J Woodhouse

E K Y Manuel

L A Smith

A J C Wheble

C Fritz

Perceptual Thresholds forAcoustical Guitar Models

Synthesised acoustic guitar sounds based on adetailed physical model are used to provide input for psycho-acoustical testing. Thresholds of perception are found for changes in the main parameters of the model. Using a three-alternative forced-choice procedure, just-noticeable differences are presented for changes in frequency and damping of the modes of the guitar body, and also for changes in the tension, bending stiffness and damping parameters of the strings. These are compared with measured data on the range of variation of these parameters in aselection of guitars.

Introduction

Assessments of the quality of musical instruments can be influenced by manyf actors, including appearance and ergonomic considerations, butthe final judgement will presumably always relate to some extent to perceivedquality of sound. Instrument makers want to knowh ow to link particular changes in physical structure to their effect on perceivedsound, so that theycan control and improve the quality of their instruments. Auseful first step in mapping out these links, which lends itself to relatively uncontroversial psychoacoustical testing, is to establish the smallest change of each structural parameter that produces an audible effect under the most favourable listening conditions. Ac omprehensive set of such just-noticeable differences (JNDs)w ould give information about the relative sensitivity of the sound to the different parameters under the control of an instrument maker,w ithout the complications of trying to describe the nature of the changes in sound. Such descriptions are important, of course, butthey require different methods to study them and can raise more difficult challenges in experimental methodology.This paper concentrates on the first stage, finding JNDs.

Ar ecent research project [START_REF] Fritz | Perceptual thresholds for detecting modifications applied to the acoustical properties of aviolin[END_REF] has investigated some JNDs for violin acoustics. While that work wasgoing on, as eries of undergraduate projects wasu sed to perform a parallel investigation of the acoustical guitar,using similar psychoacoustical methodology.This paper summarises the main findings of those projects. As well as psychoacoustical results, measurements on av ariety of guitars will be presented to provide information about the actual range of variation in the parameters investigated.

In one important respect the guitar is very different from the violin. In the violin, the player can produce asustained sound by feeding energy into the vibrating string from the bow. This involves nonlinear interactions (from the friction force at the bow-string contact: see e.g. [START_REF] Woodhouse | The bowed string as we knowit today[END_REF])that make theoretical modelling difficult, and it also means that the player can influence the details of the sound of each note throughout its duration. The guitar is quite different: the instrument is, to ag ood approximation, al inear vibrating system. The player starts aparticular note with control overthe details of the pluck, butonce the string has been released the subsequent sound is not under the player's control. (This study excludes anyuse of vibrato.) This distinction leads to ad i ff erence in the wayt hat sounds for psychoacoustical testing can most usefully be generated. Forthe violin, it is remarkably hard to synthesise sound that is convincingly realistic because of the allpervading influence of the human performer.Instead, ahybrid strategy wasa dopted for the psychoacoustical tests, in which real playing wascombined with digital-filter emulation of the vibration response of the violin body.T he fluctuating force exerted by avibrating string on the violin bridge during normal playing wasm easured with an embedded piezoelectric sensor,a nd recorded. It wasa rgued that, to au seful first approximation, ag iven bowg esture on ag iven string will produce essentially the same force waveform regardless of the acoustical response of the particular violin body.For example, during asteady note the player will ensure that the string vibrates in Helmholtz motion, producing as awtooth force waveform at the bridge with relatively subtle modifications due to the bowing position and force, and the string parameters such as damping and bending stiffness [START_REF] Woodhouse | The bowed string as we knowit today[END_REF].

It wast hus argued that the main effects of ap articular violin body on the perceiveds ound should arise from the filtering of this bridge-force signal by the linear vibration response and sound radiation characteristics of the violin. Linear vibration response can be analysed into a sum of modal contributions, and can then relatively easily be emulated by as uitable digital filter.S o, for example, to investigate the JNDfor ashift in frequencyofone particular vibration mode of the body,aseries of sound files could be generated using the same recorded bridge force signal, modified by ar ange of digital filters initially calibrated against ap articular violin body butw ith the one desired mode frequencysystematically altered. The use of recorded bridge-force signal from ahuman performer gets around the problem of realism, without giving the player anyopportunity to alter their bowing in response to the parameter being varied because the same input signal is used in every case.

Forthe guitar,this approach will not work convincingly. Part of the sound quality of ag uitar note comes from the frequency-dependent decay rates of the different overtone components of the sound. These are determined by interaction of the string and the guitar body through the coupling at the bridge. The vibration response of aguitar body could be emulated by digital filter just as easily as aviolin body,b ut if the hybrid approach were to be used for the guitar,t he mechanical coupling of string to body would be missing and it is likely that ac rucial part of the influence of the body acoustics on the sound would be lost. Luckily,t he issue of non-realism of synthesised sound is much less challenging for the guitar.As ynthesis model for guitar notes based on ad etailed physical model has been described previously [START_REF] Woodhouse | On the synthesis of guitar plucks[END_REF][START_REF] Woodhouse | Plucked guitar transients: comparison of measurements and synthesis[END_REF], and this has been found to produce quite plausible sounds. Indeed, several participants in the studies to be described here expressed surprise when told that theyw ere hearing synthesised sounds. In consequence, the JNDstudies for guitars were carried out entirely using synthesised sounds.

There is very little existing literature that is directly relevant to this study.While alot has been written about the vibration and sound radiation characteristics of guitar bodies, published psychoacoustical studies based on the modelling are very rare: the only example seems to be the doctoral work of Wright [START_REF] Wright | The acoustics and psychoacoustics of the guitar[END_REF]. There is, however, some literature relating to guitar strings, specifically to the perception of inharmonicity associated with bending stiffness. Two separate studies have been published by Järveläinen and co-workers [START_REF] Järveläinen | Audibility of the timbral effects of inharmonicity in stringed instrument tones[END_REF][START_REF] Järveläinen | Perceptibility of inharmonicity in the acoustic guitar[END_REF] using synthesised sounds of differing degrees of realism to obtain thresholds for discrimination between harmonic and inharmonic sounds. Theyf ound that thresholds for highly idealised sounds were lower than for more realistic sounds including the initial transient of a guitar pluck, presumably because of some degree of informational masking in the more complexsounds.

The outline of this paper is as follows. The synthesis model is briefly described, and the parameters to be explored in JNDs tudies are defined. The technical details of the psychoacoustical test procedure are then given, followed by the acoustical test methods for establishing physical parameter values for the guitar body and for the strings investigated. Results are then presented, first for the properties of the guitar body.Several tests will be presented in which modal frequencies are shifted: all together,orconfined to particular frequencyb ands. The influence of the choice of musical passage for the test stimulus is also discussed. After this, amore restricted study wasundertaken in which the modal damping factors were scaled. The final section of test results concerns properties of the strings, especially the damping properties. These results will be related to the perception of "new" versus "old" strings by achange in their characteristic sound, in the light of measurements of what actually changes as steel-cored guitar strings age.

Outline of synthesis model

The synthesis model to be used here is primarily aimed at achieving an accurate representation of the coupled mechanical vibration of the strings and guitar body.I td oes not attempt ac omparably full treatment of the details of the player'sp luck gesture, or of the sound radiation behaviour: these are both taken into account in asimple way only,aswill be described shortly.The input to the model is as et of parameter values describing separately the behaviour of the guitar body and the strings. Foreach body mode four things are required: the natural frequency, the modal damping factor,t he modal amplitude at the position on the bridge where the string is coupled, and an angle describing the orientation of motion at that coupling point. The modal amplitude can be equivalently expressed via an effective mass: the amplitudes deduced by the techniques of experimental modal analysis correspond to mass-normalised modes (see for example Ewins [START_REF] Ewins | Modal testing: Theory,practice and application[END_REF]), and the square of the measured amplitude at agiven point is the inverse of the effective mass for representing that mode at that point by amass-spring-damper combination. The angle of motion at the coupling point is needed so that both polarisations of string motion can be included in the synthesis.

Ideally,t he amplitudes and angles of each body mode would be separately determined at the six positions on the bridge saddle where the strings makecontact. This would be possible butlaborious, and for the present study it has not been done. The body modal parameters are deduced from am easurement of the input admittance (also called the driving point mobility)a tap osition between the 5th and 6th strings (the twotuned to the lowest frequencies). The same body response is used for all strings in the synthesis process. This position waschosen mainly because it is easier to perform the measurements necessary to establish the modal angles at the ends of the bridge rather than in the middle for practical reasons of access. Other positions on the bridge would of course give the same natural frequencies and modal damping factors, butw ith different amplitudes and angles. Since the aspects of body behaviour studied here involveo nly frequencies and damping, one might hope that the conclusions would not be significantly changed by amore thorough synthesis allowing different body behaviour for each string.

The string is treated as ac ontinuous system rather than being described in modal terms: this wass hown to give am ore efficient strategy for computing the coupled string/body motion with correct allowance for the very different damping levels of the twos ystems, which causes the damping of the coupled system to be strongly nonproportional [START_REF] Woodhouse | On the synthesis of guitar plucks[END_REF]. Ag iven string is specified by its length L,t ension T ,m ass per unit length m,b ending stiffness EI,a nd am easure of its damping. The bending stiffness is expressed as the product of the Yo ung'smodulus E and the second moment of area I,w hich for ah omogeneous monofilament string of diameter d is givenby

I =πd 4 /64. ( 1
)
Inharmonicity due to bending stiffness is often expressed in terms of the dimensionless parameter B that appears in the approximate expression for the frequencyo ft he nth overtone,

f n ≈ nf 0 1 + Bn 2 , ( 2
)
where f 0 is the fundamental frequency. In terms of the notation introduced here,

B = EIπ 2 TL 2 . ( 3
)
Foro verwound strings, as normally used for the lower strings of the guitar,t he effective value of EI or B cannot be reliably predicted, and must be measured. Musical strings in general have been shown to exhibit strongly frequency-dependent damping, and the measured behaviour can be fitted to at hree-parameter model proposed by Va lette [START_REF] Va Lette | The mechanics of vibrating strings. -I n: Mechanics of Musical Instruments[END_REF]. The loss factor of the nth mode is assumed to takethe form

η n = T (η F + η A /ω n ) + EIη B (nπ/L) 2 T + EI(nπ/L) 2 . ( 4
)
The damping is lowest in the mid range of frequencies, with aminimum levelset by η F .Atv ery lowfrequencies the damping increases, an effect attributed by Va lette to air damping and quantified by the parameter η A .Athigh frequencies the damping increases again, because energy dissipation associated with the complexb ending stiffness of the string comes to dominate, especially for the polymerbased strings of classical guitars. This effect is characterised by the parameter η B . The chosen synthesis method works in the frequency domain, then uses an inverse FFT to obtain the time response. Full details are givenin [START_REF] Woodhouse | On the synthesis of guitar plucks[END_REF][START_REF] Woodhouse | Plucked guitar transients: comparison of measurements and synthesis[END_REF]. The string and body are coupled at the bridge using the standard point-coupling formula

Y -1 coup = Y -1 body + Y -1 str , ( 5
)
where Y coup , Y body , Y str are the 2 × 2matrices of admittance at the bridge notch for the coupled system, the body alone and the string alone, respectively.E ach synthesised note begins with an idealised pluck, which is astep function of force as the finger or plectrum loses contact with the string. The string is assumed to be released from rest at the instant of the note starting. Advantage is taken of the reciprocity principle: it is convenient to apply the force step at the bridge and calculate the response at the point on the string where the pluck acts. As well as the plucking position on the string, it is possible to specify an angle of pluck (which determines the mix of the twopolarisations of string motion excited). The player'sfi nger or plectrum will have a finite size, and this will have the effect of alow-pass filter on the excitation of string modes. This is represented in a simple waybyafrequency-domain Gaussian filter proportional to exp(-ω 2 /K 2 ). Here, K = v/h where v = T/m is the wave speed of the string and the width h wasgiven the value 7.5 mm in order to obtain ar easonably realistic sound.

The synthesis model calculates the transient waveform of acceleration at the coupling point between the string and body.Afull model would then need to calculate the body motion at all other positions, and perform ar ather complicated numerical calculation to give the resulting sound radiation. The present model uses aseverely simplified approach: the computed bridge acceleration is regarded as being applied to as pherical radiating body of radius a, vibrating in the symmetric breathing mode only.F or that problem, the sound pressure in the farfi eld is givens imply by filtering the acceleration waveform with alow-pass filter that, apart from an overall scale factor,takes the form

R(ω) = ika 1 + ika = iω/ω c 1 + iω/ω c , (6)
where k = ω/c is the wavenumber for sound in air, c is the speed of sound, and ω c = c/a is the filter roll-off frequency. Setting this roll-off at 250 Hz, corresponding to asphere radius around 0.2 m, provedtogivesatisfactorysounding results. This synthesis procedure has been implemented in Matlab, in aprogram that allows the user to specify amusical fragment entered in aform of guitar tablature. The resulting "tune" is savedasasound file.

Experimental methodology

Psychoacoustical tests

Using synthesised sounds generated by this method, experiments were conducted to give some JNDinformation relating to the major parameters. Forthe body model, experiments were done in which all the mode frequencies were shifted by agiven factor,and also groups of modes in particular frequencybands were shifted, leaving the other modes unchanged. In aseparate pair of tests, the damping factors of all modes were modified by ag iven factor,e ither upwards or downwards from the reference values in case there wasanasymmetry of perception. Forthe string model, the bending stiffness and the damping parameters η F ,η B were changed. The parameter η A wasfound to have so little influence on the chosen test sound that it wasnot possible to obtain am eaningful JNDf or changes to its value. Ag eneral comment should be noted: all the tests to be reported here were constrained by the time limits imposed by the format of undergraduate projects. In some cases there is no doubt that more could have been learned had more time been available, butitisfelt that the results have value even with these limitations.

JNDthresholds were estimated using athree-alternative forced-choice procedure (3AFC). This is not the only possible procedure that could have been used, buti th ad already been chosen for the violin studies and software to implement 3AFC wasc onveniently to hand. At hreedown one-up adaptive tracking rule wasu sed that estimated the 79% correct point on the psychometric function [START_REF] Levitt | Transformed up-down methods in psychoacoustics[END_REF]. It should be noted that this is the definition assumed for "JND" throughout this paper: it is arather conservative value, significantly higher than that which would correspond to the 50% point on the psychometric function, which perhaps better matches the intuitive notion of "threshold of perception". Three sounds -t wo the same, the reference sound, and one different, the modified sound -were played in arandom sequence, and the subject was asked to choose which one wasd i ff erent. The amount of parameter modification between the stimuli in agiven trial waschanged by acertain factor,the step size. The step size decreased after three correct answers and increased after one wrong answer.The test stopped after eight turnpoints. Arelatively large initial step size wasapplied until the second turnpoint wasreached, in order to allowrapid convergence toward the threshold region. After the second turnpoint the step size wasr educed. The threshold wast aken as the mean of the values of the amount of modification at the last six turnpoints. The values of the twostep sizes, and the starting levelo fp arameter modification, were chosen differently for each test on the basis of preliminary trials. Each trial took no more than about 10 minutes, and subjects had abreak between tests.

In all tests the reference sound waskept constant, in order to increase discriminability: subjects "learnt" to recognize the reference sound. It is argued that for the purpose of relating to musical judgments, which in practice are made by musicians who may spend several hours every day concentrating on subtle details of the sound of their instrument, the threshold of most interest is the one obtained under the most favourable conditions for discrimination. It is accepted that the test circumstances are quite remote from normal musical experience, as seems to be inevitable in the design of anys cientifically-acceptable test procedure, and the approach adopted seems to the authors the best that can easily be done. The 3AFC software gave visual feedback on right/wrong answers during the experiment, although some subjects performed best by keeping their eyes shut and ignoring it. Subjects did not receive anyt raining beforehand, buti ft heyp erformed erratically on the first run theyw ere able to repeat the run. This applied only to afew subjects, and theywere all able to perform the task at the second attempt. The sounds were presented diotically via Sennheiser HD580 headphones, cho- These are all longer than the 300 ms length that yielded the lowest thresholds in the violin tests, and perhaps some advantages of echoic memory are sacrificed by using these longer stimuli. However, it wasf ound that the sense of a realistic guitar sound wascompromised if the duration was too short, and it seemed more appropriate to use the longer, more "guitarish" sounds.

The test subjects were all "musical" in one sense or another,b ut for the purposes of the presentation of results theyare divided into twogroups using the same criterion as in the work on the violin [START_REF] Fritz | Perceptual thresholds for detecting modifications applied to the acoustical properties of aviolin[END_REF]: those labelled "musicians" had at least 8years of formal training and were currently active in music-making at least once per week, while the others are labelled "non-musicians". The first author was asubject in each test, while all the other subjects were of student age. The results for the first author did not seem in anyw ay atypical of the rest, despite his relatively advanced age.

Physical measurements

Physical measurements on the body and strings of several guitars are used in twod i ff erent ways in this study.F irst, measurements are needed to calibrate the synthesis model, as has been described in detail previously [START_REF] Woodhouse | Plucked guitar transients: comparison of measurements and synthesis[END_REF]. Second, the same range of measurements can be analysed to give estimates of the variability in practice of the parameters for which JNDs will be found. By comparing the actual variation with the JNDfor perception, it is hoped to give afirst estimate of the relative sensitivity and importance of different kinds of variation in guitar construction and stringing.

Input admittance Y (ω)o ft he guitar body is measured by applying acontrolled force either with aminiature impulse hammer (PCB 086D80)orbygently pulling alength of finec opper wire, looped around as tring at the bridge saddle, until it snaps. The latter method is less accurate, buthas the advantage that it can be used to apply force in different controlled directions to give information on the modal angles. In all cases the velocity response is measured with alaser-Doppler vibrometer (Polytec OFV-056), directed to aspot as close as possible to the forcing point. Signals are collected at as ampling rate of 40 kHz by a digital data-logger based on aP Cr unning Matlab, with aN ational Instruments 6220 data acquisition card. The strings of the guitar were damped during admittance measurements.

Input admittances in the direction normal to the plane of the top plate have been measured on aw ide range of acoustic guitars, and ar epresentative selection of 15 are used in the present study.F igure 2s hows results for four of these, all hand-made instruments of high quality.T he heavy curveisthe datum guitar used in the psychoacoustical tests, aflamenco guitar by Martin Woodhouse studied in the previous work [START_REF] Woodhouse | Plucked guitar transients: comparison of measurements and synthesis[END_REF]. The others are afl amenco guitar by Jose Ramirez, and classical guitars by GregSmallman and Michael J. O'Leary.I ti si mmediately clear that all these guitars, although different in detail, showal ot of common features. All have as trong peak in the range 80-100 Hz, which is aH elmholtz resonance associated with the internal air cavity and the soundhole, modified by the compliance of the top and back plates. All the guitars then showadeep antiresonance (marking the frequencyat which the Helmholtz resonance would occur with ar igid cavity,s ee equation (10.19)o fC remer [START_REF] Cremer | The physics of the violin[END_REF])f ollowed by ac luster of strong peaks in the range 150-250 Hz. Most then showas teady decline to ad ip around 400-500 Hz, followed by fluctuations around arather steady levelwith gradually reducing peak amplitudes as the modal overlap becomes significant. One guitar (the O'Leary)h as an additional strong peak around 300 Hz interrupting the decline to the dip. The twohorizontal lines mark frequency bands that will be used in frequency-shifting tests in the next section. The first of these bands (150-250 Hz)covers the range where the datum guitar has its strongest peaks. The second band (500-1000 Hz)covers arange where the modal overlap is becoming significant, butfairly clear individual peaks can still be seen.

The other physical measurements used in this study are aimed at establishing the properties of strings. Tension and mass per unit length are readily established, butthe bending stiffness and damping properties require careful measurement. Such measurements have previously been made on atypical set of nylon strings for aclassical guitar,and in the course of this project corresponding measurements were made on as et of steel strings as commonly used on folk guitars. The chosen strings were Martin "80/20 Bronze". It is af amiliar experience to guitarists that the sound changes alot between when the strings are newand when theyare well-played, so the strings studied here were measured in both states to give quantitative data on what changes in the ageing process.

The bending stiffness and damping properties can be estimated by analysis of sounds recorded from notes played on the guitar in the usual way [START_REF] Woodhouse | Plucked guitar transients: comparison of measurements and synthesis[END_REF]. The frequencies of the strongest spectral components, corresponding to "string" modes, can be best-fitted to equation [START_REF] Woodhouse | The bowed string as we knowit today[END_REF] to yield values of B and hence EI.T he associated decay rate of each of these "string" modes can also be found using timefrequencya nalysis, and this givesi nformation about the string damping. The procedure for obtaining damping information from the decay rates is alittle indirect, because in most cases the decay rate is dominated by losses into the guitar body,rather than by internal damping in the string.

The approach adopted is to analyse the frequencies for aplucked note on every fret of agiven string, and combine them all into ac loud plot of damping factor against frequency: examples are shown in Figure 3. This shows data for the 3rd string (G 3 ), for the nylon string measured previously and the news teel-cored string. The inherent damping of the string, as af unction of frequency, determines the "floor level" of this cloud plot. Approximate values of η A ,η F ,η B can be estimated by fitting eqquation (4) to the trend of this floor level, as illustrated by the curves in Figure 3. The fitting process wasc arried out manually,w ith some regard to the error tolerances of the data points. Note that the high-frequencyfi tl ine for the nylon string was influenced by additional data obtained by ad i ff erent approach, not shown here: see [START_REF] Woodhouse | Plucked guitar transients: comparison of measurements and synthesis[END_REF] for more detail. It is clear from the figure that the nylon string has higher damping than the steel string. The measured string properties are summarised in Table I. The differences between the nylon and steel strings in newcondition are broadly as one would expect. The nylon strings have lower tensions and adifferent pattern of bending stiffness because of the difference in construction: for the nylon set the top three strings are monofilaments while the lower three have multifilament cores with metal windings. The steel strings have monofilament cores throughout, and only the top twos trings are not wrapped. The monofilament nylon strings have higher internal damping than the steel, butt he difference is much less marked for the lower three strings.

The effect of age and use on the steel strings can also be seen. Previous investigators have reported increased damping between newand old guitar strings [START_REF] Allen | On the aging of steel guitar strings[END_REF][START_REF] Hanson | Analysis of "live"and "dead" guitar strings[END_REF], but without sufficient detail to allowafi tt oe quation (4).I t is generally accepted that the mechanism for this ageing effect is the accumulation of dirt and corrosion products between the windings of wrapped strings [START_REF] Hanson | Analysis of "live"and "dead" guitar strings[END_REF], perhaps with some additional cumulative wear of the outer wrappings where the strings makec ontact with the frets. The measurements reported here support this picture: there is very little change in the twohighest strings, which are not wrapped. Fort he wrapped strings there is am arked increase in damping, especially in η B which determines the decay at the highest frequencies. Also, not noted by previous authors, there is am arked increase in the effective bending stiffness of some of the strings. This seems perfectly credible since the contact conditions between the windings in the tensioned string will contribute to the bending stiffness, and material col-lecting between the windings will surely change those conditions. This is consistent with the observed increase in η B , which is the loss factor associated with the bending stiffness term in the governing equation. It may be noted that the bending stiffness values givenh ere are in good general agreement with values reported by Järveläinen et al. [START_REF] Järveläinen | Perceptibility of inharmonicity in the acoustic guitar[END_REF], noting that their steel-string results match the "used" results more closely than the "new" results. Their paper does not report the condition of the strings as measured.

Results forp roperties of the guitar body

Psychoacoustical tests

The first series of psychoacoustical tests concerned the shifting of all body mode frequencies by ag iven factor, keeping other simulation parameters fixed. This modification is of course somewhat artificial, butapossible physical interpretation will be discussed in the next subsection. The datum for this test wasb ased on the modal properties of the flamenco guitar by Martin Woodhouse. This test wasr un three times, using the three different musical inputs described above.The three tests were completed by 8 musicians and 5non-musicians (according to the criterion described above). The results are summarised in Figure 4. The program that runs the 3AFC protocol givesasoutput the mean and standard deviation overthe range defined by the final 6turnpoints. The plots show, for each test subject, the mean on the horizontal axis and the relative standard deviation (standard deviation divided by mean)onthe vertical axis. The numbers on the horizontal axis indicate the The open square symbols showthe average of the four subjects who achievedthe lowest mean levels. As wassuggested earlier,w ithin the scope of the present study the most interesting threshold to compare with serious musical judgments is that achievedb yt he best listeners under the most favourable conditions. In the violin tests, the best fiveresults were presented for this purpose [START_REF] Fritz | Perceptual thresholds for detecting modifications applied to the acoustical properties of aviolin[END_REF]. In the tests reported here the number of subjects is sometimes rather low, and it wasd ecided to average the best four.T he numerical results of this best-four measure for all the 3AFC tests presented here are giveninthe figure captions.

It can be seen that for each of the three musical inputs the best subjects could discriminate ashift of 1% or less. Interesting patterns can be seen in the detailed results, despite the small number of subjects that would makea ny elaborate statistical analysis of limited value. The general spread of results, and the twoa verages, look very similar for the single note and the Huntsuppe extract. Fort he single note the musicians and non-musicians are intermingled, while for Huntsuppe all butone of the non-musicians has aw orse result than all the musicians. Perhapsn onmusicians have their performance degraded by informa- tional masking while experienced musicians do not, with such an obviously musical sound sample as "Huntsuppe".

The chord reveals ad i ff erent picture. Four of the musicians achievedv ery lowt hresholds with this sound input, buta ll other subjects seem to have been seriously distracted by this rather non-musical sound, and achieved only rather high thresholds. One of the musicians comes out worst of all in this test, whereas for Huntsuppe all the musicians performed well. This is as triking example of informational masking, especially since the group of four subjects who did well actually produced al ower average than in either of the other tests. That is as one would hope: this signal contains, deliberately,aw ider range of frequencycomponents so that analytical listeners with sufficient skill and experience to takeadvantage of it can make very subtle judgements down to 0.32% shift in the best case. This corresponds to shifting the mode frequencies about 6cents or 1/20 of asemitone, an impressively small amount that is not very much bigger than the threshold for pitch discrimination tasks (see e.g. [START_REF] Moore | An introduction to the psychology of hearing[END_REF]).

The thresholds are of the same order as the "best five" thresholds in the corresponding violin tests, butt he detailed pattern seems different. The choice of input had a more marked effect in the violin, and to achieve the lowest thresholds av ery short, and consequently rather unmusical, sound sample wasused.

The best-performing 7s ubjects of the first test series were asked to undertaket wo further tests, in which fre-quencies were shifted only in limited bands. The two bands studied were shown in Figure 2, and this time the only input sound tested wasthe single-note open Astring. The results of the tests are shown in Figure 5inthe same format as Figure 4. Rather unexpectedly,t he "best four" threshold for the first band, containing the strongest peaks of the datum guitar,w as much higher than for the second band. Related tests for the violin, using the so-called "Dünnwald bands", gave thresholds around 2% for the first three bands (see Figure 6ofFritz et al. [START_REF] Fritz | Perceptual thresholds for detecting modifications applied to the acoustical properties of aviolin[END_REF]). Band 1inthe present study gave acomparable result, butband 2gavea lower result than waso bserved in anyo ft he violin tests. The best-four JNDinthis case is virtually the same as for the corresponding test when all frequencies were shifted. Further studies might be worth conducting to add more detail to this observation.

The final pair of tests relating to the perception of body mode changes investigated the effect of damping factors. This time the mode frequencies were kept fixed butt he modal Qf actors were all scaled by ag iven factor: Qf actor is the inverse of damping factor,sohigh Qcorresponds to lowd amping. In twos eparate tests the Qf actors were increased and decreased, to look for anyasymmetry in the perception. Thirteen test subjects took these twotests, 10 musicians and 3non-musicians. Results are shown in Figure 6inthe same format as Figs. 4and 5, except that the numbers on the horizontal axis nowindicate the scale factor applied rather than the percentage change. This means that "no change" corresponds to the value 1rather than 0 as in the previous results, which has the effect of making the relative standard deviations numerically smaller.

An asymmetry is seen between the twotests, butitdoes not significantly affect the threshold value for the best four listeners. Theya chievedav alue around 1.2, i.e. a 20% change in Qf actors, in both cases. However, across the whole set of test subjects the spread is much greater when the Qf actors were decreased compared to the increasing case. Manyparticipants reported initial difficulty in hearing the difference when Qf actors were decreased, although once theyh ad learned to recognise the sound change theyoften achievedquite lowthresholds.

It wasnot thought worthwhile to test separate frequency bands in this case, for tworeasons. First, body damping is most readily altered by aphysical change of material, and this is likely to have an effect overabroad frequencyrange. Second, it will be shown shortly that the JNDfor perception of ac hange in body damping is of the same order as the measured standard deviation among real guitars, so that the perceptual significance is probably marginal and digging deeper into the details is likely to be of limited importance.

Physical measurements

The next step is to look at physical measurements to assess howthese thresholds of perception compare with the actual extent of variation between guitars. Of course, one would neverfi nd twor eal instruments which differed by having all the body frequencies simultaneously scaled by exactly the same factor.However,itiseasy to visualise a change which would in principle have this effect: it would happen if the dimensions of the guitar were all scaled, keeping the same materials and other details. Some simple deductions can then be drawn by scaling analysis. If the main vibrating elements of the guitar body are treated as thin, flat plates deforming in bending, then the natural frequencies scale proportional to h/d 2 where d is the scale factor for dimensions in the plane and h is the scale factor for thickness. Taking 1% as around number for the order of magnitude of the threshold of detection for shifting frequencies, it follows that the JNDs for dimensional scaling are approximately 1% if d and h are both scaled together (full geometric scaling), 1% again if thickness is changed keeping the in-plane dimensions fixed, and only 0.5% if in-plane dimensions are scaled keeping the thickness the same. These are all very small numbers: theya mount to changing in-plane dimensions no more than afew mm, and to changing atypical top plate thickness by amere 25 µm. Real instruments have dimensional variations which are considerably bigger than this, especially in thickness, so on this ground alone it is no surprise that guitars can sound different.

The simplest approach to obtaining quantitative information from admittance measurements on as et of guitars is to perform explicit modal analysis. This can only be expected to give reliable results at relatively lowf requencies: pole-residue extraction, whatevert he method, becomes sensitive once the modal overlap begins to approach unity.Toillustrate the kind of values which emerge from such an analysis, Figure 7s hows results for modal frequencyand Qfactor for 15 guitars, for the highest peak in each of the twolow-frequencybands mentioned in the discussion of Figure 2: 75-100 Hz, the "Helmholtz resonance", and 150-250 Hz, the "first top plate mode".

Ac omplicating factor needs to be noted. The admittance measurements are made with the strings well damped, because the purpose is to obtain information about the body behaviour without the effect of the strings. However, the damped strings remove some energy from the body vibration, and this will tend to reduce the measured Qf actors. This effect can be compensated, at least approximately.The damped strings will act essentially as semi-infinite strings and will thus present an impedance to the bridge which is real and, if the strings are all assumed to be attached at the same point, equal to the sum of the string impedances givenbyZ = √ Tm.From the data for nylon strings in Table I, the sum of these impedances is 2.12 Ns/m. To compensate, the inverse of Y (ω)i st aken, this real quantity is subtracted, then the result re-inverted to give anew admittance. This approach can be expected to work well at higher frequencies, butitishard to be sure if the strings are well enough damped at lowf requencies to justify the semi-infinite approximation.

Figure 7shows results before and after this compensation. Fore ach case the mean and scatter are indicated by lines showing one standard deviation above and belowthe mean for frequencyand Q. Interestingly,the standard deviations have asimilar pattern for both cases. The relative standard deviations for frequencyare 5.5% and 6% for the twocases, and those for Qfactor about 33% in both cases before compensation.

To obtain good comparative information about modal frequencies overab roader bandwidth is more challenging. It is not possible to identify the "same" mode in different guitars except at very lowf requencies: sensitivity to small structural changes increases with frequency, as is familiar in other vibration problems (see e.g. [START_REF] Manohar | Statistics of energy flows in spring-coupled one-dimensional subsystems[END_REF]). For mode frequencies, the threshold for audible changes is so small that it is hard to obtain anym easure which is good enough: scaling frequencies by 1% means that one guitar body might have 101 modes in af requencyr ange where another guitar had 100, and there is no known analysis method which can give robust discrimination at this level. Anymeasure which can be computed shows much bigger variability than this, and there is always astrong suspicion that it is revealing the vagaries of the analysis method more than the real differences between guitars.

Things are alittle more promising in the case of damping estimation overaw ide frequencyr ange. Taking the inverse FFT of an input admittance givesa ni mpulse response function of the guitar body,a nd this can be processed using time-frequencya nalysis to give as onogram representation. In each frequencyb and, ab est-fitting exponential decay can then be obtained. Combining results from several different choices of the time/frequencyr esolution tradeoff improvest he robustness of the results. The process is described in more detail elsewhere [START_REF] Woodhouse | Interpreting the input admittance of violins and guitars[END_REF]. Some results are shown in Figure 8: compensation for string damping has been included. In Figure 8a the discrete points represent, for the datum guitar,f requency/decay combinations which satisfied an assumed criterion for good fit quality.F or compatibility with other plots, decay rate is represented as an equivalent Qfactor.The continuous line is simply amoving band average of these discrete points.

Figure 8b then shows the band-averaged lines for the 15 guitars in the test set. Finally,F igure 8c shows as catter plot of the mean and standard deviation of these bandaveraged lines for the 15 guitars. As in Figure 7, the large cross represents one standard deviation above and belowt he mean on each axis. The main result is that this frequency-averaged measure of damping has ar elative standard deviation of the order of 20%, comparable to the JNDfound earlier for perception of damping changes. This suggests that the extremes of mean damping in the measured set of guitars are sufficiently different that the effect should be audible, butt he results do not lead one to expect that this will be av ery strong component of the perception of tonal differences among guitars.

Results forp roperties of the strings

The final set of JNDe xperiments relates to changes in string properties, including those properties that changed as the string aged: one objective wast ofi nd out which of those changes are likely to contribute most to the perception of ageing. Using as datum the properties of the newsteel strings listed in Table I, changes to the damping properties of the 5th string were investigated using singlenote test sounds. These 3AFC tests were completed by 11 musicians and 5n on-musicians. One test wasa lso conducted relating to changes in the bending stiffness of this string: it gave results in line with those of Järveläinen et al. [START_REF] Järveläinen | Perceptibility of inharmonicity in the acoustic guitar[END_REF] and the details are not reproduced here. The actual bending stiffness of this string wasf ound to be close to the threshold for detecting an increase in bending stiffness when starting from amuch lower level. In the aged string, the bending stiffness only increased by af actor 1.6 (see Table I).Ifthat change is audible at all the effect is likely to be small. The results for the damping tests are shown in Figure 9, in as imilar format to earlier 3AFC results. As noted earlier,t he perceptual effects of changing η A for this string were found to be so weak that it wasnot possible to conduct ameaningful test. However, both η B and η F gave clear results. The parameter η B ,a ssociated with bending stiffness and mainly affecting higher frequencies, wast ested using twodifferent notes "played" on the same string. The results are shown in Figures 9a,b, and showag ood level of consistencyb etween the twon otes. The spread, overall mean and best-four mean are all very similar for both cases. The best listeners need about afactor 3increase in this parameter to perceive the change. Note from Table I that in the aged string this parameter increased from 0.004 to 0.029: that change should be very clearly audible.

Figure 9c shows corresponding results for η F .All subjects gave asomewhat lower threshold for this parameter, compared to η B ,and the very best listener achievedaJND of less than af actor 2. In the aged string η F increased in the ratio 17:11 for this string, from Table I. This effect is probably not audible to most listeners. It can be concluded that the factor most likely to account for the difference in sound of the aged string is η B .

Conclusions

Psychoacoustical experiments have been reported to establish JNDl evels for the perception of changes to various parameters of amodel of synthesised guitar sound [START_REF] Woodhouse | On the synthesis of guitar plucks[END_REF][START_REF] Woodhouse | Plucked guitar transients: comparison of measurements and synthesis[END_REF]. These investigations parallel earlier work exploring related JNDs for violin sounds [START_REF] Fritz | Perceptual thresholds for detecting modifications applied to the acoustical properties of aviolin[END_REF]. The parameters investigated here relate to the behaviour of the guitar body (modal frequencies and damping factors)and of the strings (bending stiffness and damping model parameters). All tests were carried out using athree-alternative forced-choice methodology.I ti sa rgued that for application to musical judgements, which can be very subtle, the most interesting perceptual thresholds are those attainable by the best listeners under the most favourable conditions. To give an estimate of these, the results for the best four subjects in each test were averaged.

Results of physical measurements have also been reported, to give an indication of the spread of each parameter among real guitar bodies or strings. Acomparison with the psychoacoustical results givesafi rst impression of which quantities might be responsible for the biggest perceptual differences between instruments. By fart he most sensitive parameter of all those tested relates to shifting the frequencies of body modes. The best four listeners could perceive a1%shift when all frequencies were scaled together,as mall change compared to the actual variation among guitars. This performance waso nly degraded slightly when the shifted frequencies were band-limited in twod i ff erent bands tested. The results were of the same order as the corresponding JNDs found for violin sounds [START_REF] Fritz | Perceptual thresholds for detecting modifications applied to the acoustical properties of aviolin[END_REF]: if anything, the JNDs seem to be somewhat lower for the guitar sounds, perhaps because the fully-coupled synthesis method givest he listener additional auditory cues with transient decay rates of the string overtones. Body mode damping has to be changed by about 20% for perception according to the definition used here. Actual variations of modal damping exceed this level, but not by alarge factor,sobody damping is probably quite a minor contributor to tonal perception.

String damping parameters are even less sensitive:they have to be changed by af actor of the order of 3. However,t his large factor does not mean that the effects are neverheard. Measurements have been presented on strings of different materials, and on steel-cored strings in new and "aged" conditions. The damping parameters vary between these cases by factors significantly bigger than the JNDs. The results strongly indicate that the most important parameter for the perception of the degradation of sound in well-used strings is η B ,t he damping factor associated with the bending stiffness of the string. This parameter increases greatly with age of strings, probably from the damping effect of dirt and corrosion products accumulating between the windings of the lower strings of the guitar [START_REF] Hanson | Analysis of "live"and "dead" guitar strings[END_REF]. Strings without over-winding do not deteriorate in sound to anywhere near the same extent.

The correct interpretation of the high sensitivity to shifting body frequencies is not obvious. The actual change investigated here, moving all frequencies in step, is very artificial. Listeners might be responding to the actual frequencies, or perhaps to the change in modal density (the inverse of the mean spacing between resonant frequencies), or perhaps to the shifting amplitude envelope of the body response. Modal amplitudes were not changed when the frequencies were scaled, so the amplitude pattern moves along the frequencya xis with the scaling. It would require further testing to establish whether one of these factors dominates or whether all contribute significantly.Such testing would be agood target for future research, to bring the results closer to the detailed concerns of instrument makers.

Figure 1 .

 1 Figure 1. Guitar tablature for sound examples used in psychoacoustical tests: first the "Huntsuppe" extract, second the strummed chord with notes E 2 B 2 D# 3 B 3 C# 4 A 4 .

Figure 2 .

 2 Figure 2. Magnitude of the input admittances of four guitars, measured normal to the top plate at the bridge saddle near the lowest string (E 2 ), plotted in decibels relative to 1m/Ns. Heavy line: flamenco guitar by Martin Woodhouse; thin solid line: classical guitar by Michael JO'Leary; dashed line: classical guitar by GregSmallman; dash-dot line: flamenco guitar by José Ramirez. Horizontal lines showbands used in testing: see text.

Figure 3 .

 3 Figure 3. Damping measurements on (top)anylon guitar string (3rd string, G 3 )a nd (bottom)t he corresponding string from a steel-cored set. Discrete points showindividual measured decay rates, curves showfits to the floor levelusing equation (4).

Figure 4 .

 4 Figure 4. Results from 3AFC studies for shifting all mode frequencies of the datum guitar,u sing as input (a) the Huntsuppe fragment; (b) as ingle note (open As tring); (c) the discordant chord. Horizontal axis shows percentage frequencys hift on a logarithmic scale, vertical axis shows relative standard deviation. Points *denote musicians, points + denote non-musicians. Open circle: average of all subjects; open square: average of best four subjects. The percentage shifts corresponding to the best-four averages are (a) 1.13; (b) 0.92; (c) 0.59.

Figure 5 .

 5 Figure 5. Results in the format of Figure 4from 3AFC studies for shifting all mode frequencies of the datum guitar lying in (a) the band 150-250 Hz; and (b) the band 500-1000 Hz, using as input asingle note (open Astring). The percentage shifts corresponding to the best-four averages are (a) 1.87; (b) 1.53.

Figure 6 .

 6 Figure 6. Results in the format of Figure 4f rom 3AFC studies for scaling all modal Qfactors of the datum guitar (a) upwards; and (b) downwards, using as input asingle note (open Astring). Horizontal axis shows scale factor applied by multiplication or division to Qfactors. The scale factors corresponding to the bestfour averages are (a) 1.19; (b) 1.22.

Figure 7 .

 7 Figure 7. Modal frequencies and Qf actors for 15 guitars, for the highest peak in the frequencyb ands (a) 75-100 Hz, the "Helmholtz resonance"; (b) 150-250 Hz, the "first top plate mode". Plus and cross symbols showfits without and with compensation for string damping, respectively.D iamond symbols mark the datum guitar for the listening tests. Open squares show the means, and range bars showo ne standard deviation above and belowthe means, for the twosets of fitted data. The ranges for the compensated results can be recognised by their higher Q values. Dashed lines near the axes showa pproximately 1J ND above and belowt he uncompensated means from the results of section 4.1.

Figure 8 .

 8 Figure 8. Damping trends for guitar bodies determined from time-frequencya nalysis. (a):i ndividual measurements of frequencya nd equivalent Qf actor (circles)a nd the band average of these (solid line)f or the datum guitar; (b):t he solid lines as in (a),for the set of 15 guitars; (c) the mean and standard deviation of the results from (b) across the frequencyrange, with range bars indicating one standard deviation above and belowthe mean on each axis. The circle marks the datum guitar for the listening tests.

Figure 9 .

 9 Figure 9. Results in the format of Figure 4f rom 3AFC studies for changing the damping parameters of the 5th string, based on datum values for the newsteel strings. Case (a):varying η B ,test note E 2 ;( b):v arying η B ,t est note C 3 ;( c):v arying η F ,t est note C 3 .The scale factors corresponding to the best-four averages are (a) 2.99; (b) 3.13; (c) 2.41.

Table I .

 I Properties of guitar strings: (a) D'Addario Pro Arté "Composites, hard tension" nylon strings; (b) Martin "80/20 bronze" steel strings in newcondition; (c) as (b) butinused condition.

	String:				123456
	Tuning note				E 4	B 3	G 3	D 3	A 2	E 2
	Frequency		f n	(Hz)	329.6	246.9	196.0	146.8	110.0	82.4
	Tension	(a)	T	(N)	70.3	53.4	58.3	71.2	73.9	71.6
		(b)			121	135	182	185	169	147
	Mass/unit length	(a)	m	(g/m)0 .38	0.52	0.90	1.95	3.61	6.24
		(b)			0.57	1.13	2.42	4.37	7.13	11.02
	Bending stiffness	(a)	EI	(10 -6 N/m 2 )1 30	160	310	51	40	57
		(b)			108	336	234	255	327	667
		(c)			103	394	343	418	480	719
	Inharmonicity	(a)	B	(10 -6 )4 37 01 24	17	13	19
		(b)			21	58	30	32	45	106
		(c)			20	68	44	53	66	114
	Loss coefficients	(a)	η F	(10 -5 )4 04 01 45	72
		(b)			7611	91 15
		(c)			6717	33	17	10
		(a)	η B	(10 -2 )2 .4	2.0	2.0	2.0	2.5	2.0
		(b)			0.7	1.1	1.8	1.1	0.4	0.3
		(c)			0.7	1.1	6.8	4.1	2.9	3.8
		(a)	η A	(1/s)1 .5	1.2	1.7	1.2	0.9	1.2
		(b)			2.5	1.0	1.2	1.3	1.2	1.7
		(c)			1.5	1.0	0.6	0.7	0.6	1.4

Acknowledgement

The authors thank Brian Moore, Ian Cross and Alan Blackwell for manyhelpful discussions on this work, and Martin Woodhouse, Stewart French, Ian Cross and James Westbrook for providing access to guitars for testing. Two anonymous reviewers are thanked for constructive suggestions.