the convenience and practicality of synthetic strings. Theysome- times opt for fluorocarbon rather than nylon, especially for the 3rd string, just as the charts suggest.
Harpists are also showing ag rowing enthusiasm for fluorocarbon strings, saying that they" sound more like gut". The analysis presented here sheds light on this claim. This choice of string material is sufficiently newthat there is not yet awell-established choice of string gauges for the harp. Results are shown here for aparticular choice. Starting from the specification of the gut strings, string diameters d for fluorocarbon can be chosen by scaling the gut values by afactor 0.86, which is the square root of the density ratio of the twomaterials. That has the result of giving the twosets of strings the same mass per unit length, and hence the same tension. It follows that the impedance and the feel will be identical (atl east in this limited sense of the word "feel").
The dashed lines in Figure 11 showt he results of this choice. The lower twosubplots showthat the fluorocarbon strings consistently beat nylon strings in terms of damping roll-off.The values fall quite close to those for the gut strings, and for the lower strings theye venb eat them. It seems that fluorocarbon strings, with this choice of gauges, should indeed sound "more likeg ut", and for the lowerfrequencystrings theyshould actually sound brighter than gut. Of course, one would not in fact want to use fluorocarbon for the highest strings of the harp: nylon would still be used on grounds of strength. However, there is no problem with brightness of nylon for the highest strings: all three curves showabandwidth which exceeds the limits of human hearing, so that in practice the bandwidth will be determined by the details of the player'spluck gesture. There might even be advantages in the higher damping of nylon if there is ad anger of harshness in these highest notes.

Discussion and Conclusions
The choice of string materials and gauges for musical instrument use is influenced by several factors. There are limits on tension arising from practical considerations of strength and playability,a nd there is an upper limit on stress so that the string does not break. These are very familiar,b ut it has been shown here that there is also aless familiar limit arising from the influence of material damping. Fort he particular case of monofilament strings of a givenmaterial, it has been shown that all these factors can be represented in asingle design chart. Such charts give a synoptic view, shedding newlight on the selection process. It must be emphasised again that all the "limits" shown in these charts should be regarded as approximate. The intention is to showtrends to guide the process of selecting ac oherent set of strings for an instrument, not to show ultra-precise numerical values. Ultimately,the musician's ears are still the most important item of test equipment.
Examples of the design charts have been shown for nylon, fluorocarbon and natural gut strings and, based on these charts, detailed case studies for the stringing of a lute and ah arp have been presented. In both cases, the selection process has been shown to be significantly influenced by the newl imit associated with damping. This has givenpossible explanations for twoobservations: why manyh arpists still prefer gut overn ylon whereas classical guitarists have almost universally switched from gut to nylon, and whyfl uorocarbon strings are gaining in popularity overnylon in certain contexts because theyare said to sound more likegut.
The damping limit has been expressed in terms of a threshold value of the parameter λ,d efined in Equation (4),quantifying the influence of the string'sbending stiffness. Interestingly,the same parameter governs the extent of inharmonicity in the string'sovertones, also associated with the effect of bending stiffness. The fact that these two effects are directly connected in this waymay have perceptual significance, and it may shed newlight on some previously published perceptual experiments relating to inharmonicity.Järveläinen et al. [26] used synthesised tones to establish perceptual thresholds for the timbral effects of inharmonicity.T heyc oncluded that the inharmonicity effect should be clearly audible in standard classical guitar strings, especially the 3rd and 6th strings. But alater study by the same group [27] used manipulated sounds based on "authentic guitar sounds", and reached adifferent conclusion: theystate that inharmonicity should be barely audible in classical guitar sounds.
This second study differed from the first in howt he damping of the string overtones wasm odelled. The reference does not give very much detail, buti ts eems clear that the authors were not aware of the direct linkage between inharmonicity and damping behaviour.T he results presented here suggest that it might be worthwhile to carry out an ew perceptual test employing synthesised guitarlikeo rh arp-liket ones for strings of different materials and different gauges. The object of the test would be to establish the threshold of perception for ac hange in the value of λ,taking into account both the inharmonicity and the damping effect. Such at est would yield results of direct applicability to musical strings of the kind discussed here. Rather than being restricted to strings on aparticular instrument by manipulating measured sounds, the results should generalise to strings of agiven type in anymusical context.
The detailed results presented in this work have all related to monofilament strings of synthetic polymer or natural gut. But af ew comments can be added in relation to the other main types of musical string: monofilament metal strings, and metal-overwrapped strings with either metal or polymer cores. The methods used here would carry across directly to monofilament metal strings, butthe loss factor η E for relevant metals such as piano wire usually has av alue at least an order of magnitude smaller than those seen earlier for polymeric materials. Va lette has presented measurements on metal strings [3] which demonstrate that the damping model still works well, buts uch strings do not exhibit a" damping roll-off"i nt he sense explored in this work: the total loss factor η n at high frequencyn ever falls below η E whateverh appens, and so modal damping is always lowenough to be "musically acceptable".
Overwrapped strings are more complicated, and detailed analysis lies outside the scope of this article. The rationale of overwrapped strings is to increase the mass per unit length while limiting the effects of bending stiffness. Va rious details of the construction of such strings contribute to the sound theym ake: the choice of core material and of wrapping material(s),t he relative thicknesses of core and wrapping, and the details of the wrapping procedure all play ar ole. Va lette [3] has shown direct evidence that the tightness of the windings of ametalwrapped string can have alarge effect on the damping behaviour,because tight windings introduce anew damping mechanism associated with dry friction, whereas an open winding eliminates this effect.
Initially tight windings will loosen al ittle as the core stretches under tension, particularly with polymer-cored strings. This suggests that the sound of the string will change alittle as the string stretches and settles. However, this is not the only mechanism for the sound of wrapped strings changing overtime. It is very familiar to guitarists that news trings lose their "twang" as theya ge. Convincing evidence has been shown that suggests am echanism for this change: dirt and grease from the player'sfi ngers gradually penetrates between the windings of the string, changing the damping behaviour [28]. It has been shown that this effect can be represented within adamping model of the kind used here by an increase in the effective value of η E ,exactly as one would guess from the mechanism just described [29]. That brings the string-ageing phenomenon into contact with the earlier discussion in this article: the change in sound of an ageing guitar string results from ag radual decrease in the damping roll-off frequency, not because λ is changing butbecause the effective threshold value of λ falls as η E increases.