ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012) 475 ndash 486

DOI 103813AAA918531

Perceptual Thresholds for Acoustical GuitarModels

J Woodhouse1) E K Y Manuel1) L A Smith1) A J C Wheble1) C Fritz2)

1) Cambridge University Engineering Department Trumpington St Cambridge CB2 1PZ UK jw12camacuk2) LAM Institut Jean le Rond drsquoAlembert UMR 7190 CNRS UPMC Univ Paris 06 11 rue de Lourmel Paris

France

SummarySynthesised acoustic guitar sounds based on a detailed 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 athree-alternative forced-choice procedure just-noticeable differences are presented for changes in frequency anddamping of the modes of the guitar body and also for changes in the tension bending stiffness and dampingparameters of the strings These are compared with measured data on the range of variation of these parametersin a selection of guitarsPACS no 4366Jh 4375Gh

1 Introduction

Assessments of the quality of musical instruments can beinfluenced by many factors including appearance and er-gonomic considerations but the final judgement will pre-sumably always relate to some extent to perceived qualityof sound Instrument makers want to know how to linkparticular changes in physical structure to their effect onperceived sound so that they can control and improve thequality of their instruments A useful first step in mappingout these links which lends itself to relatively uncontro-versial psychoacoustical testing is to establish the small-est change of each structural parameter that produces anaudible effect under the most favourable listening condi-tions A comprehensive set of such just-noticeable differ-ences (JNDs) would give information about the relativesensitivity of the sound to the different parameters underthe control of an instrument maker without the compli-cations of trying to describe the nature of the changes insound Such descriptions are important of course but theyrequire different methods to study them and can raise moredifficult challenges in experimental methodology This pa-per concentrates on the first stage finding JNDs

A recent research project [1] has investigated someJNDs for violin acoustics While that work was going ona series of undergraduate projects was used to perform aparallel investigation of the acoustical guitar using similarpsychoacoustical methodology This paper summarises themain findings of those projects As well as psychoacous-tical results measurements on a variety of guitars will bepresented to provide information about the actual range ofvariation in the parameters investigated

Received 2 September 2011accepted 19 February 2012

In one important respect the guitar is very different fromthe violin In the violin the player can produce a sustainedsound by feeding energy into the vibrating string from thebow This involves nonlinear interactions (from the fric-tion force at the bow-string contact see eg [2]) that maketheoretical modelling difficult and it also means that theplayer can influence the details of the sound of each notethroughout its duration The guitar is quite different theinstrument is to a good approximation a linear vibrat-ing system The player starts a particular note with controlover the details of the pluck but once the string has beenreleased the subsequent sound is not under the playerrsquoscontrol (This study excludes any use of vibrato)

This distinction leads to a difference in the way thatsounds for psychoacoustical testing can most usefully begenerated For the violin it is remarkably hard to synthe-sise sound that is convincingly realistic because of the all-pervading influence of the human performer Instead a hy-brid strategy was adopted for the psychoacoustical testsin which real playing was combined with digital-filter em-ulation of the vibration response of the violin body Thefluctuating force exerted by a vibrating string on the violinbridge during normal playing was measured with an em-bedded piezoelectric sensor and recorded It was arguedthat to a useful first approximation a given bow gestureon a given string will produce essentially the same forcewaveform regardless of the acoustical response of the par-ticular violin body For example during a steady note theplayer will ensure that the string vibrates in Helmholtz mo-tion producing a sawtooth force waveform at the bridgewith relatively subtle modifications due to the bowing po-sition and force and the string parameters such as dampingand bending stiffness [2]

It was thus argued that the main effects of a particu-lar violin body on the perceived sound should arise from

copy S Hirzel Verlag middot EAA 475

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

the filtering of this bridge-force signal by the linear vi-bration response and sound radiation characteristics of theviolin Linear vibration response can be analysed into asum of modal contributions and can then relatively easilybe emulated by a suitable digital filter So for exampleto investigate the JND for a shift in frequency of one par-ticular vibration mode of the body a series of sound filescould be generated using the same recorded bridge forcesignal modified by a range of digital filters initially cal-ibrated against a particular violin body but with the onedesired mode frequency systematically altered The use ofrecorded bridge-force signal from a human performer getsaround the problem of realism without giving the playerany opportunity to alter their bowing in response to the pa-rameter being varied because the same input signal is usedin every case

For the guitar this approach will not work convincinglyPart of the sound quality of a guitar note comes from thefrequency-dependent decay rates of the different overtonecomponents of the sound These are determined by inter-action of the string and the guitar body through the cou-pling at the bridge The vibration response of a guitar bodycould be emulated by digital filter just as easily as a violinbody but if the hybrid approach were to be used for theguitar the mechanical coupling of string to body wouldbe missing and it is likely that a crucial part of the influ-ence of the body acoustics on the sound would be lostLuckily the issue of non-realism of synthesised sound ismuch less challenging for the guitar A synthesis modelfor guitar notes based on a detailed physical model hasbeen described previously [3 4] and this has been foundto produce quite plausible sounds Indeed several partici-pants in the studies to be described here expressed surprisewhen told that they were hearing synthesised sounds Inconsequence the JND studies for guitars were carried outentirely using synthesised sounds

There is very little existing literature that is directly rel-evant to this study While a lot has been written about thevibration and sound radiation characteristics of guitar bod-ies published psychoacoustical studies based on the mod-elling are very rare the only example seems to be the doc-toral work of Wright [5] There is however some litera-ture relating to guitar strings specifically to the perceptionof inharmonicity associated with bending stiffness Twoseparate studies have been published by Jaumlrvelaumlinen andco-workers [6 7] using synthesised sounds of differing de-grees of realism to obtain thresholds for discrimination be-tween harmonic and inharmonic sounds They found thatthresholds for highly idealised sounds were lower than formore realistic sounds including the initial transient of aguitar pluck presumably because of some degree of infor-mational masking in the more complex sounds

The outline of this paper is as follows The synthesismodel is briefly described and the parameters to be ex-plored in JND studies are defined The technical detailsof the psychoacoustical test procedure are then given fol-lowed by the acoustical test methods for establishing phys-ical parameter values for the guitar body and for the strings

investigated Results are then presented first for the prop-erties of the guitar body Several tests will be presented inwhich modal frequencies are shifted all together or con-fined to particular frequency bands The influence of thechoice of musical passage for the test stimulus is also dis-cussed After this a more restricted study was undertakenin which the modal damping factors were scaled The finalsection of test results concerns properties of the stringsespecially the damping properties These results will berelated to the perception of ldquonewrdquo versus ldquooldrdquo strings bya change in their characteristic sound in the light of mea-surements of what actually changes as steel-cored guitarstrings age

2 Outline of synthesis model

The synthesis model to be used here is primarily aimed atachieving an accurate representation of the coupled me-chanical vibration of the strings and guitar body It doesnot attempt a comparably full treatment of the details ofthe playerrsquos pluck gesture or of the sound radiation be-haviour these are both taken into account in a simple wayonly as will be described shortly The input to the modelis a set of parameter values describing separately the be-haviour of the guitar body and the strings For each bodymode four things are required the natural frequency themodal damping factor the modal amplitude at the posi-tion on the bridge where the string is coupled and anangle describing the orientation of motion at that cou-pling point The modal amplitude can be equivalently ex-pressed via an effective mass the amplitudes deduced bythe techniques of experimental modal analysis correspondto mass-normalised modes (see for example Ewins [8])and the square of the measured amplitude at a given pointis the inverse of the effective mass for representing thatmode at that point by a mass-spring-damper combinationThe angle of motion at the coupling point is needed so thatboth polarisations of string motion can be included in thesynthesis

Ideally the amplitudes and angles of each body modewould be separately determined at the six positions on thebridge saddle where the strings make contact This wouldbe possible but laborious and for the present study it hasnot been done The body modal parameters are deducedfrom a measurement of the input admittance (also calledthe driving point mobility) at a position between the 5thand 6th strings (the two tuned to the lowest frequencies)The same body response is used for all strings in the syn-thesis process This position was chosen mainly because itis easier to perform the measurements necessary to estab-lish the modal angles at the ends of the bridge rather thanin the middle for practical reasons of access Other posi-tions on the bridge would of course give the same naturalfrequencies and modal damping factors but with differ-ent amplitudes and angles Since the aspects of body be-haviour studied here involve only frequencies and damp-ing one might hope that the conclusions would not be sig-nificantly changed by a more thorough synthesis allowingdifferent body behaviour for each string

476

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

The string is treated as a continuous system ratherthan being described in modal terms this was shown togive a more efficient strategy for computing the coupledstringbody motion with correct allowance for the very dif-ferent damping levels of the two systems which causesthe damping of the coupled system to be strongly non-proportional [3] A given string is specified by its lengthL tension T mass per unit length m bending stiffnessEI and a measure of its damping The bending stiffnessis expressed as the product of the Youngrsquos modulus E andthe second moment of area I which for a homogeneousmonofilament string of diameter d is given by

I = πd464 (1)

Inharmonicity due to bending stiffness is often expressedin terms of the dimensionless parameter B that appears inthe approximate expression for the frequency of the nthovertone

fn asymp nf0 1 + Bn2 (2)

where f0 is the fundamental frequency In terms of the no-tation introduced here

B =EIπ2

TL2 (3)

For overwound strings as normally used for the lowerstrings of the guitar the effective value of EI or B can-not be reliably predicted and must be measured

Musical strings in general have been shown to exhibitstrongly frequency-dependent damping and the measuredbehaviour can be fitted to a three-parameter model pro-posed by Valette [9] The loss factor of the nth mode isassumed to take the form

ηn =T (ηF + ηAωn) + EIηB (nπL)2

T + EI (nπL)2 (4)

The damping is lowest in the mid range of frequencieswith a minimum level set by ηF At very low frequenciesthe damping increases an effect attributed by Valette to airdamping and quantified by the parameter ηA At high fre-quencies the damping increases again because energy dis-sipation associated with the complex bending stiffness ofthe string comes to dominate especially for the polymer-based strings of classical guitars This effect is charac-terised by the parameter ηB

The chosen synthesis method works in the frequencydomain then uses an inverse FFT to obtain the time re-sponse Full details are given in [3 4] The string and bodyare coupled at the bridge using the standard point-couplingformula

Y minus1coup = Y minus1

body + Y minus1str (5)

where Ycoup Ybody Ystr are the 2times2 matrices of admittanceat the bridge notch for the coupled system the body aloneand the string alone respectively Each synthesised notebegins with an idealised pluck which is a step function of

force as the finger or plectrum loses contact with the stringThe string is assumed to be released from rest at the instantof the note starting Advantage is taken of the reciprocityprinciple it is convenient to apply the force step at thebridge and calculate the response at the point on the stringwhere the pluck acts As well as the plucking position onthe string it is possible to specify an angle of pluck (whichdetermines the mix of the two polarisations of string mo-tion excited) The playerrsquos finger or plectrum will have afinite size and this will have the effect of a low-pass filteron the excitation of string modes This is represented in asimple way by a frequency-domain Gaussian filter propor-tional to exp(minusω2K2) Here K = vh where v = Tmis the wave speed of the string and the width h was giventhe value 75 mm in order to obtain a reasonably realisticsound

The synthesis model calculates the transient waveformof acceleration at the coupling point between the string andbody A full model would then need to calculate the bodymotion at all other positions and perform a rather com-plicated numerical calculation to give the resulting soundradiation The present model uses a severely simplified ap-proach the computed bridge acceleration is regarded asbeing applied to a spherical radiating body of radius avibrating in the symmetric breathing mode only For thatproblem the sound pressure in the far field is given sim-ply by filtering the acceleration waveform with a low-passfilter that apart from an overall scale factor takes the form

R(ω) =ika

1 + ika=

iωωc

1 + iωωc (6)

where k = ωc is the wavenumber for sound in air c isthe speed of sound and ωc = ca is the filter roll-off fre-quency Setting this roll-off at 250 Hz corresponding toa sphere radius around 02 m proved to give satisfactory-sounding results

This synthesis procedure has been implemented in Mat-lab in a program that allows the user to specify a musicalfragment entered in a form of guitar tablature The result-ing ldquotunerdquo is saved as a sound file

3 Experimental methodology

31 Psychoacoustical tests

Using synthesised sounds generated by this method ex-periments were conducted to give some JND informationrelating to the major parameters For the body model ex-periments were done in which all the mode frequencieswere shifted by a given factor and also groups of modes inparticular frequency bands were shifted leaving the othermodes unchanged In a separate pair of tests the dampingfactors of all modes were modified by a given factor ei-ther upwards or downwards from the reference values incase there was an asymmetry of perception For the stringmodel the bending stiffness and the damping parametersηF ηB were changed The parameter ηA was found to haveso little influence on the chosen test sound that it was not

477

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

possible to obtain a meaningful JND for changes to itsvalue A general comment should be noted all the teststo be reported here were constrained by the time limitsimposed by the format of undergraduate projects In somecases there is no doubt that more could have been learnedhad more time been available but it is felt that the resultshave value even with these limitations

JND thresholds were estimated using a three-alternativeforced-choice procedure (3AFC) This is not the only pos-sible procedure that could have been used but it had al-ready been chosen for the violin studies and softwareto implement 3AFC was conveniently to hand A three-down one-up adaptive tracking rule was used that esti-mated the 79 correct point on the psychometric func-tion [10] It should be noted that this is the definition as-sumed for ldquoJNDrdquo throughout this paper it is a rather con-servative value significantly higher than that which wouldcorrespond to the 50 point on the psychometric func-tion which perhaps better matches the intuitive notion ofldquothreshold of perceptionrdquo Three sounds ndash two the samethe reference sound and one different the modified soundndash were played in a random sequence and the subject wasasked to choose which one was different The amount ofparameter modification between the stimuli in a given trialwas changed by a certain factor the step size The step sizedecreased after three correct answers and increased afterone wrong answer The test stopped after eight turnpointsA relatively large initial step size was applied until the sec-ond turnpoint was reached in order to allow rapid conver-gence toward the threshold region After the second turn-point the step size was reduced The threshold was takenas the mean of the values of the amount of modification atthe last six turnpoints The values of the two step sizes andthe starting level of parameter modification were chosendifferently for each test on the basis of preliminary trialsEach trial took no more than about 10 minutes and sub-jects had a break between tests

In all tests the reference sound was kept constant in or-der to increase discriminability subjects ldquolearntrdquo to rec-ognize the reference sound It is argued that for the pur-pose of relating to musical judgments which in practiceare made by musicians who may spend several hours everyday concentrating on subtle details of the sound of their in-strument the threshold of most interest is the one obtainedunder the most favourable conditions for discriminationIt is accepted that the test circumstances are quite remotefrom normal musical experience as seems to be inevitablein the design of any scientifically-acceptable test proce-dure and the approach adopted seems to the authors thebest that can easily be done The 3AFC software gave vi-sual feedback on rightwrong answers during the experi-ment although some subjects performed best by keepingtheir eyes shut and ignoring it Subjects did not receiveany training beforehand but if they performed erraticallyon the first run they were able to repeat the run This ap-plied only to a few subjects and they were all able to per-form the task at the second attempt The sounds were pre-sented diotically via Sennheiser HD580 headphones cho-

Figure 1 Guitar tablature for sound examples used in psy-choacoustical tests first the ldquoHuntsupperdquo extract second thestrummed chord with notes E2 B2 D3 B3 C4 A4

sen because of their diffuse field response in a relativelyquiet environment The sampling rate was 441 kHz andthe number of bits was 16

The choice of musical input for the test sounds canhave a significant influence on the JND obtained In thefirst tests on mode frequency shifting three different sam-ple sounds were tested a single note (the open A string110 Hz) a short musical fragment and a strummed ldquodis-cordant chordrdquo with notes E2 B2 D3 B3 C4 A4 selectedto give input at a wide range of frequencies The musi-cal fragment consisted of the first bar of the lute pieceldquoThe English Huntsupperdquo by John Whitfelde This andthe chord are shown in tablature notation in Figure 1 thetablature applies to a guitar in regular EADGBE tuningFor the remaining tests a single note was used for the testsound the open A string (110 Hz) for the damping factortests and C3 on the A string (131 Hz) for the string param-eter tests The total stimulus length for the single notes andthe chord was 1 s while the Huntsuppe fragment took 2 sThese are all longer than the 300 ms length that yieldedthe lowest thresholds in the violin tests and perhaps someadvantages of echoic memory are sacrificed by using theselonger stimuli However it was found that the sense of arealistic guitar sound was compromised if the duration wastoo short and it seemed more appropriate to use the longermore ldquoguitarishrdquo sounds

The test subjects were all ldquomusicalrdquo in one sense or an-other but for the purposes of the presentation of resultsthey are divided into two groups using the same criterionas in the work on the violin [1] those labelled ldquomusiciansrdquohad at least 8 years of formal training and were currentlyactive in music-making at least once per week while theothers are labelled ldquonon-musiciansrdquo The first author wasa subject in each test while all the other subjects were ofstudent age The results for the first author did not seemin any way atypical of the rest despite his relatively ad-vanced age

32 Physical measurements

Physical measurements on the body and strings of severalguitars are used in two different ways in this study Firstmeasurements are needed to calibrate the synthesis modelas has been described in detail previously [4] Second thesame range of measurements can be analysed to give es-timates of the variability in practice of the parameters forwhich JNDs will be found By comparing the actual varia-tion with the JND for perception it is hoped to give a first

478

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

80 100 200 500 1000 2000 5000-70

-60

-50

-40

-30

-20

-10

Frequency

|Y|(dB)

Figure 2 Magnitude of the input admittances of four guitarsmeasured normal to the top plate at the bridge saddle near thelowest string (E2) plotted in decibels relative to 1 mNs Heavyline flamenco guitar by Martin Woodhouse thin solid line clas-sical guitar by Michael J OrsquoLeary dashed line classical guitar byGreg Smallman dash-dot line flamenco guitar by Joseacute RamirezHorizontal lines show bands used in testing see text

estimate of the relative sensitivity and importance of dif-ferent kinds of variation in guitar construction and string-ing

Input admittance Y (ω) of the guitar body is measuredby applying a controlled force either with a miniature im-pulse hammer (PCB 086D80) or by gently pulling a lengthof fine copper wire looped around a string at the bridgesaddle until it snaps The latter method is less accuratebut has the advantage that it can be used to apply force indifferent controlled directions to give information on themodal angles In all cases the velocity response is mea-sured with a laser-Doppler vibrometer (Polytec OFV-056)directed to a spot as close as possible to the forcing pointSignals are collected at a sampling rate of 40 kHz by adigital data-logger based on a PC running Matlab witha National Instruments 6220 data acquisition card Thestrings of the guitar were damped during admittance mea-surements

Input admittances in the direction normal to the planeof the top plate have been measured on a wide range ofacoustic guitars and a representative selection of 15 areused in the present study Figure 2 shows results for fourof these all hand-made instruments of high quality Theheavy curve is the datum guitar used in the psychoacousti-cal tests a flamenco guitar by Martin Woodhouse studiedin the previous work [4] The others are a flamenco gui-tar by Jose Ramirez and classical guitars by Greg Small-man and Michael J OrsquoLeary It is immediately clear thatall these guitars although different in detail show a lotof common features All have a strong peak in the range80ndash100 Hz which is a Helmholtz resonance associatedwith the internal air cavity and the soundhole modified bythe compliance of the top and back plates All the guitarsthen show a deep antiresonance (marking the frequency atwhich the Helmholtz resonance would occur with a rigid

cavity see equation (1019) of Cremer [11]) followed bya cluster of strong peaks in the range 150ndash250 Hz Mostthen show a steady decline to a dip around 400ndash500 Hzfollowed by fluctuations around a rather steady level withgradually reducing peak amplitudes as the modal overlapbecomes significant One guitar (the OrsquoLeary) has an ad-ditional strong peak around 300 Hz interrupting the de-cline to the dip The two horizontal lines mark frequencybands that will be used in frequency-shifting tests in thenext section The first of these bands (150ndash250 Hz) coversthe range where the datum guitar has its strongest peaksThe second band (500ndash1000 Hz) covers a range where themodal overlap is becoming significant but fairly clear in-dividual peaks can still be seen

The other physical measurements used in this study areaimed at establishing the properties of strings Tension andmass per unit length are readily established but the bend-ing stiffness and damping properties require careful mea-surement Such measurements have previously been madeon a typical set of nylon strings for a classical guitar andin the course of this project corresponding measurementswere made on a set of steel strings as commonly usedon folk guitars The chosen strings were Martin ldquo8020Bronzerdquo It is a familiar experience to guitarists that thesound changes a lot between when the strings are new andwhen they are well-played so the strings studied here weremeasured in both states to give quantitative data on whatchanges in the ageing process

The bending stiffness and damping properties can be es-timated by analysis of sounds recorded from notes playedon the guitar in the usual way [4] The frequencies of thestrongest spectral components corresponding to ldquostringrdquomodes can be best-fitted to equation (2) to yield valuesof B and hence EI The associated decay rate of eachof these ldquostringrdquo modes can also be found using time-frequency analysis and this gives information about thestring damping The procedure for obtaining damping in-formation from the decay rates is a little indirect becausein most cases the decay rate is dominated by losses into theguitar body rather than by internal damping in the string

The approach adopted is to analyse the frequencies fora plucked note on every fret of a given string and combinethem all into a cloud plot of damping factor against fre-quency examples are shown in Figure 3 This shows datafor the 3rd string (G3) for the nylon string measured previ-ously and the new steel-cored string The inherent damp-ing of the string as a function of frequency determinesthe ldquofloor levelrdquo of this cloud plot Approximate values ofηA ηF ηB can be estimated by fitting eqquation (4) to thetrend of this floor level as illustrated by the curves in Fig-ure 3 The fitting process was carried out manually withsome regard to the error tolerances of the data points Notethat the high-frequency fit line for the nylon string wasinfluenced by additional data obtained by a different ap-proach not shown here see [4] for more detail It is clearfrom the figure that the nylon string has higher dampingthan the steel string The measured string properties aresummarised in Table I

479

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

Table I Properties of guitar strings (a) DrsquoAddario Pro Arteacute ldquoComposites hard tensionrdquo nylon strings (b) Martin ldquo8020 bronzerdquo steelstrings in new condition (c) as (b) but in used condition

String 1 2 3 4 5 6

Tuning note E4 B3 G3 D3 A2 E2

Frequency fn (Hz) 3296 2469 1960 1468 1100 824

Tension (a) T (N) 703 534 583 712 739 716(b) 121 135 182 185 169 147

Massunit length (a) m (gm) 038 052 090 195 361 624(b) 057 113 242 437 713 1102

Bending stiffness (a) EI (10minus6Nm2) 130 160 310 51 40 57(b) 108 336 234 255 327 667(c) 103 394 343 418 480 719

Inharmonicity (a) B (10minus6) 43 70 124 17 13 19(b) 21 58 30 32 45 106(c) 20 68 44 53 66 114

Loss coefficients (a) ηF (10minus5) 40 40 14 5 7 2(b) 7 6 11 9 11 5(c) 6 7 17 33 17 10

(a) ηB (10minus2) 24 20 20 20 25 20(b) 07 11 18 11 04 03(c) 07 11 68 41 29 38

(a) ηA (1s) 15 12 17 12 09 12(b) 25 10 12 13 12 17(c) 15 10 06 07 06 14

The differences between the nylon and steel strings innew condition are broadly as one would expect The nylonstrings have lower tensions and a different pattern of bend-ing stiffness because of the difference in construction forthe nylon set the top three strings are monofilaments whilethe lower three have multifilament cores with metal wind-ings The steel strings have monofilament cores through-out and only the top two strings are not wrapped Themonofilament nylon strings have higher internal dampingthan the steel but the difference is much less marked forthe lower three strings

The effect of age and use on the steel strings can alsobe seen Previous investigators have reported increaseddamping between new and old guitar strings [12 13] butwithout sufficient detail to allow a fit to equation (4) Itis generally accepted that the mechanism for this ageingeffect is the accumulation of dirt and corrosion productsbetween the windings of wrapped strings [13] perhapswith some additional cumulative wear of the outer wrap-pings where the strings make contact with the frets Themeasurements reported here support this picture there isvery little change in the two highest strings which are notwrapped For the wrapped strings there is a marked in-crease in damping especially in ηB which determines thedecay at the highest frequencies

Also not noted by previous authors there is a markedincrease in the effective bending stiffness of some of thestrings This seems perfectly credible since the contactconditions between the windings in the tensioned stringwill contribute to the bending stiffness and material col-

lecting between the windings will surely change those con-ditions This is consistent with the observed increase in ηB which is the loss factor associated with the bending stiff-ness term in the governing equation It may be noted thatthe bending stiffness values given here are in good gen-eral agreement with values reported by Jaumlrvelaumlinen et al[7] noting that their steel-string results match the ldquousedrdquoresults more closely than the ldquonewrdquo results Their paperdoes not report the condition of the strings as measured

4 Results for properties of the guitar body

41 Psychoacoustical tests

The first series of psychoacoustical tests concerned theshifting of all body mode frequencies by a given factorkeeping other simulation parameters fixed This modifica-tion is of course somewhat artificial but a possible physi-cal interpretation will be discussed in the next subsectionThe datum for this test was based on the modal proper-ties of the flamenco guitar by Martin Woodhouse This testwas run three times using the three different musical in-puts described above The three tests were completed by 8musicians and 5 non-musicians (according to the criteriondescribed above) The results are summarised in Figure 4The program that runs the 3AFC protocol gives as outputthe mean and standard deviation over the range defined bythe final 6 turnpoints The plots show for each test subjectthe mean on the horizontal axis and the relative standarddeviation (standard deviation divided by mean) on the ver-tical axis The numbers on the horizontal axis indicate the

480

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

100 200 500 1000 2000 500010

-4

10-3

10-2

Frequency (Hz)

Dampingfactor

100 200 500 1000 2000 500010

-4

10-3

10-2

Frequency (Hz)

Dampingfactor

Figure 3 Damping measurements on (top) a nylon guitar string(3rd string G3) and (bottom) the corresponding string from asteel-cored set Discrete points show individual measured decayrates curves show fits to the floor level using equation (4)

percentage shift in mode frequency Musicians are indi-cated by stars non-musicians by plus symbols The aver-ages over all test subjects are shown by the open circles

The open square symbols show the average of the foursubjects who achieved the lowest mean levels As was sug-gested earlier within the scope of the present study themost interesting threshold to compare with serious musi-cal judgments is that achieved by the best listeners underthe most favourable conditions In the violin tests the bestfive results were presented for this purpose [1] In the testsreported here the number of subjects is sometimes ratherlow and it was decided to average the best four The nu-merical results of this best-four measure for all the 3AFCtests presented here are given in the figure captions

It can be seen that for each of the three musical inputsthe best subjects could discriminate a shift of 1 or lessInteresting patterns can be seen in the detailed results de-spite the small number of subjects that would make anyelaborate statistical analysis of limited value The generalspread of results and the two averages look very simi-lar for the single note and the Huntsuppe extract For thesingle note the musicians and non-musicians are intermin-gled while for Huntsuppe all but one of the non-musicianshas a worse result than all the musicians Perhaps non-musicians have their performance degraded by informa-

03 05 1 2 3 5 100

05

1

(a) Huntsuppe

Percentage shift

RelSTD

03 05 1 2 3 5 100

05

1

(b) Single note

Percentage shift

RelSTD

03 05 1 2 3 5 100

05

1

(c) Discordant chord

Percentage shiftRelSTD

Figure 4 Results from 3AFC studies for shifting all mode fre-quencies of the datum guitar using as input (a) the Huntsuppefragment (b) a single note (open A string) (c) the discordantchord Horizontal axis shows percentage frequency shift on alogarithmic scale vertical axis shows relative standard deviationPoints denote musicians points + denote non-musicians Opencircle average of all subjects open square average of best foursubjects The percentage shifts corresponding to the best-four av-erages are (a) 113 (b) 092 (c) 059

tional masking while experienced musicians do not withsuch an obviously musical sound sample as ldquoHuntsupperdquo

The chord reveals a different picture Four of the mu-sicians achieved very low thresholds with this sound in-put but all other subjects seem to have been seriouslydistracted by this rather non-musical sound and achievedonly rather high thresholds One of the musicians comesout worst of all in this test whereas for Huntsuppe all themusicians performed well This is a striking example ofinformational masking especially since the group of foursubjects who did well actually produced a lower averagethan in either of the other tests That is as one would hopethis signal contains deliberately a wider range of fre-quency components so that analytical listeners with suffi-cient skill and experience to take advantage of it can makevery subtle judgements down to 032 shift in the bestcase This corresponds to shifting the mode frequenciesabout 6 cents or 120 of a semitone an impressively smallamount that is not very much bigger than the threshold forpitch discrimination tasks (see eg [14])

The thresholds are of the same order as the ldquobest fiverdquothresholds in the corresponding violin tests but the de-tailed pattern seems different The choice of input had amore marked effect in the violin and to achieve the lowestthresholds a very short and consequently rather unmusi-cal sound sample was used

The best-performing 7 subjects of the first test serieswere asked to undertake two further tests in which fre-

481

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

quencies were shifted only in limited bands The twobands studied were shown in Figure 2 and this time theonly input sound tested was the single-note open A stringThe results of the tests are shown in Figure 5 in the sameformat as Figure 4 Rather unexpectedly the ldquobest fourrdquothreshold for the first band containing the strongest peaksof the datum guitar was much higher than for the sec-ond band Related tests for the violin using the so-calledldquoDuumlnnwald bandsrdquo gave thresholds around 2 for the firstthree bands (see Figure 6 of Fritz et al [1]) Band 1 in thepresent study gave a comparable result but band 2 gave alower result than was observed in any of the violin testsThe best-four JND in this case is virtually the same as forthe corresponding test when all frequencies were shiftedFurther studies might be worth conducting to add moredetail to this observation

The final pair of tests relating to the perception of bodymode changes investigated the effect of damping factorsThis time the mode frequencies were kept fixed but themodal Q factors were all scaled by a given factor Q fac-tor is the inverse of damping factor so high Q correspondsto low damping In two separate tests the Q factors wereincreased and decreased to look for any asymmetry in theperception Thirteen test subjects took these two tests 10musicians and 3 non-musicians Results are shown in Fig-ure 6 in the same format as Figs 4 and 5 except that thenumbers on the horizontal axis now indicate the scale fac-tor applied rather than the percentage change This meansthat ldquono changerdquo corresponds to the value 1 rather than 0as in the previous results which has the effect of makingthe relative standard deviations numerically smaller

An asymmetry is seen between the two tests but it doesnot significantly affect the threshold value for the bestfour listeners They achieved a value around 12 ie a20 change in Q factors in both cases However acrossthe whole set of test subjects the spread is much greaterwhen the Q factors were decreased compared to the in-creasing case Many participants reported initial difficultyin hearing the difference when Q factors were decreasedalthough once they had learned to recognise the soundchange they often achieved quite low thresholds

It was not thought worthwhile to test separate frequencybands in this case for two reasons First body damping ismost readily altered by a physical change of material andthis is likely to have an effect over a broad frequency rangeSecond it will be shown shortly that the JND for percep-tion of a change in body damping is of the same orderas the measured standard deviation among real guitars sothat the perceptual significance is probably marginal anddigging deeper into the details is likely to be of limitedimportance

42 Physical measurements

The next step is to look at physical measurements to as-sess how these thresholds of perception compare with theactual extent of variation between guitars Of course onewould never find two real instruments which differed byhaving all the body frequencies simultaneously scaled by

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(a) 150-250 Hz

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(b) 500-1000 Hz

Figure 5 Results in the format of Figure 4 from 3AFC studies forshifting all mode frequencies of the datum guitar lying in (a) theband 150ndash250 Hz and (b) the band 500ndash1000 Hz using as inputa single note (open A string) The percentage shifts correspond-ing to the best-four averages are (a) 187 (b) 153

1 12 15 2 25 30

005

01

015

02

(a)

Scale factor

RelSTD

1 12 15 2 25 30

005

01

015

02

(b)

Scale factor

RelSTD

Figure 6 Results in the format of Figure 4 from 3AFC studiesfor scaling all modal Q factors of the datum guitar (a) upwardsand (b) downwards using as input a single note (open A string)Horizontal axis shows scale factor applied by multiplication ordivision to Q factors The scale factors corresponding to the best-four averages are (a) 119 (b) 122

exactly the same factor However it is easy to visualise achange which would in principle have this effect it wouldhappen if the dimensions of the guitar were all scaledkeeping the same materials and other details Some sim-ple deductions can then be drawn by scaling analysis Ifthe main vibrating elements of the guitar body are treatedas thin flat plates deforming in bending then the naturalfrequencies scale proportional to hd2 where d is the scalefactor for dimensions in the plane and h is the scale factorfor thickness Taking 1 as a round number for the orderof magnitude of the threshold of detection for shifting fre-quencies it follows that the JNDs for dimensional scalingare approximately 1 if d and h are both scaled together(full geometric scaling) 1 again if thickness is changedkeeping the in-plane dimensions fixed and only 05 ifin-plane dimensions are scaled keeping the thickness thesame These are all very small numbers they amount tochanging in-plane dimensions no more than a few mm and

482

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

75 80 85 90 95 1000

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(a)

160 170 180 190 200 210 2200

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(b)

Figure 7 Modal frequencies and Q factors for 15 guitars forthe highest peak in the frequency bands (a) 75ndash100 Hz theldquoHelmholtz resonancerdquo (b) 150ndash250 Hz the ldquofirst top platemoderdquo Plus and cross symbols show fits without and with com-pensation for string damping respectively Diamond symbolsmark the datum guitar for the listening tests Open squares showthe means and range bars show one standard deviation aboveand below the means for the two sets of fitted data The rangesfor the compensated results can be recognised by their higher Qvalues Dashed lines near the axes show approximately 1 JNDabove and below the uncompensated means from the results ofsection 41

to changing a typical top plate thickness by a mere 25 micromReal instruments have dimensional variations which areconsiderably bigger than this especially in thickness soon this ground alone it is no surprise that guitars can sounddifferent

The simplest approach to obtaining quantitative infor-mation from admittance measurements on a set of gui-tars is to perform explicit modal analysis This can onlybe expected to give reliable results at relatively low fre-quencies pole-residue extraction whatever the methodbecomes sensitive once the modal overlap begins to ap-proach unity To illustrate the kind of values which emergefrom such an analysis Figure 7 shows results for modalfrequency and Q factor for 15 guitars for the highest peakin each of the two low-frequency bands mentioned in thediscussion of Figure 2 75ndash100 Hz the ldquoHelmholtz reso-nancerdquo and 150ndash250 Hz the ldquofirst top plate moderdquo

A complicating factor needs to be noted The ad-mittance measurements are made with the strings welldamped because the purpose is to obtain informationabout the body behaviour without the effect of the stringsHowever the damped strings remove some energy fromthe body vibration and this will tend to reduce the mea-sured Q factors This effect can be compensated at leastapproximately The damped strings will act essentially assemi-infinite strings and will thus present an impedance tothe bridge which is real and if the strings are all assumedto be attached at the same point equal to the sum of thestring impedances given by Z =

radicTm From the data for

nylon strings in Table I the sum of these impedances is212 Nsm To compensate the inverse of Y (ω) is takenthis real quantity is subtracted then the result re-invertedto give a new admittance This approach can be expectedto work well at higher frequencies but it is hard to be sureif the strings are well enough damped at low frequenciesto justify the semi-infinite approximation

Figure 7 shows results before and after this compensa-tion For each case the mean and scatter are indicated bylines showing one standard deviation above and below themean for frequency and Q Interestingly the standard de-viations have a similar pattern for both cases The relativestandard deviations for frequency are 55 and 6 for thetwo cases and those for Q factor about 33 in both casesbefore compensation

To obtain good comparative information about modalfrequencies over a broader bandwidth is more challeng-ing It is not possible to identify the ldquosamerdquo mode in dif-ferent guitars except at very low frequencies sensitivityto small structural changes increases with frequency as isfamiliar in other vibration problems (see eg [15]) Formode frequencies the threshold for audible changes is sosmall that it is hard to obtain any measure which is goodenough scaling frequencies by 1 means that one guitarbody might have 101 modes in a frequency range whereanother guitar had 100 and there is no known analysismethod which can give robust discrimination at this levelAny measure which can be computed shows much biggervariability than this and there is always a strong suspicionthat it is revealing the vagaries of the analysis method morethan the real differences between guitars

Things are a little more promising in the case of damp-ing estimation over a wide frequency range Taking theinverse FFT of an input admittance gives an impulse re-sponse function of the guitar body and this can be pro-cessed using time-frequency analysis to give a sonogramrepresentation In each frequency band a best-fitting ex-ponential decay can then be obtained Combining resultsfrom several different choices of the timefrequency res-olution tradeoff improves the robustness of the resultsThe process is described in more detail elsewhere [16]Some results are shown in Figure 8 compensation forstring damping has been included In Figure 8a the discretepoints represent for the datum guitar frequencydecaycombinations which satisfied an assumed criterion forgood fit quality For compatibility with other plots decay

483

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70

80

90

100

Frequency

Qfactor

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100

120

Frequency

Qfactor

(b)

30 35 40 45 50 55 60 6515

20

25

30

35

40

Mean Q

Relativestandard

deviation(

)

(c)

Figure 8 Damping trends for guitar bodies determined fromtime-frequency analysis (a) individual measurements of fre-quency and equivalent Q factor (circles) and the band averageof these (solid line) for the datum guitar (b) the solid lines asin (a) for the set of 15 guitars (c) the mean and standard devia-tion of the results from (b) across the frequency range with rangebars indicating one standard deviation above and below the meanon each axis The circle marks the datum guitar for the listeningtests

rate is represented as an equivalent Q factor The continu-ous line is simply a moving band average of these discretepoints

Figure 8b then shows the band-averaged lines for the 15guitars in the test set Finally Figure 8c shows a scatterplot of the mean and standard deviation of these band-averaged lines for the 15 guitars As in Figure 7 thelarge cross represents one standard deviation above and

2 3 5 8 100

01

02

03

04 (a)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (b)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (c)

Scale factor

RelSTD

Figure 9 Results in the format of Figure 4 from 3AFC studiesfor changing the damping parameters of the 5th string based ondatum values for the new steel strings Case (a) varying ηB testnote E2 (b) varying ηB test note C3 (c) varying ηF test noteC3 The scale factors corresponding to the best-four averages are(a) 299 (b) 313 (c) 241

below the mean on each axis The main result is thatthis frequency-averaged measure of damping has a rela-tive standard deviation of the order of 20 comparable tothe JND found earlier for perception of damping changesThis suggests that the extremes of mean damping in themeasured set of guitars are sufficiently different that theeffect should be audible but the results do not lead oneto expect that this will be a very strong component of theperception of tonal differences among guitars

5 Results for properties of the strings

The final set of JND experiments relates to changes instring properties including those properties that changedas the string aged one objective was to find out whichof those changes are likely to contribute most to the per-ception of ageing Using as datum the properties of thenew steel strings listed in Table I changes to the dampingproperties of the 5th string were investigated using single-note test sounds These 3AFC tests were completed by 11musicians and 5 non-musicians One test was also con-ducted relating to changes in the bending stiffness of thisstring it gave results in line with those of Jaumlrvelaumlinen etal [7] and the details are not reproduced here The actualbending stiffness of this string was found to be close tothe threshold for detecting an increase in bending stiffnesswhen starting from a much lower level In the aged stringthe bending stiffness only increased by a factor 16 (seeTable I) If that change is audible at all the effect is likelyto be small

The results for the damping tests are shown in Figure 9in a similar format to earlier 3AFC results As noted ear-lier the perceptual effects of changing ηA for this string

484

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

were found to be so weak that it was not possible to con-duct a meaningful test However both ηB and ηF gave clearresults The parameter ηB associated with bending stiff-ness and mainly affecting higher frequencies was testedusing two different notes ldquoplayedrdquo on the same string Theresults are shown in Figures 9ab and show a good levelof consistency between the two notes The spread over-all mean and best-four mean are all very similar for bothcases The best listeners need about a factor 3 increase inthis parameter to perceive the change Note from Table Ithat in the aged string this parameter increased from 0004to 0029 that change should be very clearly audible

Figure 9c shows corresponding results for ηF All sub-jects gave a somewhat lower threshold for this parametercompared to ηB and the very best listener achieved a JNDof less than a factor 2 In the aged string ηF increased inthe ratio 1711 for this string from Table I This effect isprobably not audible to most listeners It can be concludedthat the factor most likely to account for the difference insound of the aged string is ηB

6 Conclusions

Psychoacoustical experiments have been reported to estab-lish JND levels for the perception of changes to variousparameters of a model of synthesised guitar sound [3 4]These investigations parallel earlier work exploring relatedJNDs for violin sounds [1] The parameters investigatedhere relate to the behaviour of the guitar body (modal fre-quencies and damping factors) and of the strings (bendingstiffness and damping model parameters) All tests werecarried out using a three-alternative forced-choice method-ology It is argued that for application to musical judge-ments which can be very subtle the most interesting per-ceptual thresholds are those attainable by the best listenersunder the most favourable conditions To give an estimateof these the results for the best four subjects in each testwere averaged

Results of physical measurements have also been re-ported to give an indication of the spread of each parame-ter among real guitar bodies or strings A comparison withthe psychoacoustical results gives a first impression ofwhich quantities might be responsible for the biggest per-ceptual differences between instruments By far the mostsensitive parameter of all those tested relates to shiftingthe frequencies of body modes The best four listenerscould perceive a 1 shift when all frequencies were scaledtogether a small change compared to the actual varia-tion among guitars This performance was only degradedslightly when the shifted frequencies were band-limited intwo different bands tested The results were of the sameorder as the corresponding JNDs found for violin sounds[1] if anything the JNDs seem to be somewhat lower forthe guitar sounds perhaps because the fully-coupled syn-thesis method gives the listener additional auditory cuesassociated with transient decay rates of the string over-tones Body mode damping has to be changed by about20 for perception according to the definition used hereActual variations of modal damping exceed this level but

not by a large factor so body damping is probably quite aminor contributor to tonal perception

String damping parameters are even less sensitive theyhave to be changed by a factor of the order of 3 How-ever this large factor does not mean that the effects arenever heard Measurements have been presented on stringsof different materials and on steel-cored strings in newand ldquoagedrdquo conditions The damping parameters vary be-tween these cases by factors significantly bigger than theJNDs The results strongly indicate that the most importantparameter for the perception of the degradation of soundin well-used strings is ηB the damping factor associatedwith the bending stiffness of the string This parameterincreases greatly with age of strings probably from thedamping effect of dirt and corrosion products accumulat-ing between the windings of the lower strings of the guitar[13] Strings without over-winding do not deteriorate insound to anywhere near the same extent

The correct interpretation of the high sensitivity to shift-ing body frequencies is not obvious The actual change in-vestigated here moving all frequencies in step is very arti-ficial Listeners might be responding to the actual frequen-cies or perhaps to the change in modal density (the in-verse of the mean spacing between resonant frequencies)or perhaps to the shifting amplitude envelope of the bodyresponse Modal amplitudes were not changed when thefrequencies were scaled so the amplitude pattern movesalong the frequency axis with the scaling It would re-quire further testing to establish whether one of these fac-tors dominates or whether all contribute significantly Suchtesting would be a good target for future research to bringthe results closer to the detailed concerns of instrumentmakers

AcknowledgementThe authors thank Brian Moore Ian Cross and Alan Black-well for many helpful discussions on this work and MartinWoodhouse Stewart French Ian Cross and James West-brook for providing access to guitars for testing Twoanonymous reviewers are thanked for constructive sugges-tions

References

[1] C Fritz I Cross B C J Moore J Woodhouse Perceptualthresholds for detecting modifications applied to the acous-tical properties of a violin J Acoust Soc Am 122 (2007)3640ndash3650

[2] J Woodhouse P Galluzzo The bowed string as we know ittoday Acta Acustica united with Acustica 90 (2004) 579ndash589

[3] J Woodhouse On the synthesis of guitar plucks ActaAcustica united with Acustica 90 (2004) 928ndash944

[4] J Woodhouse Plucked guitar transients comparison ofmeasurements and synthesis Acta Acustica united withAcustica 90 (2004) 945ndash965

[5] H A K Wright The acoustics and psychoacoustics of theguitar PhD Thesis Cardiff University 1996

[6] H Jaumlrvelaumlinen V Vaumllimaumlki M Karjalainen Audibility ofthe timbral effects of inharmonicity in stringed instrumenttones Acoustics Research Letters Online 2 (2001) 79ndash84

485

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

the filtering of this bridge-force signal by the linear vi-bration response and sound radiation characteristics of theviolin Linear vibration response can be analysed into asum of modal contributions and can then relatively easilybe emulated by a suitable digital filter So for exampleto investigate the JND for a shift in frequency of one par-ticular vibration mode of the body a series of sound filescould be generated using the same recorded bridge forcesignal modified by a range of digital filters initially cal-ibrated against a particular violin body but with the onedesired mode frequency systematically altered The use ofrecorded bridge-force signal from a human performer getsaround the problem of realism without giving the playerany opportunity to alter their bowing in response to the pa-rameter being varied because the same input signal is usedin every case

For the guitar this approach will not work convincinglyPart of the sound quality of a guitar note comes from thefrequency-dependent decay rates of the different overtonecomponents of the sound These are determined by inter-action of the string and the guitar body through the cou-pling at the bridge The vibration response of a guitar bodycould be emulated by digital filter just as easily as a violinbody but if the hybrid approach were to be used for theguitar the mechanical coupling of string to body wouldbe missing and it is likely that a crucial part of the influ-ence of the body acoustics on the sound would be lostLuckily the issue of non-realism of synthesised sound ismuch less challenging for the guitar A synthesis modelfor guitar notes based on a detailed physical model hasbeen described previously [3 4] and this has been foundto produce quite plausible sounds Indeed several partici-pants in the studies to be described here expressed surprisewhen told that they were hearing synthesised sounds Inconsequence the JND studies for guitars were carried outentirely using synthesised sounds

There is very little existing literature that is directly rel-evant to this study While a lot has been written about thevibration and sound radiation characteristics of guitar bod-ies published psychoacoustical studies based on the mod-elling are very rare the only example seems to be the doc-toral work of Wright [5] There is however some litera-ture relating to guitar strings specifically to the perceptionof inharmonicity associated with bending stiffness Twoseparate studies have been published by Jaumlrvelaumlinen andco-workers [6 7] using synthesised sounds of differing de-grees of realism to obtain thresholds for discrimination be-tween harmonic and inharmonic sounds They found thatthresholds for highly idealised sounds were lower than formore realistic sounds including the initial transient of aguitar pluck presumably because of some degree of infor-mational masking in the more complex sounds

The outline of this paper is as follows The synthesismodel is briefly described and the parameters to be ex-plored in JND studies are defined The technical detailsof the psychoacoustical test procedure are then given fol-lowed by the acoustical test methods for establishing phys-ical parameter values for the guitar body and for the strings

investigated Results are then presented first for the prop-erties of the guitar body Several tests will be presented inwhich modal frequencies are shifted all together or con-fined to particular frequency bands The influence of thechoice of musical passage for the test stimulus is also dis-cussed After this a more restricted study was undertakenin which the modal damping factors were scaled The finalsection of test results concerns properties of the stringsespecially the damping properties These results will berelated to the perception of ldquonewrdquo versus ldquooldrdquo strings bya change in their characteristic sound in the light of mea-surements of what actually changes as steel-cored guitarstrings age

2 Outline of synthesis model

The synthesis model to be used here is primarily aimed atachieving an accurate representation of the coupled me-chanical vibration of the strings and guitar body It doesnot attempt a comparably full treatment of the details ofthe playerrsquos pluck gesture or of the sound radiation be-haviour these are both taken into account in a simple wayonly as will be described shortly The input to the modelis a set of parameter values describing separately the be-haviour of the guitar body and the strings For each bodymode four things are required the natural frequency themodal damping factor the modal amplitude at the posi-tion on the bridge where the string is coupled and anangle describing the orientation of motion at that cou-pling point The modal amplitude can be equivalently ex-pressed via an effective mass the amplitudes deduced bythe techniques of experimental modal analysis correspondto mass-normalised modes (see for example Ewins [8])and the square of the measured amplitude at a given pointis the inverse of the effective mass for representing thatmode at that point by a mass-spring-damper combinationThe angle of motion at the coupling point is needed so thatboth polarisations of string motion can be included in thesynthesis

Ideally the amplitudes and angles of each body modewould be separately determined at the six positions on thebridge saddle where the strings make contact This wouldbe possible but laborious and for the present study it hasnot been done The body modal parameters are deducedfrom a measurement of the input admittance (also calledthe driving point mobility) at a position between the 5thand 6th strings (the two tuned to the lowest frequencies)The same body response is used for all strings in the syn-thesis process This position was chosen mainly because itis easier to perform the measurements necessary to estab-lish the modal angles at the ends of the bridge rather thanin the middle for practical reasons of access Other posi-tions on the bridge would of course give the same naturalfrequencies and modal damping factors but with differ-ent amplitudes and angles Since the aspects of body be-haviour studied here involve only frequencies and damp-ing one might hope that the conclusions would not be sig-nificantly changed by a more thorough synthesis allowingdifferent body behaviour for each string

476

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

The string is treated as a continuous system ratherthan being described in modal terms this was shown togive a more efficient strategy for computing the coupledstringbody motion with correct allowance for the very dif-ferent damping levels of the two systems which causesthe damping of the coupled system to be strongly non-proportional [3] A given string is specified by its lengthL tension T mass per unit length m bending stiffnessEI and a measure of its damping The bending stiffnessis expressed as the product of the Youngrsquos modulus E andthe second moment of area I which for a homogeneousmonofilament string of diameter d is given by

I = πd464 (1)

Inharmonicity due to bending stiffness is often expressedin terms of the dimensionless parameter B that appears inthe approximate expression for the frequency of the nthovertone

fn asymp nf0 1 + Bn2 (2)

where f0 is the fundamental frequency In terms of the no-tation introduced here

B =EIπ2

TL2 (3)

For overwound strings as normally used for the lowerstrings of the guitar the effective value of EI or B can-not be reliably predicted and must be measured

Musical strings in general have been shown to exhibitstrongly frequency-dependent damping and the measuredbehaviour can be fitted to a three-parameter model pro-posed by Valette [9] The loss factor of the nth mode isassumed to take the form

ηn =T (ηF + ηAωn) + EIηB (nπL)2

T + EI (nπL)2 (4)

The damping is lowest in the mid range of frequencieswith a minimum level set by ηF At very low frequenciesthe damping increases an effect attributed by Valette to airdamping and quantified by the parameter ηA At high fre-quencies the damping increases again because energy dis-sipation associated with the complex bending stiffness ofthe string comes to dominate especially for the polymer-based strings of classical guitars This effect is charac-terised by the parameter ηB

The chosen synthesis method works in the frequencydomain then uses an inverse FFT to obtain the time re-sponse Full details are given in [3 4] The string and bodyare coupled at the bridge using the standard point-couplingformula

Y minus1coup = Y minus1

body + Y minus1str (5)

where Ycoup Ybody Ystr are the 2times2 matrices of admittanceat the bridge notch for the coupled system the body aloneand the string alone respectively Each synthesised notebegins with an idealised pluck which is a step function of

force as the finger or plectrum loses contact with the stringThe string is assumed to be released from rest at the instantof the note starting Advantage is taken of the reciprocityprinciple it is convenient to apply the force step at thebridge and calculate the response at the point on the stringwhere the pluck acts As well as the plucking position onthe string it is possible to specify an angle of pluck (whichdetermines the mix of the two polarisations of string mo-tion excited) The playerrsquos finger or plectrum will have afinite size and this will have the effect of a low-pass filteron the excitation of string modes This is represented in asimple way by a frequency-domain Gaussian filter propor-tional to exp(minusω2K2) Here K = vh where v = Tmis the wave speed of the string and the width h was giventhe value 75 mm in order to obtain a reasonably realisticsound

The synthesis model calculates the transient waveformof acceleration at the coupling point between the string andbody A full model would then need to calculate the bodymotion at all other positions and perform a rather com-plicated numerical calculation to give the resulting soundradiation The present model uses a severely simplified ap-proach the computed bridge acceleration is regarded asbeing applied to a spherical radiating body of radius avibrating in the symmetric breathing mode only For thatproblem the sound pressure in the far field is given sim-ply by filtering the acceleration waveform with a low-passfilter that apart from an overall scale factor takes the form

R(ω) =ika

1 + ika=

iωωc

1 + iωωc (6)

where k = ωc is the wavenumber for sound in air c isthe speed of sound and ωc = ca is the filter roll-off fre-quency Setting this roll-off at 250 Hz corresponding toa sphere radius around 02 m proved to give satisfactory-sounding results

This synthesis procedure has been implemented in Mat-lab in a program that allows the user to specify a musicalfragment entered in a form of guitar tablature The result-ing ldquotunerdquo is saved as a sound file

3 Experimental methodology

31 Psychoacoustical tests

Using synthesised sounds generated by this method ex-periments were conducted to give some JND informationrelating to the major parameters For the body model ex-periments were done in which all the mode frequencieswere shifted by a given factor and also groups of modes inparticular frequency bands were shifted leaving the othermodes unchanged In a separate pair of tests the dampingfactors of all modes were modified by a given factor ei-ther upwards or downwards from the reference values incase there was an asymmetry of perception For the stringmodel the bending stiffness and the damping parametersηF ηB were changed The parameter ηA was found to haveso little influence on the chosen test sound that it was not

477

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

possible to obtain a meaningful JND for changes to itsvalue A general comment should be noted all the teststo be reported here were constrained by the time limitsimposed by the format of undergraduate projects In somecases there is no doubt that more could have been learnedhad more time been available but it is felt that the resultshave value even with these limitations

JND thresholds were estimated using a three-alternativeforced-choice procedure (3AFC) This is not the only pos-sible procedure that could have been used but it had al-ready been chosen for the violin studies and softwareto implement 3AFC was conveniently to hand A three-down one-up adaptive tracking rule was used that esti-mated the 79 correct point on the psychometric func-tion [10] It should be noted that this is the definition as-sumed for ldquoJNDrdquo throughout this paper it is a rather con-servative value significantly higher than that which wouldcorrespond to the 50 point on the psychometric func-tion which perhaps better matches the intuitive notion ofldquothreshold of perceptionrdquo Three sounds ndash two the samethe reference sound and one different the modified soundndash were played in a random sequence and the subject wasasked to choose which one was different The amount ofparameter modification between the stimuli in a given trialwas changed by a certain factor the step size The step sizedecreased after three correct answers and increased afterone wrong answer The test stopped after eight turnpointsA relatively large initial step size was applied until the sec-ond turnpoint was reached in order to allow rapid conver-gence toward the threshold region After the second turn-point the step size was reduced The threshold was takenas the mean of the values of the amount of modification atthe last six turnpoints The values of the two step sizes andthe starting level of parameter modification were chosendifferently for each test on the basis of preliminary trialsEach trial took no more than about 10 minutes and sub-jects had a break between tests

In all tests the reference sound was kept constant in or-der to increase discriminability subjects ldquolearntrdquo to rec-ognize the reference sound It is argued that for the pur-pose of relating to musical judgments which in practiceare made by musicians who may spend several hours everyday concentrating on subtle details of the sound of their in-strument the threshold of most interest is the one obtainedunder the most favourable conditions for discriminationIt is accepted that the test circumstances are quite remotefrom normal musical experience as seems to be inevitablein the design of any scientifically-acceptable test proce-dure and the approach adopted seems to the authors thebest that can easily be done The 3AFC software gave vi-sual feedback on rightwrong answers during the experi-ment although some subjects performed best by keepingtheir eyes shut and ignoring it Subjects did not receiveany training beforehand but if they performed erraticallyon the first run they were able to repeat the run This ap-plied only to a few subjects and they were all able to per-form the task at the second attempt The sounds were pre-sented diotically via Sennheiser HD580 headphones cho-

Figure 1 Guitar tablature for sound examples used in psy-choacoustical tests first the ldquoHuntsupperdquo extract second thestrummed chord with notes E2 B2 D3 B3 C4 A4

sen because of their diffuse field response in a relativelyquiet environment The sampling rate was 441 kHz andthe number of bits was 16

The choice of musical input for the test sounds canhave a significant influence on the JND obtained In thefirst tests on mode frequency shifting three different sam-ple sounds were tested a single note (the open A string110 Hz) a short musical fragment and a strummed ldquodis-cordant chordrdquo with notes E2 B2 D3 B3 C4 A4 selectedto give input at a wide range of frequencies The musi-cal fragment consisted of the first bar of the lute pieceldquoThe English Huntsupperdquo by John Whitfelde This andthe chord are shown in tablature notation in Figure 1 thetablature applies to a guitar in regular EADGBE tuningFor the remaining tests a single note was used for the testsound the open A string (110 Hz) for the damping factortests and C3 on the A string (131 Hz) for the string param-eter tests The total stimulus length for the single notes andthe chord was 1 s while the Huntsuppe fragment took 2 sThese are all longer than the 300 ms length that yieldedthe lowest thresholds in the violin tests and perhaps someadvantages of echoic memory are sacrificed by using theselonger stimuli However it was found that the sense of arealistic guitar sound was compromised if the duration wastoo short and it seemed more appropriate to use the longermore ldquoguitarishrdquo sounds

The test subjects were all ldquomusicalrdquo in one sense or an-other but for the purposes of the presentation of resultsthey are divided into two groups using the same criterionas in the work on the violin [1] those labelled ldquomusiciansrdquohad at least 8 years of formal training and were currentlyactive in music-making at least once per week while theothers are labelled ldquonon-musiciansrdquo The first author wasa subject in each test while all the other subjects were ofstudent age The results for the first author did not seemin any way atypical of the rest despite his relatively ad-vanced age

32 Physical measurements

Physical measurements on the body and strings of severalguitars are used in two different ways in this study Firstmeasurements are needed to calibrate the synthesis modelas has been described in detail previously [4] Second thesame range of measurements can be analysed to give es-timates of the variability in practice of the parameters forwhich JNDs will be found By comparing the actual varia-tion with the JND for perception it is hoped to give a first

478

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

80 100 200 500 1000 2000 5000-70

-60

-50

-40

-30

-20

-10

Frequency

|Y|(dB)

Figure 2 Magnitude of the input admittances of four guitarsmeasured normal to the top plate at the bridge saddle near thelowest string (E2) plotted in decibels relative to 1 mNs Heavyline flamenco guitar by Martin Woodhouse thin solid line clas-sical guitar by Michael J OrsquoLeary dashed line classical guitar byGreg Smallman dash-dot line flamenco guitar by Joseacute RamirezHorizontal lines show bands used in testing see text

estimate of the relative sensitivity and importance of dif-ferent kinds of variation in guitar construction and string-ing

Input admittance Y (ω) of the guitar body is measuredby applying a controlled force either with a miniature im-pulse hammer (PCB 086D80) or by gently pulling a lengthof fine copper wire looped around a string at the bridgesaddle until it snaps The latter method is less accuratebut has the advantage that it can be used to apply force indifferent controlled directions to give information on themodal angles In all cases the velocity response is mea-sured with a laser-Doppler vibrometer (Polytec OFV-056)directed to a spot as close as possible to the forcing pointSignals are collected at a sampling rate of 40 kHz by adigital data-logger based on a PC running Matlab witha National Instruments 6220 data acquisition card Thestrings of the guitar were damped during admittance mea-surements

Input admittances in the direction normal to the planeof the top plate have been measured on a wide range ofacoustic guitars and a representative selection of 15 areused in the present study Figure 2 shows results for fourof these all hand-made instruments of high quality Theheavy curve is the datum guitar used in the psychoacousti-cal tests a flamenco guitar by Martin Woodhouse studiedin the previous work [4] The others are a flamenco gui-tar by Jose Ramirez and classical guitars by Greg Small-man and Michael J OrsquoLeary It is immediately clear thatall these guitars although different in detail show a lotof common features All have a strong peak in the range80ndash100 Hz which is a Helmholtz resonance associatedwith the internal air cavity and the soundhole modified bythe compliance of the top and back plates All the guitarsthen show a deep antiresonance (marking the frequency atwhich the Helmholtz resonance would occur with a rigid

cavity see equation (1019) of Cremer [11]) followed bya cluster of strong peaks in the range 150ndash250 Hz Mostthen show a steady decline to a dip around 400ndash500 Hzfollowed by fluctuations around a rather steady level withgradually reducing peak amplitudes as the modal overlapbecomes significant One guitar (the OrsquoLeary) has an ad-ditional strong peak around 300 Hz interrupting the de-cline to the dip The two horizontal lines mark frequencybands that will be used in frequency-shifting tests in thenext section The first of these bands (150ndash250 Hz) coversthe range where the datum guitar has its strongest peaksThe second band (500ndash1000 Hz) covers a range where themodal overlap is becoming significant but fairly clear in-dividual peaks can still be seen

The other physical measurements used in this study areaimed at establishing the properties of strings Tension andmass per unit length are readily established but the bend-ing stiffness and damping properties require careful mea-surement Such measurements have previously been madeon a typical set of nylon strings for a classical guitar andin the course of this project corresponding measurementswere made on a set of steel strings as commonly usedon folk guitars The chosen strings were Martin ldquo8020Bronzerdquo It is a familiar experience to guitarists that thesound changes a lot between when the strings are new andwhen they are well-played so the strings studied here weremeasured in both states to give quantitative data on whatchanges in the ageing process

The bending stiffness and damping properties can be es-timated by analysis of sounds recorded from notes playedon the guitar in the usual way [4] The frequencies of thestrongest spectral components corresponding to ldquostringrdquomodes can be best-fitted to equation (2) to yield valuesof B and hence EI The associated decay rate of eachof these ldquostringrdquo modes can also be found using time-frequency analysis and this gives information about thestring damping The procedure for obtaining damping in-formation from the decay rates is a little indirect becausein most cases the decay rate is dominated by losses into theguitar body rather than by internal damping in the string

The approach adopted is to analyse the frequencies fora plucked note on every fret of a given string and combinethem all into a cloud plot of damping factor against fre-quency examples are shown in Figure 3 This shows datafor the 3rd string (G3) for the nylon string measured previ-ously and the new steel-cored string The inherent damp-ing of the string as a function of frequency determinesthe ldquofloor levelrdquo of this cloud plot Approximate values ofηA ηF ηB can be estimated by fitting eqquation (4) to thetrend of this floor level as illustrated by the curves in Fig-ure 3 The fitting process was carried out manually withsome regard to the error tolerances of the data points Notethat the high-frequency fit line for the nylon string wasinfluenced by additional data obtained by a different ap-proach not shown here see [4] for more detail It is clearfrom the figure that the nylon string has higher dampingthan the steel string The measured string properties aresummarised in Table I

479

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

Table I Properties of guitar strings (a) DrsquoAddario Pro Arteacute ldquoComposites hard tensionrdquo nylon strings (b) Martin ldquo8020 bronzerdquo steelstrings in new condition (c) as (b) but in used condition

String 1 2 3 4 5 6

Tuning note E4 B3 G3 D3 A2 E2

Frequency fn (Hz) 3296 2469 1960 1468 1100 824

Tension (a) T (N) 703 534 583 712 739 716(b) 121 135 182 185 169 147

Massunit length (a) m (gm) 038 052 090 195 361 624(b) 057 113 242 437 713 1102

Bending stiffness (a) EI (10minus6Nm2) 130 160 310 51 40 57(b) 108 336 234 255 327 667(c) 103 394 343 418 480 719

Inharmonicity (a) B (10minus6) 43 70 124 17 13 19(b) 21 58 30 32 45 106(c) 20 68 44 53 66 114

Loss coefficients (a) ηF (10minus5) 40 40 14 5 7 2(b) 7 6 11 9 11 5(c) 6 7 17 33 17 10

(a) ηB (10minus2) 24 20 20 20 25 20(b) 07 11 18 11 04 03(c) 07 11 68 41 29 38

(a) ηA (1s) 15 12 17 12 09 12(b) 25 10 12 13 12 17(c) 15 10 06 07 06 14

The differences between the nylon and steel strings innew condition are broadly as one would expect The nylonstrings have lower tensions and a different pattern of bend-ing stiffness because of the difference in construction forthe nylon set the top three strings are monofilaments whilethe lower three have multifilament cores with metal wind-ings The steel strings have monofilament cores through-out and only the top two strings are not wrapped Themonofilament nylon strings have higher internal dampingthan the steel but the difference is much less marked forthe lower three strings

The effect of age and use on the steel strings can alsobe seen Previous investigators have reported increaseddamping between new and old guitar strings [12 13] butwithout sufficient detail to allow a fit to equation (4) Itis generally accepted that the mechanism for this ageingeffect is the accumulation of dirt and corrosion productsbetween the windings of wrapped strings [13] perhapswith some additional cumulative wear of the outer wrap-pings where the strings make contact with the frets Themeasurements reported here support this picture there isvery little change in the two highest strings which are notwrapped For the wrapped strings there is a marked in-crease in damping especially in ηB which determines thedecay at the highest frequencies

Also not noted by previous authors there is a markedincrease in the effective bending stiffness of some of thestrings This seems perfectly credible since the contactconditions between the windings in the tensioned stringwill contribute to the bending stiffness and material col-

lecting between the windings will surely change those con-ditions This is consistent with the observed increase in ηB which is the loss factor associated with the bending stiff-ness term in the governing equation It may be noted thatthe bending stiffness values given here are in good gen-eral agreement with values reported by Jaumlrvelaumlinen et al[7] noting that their steel-string results match the ldquousedrdquoresults more closely than the ldquonewrdquo results Their paperdoes not report the condition of the strings as measured

4 Results for properties of the guitar body

41 Psychoacoustical tests

The first series of psychoacoustical tests concerned theshifting of all body mode frequencies by a given factorkeeping other simulation parameters fixed This modifica-tion is of course somewhat artificial but a possible physi-cal interpretation will be discussed in the next subsectionThe datum for this test was based on the modal proper-ties of the flamenco guitar by Martin Woodhouse This testwas run three times using the three different musical in-puts described above The three tests were completed by 8musicians and 5 non-musicians (according to the criteriondescribed above) The results are summarised in Figure 4The program that runs the 3AFC protocol gives as outputthe mean and standard deviation over the range defined bythe final 6 turnpoints The plots show for each test subjectthe mean on the horizontal axis and the relative standarddeviation (standard deviation divided by mean) on the ver-tical axis The numbers on the horizontal axis indicate the

480

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

100 200 500 1000 2000 500010

-4

10-3

10-2

Frequency (Hz)

Dampingfactor

100 200 500 1000 2000 500010

-4

10-3

10-2

Frequency (Hz)

Dampingfactor

Figure 3 Damping measurements on (top) a nylon guitar string(3rd string G3) and (bottom) the corresponding string from asteel-cored set Discrete points show individual measured decayrates curves show fits to the floor level using equation (4)

percentage shift in mode frequency Musicians are indi-cated by stars non-musicians by plus symbols The aver-ages over all test subjects are shown by the open circles

The open square symbols show the average of the foursubjects who achieved the lowest mean levels As was sug-gested earlier within the scope of the present study themost interesting threshold to compare with serious musi-cal judgments is that achieved by the best listeners underthe most favourable conditions In the violin tests the bestfive results were presented for this purpose [1] In the testsreported here the number of subjects is sometimes ratherlow and it was decided to average the best four The nu-merical results of this best-four measure for all the 3AFCtests presented here are given in the figure captions

It can be seen that for each of the three musical inputsthe best subjects could discriminate a shift of 1 or lessInteresting patterns can be seen in the detailed results de-spite the small number of subjects that would make anyelaborate statistical analysis of limited value The generalspread of results and the two averages look very simi-lar for the single note and the Huntsuppe extract For thesingle note the musicians and non-musicians are intermin-gled while for Huntsuppe all but one of the non-musicianshas a worse result than all the musicians Perhaps non-musicians have their performance degraded by informa-

03 05 1 2 3 5 100

05

1

(a) Huntsuppe

Percentage shift

RelSTD

03 05 1 2 3 5 100

05

1

(b) Single note

Percentage shift

RelSTD

03 05 1 2 3 5 100

05

1

(c) Discordant chord

Percentage shiftRelSTD

Figure 4 Results from 3AFC studies for shifting all mode fre-quencies of the datum guitar using as input (a) the Huntsuppefragment (b) a single note (open A string) (c) the discordantchord Horizontal axis shows percentage frequency shift on alogarithmic scale vertical axis shows relative standard deviationPoints denote musicians points + denote non-musicians Opencircle average of all subjects open square average of best foursubjects The percentage shifts corresponding to the best-four av-erages are (a) 113 (b) 092 (c) 059

tional masking while experienced musicians do not withsuch an obviously musical sound sample as ldquoHuntsupperdquo

The chord reveals a different picture Four of the mu-sicians achieved very low thresholds with this sound in-put but all other subjects seem to have been seriouslydistracted by this rather non-musical sound and achievedonly rather high thresholds One of the musicians comesout worst of all in this test whereas for Huntsuppe all themusicians performed well This is a striking example ofinformational masking especially since the group of foursubjects who did well actually produced a lower averagethan in either of the other tests That is as one would hopethis signal contains deliberately a wider range of fre-quency components so that analytical listeners with suffi-cient skill and experience to take advantage of it can makevery subtle judgements down to 032 shift in the bestcase This corresponds to shifting the mode frequenciesabout 6 cents or 120 of a semitone an impressively smallamount that is not very much bigger than the threshold forpitch discrimination tasks (see eg [14])

The thresholds are of the same order as the ldquobest fiverdquothresholds in the corresponding violin tests but the de-tailed pattern seems different The choice of input had amore marked effect in the violin and to achieve the lowestthresholds a very short and consequently rather unmusi-cal sound sample was used

The best-performing 7 subjects of the first test serieswere asked to undertake two further tests in which fre-

481

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

quencies were shifted only in limited bands The twobands studied were shown in Figure 2 and this time theonly input sound tested was the single-note open A stringThe results of the tests are shown in Figure 5 in the sameformat as Figure 4 Rather unexpectedly the ldquobest fourrdquothreshold for the first band containing the strongest peaksof the datum guitar was much higher than for the sec-ond band Related tests for the violin using the so-calledldquoDuumlnnwald bandsrdquo gave thresholds around 2 for the firstthree bands (see Figure 6 of Fritz et al [1]) Band 1 in thepresent study gave a comparable result but band 2 gave alower result than was observed in any of the violin testsThe best-four JND in this case is virtually the same as forthe corresponding test when all frequencies were shiftedFurther studies might be worth conducting to add moredetail to this observation

The final pair of tests relating to the perception of bodymode changes investigated the effect of damping factorsThis time the mode frequencies were kept fixed but themodal Q factors were all scaled by a given factor Q fac-tor is the inverse of damping factor so high Q correspondsto low damping In two separate tests the Q factors wereincreased and decreased to look for any asymmetry in theperception Thirteen test subjects took these two tests 10musicians and 3 non-musicians Results are shown in Fig-ure 6 in the same format as Figs 4 and 5 except that thenumbers on the horizontal axis now indicate the scale fac-tor applied rather than the percentage change This meansthat ldquono changerdquo corresponds to the value 1 rather than 0as in the previous results which has the effect of makingthe relative standard deviations numerically smaller

An asymmetry is seen between the two tests but it doesnot significantly affect the threshold value for the bestfour listeners They achieved a value around 12 ie a20 change in Q factors in both cases However acrossthe whole set of test subjects the spread is much greaterwhen the Q factors were decreased compared to the in-creasing case Many participants reported initial difficultyin hearing the difference when Q factors were decreasedalthough once they had learned to recognise the soundchange they often achieved quite low thresholds

It was not thought worthwhile to test separate frequencybands in this case for two reasons First body damping ismost readily altered by a physical change of material andthis is likely to have an effect over a broad frequency rangeSecond it will be shown shortly that the JND for percep-tion of a change in body damping is of the same orderas the measured standard deviation among real guitars sothat the perceptual significance is probably marginal anddigging deeper into the details is likely to be of limitedimportance

42 Physical measurements

The next step is to look at physical measurements to as-sess how these thresholds of perception compare with theactual extent of variation between guitars Of course onewould never find two real instruments which differed byhaving all the body frequencies simultaneously scaled by

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(a) 150-250 Hz

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(b) 500-1000 Hz

Figure 5 Results in the format of Figure 4 from 3AFC studies forshifting all mode frequencies of the datum guitar lying in (a) theband 150ndash250 Hz and (b) the band 500ndash1000 Hz using as inputa single note (open A string) The percentage shifts correspond-ing to the best-four averages are (a) 187 (b) 153

1 12 15 2 25 30

005

01

015

02

(a)

Scale factor

RelSTD

1 12 15 2 25 30

005

01

015

02

(b)

Scale factor

RelSTD

Figure 6 Results in the format of Figure 4 from 3AFC studiesfor scaling all modal Q factors of the datum guitar (a) upwardsand (b) downwards using as input a single note (open A string)Horizontal axis shows scale factor applied by multiplication ordivision to Q factors The scale factors corresponding to the best-four averages are (a) 119 (b) 122

exactly the same factor However it is easy to visualise achange which would in principle have this effect it wouldhappen if the dimensions of the guitar were all scaledkeeping the same materials and other details Some sim-ple deductions can then be drawn by scaling analysis Ifthe main vibrating elements of the guitar body are treatedas thin flat plates deforming in bending then the naturalfrequencies scale proportional to hd2 where d is the scalefactor for dimensions in the plane and h is the scale factorfor thickness Taking 1 as a round number for the orderof magnitude of the threshold of detection for shifting fre-quencies it follows that the JNDs for dimensional scalingare approximately 1 if d and h are both scaled together(full geometric scaling) 1 again if thickness is changedkeeping the in-plane dimensions fixed and only 05 ifin-plane dimensions are scaled keeping the thickness thesame These are all very small numbers they amount tochanging in-plane dimensions no more than a few mm and

482

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

75 80 85 90 95 1000

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(a)

160 170 180 190 200 210 2200

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(b)

Figure 7 Modal frequencies and Q factors for 15 guitars forthe highest peak in the frequency bands (a) 75ndash100 Hz theldquoHelmholtz resonancerdquo (b) 150ndash250 Hz the ldquofirst top platemoderdquo Plus and cross symbols show fits without and with com-pensation for string damping respectively Diamond symbolsmark the datum guitar for the listening tests Open squares showthe means and range bars show one standard deviation aboveand below the means for the two sets of fitted data The rangesfor the compensated results can be recognised by their higher Qvalues Dashed lines near the axes show approximately 1 JNDabove and below the uncompensated means from the results ofsection 41

to changing a typical top plate thickness by a mere 25 micromReal instruments have dimensional variations which areconsiderably bigger than this especially in thickness soon this ground alone it is no surprise that guitars can sounddifferent

The simplest approach to obtaining quantitative infor-mation from admittance measurements on a set of gui-tars is to perform explicit modal analysis This can onlybe expected to give reliable results at relatively low fre-quencies pole-residue extraction whatever the methodbecomes sensitive once the modal overlap begins to ap-proach unity To illustrate the kind of values which emergefrom such an analysis Figure 7 shows results for modalfrequency and Q factor for 15 guitars for the highest peakin each of the two low-frequency bands mentioned in thediscussion of Figure 2 75ndash100 Hz the ldquoHelmholtz reso-nancerdquo and 150ndash250 Hz the ldquofirst top plate moderdquo

A complicating factor needs to be noted The ad-mittance measurements are made with the strings welldamped because the purpose is to obtain informationabout the body behaviour without the effect of the stringsHowever the damped strings remove some energy fromthe body vibration and this will tend to reduce the mea-sured Q factors This effect can be compensated at leastapproximately The damped strings will act essentially assemi-infinite strings and will thus present an impedance tothe bridge which is real and if the strings are all assumedto be attached at the same point equal to the sum of thestring impedances given by Z =

radicTm From the data for

nylon strings in Table I the sum of these impedances is212 Nsm To compensate the inverse of Y (ω) is takenthis real quantity is subtracted then the result re-invertedto give a new admittance This approach can be expectedto work well at higher frequencies but it is hard to be sureif the strings are well enough damped at low frequenciesto justify the semi-infinite approximation

Figure 7 shows results before and after this compensa-tion For each case the mean and scatter are indicated bylines showing one standard deviation above and below themean for frequency and Q Interestingly the standard de-viations have a similar pattern for both cases The relativestandard deviations for frequency are 55 and 6 for thetwo cases and those for Q factor about 33 in both casesbefore compensation

To obtain good comparative information about modalfrequencies over a broader bandwidth is more challeng-ing It is not possible to identify the ldquosamerdquo mode in dif-ferent guitars except at very low frequencies sensitivityto small structural changes increases with frequency as isfamiliar in other vibration problems (see eg [15]) Formode frequencies the threshold for audible changes is sosmall that it is hard to obtain any measure which is goodenough scaling frequencies by 1 means that one guitarbody might have 101 modes in a frequency range whereanother guitar had 100 and there is no known analysismethod which can give robust discrimination at this levelAny measure which can be computed shows much biggervariability than this and there is always a strong suspicionthat it is revealing the vagaries of the analysis method morethan the real differences between guitars

Things are a little more promising in the case of damp-ing estimation over a wide frequency range Taking theinverse FFT of an input admittance gives an impulse re-sponse function of the guitar body and this can be pro-cessed using time-frequency analysis to give a sonogramrepresentation In each frequency band a best-fitting ex-ponential decay can then be obtained Combining resultsfrom several different choices of the timefrequency res-olution tradeoff improves the robustness of the resultsThe process is described in more detail elsewhere [16]Some results are shown in Figure 8 compensation forstring damping has been included In Figure 8a the discretepoints represent for the datum guitar frequencydecaycombinations which satisfied an assumed criterion forgood fit quality For compatibility with other plots decay

483

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70

80

90

100

Frequency

Qfactor

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100

120

Frequency

Qfactor

(b)

30 35 40 45 50 55 60 6515

20

25

30

35

40

Mean Q

Relativestandard

deviation(

)

(c)

Figure 8 Damping trends for guitar bodies determined fromtime-frequency analysis (a) individual measurements of fre-quency and equivalent Q factor (circles) and the band averageof these (solid line) for the datum guitar (b) the solid lines asin (a) for the set of 15 guitars (c) the mean and standard devia-tion of the results from (b) across the frequency range with rangebars indicating one standard deviation above and below the meanon each axis The circle marks the datum guitar for the listeningtests

rate is represented as an equivalent Q factor The continu-ous line is simply a moving band average of these discretepoints

Figure 8b then shows the band-averaged lines for the 15guitars in the test set Finally Figure 8c shows a scatterplot of the mean and standard deviation of these band-averaged lines for the 15 guitars As in Figure 7 thelarge cross represents one standard deviation above and

2 3 5 8 100

01

02

03

04 (a)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (b)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (c)

Scale factor

RelSTD

Figure 9 Results in the format of Figure 4 from 3AFC studiesfor changing the damping parameters of the 5th string based ondatum values for the new steel strings Case (a) varying ηB testnote E2 (b) varying ηB test note C3 (c) varying ηF test noteC3 The scale factors corresponding to the best-four averages are(a) 299 (b) 313 (c) 241

below the mean on each axis The main result is thatthis frequency-averaged measure of damping has a rela-tive standard deviation of the order of 20 comparable tothe JND found earlier for perception of damping changesThis suggests that the extremes of mean damping in themeasured set of guitars are sufficiently different that theeffect should be audible but the results do not lead oneto expect that this will be a very strong component of theperception of tonal differences among guitars

5 Results for properties of the strings

The final set of JND experiments relates to changes instring properties including those properties that changedas the string aged one objective was to find out whichof those changes are likely to contribute most to the per-ception of ageing Using as datum the properties of thenew steel strings listed in Table I changes to the dampingproperties of the 5th string were investigated using single-note test sounds These 3AFC tests were completed by 11musicians and 5 non-musicians One test was also con-ducted relating to changes in the bending stiffness of thisstring it gave results in line with those of Jaumlrvelaumlinen etal [7] and the details are not reproduced here The actualbending stiffness of this string was found to be close tothe threshold for detecting an increase in bending stiffnesswhen starting from a much lower level In the aged stringthe bending stiffness only increased by a factor 16 (seeTable I) If that change is audible at all the effect is likelyto be small

The results for the damping tests are shown in Figure 9in a similar format to earlier 3AFC results As noted ear-lier the perceptual effects of changing ηA for this string

484

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

were found to be so weak that it was not possible to con-duct a meaningful test However both ηB and ηF gave clearresults The parameter ηB associated with bending stiff-ness and mainly affecting higher frequencies was testedusing two different notes ldquoplayedrdquo on the same string Theresults are shown in Figures 9ab and show a good levelof consistency between the two notes The spread over-all mean and best-four mean are all very similar for bothcases The best listeners need about a factor 3 increase inthis parameter to perceive the change Note from Table Ithat in the aged string this parameter increased from 0004to 0029 that change should be very clearly audible

Figure 9c shows corresponding results for ηF All sub-jects gave a somewhat lower threshold for this parametercompared to ηB and the very best listener achieved a JNDof less than a factor 2 In the aged string ηF increased inthe ratio 1711 for this string from Table I This effect isprobably not audible to most listeners It can be concludedthat the factor most likely to account for the difference insound of the aged string is ηB

6 Conclusions

Psychoacoustical experiments have been reported to estab-lish JND levels for the perception of changes to variousparameters of a model of synthesised guitar sound [3 4]These investigations parallel earlier work exploring relatedJNDs for violin sounds [1] The parameters investigatedhere relate to the behaviour of the guitar body (modal fre-quencies and damping factors) and of the strings (bendingstiffness and damping model parameters) All tests werecarried out using a three-alternative forced-choice method-ology It is argued that for application to musical judge-ments which can be very subtle the most interesting per-ceptual thresholds are those attainable by the best listenersunder the most favourable conditions To give an estimateof these the results for the best four subjects in each testwere averaged

Results of physical measurements have also been re-ported to give an indication of the spread of each parame-ter among real guitar bodies or strings A comparison withthe psychoacoustical results gives a first impression ofwhich quantities might be responsible for the biggest per-ceptual differences between instruments By far the mostsensitive parameter of all those tested relates to shiftingthe frequencies of body modes The best four listenerscould perceive a 1 shift when all frequencies were scaledtogether a small change compared to the actual varia-tion among guitars This performance was only degradedslightly when the shifted frequencies were band-limited intwo different bands tested The results were of the sameorder as the corresponding JNDs found for violin sounds[1] if anything the JNDs seem to be somewhat lower forthe guitar sounds perhaps because the fully-coupled syn-thesis method gives the listener additional auditory cuesassociated with transient decay rates of the string over-tones Body mode damping has to be changed by about20 for perception according to the definition used hereActual variations of modal damping exceed this level but

not by a large factor so body damping is probably quite aminor contributor to tonal perception

String damping parameters are even less sensitive theyhave to be changed by a factor of the order of 3 How-ever this large factor does not mean that the effects arenever heard Measurements have been presented on stringsof different materials and on steel-cored strings in newand ldquoagedrdquo conditions The damping parameters vary be-tween these cases by factors significantly bigger than theJNDs The results strongly indicate that the most importantparameter for the perception of the degradation of soundin well-used strings is ηB the damping factor associatedwith the bending stiffness of the string This parameterincreases greatly with age of strings probably from thedamping effect of dirt and corrosion products accumulat-ing between the windings of the lower strings of the guitar[13] Strings without over-winding do not deteriorate insound to anywhere near the same extent

The correct interpretation of the high sensitivity to shift-ing body frequencies is not obvious The actual change in-vestigated here moving all frequencies in step is very arti-ficial Listeners might be responding to the actual frequen-cies or perhaps to the change in modal density (the in-verse of the mean spacing between resonant frequencies)or perhaps to the shifting amplitude envelope of the bodyresponse Modal amplitudes were not changed when thefrequencies were scaled so the amplitude pattern movesalong the frequency axis with the scaling It would re-quire further testing to establish whether one of these fac-tors dominates or whether all contribute significantly Suchtesting would be a good target for future research to bringthe results closer to the detailed concerns of instrumentmakers

AcknowledgementThe authors thank Brian Moore Ian Cross and Alan Black-well for many helpful discussions on this work and MartinWoodhouse Stewart French Ian Cross and James West-brook for providing access to guitars for testing Twoanonymous reviewers are thanked for constructive sugges-tions

References

[1] C Fritz I Cross B C J Moore J Woodhouse Perceptualthresholds for detecting modifications applied to the acous-tical properties of a violin J Acoust Soc Am 122 (2007)3640ndash3650

[2] J Woodhouse P Galluzzo The bowed string as we know ittoday Acta Acustica united with Acustica 90 (2004) 579ndash589

[3] J Woodhouse On the synthesis of guitar plucks ActaAcustica united with Acustica 90 (2004) 928ndash944

[4] J Woodhouse Plucked guitar transients comparison ofmeasurements and synthesis Acta Acustica united withAcustica 90 (2004) 945ndash965

[5] H A K Wright The acoustics and psychoacoustics of theguitar PhD Thesis Cardiff University 1996

[6] H Jaumlrvelaumlinen V Vaumllimaumlki M Karjalainen Audibility ofthe timbral effects of inharmonicity in stringed instrumenttones Acoustics Research Letters Online 2 (2001) 79ndash84

485

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

The string is treated as a continuous system ratherthan being described in modal terms this was shown togive a more efficient strategy for computing the coupledstringbody motion with correct allowance for the very dif-ferent damping levels of the two systems which causesthe damping of the coupled system to be strongly non-proportional [3] A given string is specified by its lengthL tension T mass per unit length m bending stiffnessEI and a measure of its damping The bending stiffnessis expressed as the product of the Youngrsquos modulus E andthe second moment of area I which for a homogeneousmonofilament string of diameter d is given by

I = πd464 (1)

Inharmonicity due to bending stiffness is often expressedin terms of the dimensionless parameter B that appears inthe approximate expression for the frequency of the nthovertone

fn asymp nf0 1 + Bn2 (2)

where f0 is the fundamental frequency In terms of the no-tation introduced here

B =EIπ2

TL2 (3)

For overwound strings as normally used for the lowerstrings of the guitar the effective value of EI or B can-not be reliably predicted and must be measured

Musical strings in general have been shown to exhibitstrongly frequency-dependent damping and the measuredbehaviour can be fitted to a three-parameter model pro-posed by Valette [9] The loss factor of the nth mode isassumed to take the form

ηn =T (ηF + ηAωn) + EIηB (nπL)2

T + EI (nπL)2 (4)

The damping is lowest in the mid range of frequencieswith a minimum level set by ηF At very low frequenciesthe damping increases an effect attributed by Valette to airdamping and quantified by the parameter ηA At high fre-quencies the damping increases again because energy dis-sipation associated with the complex bending stiffness ofthe string comes to dominate especially for the polymer-based strings of classical guitars This effect is charac-terised by the parameter ηB

The chosen synthesis method works in the frequencydomain then uses an inverse FFT to obtain the time re-sponse Full details are given in [3 4] The string and bodyare coupled at the bridge using the standard point-couplingformula

Y minus1coup = Y minus1

body + Y minus1str (5)

where Ycoup Ybody Ystr are the 2times2 matrices of admittanceat the bridge notch for the coupled system the body aloneand the string alone respectively Each synthesised notebegins with an idealised pluck which is a step function of

force as the finger or plectrum loses contact with the stringThe string is assumed to be released from rest at the instantof the note starting Advantage is taken of the reciprocityprinciple it is convenient to apply the force step at thebridge and calculate the response at the point on the stringwhere the pluck acts As well as the plucking position onthe string it is possible to specify an angle of pluck (whichdetermines the mix of the two polarisations of string mo-tion excited) The playerrsquos finger or plectrum will have afinite size and this will have the effect of a low-pass filteron the excitation of string modes This is represented in asimple way by a frequency-domain Gaussian filter propor-tional to exp(minusω2K2) Here K = vh where v = Tmis the wave speed of the string and the width h was giventhe value 75 mm in order to obtain a reasonably realisticsound

The synthesis model calculates the transient waveformof acceleration at the coupling point between the string andbody A full model would then need to calculate the bodymotion at all other positions and perform a rather com-plicated numerical calculation to give the resulting soundradiation The present model uses a severely simplified ap-proach the computed bridge acceleration is regarded asbeing applied to a spherical radiating body of radius avibrating in the symmetric breathing mode only For thatproblem the sound pressure in the far field is given sim-ply by filtering the acceleration waveform with a low-passfilter that apart from an overall scale factor takes the form

R(ω) =ika

1 + ika=

iωωc

1 + iωωc (6)

where k = ωc is the wavenumber for sound in air c isthe speed of sound and ωc = ca is the filter roll-off fre-quency Setting this roll-off at 250 Hz corresponding toa sphere radius around 02 m proved to give satisfactory-sounding results

This synthesis procedure has been implemented in Mat-lab in a program that allows the user to specify a musicalfragment entered in a form of guitar tablature The result-ing ldquotunerdquo is saved as a sound file

3 Experimental methodology

31 Psychoacoustical tests

Using synthesised sounds generated by this method ex-periments were conducted to give some JND informationrelating to the major parameters For the body model ex-periments were done in which all the mode frequencieswere shifted by a given factor and also groups of modes inparticular frequency bands were shifted leaving the othermodes unchanged In a separate pair of tests the dampingfactors of all modes were modified by a given factor ei-ther upwards or downwards from the reference values incase there was an asymmetry of perception For the stringmodel the bending stiffness and the damping parametersηF ηB were changed The parameter ηA was found to haveso little influence on the chosen test sound that it was not

477

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

possible to obtain a meaningful JND for changes to itsvalue A general comment should be noted all the teststo be reported here were constrained by the time limitsimposed by the format of undergraduate projects In somecases there is no doubt that more could have been learnedhad more time been available but it is felt that the resultshave value even with these limitations

JND thresholds were estimated using a three-alternativeforced-choice procedure (3AFC) This is not the only pos-sible procedure that could have been used but it had al-ready been chosen for the violin studies and softwareto implement 3AFC was conveniently to hand A three-down one-up adaptive tracking rule was used that esti-mated the 79 correct point on the psychometric func-tion [10] It should be noted that this is the definition as-sumed for ldquoJNDrdquo throughout this paper it is a rather con-servative value significantly higher than that which wouldcorrespond to the 50 point on the psychometric func-tion which perhaps better matches the intuitive notion ofldquothreshold of perceptionrdquo Three sounds ndash two the samethe reference sound and one different the modified soundndash were played in a random sequence and the subject wasasked to choose which one was different The amount ofparameter modification between the stimuli in a given trialwas changed by a certain factor the step size The step sizedecreased after three correct answers and increased afterone wrong answer The test stopped after eight turnpointsA relatively large initial step size was applied until the sec-ond turnpoint was reached in order to allow rapid conver-gence toward the threshold region After the second turn-point the step size was reduced The threshold was takenas the mean of the values of the amount of modification atthe last six turnpoints The values of the two step sizes andthe starting level of parameter modification were chosendifferently for each test on the basis of preliminary trialsEach trial took no more than about 10 minutes and sub-jects had a break between tests

In all tests the reference sound was kept constant in or-der to increase discriminability subjects ldquolearntrdquo to rec-ognize the reference sound It is argued that for the pur-pose of relating to musical judgments which in practiceare made by musicians who may spend several hours everyday concentrating on subtle details of the sound of their in-strument the threshold of most interest is the one obtainedunder the most favourable conditions for discriminationIt is accepted that the test circumstances are quite remotefrom normal musical experience as seems to be inevitablein the design of any scientifically-acceptable test proce-dure and the approach adopted seems to the authors thebest that can easily be done The 3AFC software gave vi-sual feedback on rightwrong answers during the experi-ment although some subjects performed best by keepingtheir eyes shut and ignoring it Subjects did not receiveany training beforehand but if they performed erraticallyon the first run they were able to repeat the run This ap-plied only to a few subjects and they were all able to per-form the task at the second attempt The sounds were pre-sented diotically via Sennheiser HD580 headphones cho-

Figure 1 Guitar tablature for sound examples used in psy-choacoustical tests first the ldquoHuntsupperdquo extract second thestrummed chord with notes E2 B2 D3 B3 C4 A4

sen because of their diffuse field response in a relativelyquiet environment The sampling rate was 441 kHz andthe number of bits was 16

The choice of musical input for the test sounds canhave a significant influence on the JND obtained In thefirst tests on mode frequency shifting three different sam-ple sounds were tested a single note (the open A string110 Hz) a short musical fragment and a strummed ldquodis-cordant chordrdquo with notes E2 B2 D3 B3 C4 A4 selectedto give input at a wide range of frequencies The musi-cal fragment consisted of the first bar of the lute pieceldquoThe English Huntsupperdquo by John Whitfelde This andthe chord are shown in tablature notation in Figure 1 thetablature applies to a guitar in regular EADGBE tuningFor the remaining tests a single note was used for the testsound the open A string (110 Hz) for the damping factortests and C3 on the A string (131 Hz) for the string param-eter tests The total stimulus length for the single notes andthe chord was 1 s while the Huntsuppe fragment took 2 sThese are all longer than the 300 ms length that yieldedthe lowest thresholds in the violin tests and perhaps someadvantages of echoic memory are sacrificed by using theselonger stimuli However it was found that the sense of arealistic guitar sound was compromised if the duration wastoo short and it seemed more appropriate to use the longermore ldquoguitarishrdquo sounds

The test subjects were all ldquomusicalrdquo in one sense or an-other but for the purposes of the presentation of resultsthey are divided into two groups using the same criterionas in the work on the violin [1] those labelled ldquomusiciansrdquohad at least 8 years of formal training and were currentlyactive in music-making at least once per week while theothers are labelled ldquonon-musiciansrdquo The first author wasa subject in each test while all the other subjects were ofstudent age The results for the first author did not seemin any way atypical of the rest despite his relatively ad-vanced age

32 Physical measurements

Physical measurements on the body and strings of severalguitars are used in two different ways in this study Firstmeasurements are needed to calibrate the synthesis modelas has been described in detail previously [4] Second thesame range of measurements can be analysed to give es-timates of the variability in practice of the parameters forwhich JNDs will be found By comparing the actual varia-tion with the JND for perception it is hoped to give a first

478

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

80 100 200 500 1000 2000 5000-70

-60

-50

-40

-30

-20

-10

Frequency

|Y|(dB)

Figure 2 Magnitude of the input admittances of four guitarsmeasured normal to the top plate at the bridge saddle near thelowest string (E2) plotted in decibels relative to 1 mNs Heavyline flamenco guitar by Martin Woodhouse thin solid line clas-sical guitar by Michael J OrsquoLeary dashed line classical guitar byGreg Smallman dash-dot line flamenco guitar by Joseacute RamirezHorizontal lines show bands used in testing see text

estimate of the relative sensitivity and importance of dif-ferent kinds of variation in guitar construction and string-ing

Input admittance Y (ω) of the guitar body is measuredby applying a controlled force either with a miniature im-pulse hammer (PCB 086D80) or by gently pulling a lengthof fine copper wire looped around a string at the bridgesaddle until it snaps The latter method is less accuratebut has the advantage that it can be used to apply force indifferent controlled directions to give information on themodal angles In all cases the velocity response is mea-sured with a laser-Doppler vibrometer (Polytec OFV-056)directed to a spot as close as possible to the forcing pointSignals are collected at a sampling rate of 40 kHz by adigital data-logger based on a PC running Matlab witha National Instruments 6220 data acquisition card Thestrings of the guitar were damped during admittance mea-surements

Input admittances in the direction normal to the planeof the top plate have been measured on a wide range ofacoustic guitars and a representative selection of 15 areused in the present study Figure 2 shows results for fourof these all hand-made instruments of high quality Theheavy curve is the datum guitar used in the psychoacousti-cal tests a flamenco guitar by Martin Woodhouse studiedin the previous work [4] The others are a flamenco gui-tar by Jose Ramirez and classical guitars by Greg Small-man and Michael J OrsquoLeary It is immediately clear thatall these guitars although different in detail show a lotof common features All have a strong peak in the range80ndash100 Hz which is a Helmholtz resonance associatedwith the internal air cavity and the soundhole modified bythe compliance of the top and back plates All the guitarsthen show a deep antiresonance (marking the frequency atwhich the Helmholtz resonance would occur with a rigid

cavity see equation (1019) of Cremer [11]) followed bya cluster of strong peaks in the range 150ndash250 Hz Mostthen show a steady decline to a dip around 400ndash500 Hzfollowed by fluctuations around a rather steady level withgradually reducing peak amplitudes as the modal overlapbecomes significant One guitar (the OrsquoLeary) has an ad-ditional strong peak around 300 Hz interrupting the de-cline to the dip The two horizontal lines mark frequencybands that will be used in frequency-shifting tests in thenext section The first of these bands (150ndash250 Hz) coversthe range where the datum guitar has its strongest peaksThe second band (500ndash1000 Hz) covers a range where themodal overlap is becoming significant but fairly clear in-dividual peaks can still be seen

The other physical measurements used in this study areaimed at establishing the properties of strings Tension andmass per unit length are readily established but the bend-ing stiffness and damping properties require careful mea-surement Such measurements have previously been madeon a typical set of nylon strings for a classical guitar andin the course of this project corresponding measurementswere made on a set of steel strings as commonly usedon folk guitars The chosen strings were Martin ldquo8020Bronzerdquo It is a familiar experience to guitarists that thesound changes a lot between when the strings are new andwhen they are well-played so the strings studied here weremeasured in both states to give quantitative data on whatchanges in the ageing process

The bending stiffness and damping properties can be es-timated by analysis of sounds recorded from notes playedon the guitar in the usual way [4] The frequencies of thestrongest spectral components corresponding to ldquostringrdquomodes can be best-fitted to equation (2) to yield valuesof B and hence EI The associated decay rate of eachof these ldquostringrdquo modes can also be found using time-frequency analysis and this gives information about thestring damping The procedure for obtaining damping in-formation from the decay rates is a little indirect becausein most cases the decay rate is dominated by losses into theguitar body rather than by internal damping in the string

The approach adopted is to analyse the frequencies fora plucked note on every fret of a given string and combinethem all into a cloud plot of damping factor against fre-quency examples are shown in Figure 3 This shows datafor the 3rd string (G3) for the nylon string measured previ-ously and the new steel-cored string The inherent damp-ing of the string as a function of frequency determinesthe ldquofloor levelrdquo of this cloud plot Approximate values ofηA ηF ηB can be estimated by fitting eqquation (4) to thetrend of this floor level as illustrated by the curves in Fig-ure 3 The fitting process was carried out manually withsome regard to the error tolerances of the data points Notethat the high-frequency fit line for the nylon string wasinfluenced by additional data obtained by a different ap-proach not shown here see [4] for more detail It is clearfrom the figure that the nylon string has higher dampingthan the steel string The measured string properties aresummarised in Table I

479

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

Table I Properties of guitar strings (a) DrsquoAddario Pro Arteacute ldquoComposites hard tensionrdquo nylon strings (b) Martin ldquo8020 bronzerdquo steelstrings in new condition (c) as (b) but in used condition

String 1 2 3 4 5 6

Tuning note E4 B3 G3 D3 A2 E2

Frequency fn (Hz) 3296 2469 1960 1468 1100 824

Tension (a) T (N) 703 534 583 712 739 716(b) 121 135 182 185 169 147

Massunit length (a) m (gm) 038 052 090 195 361 624(b) 057 113 242 437 713 1102

Bending stiffness (a) EI (10minus6Nm2) 130 160 310 51 40 57(b) 108 336 234 255 327 667(c) 103 394 343 418 480 719

Inharmonicity (a) B (10minus6) 43 70 124 17 13 19(b) 21 58 30 32 45 106(c) 20 68 44 53 66 114

Loss coefficients (a) ηF (10minus5) 40 40 14 5 7 2(b) 7 6 11 9 11 5(c) 6 7 17 33 17 10

(a) ηB (10minus2) 24 20 20 20 25 20(b) 07 11 18 11 04 03(c) 07 11 68 41 29 38

(a) ηA (1s) 15 12 17 12 09 12(b) 25 10 12 13 12 17(c) 15 10 06 07 06 14

The differences between the nylon and steel strings innew condition are broadly as one would expect The nylonstrings have lower tensions and a different pattern of bend-ing stiffness because of the difference in construction forthe nylon set the top three strings are monofilaments whilethe lower three have multifilament cores with metal wind-ings The steel strings have monofilament cores through-out and only the top two strings are not wrapped Themonofilament nylon strings have higher internal dampingthan the steel but the difference is much less marked forthe lower three strings

The effect of age and use on the steel strings can alsobe seen Previous investigators have reported increaseddamping between new and old guitar strings [12 13] butwithout sufficient detail to allow a fit to equation (4) Itis generally accepted that the mechanism for this ageingeffect is the accumulation of dirt and corrosion productsbetween the windings of wrapped strings [13] perhapswith some additional cumulative wear of the outer wrap-pings where the strings make contact with the frets Themeasurements reported here support this picture there isvery little change in the two highest strings which are notwrapped For the wrapped strings there is a marked in-crease in damping especially in ηB which determines thedecay at the highest frequencies

Also not noted by previous authors there is a markedincrease in the effective bending stiffness of some of thestrings This seems perfectly credible since the contactconditions between the windings in the tensioned stringwill contribute to the bending stiffness and material col-

lecting between the windings will surely change those con-ditions This is consistent with the observed increase in ηB which is the loss factor associated with the bending stiff-ness term in the governing equation It may be noted thatthe bending stiffness values given here are in good gen-eral agreement with values reported by Jaumlrvelaumlinen et al[7] noting that their steel-string results match the ldquousedrdquoresults more closely than the ldquonewrdquo results Their paperdoes not report the condition of the strings as measured

4 Results for properties of the guitar body

41 Psychoacoustical tests

The first series of psychoacoustical tests concerned theshifting of all body mode frequencies by a given factorkeeping other simulation parameters fixed This modifica-tion is of course somewhat artificial but a possible physi-cal interpretation will be discussed in the next subsectionThe datum for this test was based on the modal proper-ties of the flamenco guitar by Martin Woodhouse This testwas run three times using the three different musical in-puts described above The three tests were completed by 8musicians and 5 non-musicians (according to the criteriondescribed above) The results are summarised in Figure 4The program that runs the 3AFC protocol gives as outputthe mean and standard deviation over the range defined bythe final 6 turnpoints The plots show for each test subjectthe mean on the horizontal axis and the relative standarddeviation (standard deviation divided by mean) on the ver-tical axis The numbers on the horizontal axis indicate the

480

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

100 200 500 1000 2000 500010

-4

10-3

10-2

Frequency (Hz)

Dampingfactor

100 200 500 1000 2000 500010

-4

10-3

10-2

Frequency (Hz)

Dampingfactor

Figure 3 Damping measurements on (top) a nylon guitar string(3rd string G3) and (bottom) the corresponding string from asteel-cored set Discrete points show individual measured decayrates curves show fits to the floor level using equation (4)

percentage shift in mode frequency Musicians are indi-cated by stars non-musicians by plus symbols The aver-ages over all test subjects are shown by the open circles

The open square symbols show the average of the foursubjects who achieved the lowest mean levels As was sug-gested earlier within the scope of the present study themost interesting threshold to compare with serious musi-cal judgments is that achieved by the best listeners underthe most favourable conditions In the violin tests the bestfive results were presented for this purpose [1] In the testsreported here the number of subjects is sometimes ratherlow and it was decided to average the best four The nu-merical results of this best-four measure for all the 3AFCtests presented here are given in the figure captions

It can be seen that for each of the three musical inputsthe best subjects could discriminate a shift of 1 or lessInteresting patterns can be seen in the detailed results de-spite the small number of subjects that would make anyelaborate statistical analysis of limited value The generalspread of results and the two averages look very simi-lar for the single note and the Huntsuppe extract For thesingle note the musicians and non-musicians are intermin-gled while for Huntsuppe all but one of the non-musicianshas a worse result than all the musicians Perhaps non-musicians have their performance degraded by informa-

03 05 1 2 3 5 100

05

1

(a) Huntsuppe

Percentage shift

RelSTD

03 05 1 2 3 5 100

05

1

(b) Single note

Percentage shift

RelSTD

03 05 1 2 3 5 100

05

1

(c) Discordant chord

Percentage shiftRelSTD

Figure 4 Results from 3AFC studies for shifting all mode fre-quencies of the datum guitar using as input (a) the Huntsuppefragment (b) a single note (open A string) (c) the discordantchord Horizontal axis shows percentage frequency shift on alogarithmic scale vertical axis shows relative standard deviationPoints denote musicians points + denote non-musicians Opencircle average of all subjects open square average of best foursubjects The percentage shifts corresponding to the best-four av-erages are (a) 113 (b) 092 (c) 059

tional masking while experienced musicians do not withsuch an obviously musical sound sample as ldquoHuntsupperdquo

The chord reveals a different picture Four of the mu-sicians achieved very low thresholds with this sound in-put but all other subjects seem to have been seriouslydistracted by this rather non-musical sound and achievedonly rather high thresholds One of the musicians comesout worst of all in this test whereas for Huntsuppe all themusicians performed well This is a striking example ofinformational masking especially since the group of foursubjects who did well actually produced a lower averagethan in either of the other tests That is as one would hopethis signal contains deliberately a wider range of fre-quency components so that analytical listeners with suffi-cient skill and experience to take advantage of it can makevery subtle judgements down to 032 shift in the bestcase This corresponds to shifting the mode frequenciesabout 6 cents or 120 of a semitone an impressively smallamount that is not very much bigger than the threshold forpitch discrimination tasks (see eg [14])

The thresholds are of the same order as the ldquobest fiverdquothresholds in the corresponding violin tests but the de-tailed pattern seems different The choice of input had amore marked effect in the violin and to achieve the lowestthresholds a very short and consequently rather unmusi-cal sound sample was used

The best-performing 7 subjects of the first test serieswere asked to undertake two further tests in which fre-

481

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

quencies were shifted only in limited bands The twobands studied were shown in Figure 2 and this time theonly input sound tested was the single-note open A stringThe results of the tests are shown in Figure 5 in the sameformat as Figure 4 Rather unexpectedly the ldquobest fourrdquothreshold for the first band containing the strongest peaksof the datum guitar was much higher than for the sec-ond band Related tests for the violin using the so-calledldquoDuumlnnwald bandsrdquo gave thresholds around 2 for the firstthree bands (see Figure 6 of Fritz et al [1]) Band 1 in thepresent study gave a comparable result but band 2 gave alower result than was observed in any of the violin testsThe best-four JND in this case is virtually the same as forthe corresponding test when all frequencies were shiftedFurther studies might be worth conducting to add moredetail to this observation

The final pair of tests relating to the perception of bodymode changes investigated the effect of damping factorsThis time the mode frequencies were kept fixed but themodal Q factors were all scaled by a given factor Q fac-tor is the inverse of damping factor so high Q correspondsto low damping In two separate tests the Q factors wereincreased and decreased to look for any asymmetry in theperception Thirteen test subjects took these two tests 10musicians and 3 non-musicians Results are shown in Fig-ure 6 in the same format as Figs 4 and 5 except that thenumbers on the horizontal axis now indicate the scale fac-tor applied rather than the percentage change This meansthat ldquono changerdquo corresponds to the value 1 rather than 0as in the previous results which has the effect of makingthe relative standard deviations numerically smaller

An asymmetry is seen between the two tests but it doesnot significantly affect the threshold value for the bestfour listeners They achieved a value around 12 ie a20 change in Q factors in both cases However acrossthe whole set of test subjects the spread is much greaterwhen the Q factors were decreased compared to the in-creasing case Many participants reported initial difficultyin hearing the difference when Q factors were decreasedalthough once they had learned to recognise the soundchange they often achieved quite low thresholds

It was not thought worthwhile to test separate frequencybands in this case for two reasons First body damping ismost readily altered by a physical change of material andthis is likely to have an effect over a broad frequency rangeSecond it will be shown shortly that the JND for percep-tion of a change in body damping is of the same orderas the measured standard deviation among real guitars sothat the perceptual significance is probably marginal anddigging deeper into the details is likely to be of limitedimportance

42 Physical measurements

The next step is to look at physical measurements to as-sess how these thresholds of perception compare with theactual extent of variation between guitars Of course onewould never find two real instruments which differed byhaving all the body frequencies simultaneously scaled by

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(a) 150-250 Hz

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(b) 500-1000 Hz

Figure 5 Results in the format of Figure 4 from 3AFC studies forshifting all mode frequencies of the datum guitar lying in (a) theband 150ndash250 Hz and (b) the band 500ndash1000 Hz using as inputa single note (open A string) The percentage shifts correspond-ing to the best-four averages are (a) 187 (b) 153

1 12 15 2 25 30

005

01

015

02

(a)

Scale factor

RelSTD

1 12 15 2 25 30

005

01

015

02

(b)

Scale factor

RelSTD

Figure 6 Results in the format of Figure 4 from 3AFC studiesfor scaling all modal Q factors of the datum guitar (a) upwardsand (b) downwards using as input a single note (open A string)Horizontal axis shows scale factor applied by multiplication ordivision to Q factors The scale factors corresponding to the best-four averages are (a) 119 (b) 122

exactly the same factor However it is easy to visualise achange which would in principle have this effect it wouldhappen if the dimensions of the guitar were all scaledkeeping the same materials and other details Some sim-ple deductions can then be drawn by scaling analysis Ifthe main vibrating elements of the guitar body are treatedas thin flat plates deforming in bending then the naturalfrequencies scale proportional to hd2 where d is the scalefactor for dimensions in the plane and h is the scale factorfor thickness Taking 1 as a round number for the orderof magnitude of the threshold of detection for shifting fre-quencies it follows that the JNDs for dimensional scalingare approximately 1 if d and h are both scaled together(full geometric scaling) 1 again if thickness is changedkeeping the in-plane dimensions fixed and only 05 ifin-plane dimensions are scaled keeping the thickness thesame These are all very small numbers they amount tochanging in-plane dimensions no more than a few mm and

482

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

75 80 85 90 95 1000

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(a)

160 170 180 190 200 210 2200

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(b)

Figure 7 Modal frequencies and Q factors for 15 guitars forthe highest peak in the frequency bands (a) 75ndash100 Hz theldquoHelmholtz resonancerdquo (b) 150ndash250 Hz the ldquofirst top platemoderdquo Plus and cross symbols show fits without and with com-pensation for string damping respectively Diamond symbolsmark the datum guitar for the listening tests Open squares showthe means and range bars show one standard deviation aboveand below the means for the two sets of fitted data The rangesfor the compensated results can be recognised by their higher Qvalues Dashed lines near the axes show approximately 1 JNDabove and below the uncompensated means from the results ofsection 41

to changing a typical top plate thickness by a mere 25 micromReal instruments have dimensional variations which areconsiderably bigger than this especially in thickness soon this ground alone it is no surprise that guitars can sounddifferent

The simplest approach to obtaining quantitative infor-mation from admittance measurements on a set of gui-tars is to perform explicit modal analysis This can onlybe expected to give reliable results at relatively low fre-quencies pole-residue extraction whatever the methodbecomes sensitive once the modal overlap begins to ap-proach unity To illustrate the kind of values which emergefrom such an analysis Figure 7 shows results for modalfrequency and Q factor for 15 guitars for the highest peakin each of the two low-frequency bands mentioned in thediscussion of Figure 2 75ndash100 Hz the ldquoHelmholtz reso-nancerdquo and 150ndash250 Hz the ldquofirst top plate moderdquo

A complicating factor needs to be noted The ad-mittance measurements are made with the strings welldamped because the purpose is to obtain informationabout the body behaviour without the effect of the stringsHowever the damped strings remove some energy fromthe body vibration and this will tend to reduce the mea-sured Q factors This effect can be compensated at leastapproximately The damped strings will act essentially assemi-infinite strings and will thus present an impedance tothe bridge which is real and if the strings are all assumedto be attached at the same point equal to the sum of thestring impedances given by Z =

radicTm From the data for

nylon strings in Table I the sum of these impedances is212 Nsm To compensate the inverse of Y (ω) is takenthis real quantity is subtracted then the result re-invertedto give a new admittance This approach can be expectedto work well at higher frequencies but it is hard to be sureif the strings are well enough damped at low frequenciesto justify the semi-infinite approximation

Figure 7 shows results before and after this compensa-tion For each case the mean and scatter are indicated bylines showing one standard deviation above and below themean for frequency and Q Interestingly the standard de-viations have a similar pattern for both cases The relativestandard deviations for frequency are 55 and 6 for thetwo cases and those for Q factor about 33 in both casesbefore compensation

To obtain good comparative information about modalfrequencies over a broader bandwidth is more challeng-ing It is not possible to identify the ldquosamerdquo mode in dif-ferent guitars except at very low frequencies sensitivityto small structural changes increases with frequency as isfamiliar in other vibration problems (see eg [15]) Formode frequencies the threshold for audible changes is sosmall that it is hard to obtain any measure which is goodenough scaling frequencies by 1 means that one guitarbody might have 101 modes in a frequency range whereanother guitar had 100 and there is no known analysismethod which can give robust discrimination at this levelAny measure which can be computed shows much biggervariability than this and there is always a strong suspicionthat it is revealing the vagaries of the analysis method morethan the real differences between guitars

Things are a little more promising in the case of damp-ing estimation over a wide frequency range Taking theinverse FFT of an input admittance gives an impulse re-sponse function of the guitar body and this can be pro-cessed using time-frequency analysis to give a sonogramrepresentation In each frequency band a best-fitting ex-ponential decay can then be obtained Combining resultsfrom several different choices of the timefrequency res-olution tradeoff improves the robustness of the resultsThe process is described in more detail elsewhere [16]Some results are shown in Figure 8 compensation forstring damping has been included In Figure 8a the discretepoints represent for the datum guitar frequencydecaycombinations which satisfied an assumed criterion forgood fit quality For compatibility with other plots decay

483

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70

80

90

100

Frequency

Qfactor

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100

120

Frequency

Qfactor

(b)

30 35 40 45 50 55 60 6515

20

25

30

35

40

Mean Q

Relativestandard

deviation(

)

(c)

Figure 8 Damping trends for guitar bodies determined fromtime-frequency analysis (a) individual measurements of fre-quency and equivalent Q factor (circles) and the band averageof these (solid line) for the datum guitar (b) the solid lines asin (a) for the set of 15 guitars (c) the mean and standard devia-tion of the results from (b) across the frequency range with rangebars indicating one standard deviation above and below the meanon each axis The circle marks the datum guitar for the listeningtests

rate is represented as an equivalent Q factor The continu-ous line is simply a moving band average of these discretepoints

Figure 8b then shows the band-averaged lines for the 15guitars in the test set Finally Figure 8c shows a scatterplot of the mean and standard deviation of these band-averaged lines for the 15 guitars As in Figure 7 thelarge cross represents one standard deviation above and

2 3 5 8 100

01

02

03

04 (a)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (b)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (c)

Scale factor

RelSTD

Figure 9 Results in the format of Figure 4 from 3AFC studiesfor changing the damping parameters of the 5th string based ondatum values for the new steel strings Case (a) varying ηB testnote E2 (b) varying ηB test note C3 (c) varying ηF test noteC3 The scale factors corresponding to the best-four averages are(a) 299 (b) 313 (c) 241

below the mean on each axis The main result is thatthis frequency-averaged measure of damping has a rela-tive standard deviation of the order of 20 comparable tothe JND found earlier for perception of damping changesThis suggests that the extremes of mean damping in themeasured set of guitars are sufficiently different that theeffect should be audible but the results do not lead oneto expect that this will be a very strong component of theperception of tonal differences among guitars

5 Results for properties of the strings

The final set of JND experiments relates to changes instring properties including those properties that changedas the string aged one objective was to find out whichof those changes are likely to contribute most to the per-ception of ageing Using as datum the properties of thenew steel strings listed in Table I changes to the dampingproperties of the 5th string were investigated using single-note test sounds These 3AFC tests were completed by 11musicians and 5 non-musicians One test was also con-ducted relating to changes in the bending stiffness of thisstring it gave results in line with those of Jaumlrvelaumlinen etal [7] and the details are not reproduced here The actualbending stiffness of this string was found to be close tothe threshold for detecting an increase in bending stiffnesswhen starting from a much lower level In the aged stringthe bending stiffness only increased by a factor 16 (seeTable I) If that change is audible at all the effect is likelyto be small

The results for the damping tests are shown in Figure 9in a similar format to earlier 3AFC results As noted ear-lier the perceptual effects of changing ηA for this string

484

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

were found to be so weak that it was not possible to con-duct a meaningful test However both ηB and ηF gave clearresults The parameter ηB associated with bending stiff-ness and mainly affecting higher frequencies was testedusing two different notes ldquoplayedrdquo on the same string Theresults are shown in Figures 9ab and show a good levelof consistency between the two notes The spread over-all mean and best-four mean are all very similar for bothcases The best listeners need about a factor 3 increase inthis parameter to perceive the change Note from Table Ithat in the aged string this parameter increased from 0004to 0029 that change should be very clearly audible

Figure 9c shows corresponding results for ηF All sub-jects gave a somewhat lower threshold for this parametercompared to ηB and the very best listener achieved a JNDof less than a factor 2 In the aged string ηF increased inthe ratio 1711 for this string from Table I This effect isprobably not audible to most listeners It can be concludedthat the factor most likely to account for the difference insound of the aged string is ηB

6 Conclusions

Psychoacoustical experiments have been reported to estab-lish JND levels for the perception of changes to variousparameters of a model of synthesised guitar sound [3 4]These investigations parallel earlier work exploring relatedJNDs for violin sounds [1] The parameters investigatedhere relate to the behaviour of the guitar body (modal fre-quencies and damping factors) and of the strings (bendingstiffness and damping model parameters) All tests werecarried out using a three-alternative forced-choice method-ology It is argued that for application to musical judge-ments which can be very subtle the most interesting per-ceptual thresholds are those attainable by the best listenersunder the most favourable conditions To give an estimateof these the results for the best four subjects in each testwere averaged

Results of physical measurements have also been re-ported to give an indication of the spread of each parame-ter among real guitar bodies or strings A comparison withthe psychoacoustical results gives a first impression ofwhich quantities might be responsible for the biggest per-ceptual differences between instruments By far the mostsensitive parameter of all those tested relates to shiftingthe frequencies of body modes The best four listenerscould perceive a 1 shift when all frequencies were scaledtogether a small change compared to the actual varia-tion among guitars This performance was only degradedslightly when the shifted frequencies were band-limited intwo different bands tested The results were of the sameorder as the corresponding JNDs found for violin sounds[1] if anything the JNDs seem to be somewhat lower forthe guitar sounds perhaps because the fully-coupled syn-thesis method gives the listener additional auditory cuesassociated with transient decay rates of the string over-tones Body mode damping has to be changed by about20 for perception according to the definition used hereActual variations of modal damping exceed this level but

not by a large factor so body damping is probably quite aminor contributor to tonal perception

String damping parameters are even less sensitive theyhave to be changed by a factor of the order of 3 How-ever this large factor does not mean that the effects arenever heard Measurements have been presented on stringsof different materials and on steel-cored strings in newand ldquoagedrdquo conditions The damping parameters vary be-tween these cases by factors significantly bigger than theJNDs The results strongly indicate that the most importantparameter for the perception of the degradation of soundin well-used strings is ηB the damping factor associatedwith the bending stiffness of the string This parameterincreases greatly with age of strings probably from thedamping effect of dirt and corrosion products accumulat-ing between the windings of the lower strings of the guitar[13] Strings without over-winding do not deteriorate insound to anywhere near the same extent

The correct interpretation of the high sensitivity to shift-ing body frequencies is not obvious The actual change in-vestigated here moving all frequencies in step is very arti-ficial Listeners might be responding to the actual frequen-cies or perhaps to the change in modal density (the in-verse of the mean spacing between resonant frequencies)or perhaps to the shifting amplitude envelope of the bodyresponse Modal amplitudes were not changed when thefrequencies were scaled so the amplitude pattern movesalong the frequency axis with the scaling It would re-quire further testing to establish whether one of these fac-tors dominates or whether all contribute significantly Suchtesting would be a good target for future research to bringthe results closer to the detailed concerns of instrumentmakers

AcknowledgementThe authors thank Brian Moore Ian Cross and Alan Black-well for many helpful discussions on this work and MartinWoodhouse Stewart French Ian Cross and James West-brook for providing access to guitars for testing Twoanonymous reviewers are thanked for constructive sugges-tions

References

[1] C Fritz I Cross B C J Moore J Woodhouse Perceptualthresholds for detecting modifications applied to the acous-tical properties of a violin J Acoust Soc Am 122 (2007)3640ndash3650

[2] J Woodhouse P Galluzzo The bowed string as we know ittoday Acta Acustica united with Acustica 90 (2004) 579ndash589

[3] J Woodhouse On the synthesis of guitar plucks ActaAcustica united with Acustica 90 (2004) 928ndash944

[4] J Woodhouse Plucked guitar transients comparison ofmeasurements and synthesis Acta Acustica united withAcustica 90 (2004) 945ndash965

[5] H A K Wright The acoustics and psychoacoustics of theguitar PhD Thesis Cardiff University 1996

[6] H Jaumlrvelaumlinen V Vaumllimaumlki M Karjalainen Audibility ofthe timbral effects of inharmonicity in stringed instrumenttones Acoustics Research Letters Online 2 (2001) 79ndash84

485

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

possible to obtain a meaningful JND for changes to itsvalue A general comment should be noted all the teststo be reported here were constrained by the time limitsimposed by the format of undergraduate projects In somecases there is no doubt that more could have been learnedhad more time been available but it is felt that the resultshave value even with these limitations

JND thresholds were estimated using a three-alternativeforced-choice procedure (3AFC) This is not the only pos-sible procedure that could have been used but it had al-ready been chosen for the violin studies and softwareto implement 3AFC was conveniently to hand A three-down one-up adaptive tracking rule was used that esti-mated the 79 correct point on the psychometric func-tion [10] It should be noted that this is the definition as-sumed for ldquoJNDrdquo throughout this paper it is a rather con-servative value significantly higher than that which wouldcorrespond to the 50 point on the psychometric func-tion which perhaps better matches the intuitive notion ofldquothreshold of perceptionrdquo Three sounds ndash two the samethe reference sound and one different the modified soundndash were played in a random sequence and the subject wasasked to choose which one was different The amount ofparameter modification between the stimuli in a given trialwas changed by a certain factor the step size The step sizedecreased after three correct answers and increased afterone wrong answer The test stopped after eight turnpointsA relatively large initial step size was applied until the sec-ond turnpoint was reached in order to allow rapid conver-gence toward the threshold region After the second turn-point the step size was reduced The threshold was takenas the mean of the values of the amount of modification atthe last six turnpoints The values of the two step sizes andthe starting level of parameter modification were chosendifferently for each test on the basis of preliminary trialsEach trial took no more than about 10 minutes and sub-jects had a break between tests

In all tests the reference sound was kept constant in or-der to increase discriminability subjects ldquolearntrdquo to rec-ognize the reference sound It is argued that for the pur-pose of relating to musical judgments which in practiceare made by musicians who may spend several hours everyday concentrating on subtle details of the sound of their in-strument the threshold of most interest is the one obtainedunder the most favourable conditions for discriminationIt is accepted that the test circumstances are quite remotefrom normal musical experience as seems to be inevitablein the design of any scientifically-acceptable test proce-dure and the approach adopted seems to the authors thebest that can easily be done The 3AFC software gave vi-sual feedback on rightwrong answers during the experi-ment although some subjects performed best by keepingtheir eyes shut and ignoring it Subjects did not receiveany training beforehand but if they performed erraticallyon the first run they were able to repeat the run This ap-plied only to a few subjects and they were all able to per-form the task at the second attempt The sounds were pre-sented diotically via Sennheiser HD580 headphones cho-

Figure 1 Guitar tablature for sound examples used in psy-choacoustical tests first the ldquoHuntsupperdquo extract second thestrummed chord with notes E2 B2 D3 B3 C4 A4

sen because of their diffuse field response in a relativelyquiet environment The sampling rate was 441 kHz andthe number of bits was 16

The choice of musical input for the test sounds canhave a significant influence on the JND obtained In thefirst tests on mode frequency shifting three different sam-ple sounds were tested a single note (the open A string110 Hz) a short musical fragment and a strummed ldquodis-cordant chordrdquo with notes E2 B2 D3 B3 C4 A4 selectedto give input at a wide range of frequencies The musi-cal fragment consisted of the first bar of the lute pieceldquoThe English Huntsupperdquo by John Whitfelde This andthe chord are shown in tablature notation in Figure 1 thetablature applies to a guitar in regular EADGBE tuningFor the remaining tests a single note was used for the testsound the open A string (110 Hz) for the damping factortests and C3 on the A string (131 Hz) for the string param-eter tests The total stimulus length for the single notes andthe chord was 1 s while the Huntsuppe fragment took 2 sThese are all longer than the 300 ms length that yieldedthe lowest thresholds in the violin tests and perhaps someadvantages of echoic memory are sacrificed by using theselonger stimuli However it was found that the sense of arealistic guitar sound was compromised if the duration wastoo short and it seemed more appropriate to use the longermore ldquoguitarishrdquo sounds

The test subjects were all ldquomusicalrdquo in one sense or an-other but for the purposes of the presentation of resultsthey are divided into two groups using the same criterionas in the work on the violin [1] those labelled ldquomusiciansrdquohad at least 8 years of formal training and were currentlyactive in music-making at least once per week while theothers are labelled ldquonon-musiciansrdquo The first author wasa subject in each test while all the other subjects were ofstudent age The results for the first author did not seemin any way atypical of the rest despite his relatively ad-vanced age

32 Physical measurements

Physical measurements on the body and strings of severalguitars are used in two different ways in this study Firstmeasurements are needed to calibrate the synthesis modelas has been described in detail previously [4] Second thesame range of measurements can be analysed to give es-timates of the variability in practice of the parameters forwhich JNDs will be found By comparing the actual varia-tion with the JND for perception it is hoped to give a first

478

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

80 100 200 500 1000 2000 5000-70

-60

-50

-40

-30

-20

-10

Frequency

|Y|(dB)

Figure 2 Magnitude of the input admittances of four guitarsmeasured normal to the top plate at the bridge saddle near thelowest string (E2) plotted in decibels relative to 1 mNs Heavyline flamenco guitar by Martin Woodhouse thin solid line clas-sical guitar by Michael J OrsquoLeary dashed line classical guitar byGreg Smallman dash-dot line flamenco guitar by Joseacute RamirezHorizontal lines show bands used in testing see text

estimate of the relative sensitivity and importance of dif-ferent kinds of variation in guitar construction and string-ing

Input admittance Y (ω) of the guitar body is measuredby applying a controlled force either with a miniature im-pulse hammer (PCB 086D80) or by gently pulling a lengthof fine copper wire looped around a string at the bridgesaddle until it snaps The latter method is less accuratebut has the advantage that it can be used to apply force indifferent controlled directions to give information on themodal angles In all cases the velocity response is mea-sured with a laser-Doppler vibrometer (Polytec OFV-056)directed to a spot as close as possible to the forcing pointSignals are collected at a sampling rate of 40 kHz by adigital data-logger based on a PC running Matlab witha National Instruments 6220 data acquisition card Thestrings of the guitar were damped during admittance mea-surements

Input admittances in the direction normal to the planeof the top plate have been measured on a wide range ofacoustic guitars and a representative selection of 15 areused in the present study Figure 2 shows results for fourof these all hand-made instruments of high quality Theheavy curve is the datum guitar used in the psychoacousti-cal tests a flamenco guitar by Martin Woodhouse studiedin the previous work [4] The others are a flamenco gui-tar by Jose Ramirez and classical guitars by Greg Small-man and Michael J OrsquoLeary It is immediately clear thatall these guitars although different in detail show a lotof common features All have a strong peak in the range80ndash100 Hz which is a Helmholtz resonance associatedwith the internal air cavity and the soundhole modified bythe compliance of the top and back plates All the guitarsthen show a deep antiresonance (marking the frequency atwhich the Helmholtz resonance would occur with a rigid

cavity see equation (1019) of Cremer [11]) followed bya cluster of strong peaks in the range 150ndash250 Hz Mostthen show a steady decline to a dip around 400ndash500 Hzfollowed by fluctuations around a rather steady level withgradually reducing peak amplitudes as the modal overlapbecomes significant One guitar (the OrsquoLeary) has an ad-ditional strong peak around 300 Hz interrupting the de-cline to the dip The two horizontal lines mark frequencybands that will be used in frequency-shifting tests in thenext section The first of these bands (150ndash250 Hz) coversthe range where the datum guitar has its strongest peaksThe second band (500ndash1000 Hz) covers a range where themodal overlap is becoming significant but fairly clear in-dividual peaks can still be seen

The other physical measurements used in this study areaimed at establishing the properties of strings Tension andmass per unit length are readily established but the bend-ing stiffness and damping properties require careful mea-surement Such measurements have previously been madeon a typical set of nylon strings for a classical guitar andin the course of this project corresponding measurementswere made on a set of steel strings as commonly usedon folk guitars The chosen strings were Martin ldquo8020Bronzerdquo It is a familiar experience to guitarists that thesound changes a lot between when the strings are new andwhen they are well-played so the strings studied here weremeasured in both states to give quantitative data on whatchanges in the ageing process

The bending stiffness and damping properties can be es-timated by analysis of sounds recorded from notes playedon the guitar in the usual way [4] The frequencies of thestrongest spectral components corresponding to ldquostringrdquomodes can be best-fitted to equation (2) to yield valuesof B and hence EI The associated decay rate of eachof these ldquostringrdquo modes can also be found using time-frequency analysis and this gives information about thestring damping The procedure for obtaining damping in-formation from the decay rates is a little indirect becausein most cases the decay rate is dominated by losses into theguitar body rather than by internal damping in the string

The approach adopted is to analyse the frequencies fora plucked note on every fret of a given string and combinethem all into a cloud plot of damping factor against fre-quency examples are shown in Figure 3 This shows datafor the 3rd string (G3) for the nylon string measured previ-ously and the new steel-cored string The inherent damp-ing of the string as a function of frequency determinesthe ldquofloor levelrdquo of this cloud plot Approximate values ofηA ηF ηB can be estimated by fitting eqquation (4) to thetrend of this floor level as illustrated by the curves in Fig-ure 3 The fitting process was carried out manually withsome regard to the error tolerances of the data points Notethat the high-frequency fit line for the nylon string wasinfluenced by additional data obtained by a different ap-proach not shown here see [4] for more detail It is clearfrom the figure that the nylon string has higher dampingthan the steel string The measured string properties aresummarised in Table I

479

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

Table I Properties of guitar strings (a) DrsquoAddario Pro Arteacute ldquoComposites hard tensionrdquo nylon strings (b) Martin ldquo8020 bronzerdquo steelstrings in new condition (c) as (b) but in used condition

String 1 2 3 4 5 6

Tuning note E4 B3 G3 D3 A2 E2

Frequency fn (Hz) 3296 2469 1960 1468 1100 824

Tension (a) T (N) 703 534 583 712 739 716(b) 121 135 182 185 169 147

Massunit length (a) m (gm) 038 052 090 195 361 624(b) 057 113 242 437 713 1102

Bending stiffness (a) EI (10minus6Nm2) 130 160 310 51 40 57(b) 108 336 234 255 327 667(c) 103 394 343 418 480 719

Inharmonicity (a) B (10minus6) 43 70 124 17 13 19(b) 21 58 30 32 45 106(c) 20 68 44 53 66 114

Loss coefficients (a) ηF (10minus5) 40 40 14 5 7 2(b) 7 6 11 9 11 5(c) 6 7 17 33 17 10

(a) ηB (10minus2) 24 20 20 20 25 20(b) 07 11 18 11 04 03(c) 07 11 68 41 29 38

(a) ηA (1s) 15 12 17 12 09 12(b) 25 10 12 13 12 17(c) 15 10 06 07 06 14

The differences between the nylon and steel strings innew condition are broadly as one would expect The nylonstrings have lower tensions and a different pattern of bend-ing stiffness because of the difference in construction forthe nylon set the top three strings are monofilaments whilethe lower three have multifilament cores with metal wind-ings The steel strings have monofilament cores through-out and only the top two strings are not wrapped Themonofilament nylon strings have higher internal dampingthan the steel but the difference is much less marked forthe lower three strings

The effect of age and use on the steel strings can alsobe seen Previous investigators have reported increaseddamping between new and old guitar strings [12 13] butwithout sufficient detail to allow a fit to equation (4) Itis generally accepted that the mechanism for this ageingeffect is the accumulation of dirt and corrosion productsbetween the windings of wrapped strings [13] perhapswith some additional cumulative wear of the outer wrap-pings where the strings make contact with the frets Themeasurements reported here support this picture there isvery little change in the two highest strings which are notwrapped For the wrapped strings there is a marked in-crease in damping especially in ηB which determines thedecay at the highest frequencies

Also not noted by previous authors there is a markedincrease in the effective bending stiffness of some of thestrings This seems perfectly credible since the contactconditions between the windings in the tensioned stringwill contribute to the bending stiffness and material col-

lecting between the windings will surely change those con-ditions This is consistent with the observed increase in ηB which is the loss factor associated with the bending stiff-ness term in the governing equation It may be noted thatthe bending stiffness values given here are in good gen-eral agreement with values reported by Jaumlrvelaumlinen et al[7] noting that their steel-string results match the ldquousedrdquoresults more closely than the ldquonewrdquo results Their paperdoes not report the condition of the strings as measured

4 Results for properties of the guitar body

41 Psychoacoustical tests

The first series of psychoacoustical tests concerned theshifting of all body mode frequencies by a given factorkeeping other simulation parameters fixed This modifica-tion is of course somewhat artificial but a possible physi-cal interpretation will be discussed in the next subsectionThe datum for this test was based on the modal proper-ties of the flamenco guitar by Martin Woodhouse This testwas run three times using the three different musical in-puts described above The three tests were completed by 8musicians and 5 non-musicians (according to the criteriondescribed above) The results are summarised in Figure 4The program that runs the 3AFC protocol gives as outputthe mean and standard deviation over the range defined bythe final 6 turnpoints The plots show for each test subjectthe mean on the horizontal axis and the relative standarddeviation (standard deviation divided by mean) on the ver-tical axis The numbers on the horizontal axis indicate the

480

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

100 200 500 1000 2000 500010

-4

10-3

10-2

Frequency (Hz)

Dampingfactor

100 200 500 1000 2000 500010

-4

10-3

10-2

Frequency (Hz)

Dampingfactor

Figure 3 Damping measurements on (top) a nylon guitar string(3rd string G3) and (bottom) the corresponding string from asteel-cored set Discrete points show individual measured decayrates curves show fits to the floor level using equation (4)

percentage shift in mode frequency Musicians are indi-cated by stars non-musicians by plus symbols The aver-ages over all test subjects are shown by the open circles

The open square symbols show the average of the foursubjects who achieved the lowest mean levels As was sug-gested earlier within the scope of the present study themost interesting threshold to compare with serious musi-cal judgments is that achieved by the best listeners underthe most favourable conditions In the violin tests the bestfive results were presented for this purpose [1] In the testsreported here the number of subjects is sometimes ratherlow and it was decided to average the best four The nu-merical results of this best-four measure for all the 3AFCtests presented here are given in the figure captions

It can be seen that for each of the three musical inputsthe best subjects could discriminate a shift of 1 or lessInteresting patterns can be seen in the detailed results de-spite the small number of subjects that would make anyelaborate statistical analysis of limited value The generalspread of results and the two averages look very simi-lar for the single note and the Huntsuppe extract For thesingle note the musicians and non-musicians are intermin-gled while for Huntsuppe all but one of the non-musicianshas a worse result than all the musicians Perhaps non-musicians have their performance degraded by informa-

03 05 1 2 3 5 100

05

1

(a) Huntsuppe

Percentage shift

RelSTD

03 05 1 2 3 5 100

05

1

(b) Single note

Percentage shift

RelSTD

03 05 1 2 3 5 100

05

1

(c) Discordant chord

Percentage shiftRelSTD

Figure 4 Results from 3AFC studies for shifting all mode fre-quencies of the datum guitar using as input (a) the Huntsuppefragment (b) a single note (open A string) (c) the discordantchord Horizontal axis shows percentage frequency shift on alogarithmic scale vertical axis shows relative standard deviationPoints denote musicians points + denote non-musicians Opencircle average of all subjects open square average of best foursubjects The percentage shifts corresponding to the best-four av-erages are (a) 113 (b) 092 (c) 059

tional masking while experienced musicians do not withsuch an obviously musical sound sample as ldquoHuntsupperdquo

The chord reveals a different picture Four of the mu-sicians achieved very low thresholds with this sound in-put but all other subjects seem to have been seriouslydistracted by this rather non-musical sound and achievedonly rather high thresholds One of the musicians comesout worst of all in this test whereas for Huntsuppe all themusicians performed well This is a striking example ofinformational masking especially since the group of foursubjects who did well actually produced a lower averagethan in either of the other tests That is as one would hopethis signal contains deliberately a wider range of fre-quency components so that analytical listeners with suffi-cient skill and experience to take advantage of it can makevery subtle judgements down to 032 shift in the bestcase This corresponds to shifting the mode frequenciesabout 6 cents or 120 of a semitone an impressively smallamount that is not very much bigger than the threshold forpitch discrimination tasks (see eg [14])

The thresholds are of the same order as the ldquobest fiverdquothresholds in the corresponding violin tests but the de-tailed pattern seems different The choice of input had amore marked effect in the violin and to achieve the lowestthresholds a very short and consequently rather unmusi-cal sound sample was used

The best-performing 7 subjects of the first test serieswere asked to undertake two further tests in which fre-

481

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

quencies were shifted only in limited bands The twobands studied were shown in Figure 2 and this time theonly input sound tested was the single-note open A stringThe results of the tests are shown in Figure 5 in the sameformat as Figure 4 Rather unexpectedly the ldquobest fourrdquothreshold for the first band containing the strongest peaksof the datum guitar was much higher than for the sec-ond band Related tests for the violin using the so-calledldquoDuumlnnwald bandsrdquo gave thresholds around 2 for the firstthree bands (see Figure 6 of Fritz et al [1]) Band 1 in thepresent study gave a comparable result but band 2 gave alower result than was observed in any of the violin testsThe best-four JND in this case is virtually the same as forthe corresponding test when all frequencies were shiftedFurther studies might be worth conducting to add moredetail to this observation

The final pair of tests relating to the perception of bodymode changes investigated the effect of damping factorsThis time the mode frequencies were kept fixed but themodal Q factors were all scaled by a given factor Q fac-tor is the inverse of damping factor so high Q correspondsto low damping In two separate tests the Q factors wereincreased and decreased to look for any asymmetry in theperception Thirteen test subjects took these two tests 10musicians and 3 non-musicians Results are shown in Fig-ure 6 in the same format as Figs 4 and 5 except that thenumbers on the horizontal axis now indicate the scale fac-tor applied rather than the percentage change This meansthat ldquono changerdquo corresponds to the value 1 rather than 0as in the previous results which has the effect of makingthe relative standard deviations numerically smaller

An asymmetry is seen between the two tests but it doesnot significantly affect the threshold value for the bestfour listeners They achieved a value around 12 ie a20 change in Q factors in both cases However acrossthe whole set of test subjects the spread is much greaterwhen the Q factors were decreased compared to the in-creasing case Many participants reported initial difficultyin hearing the difference when Q factors were decreasedalthough once they had learned to recognise the soundchange they often achieved quite low thresholds

It was not thought worthwhile to test separate frequencybands in this case for two reasons First body damping ismost readily altered by a physical change of material andthis is likely to have an effect over a broad frequency rangeSecond it will be shown shortly that the JND for percep-tion of a change in body damping is of the same orderas the measured standard deviation among real guitars sothat the perceptual significance is probably marginal anddigging deeper into the details is likely to be of limitedimportance

42 Physical measurements

The next step is to look at physical measurements to as-sess how these thresholds of perception compare with theactual extent of variation between guitars Of course onewould never find two real instruments which differed byhaving all the body frequencies simultaneously scaled by

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(a) 150-250 Hz

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(b) 500-1000 Hz

Figure 5 Results in the format of Figure 4 from 3AFC studies forshifting all mode frequencies of the datum guitar lying in (a) theband 150ndash250 Hz and (b) the band 500ndash1000 Hz using as inputa single note (open A string) The percentage shifts correspond-ing to the best-four averages are (a) 187 (b) 153

1 12 15 2 25 30

005

01

015

02

(a)

Scale factor

RelSTD

1 12 15 2 25 30

005

01

015

02

(b)

Scale factor

RelSTD

Figure 6 Results in the format of Figure 4 from 3AFC studiesfor scaling all modal Q factors of the datum guitar (a) upwardsand (b) downwards using as input a single note (open A string)Horizontal axis shows scale factor applied by multiplication ordivision to Q factors The scale factors corresponding to the best-four averages are (a) 119 (b) 122

exactly the same factor However it is easy to visualise achange which would in principle have this effect it wouldhappen if the dimensions of the guitar were all scaledkeeping the same materials and other details Some sim-ple deductions can then be drawn by scaling analysis Ifthe main vibrating elements of the guitar body are treatedas thin flat plates deforming in bending then the naturalfrequencies scale proportional to hd2 where d is the scalefactor for dimensions in the plane and h is the scale factorfor thickness Taking 1 as a round number for the orderof magnitude of the threshold of detection for shifting fre-quencies it follows that the JNDs for dimensional scalingare approximately 1 if d and h are both scaled together(full geometric scaling) 1 again if thickness is changedkeeping the in-plane dimensions fixed and only 05 ifin-plane dimensions are scaled keeping the thickness thesame These are all very small numbers they amount tochanging in-plane dimensions no more than a few mm and

482

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

75 80 85 90 95 1000

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(a)

160 170 180 190 200 210 2200

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(b)

Figure 7 Modal frequencies and Q factors for 15 guitars forthe highest peak in the frequency bands (a) 75ndash100 Hz theldquoHelmholtz resonancerdquo (b) 150ndash250 Hz the ldquofirst top platemoderdquo Plus and cross symbols show fits without and with com-pensation for string damping respectively Diamond symbolsmark the datum guitar for the listening tests Open squares showthe means and range bars show one standard deviation aboveand below the means for the two sets of fitted data The rangesfor the compensated results can be recognised by their higher Qvalues Dashed lines near the axes show approximately 1 JNDabove and below the uncompensated means from the results ofsection 41

to changing a typical top plate thickness by a mere 25 micromReal instruments have dimensional variations which areconsiderably bigger than this especially in thickness soon this ground alone it is no surprise that guitars can sounddifferent

The simplest approach to obtaining quantitative infor-mation from admittance measurements on a set of gui-tars is to perform explicit modal analysis This can onlybe expected to give reliable results at relatively low fre-quencies pole-residue extraction whatever the methodbecomes sensitive once the modal overlap begins to ap-proach unity To illustrate the kind of values which emergefrom such an analysis Figure 7 shows results for modalfrequency and Q factor for 15 guitars for the highest peakin each of the two low-frequency bands mentioned in thediscussion of Figure 2 75ndash100 Hz the ldquoHelmholtz reso-nancerdquo and 150ndash250 Hz the ldquofirst top plate moderdquo

A complicating factor needs to be noted The ad-mittance measurements are made with the strings welldamped because the purpose is to obtain informationabout the body behaviour without the effect of the stringsHowever the damped strings remove some energy fromthe body vibration and this will tend to reduce the mea-sured Q factors This effect can be compensated at leastapproximately The damped strings will act essentially assemi-infinite strings and will thus present an impedance tothe bridge which is real and if the strings are all assumedto be attached at the same point equal to the sum of thestring impedances given by Z =

radicTm From the data for

nylon strings in Table I the sum of these impedances is212 Nsm To compensate the inverse of Y (ω) is takenthis real quantity is subtracted then the result re-invertedto give a new admittance This approach can be expectedto work well at higher frequencies but it is hard to be sureif the strings are well enough damped at low frequenciesto justify the semi-infinite approximation

Figure 7 shows results before and after this compensa-tion For each case the mean and scatter are indicated bylines showing one standard deviation above and below themean for frequency and Q Interestingly the standard de-viations have a similar pattern for both cases The relativestandard deviations for frequency are 55 and 6 for thetwo cases and those for Q factor about 33 in both casesbefore compensation

To obtain good comparative information about modalfrequencies over a broader bandwidth is more challeng-ing It is not possible to identify the ldquosamerdquo mode in dif-ferent guitars except at very low frequencies sensitivityto small structural changes increases with frequency as isfamiliar in other vibration problems (see eg [15]) Formode frequencies the threshold for audible changes is sosmall that it is hard to obtain any measure which is goodenough scaling frequencies by 1 means that one guitarbody might have 101 modes in a frequency range whereanother guitar had 100 and there is no known analysismethod which can give robust discrimination at this levelAny measure which can be computed shows much biggervariability than this and there is always a strong suspicionthat it is revealing the vagaries of the analysis method morethan the real differences between guitars

Things are a little more promising in the case of damp-ing estimation over a wide frequency range Taking theinverse FFT of an input admittance gives an impulse re-sponse function of the guitar body and this can be pro-cessed using time-frequency analysis to give a sonogramrepresentation In each frequency band a best-fitting ex-ponential decay can then be obtained Combining resultsfrom several different choices of the timefrequency res-olution tradeoff improves the robustness of the resultsThe process is described in more detail elsewhere [16]Some results are shown in Figure 8 compensation forstring damping has been included In Figure 8a the discretepoints represent for the datum guitar frequencydecaycombinations which satisfied an assumed criterion forgood fit quality For compatibility with other plots decay

483

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70

80

90

100

Frequency

Qfactor

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100

120

Frequency

Qfactor

(b)

30 35 40 45 50 55 60 6515

20

25

30

35

40

Mean Q

Relativestandard

deviation(

)

(c)

Figure 8 Damping trends for guitar bodies determined fromtime-frequency analysis (a) individual measurements of fre-quency and equivalent Q factor (circles) and the band averageof these (solid line) for the datum guitar (b) the solid lines asin (a) for the set of 15 guitars (c) the mean and standard devia-tion of the results from (b) across the frequency range with rangebars indicating one standard deviation above and below the meanon each axis The circle marks the datum guitar for the listeningtests

rate is represented as an equivalent Q factor The continu-ous line is simply a moving band average of these discretepoints

Figure 8b then shows the band-averaged lines for the 15guitars in the test set Finally Figure 8c shows a scatterplot of the mean and standard deviation of these band-averaged lines for the 15 guitars As in Figure 7 thelarge cross represents one standard deviation above and

2 3 5 8 100

01

02

03

04 (a)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (b)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (c)

Scale factor

RelSTD

Figure 9 Results in the format of Figure 4 from 3AFC studiesfor changing the damping parameters of the 5th string based ondatum values for the new steel strings Case (a) varying ηB testnote E2 (b) varying ηB test note C3 (c) varying ηF test noteC3 The scale factors corresponding to the best-four averages are(a) 299 (b) 313 (c) 241

below the mean on each axis The main result is thatthis frequency-averaged measure of damping has a rela-tive standard deviation of the order of 20 comparable tothe JND found earlier for perception of damping changesThis suggests that the extremes of mean damping in themeasured set of guitars are sufficiently different that theeffect should be audible but the results do not lead oneto expect that this will be a very strong component of theperception of tonal differences among guitars

5 Results for properties of the strings

The final set of JND experiments relates to changes instring properties including those properties that changedas the string aged one objective was to find out whichof those changes are likely to contribute most to the per-ception of ageing Using as datum the properties of thenew steel strings listed in Table I changes to the dampingproperties of the 5th string were investigated using single-note test sounds These 3AFC tests were completed by 11musicians and 5 non-musicians One test was also con-ducted relating to changes in the bending stiffness of thisstring it gave results in line with those of Jaumlrvelaumlinen etal [7] and the details are not reproduced here The actualbending stiffness of this string was found to be close tothe threshold for detecting an increase in bending stiffnesswhen starting from a much lower level In the aged stringthe bending stiffness only increased by a factor 16 (seeTable I) If that change is audible at all the effect is likelyto be small

The results for the damping tests are shown in Figure 9in a similar format to earlier 3AFC results As noted ear-lier the perceptual effects of changing ηA for this string

484

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

were found to be so weak that it was not possible to con-duct a meaningful test However both ηB and ηF gave clearresults The parameter ηB associated with bending stiff-ness and mainly affecting higher frequencies was testedusing two different notes ldquoplayedrdquo on the same string Theresults are shown in Figures 9ab and show a good levelof consistency between the two notes The spread over-all mean and best-four mean are all very similar for bothcases The best listeners need about a factor 3 increase inthis parameter to perceive the change Note from Table Ithat in the aged string this parameter increased from 0004to 0029 that change should be very clearly audible

Figure 9c shows corresponding results for ηF All sub-jects gave a somewhat lower threshold for this parametercompared to ηB and the very best listener achieved a JNDof less than a factor 2 In the aged string ηF increased inthe ratio 1711 for this string from Table I This effect isprobably not audible to most listeners It can be concludedthat the factor most likely to account for the difference insound of the aged string is ηB

6 Conclusions

Psychoacoustical experiments have been reported to estab-lish JND levels for the perception of changes to variousparameters of a model of synthesised guitar sound [3 4]These investigations parallel earlier work exploring relatedJNDs for violin sounds [1] The parameters investigatedhere relate to the behaviour of the guitar body (modal fre-quencies and damping factors) and of the strings (bendingstiffness and damping model parameters) All tests werecarried out using a three-alternative forced-choice method-ology It is argued that for application to musical judge-ments which can be very subtle the most interesting per-ceptual thresholds are those attainable by the best listenersunder the most favourable conditions To give an estimateof these the results for the best four subjects in each testwere averaged

Results of physical measurements have also been re-ported to give an indication of the spread of each parame-ter among real guitar bodies or strings A comparison withthe psychoacoustical results gives a first impression ofwhich quantities might be responsible for the biggest per-ceptual differences between instruments By far the mostsensitive parameter of all those tested relates to shiftingthe frequencies of body modes The best four listenerscould perceive a 1 shift when all frequencies were scaledtogether a small change compared to the actual varia-tion among guitars This performance was only degradedslightly when the shifted frequencies were band-limited intwo different bands tested The results were of the sameorder as the corresponding JNDs found for violin sounds[1] if anything the JNDs seem to be somewhat lower forthe guitar sounds perhaps because the fully-coupled syn-thesis method gives the listener additional auditory cuesassociated with transient decay rates of the string over-tones Body mode damping has to be changed by about20 for perception according to the definition used hereActual variations of modal damping exceed this level but

not by a large factor so body damping is probably quite aminor contributor to tonal perception

String damping parameters are even less sensitive theyhave to be changed by a factor of the order of 3 How-ever this large factor does not mean that the effects arenever heard Measurements have been presented on stringsof different materials and on steel-cored strings in newand ldquoagedrdquo conditions The damping parameters vary be-tween these cases by factors significantly bigger than theJNDs The results strongly indicate that the most importantparameter for the perception of the degradation of soundin well-used strings is ηB the damping factor associatedwith the bending stiffness of the string This parameterincreases greatly with age of strings probably from thedamping effect of dirt and corrosion products accumulat-ing between the windings of the lower strings of the guitar[13] Strings without over-winding do not deteriorate insound to anywhere near the same extent

The correct interpretation of the high sensitivity to shift-ing body frequencies is not obvious The actual change in-vestigated here moving all frequencies in step is very arti-ficial Listeners might be responding to the actual frequen-cies or perhaps to the change in modal density (the in-verse of the mean spacing between resonant frequencies)or perhaps to the shifting amplitude envelope of the bodyresponse Modal amplitudes were not changed when thefrequencies were scaled so the amplitude pattern movesalong the frequency axis with the scaling It would re-quire further testing to establish whether one of these fac-tors dominates or whether all contribute significantly Suchtesting would be a good target for future research to bringthe results closer to the detailed concerns of instrumentmakers

AcknowledgementThe authors thank Brian Moore Ian Cross and Alan Black-well for many helpful discussions on this work and MartinWoodhouse Stewart French Ian Cross and James West-brook for providing access to guitars for testing Twoanonymous reviewers are thanked for constructive sugges-tions

References

[1] C Fritz I Cross B C J Moore J Woodhouse Perceptualthresholds for detecting modifications applied to the acous-tical properties of a violin J Acoust Soc Am 122 (2007)3640ndash3650

[2] J Woodhouse P Galluzzo The bowed string as we know ittoday Acta Acustica united with Acustica 90 (2004) 579ndash589

[3] J Woodhouse On the synthesis of guitar plucks ActaAcustica united with Acustica 90 (2004) 928ndash944

[4] J Woodhouse Plucked guitar transients comparison ofmeasurements and synthesis Acta Acustica united withAcustica 90 (2004) 945ndash965

[5] H A K Wright The acoustics and psychoacoustics of theguitar PhD Thesis Cardiff University 1996

[6] H Jaumlrvelaumlinen V Vaumllimaumlki M Karjalainen Audibility ofthe timbral effects of inharmonicity in stringed instrumenttones Acoustics Research Letters Online 2 (2001) 79ndash84

485

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

80 100 200 500 1000 2000 5000-70

-60

-50

-40

-30

-20

-10

Frequency

|Y|(dB)

Figure 2 Magnitude of the input admittances of four guitarsmeasured normal to the top plate at the bridge saddle near thelowest string (E2) plotted in decibels relative to 1 mNs Heavyline flamenco guitar by Martin Woodhouse thin solid line clas-sical guitar by Michael J OrsquoLeary dashed line classical guitar byGreg Smallman dash-dot line flamenco guitar by Joseacute RamirezHorizontal lines show bands used in testing see text

estimate of the relative sensitivity and importance of dif-ferent kinds of variation in guitar construction and string-ing

Input admittance Y (ω) of the guitar body is measuredby applying a controlled force either with a miniature im-pulse hammer (PCB 086D80) or by gently pulling a lengthof fine copper wire looped around a string at the bridgesaddle until it snaps The latter method is less accuratebut has the advantage that it can be used to apply force indifferent controlled directions to give information on themodal angles In all cases the velocity response is mea-sured with a laser-Doppler vibrometer (Polytec OFV-056)directed to a spot as close as possible to the forcing pointSignals are collected at a sampling rate of 40 kHz by adigital data-logger based on a PC running Matlab witha National Instruments 6220 data acquisition card Thestrings of the guitar were damped during admittance mea-surements

Input admittances in the direction normal to the planeof the top plate have been measured on a wide range ofacoustic guitars and a representative selection of 15 areused in the present study Figure 2 shows results for fourof these all hand-made instruments of high quality Theheavy curve is the datum guitar used in the psychoacousti-cal tests a flamenco guitar by Martin Woodhouse studiedin the previous work [4] The others are a flamenco gui-tar by Jose Ramirez and classical guitars by Greg Small-man and Michael J OrsquoLeary It is immediately clear thatall these guitars although different in detail show a lotof common features All have a strong peak in the range80ndash100 Hz which is a Helmholtz resonance associatedwith the internal air cavity and the soundhole modified bythe compliance of the top and back plates All the guitarsthen show a deep antiresonance (marking the frequency atwhich the Helmholtz resonance would occur with a rigid

cavity see equation (1019) of Cremer [11]) followed bya cluster of strong peaks in the range 150ndash250 Hz Mostthen show a steady decline to a dip around 400ndash500 Hzfollowed by fluctuations around a rather steady level withgradually reducing peak amplitudes as the modal overlapbecomes significant One guitar (the OrsquoLeary) has an ad-ditional strong peak around 300 Hz interrupting the de-cline to the dip The two horizontal lines mark frequencybands that will be used in frequency-shifting tests in thenext section The first of these bands (150ndash250 Hz) coversthe range where the datum guitar has its strongest peaksThe second band (500ndash1000 Hz) covers a range where themodal overlap is becoming significant but fairly clear in-dividual peaks can still be seen

The other physical measurements used in this study areaimed at establishing the properties of strings Tension andmass per unit length are readily established but the bend-ing stiffness and damping properties require careful mea-surement Such measurements have previously been madeon a typical set of nylon strings for a classical guitar andin the course of this project corresponding measurementswere made on a set of steel strings as commonly usedon folk guitars The chosen strings were Martin ldquo8020Bronzerdquo It is a familiar experience to guitarists that thesound changes a lot between when the strings are new andwhen they are well-played so the strings studied here weremeasured in both states to give quantitative data on whatchanges in the ageing process

The bending stiffness and damping properties can be es-timated by analysis of sounds recorded from notes playedon the guitar in the usual way [4] The frequencies of thestrongest spectral components corresponding to ldquostringrdquomodes can be best-fitted to equation (2) to yield valuesof B and hence EI The associated decay rate of eachof these ldquostringrdquo modes can also be found using time-frequency analysis and this gives information about thestring damping The procedure for obtaining damping in-formation from the decay rates is a little indirect becausein most cases the decay rate is dominated by losses into theguitar body rather than by internal damping in the string

The approach adopted is to analyse the frequencies fora plucked note on every fret of a given string and combinethem all into a cloud plot of damping factor against fre-quency examples are shown in Figure 3 This shows datafor the 3rd string (G3) for the nylon string measured previ-ously and the new steel-cored string The inherent damp-ing of the string as a function of frequency determinesthe ldquofloor levelrdquo of this cloud plot Approximate values ofηA ηF ηB can be estimated by fitting eqquation (4) to thetrend of this floor level as illustrated by the curves in Fig-ure 3 The fitting process was carried out manually withsome regard to the error tolerances of the data points Notethat the high-frequency fit line for the nylon string wasinfluenced by additional data obtained by a different ap-proach not shown here see [4] for more detail It is clearfrom the figure that the nylon string has higher dampingthan the steel string The measured string properties aresummarised in Table I

479

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

Table I Properties of guitar strings (a) DrsquoAddario Pro Arteacute ldquoComposites hard tensionrdquo nylon strings (b) Martin ldquo8020 bronzerdquo steelstrings in new condition (c) as (b) but in used condition

String 1 2 3 4 5 6

Tuning note E4 B3 G3 D3 A2 E2

Frequency fn (Hz) 3296 2469 1960 1468 1100 824

Tension (a) T (N) 703 534 583 712 739 716(b) 121 135 182 185 169 147

Massunit length (a) m (gm) 038 052 090 195 361 624(b) 057 113 242 437 713 1102

Bending stiffness (a) EI (10minus6Nm2) 130 160 310 51 40 57(b) 108 336 234 255 327 667(c) 103 394 343 418 480 719

Inharmonicity (a) B (10minus6) 43 70 124 17 13 19(b) 21 58 30 32 45 106(c) 20 68 44 53 66 114

Loss coefficients (a) ηF (10minus5) 40 40 14 5 7 2(b) 7 6 11 9 11 5(c) 6 7 17 33 17 10

(a) ηB (10minus2) 24 20 20 20 25 20(b) 07 11 18 11 04 03(c) 07 11 68 41 29 38

(a) ηA (1s) 15 12 17 12 09 12(b) 25 10 12 13 12 17(c) 15 10 06 07 06 14

The differences between the nylon and steel strings innew condition are broadly as one would expect The nylonstrings have lower tensions and a different pattern of bend-ing stiffness because of the difference in construction forthe nylon set the top three strings are monofilaments whilethe lower three have multifilament cores with metal wind-ings The steel strings have monofilament cores through-out and only the top two strings are not wrapped Themonofilament nylon strings have higher internal dampingthan the steel but the difference is much less marked forthe lower three strings

The effect of age and use on the steel strings can alsobe seen Previous investigators have reported increaseddamping between new and old guitar strings [12 13] butwithout sufficient detail to allow a fit to equation (4) Itis generally accepted that the mechanism for this ageingeffect is the accumulation of dirt and corrosion productsbetween the windings of wrapped strings [13] perhapswith some additional cumulative wear of the outer wrap-pings where the strings make contact with the frets Themeasurements reported here support this picture there isvery little change in the two highest strings which are notwrapped For the wrapped strings there is a marked in-crease in damping especially in ηB which determines thedecay at the highest frequencies

Also not noted by previous authors there is a markedincrease in the effective bending stiffness of some of thestrings This seems perfectly credible since the contactconditions between the windings in the tensioned stringwill contribute to the bending stiffness and material col-

lecting between the windings will surely change those con-ditions This is consistent with the observed increase in ηB which is the loss factor associated with the bending stiff-ness term in the governing equation It may be noted thatthe bending stiffness values given here are in good gen-eral agreement with values reported by Jaumlrvelaumlinen et al[7] noting that their steel-string results match the ldquousedrdquoresults more closely than the ldquonewrdquo results Their paperdoes not report the condition of the strings as measured

4 Results for properties of the guitar body

41 Psychoacoustical tests

The first series of psychoacoustical tests concerned theshifting of all body mode frequencies by a given factorkeeping other simulation parameters fixed This modifica-tion is of course somewhat artificial but a possible physi-cal interpretation will be discussed in the next subsectionThe datum for this test was based on the modal proper-ties of the flamenco guitar by Martin Woodhouse This testwas run three times using the three different musical in-puts described above The three tests were completed by 8musicians and 5 non-musicians (according to the criteriondescribed above) The results are summarised in Figure 4The program that runs the 3AFC protocol gives as outputthe mean and standard deviation over the range defined bythe final 6 turnpoints The plots show for each test subjectthe mean on the horizontal axis and the relative standarddeviation (standard deviation divided by mean) on the ver-tical axis The numbers on the horizontal axis indicate the

480

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

100 200 500 1000 2000 500010

-4

10-3

10-2

Frequency (Hz)

Dampingfactor

100 200 500 1000 2000 500010

-4

10-3

10-2

Frequency (Hz)

Dampingfactor

Figure 3 Damping measurements on (top) a nylon guitar string(3rd string G3) and (bottom) the corresponding string from asteel-cored set Discrete points show individual measured decayrates curves show fits to the floor level using equation (4)

percentage shift in mode frequency Musicians are indi-cated by stars non-musicians by plus symbols The aver-ages over all test subjects are shown by the open circles

The open square symbols show the average of the foursubjects who achieved the lowest mean levels As was sug-gested earlier within the scope of the present study themost interesting threshold to compare with serious musi-cal judgments is that achieved by the best listeners underthe most favourable conditions In the violin tests the bestfive results were presented for this purpose [1] In the testsreported here the number of subjects is sometimes ratherlow and it was decided to average the best four The nu-merical results of this best-four measure for all the 3AFCtests presented here are given in the figure captions

It can be seen that for each of the three musical inputsthe best subjects could discriminate a shift of 1 or lessInteresting patterns can be seen in the detailed results de-spite the small number of subjects that would make anyelaborate statistical analysis of limited value The generalspread of results and the two averages look very simi-lar for the single note and the Huntsuppe extract For thesingle note the musicians and non-musicians are intermin-gled while for Huntsuppe all but one of the non-musicianshas a worse result than all the musicians Perhaps non-musicians have their performance degraded by informa-

03 05 1 2 3 5 100

05

1

(a) Huntsuppe

Percentage shift

RelSTD

03 05 1 2 3 5 100

05

1

(b) Single note

Percentage shift

RelSTD

03 05 1 2 3 5 100

05

1

(c) Discordant chord

Percentage shiftRelSTD

Figure 4 Results from 3AFC studies for shifting all mode fre-quencies of the datum guitar using as input (a) the Huntsuppefragment (b) a single note (open A string) (c) the discordantchord Horizontal axis shows percentage frequency shift on alogarithmic scale vertical axis shows relative standard deviationPoints denote musicians points + denote non-musicians Opencircle average of all subjects open square average of best foursubjects The percentage shifts corresponding to the best-four av-erages are (a) 113 (b) 092 (c) 059

tional masking while experienced musicians do not withsuch an obviously musical sound sample as ldquoHuntsupperdquo

The chord reveals a different picture Four of the mu-sicians achieved very low thresholds with this sound in-put but all other subjects seem to have been seriouslydistracted by this rather non-musical sound and achievedonly rather high thresholds One of the musicians comesout worst of all in this test whereas for Huntsuppe all themusicians performed well This is a striking example ofinformational masking especially since the group of foursubjects who did well actually produced a lower averagethan in either of the other tests That is as one would hopethis signal contains deliberately a wider range of fre-quency components so that analytical listeners with suffi-cient skill and experience to take advantage of it can makevery subtle judgements down to 032 shift in the bestcase This corresponds to shifting the mode frequenciesabout 6 cents or 120 of a semitone an impressively smallamount that is not very much bigger than the threshold forpitch discrimination tasks (see eg [14])

The thresholds are of the same order as the ldquobest fiverdquothresholds in the corresponding violin tests but the de-tailed pattern seems different The choice of input had amore marked effect in the violin and to achieve the lowestthresholds a very short and consequently rather unmusi-cal sound sample was used

The best-performing 7 subjects of the first test serieswere asked to undertake two further tests in which fre-

481

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

quencies were shifted only in limited bands The twobands studied were shown in Figure 2 and this time theonly input sound tested was the single-note open A stringThe results of the tests are shown in Figure 5 in the sameformat as Figure 4 Rather unexpectedly the ldquobest fourrdquothreshold for the first band containing the strongest peaksof the datum guitar was much higher than for the sec-ond band Related tests for the violin using the so-calledldquoDuumlnnwald bandsrdquo gave thresholds around 2 for the firstthree bands (see Figure 6 of Fritz et al [1]) Band 1 in thepresent study gave a comparable result but band 2 gave alower result than was observed in any of the violin testsThe best-four JND in this case is virtually the same as forthe corresponding test when all frequencies were shiftedFurther studies might be worth conducting to add moredetail to this observation

The final pair of tests relating to the perception of bodymode changes investigated the effect of damping factorsThis time the mode frequencies were kept fixed but themodal Q factors were all scaled by a given factor Q fac-tor is the inverse of damping factor so high Q correspondsto low damping In two separate tests the Q factors wereincreased and decreased to look for any asymmetry in theperception Thirteen test subjects took these two tests 10musicians and 3 non-musicians Results are shown in Fig-ure 6 in the same format as Figs 4 and 5 except that thenumbers on the horizontal axis now indicate the scale fac-tor applied rather than the percentage change This meansthat ldquono changerdquo corresponds to the value 1 rather than 0as in the previous results which has the effect of makingthe relative standard deviations numerically smaller

An asymmetry is seen between the two tests but it doesnot significantly affect the threshold value for the bestfour listeners They achieved a value around 12 ie a20 change in Q factors in both cases However acrossthe whole set of test subjects the spread is much greaterwhen the Q factors were decreased compared to the in-creasing case Many participants reported initial difficultyin hearing the difference when Q factors were decreasedalthough once they had learned to recognise the soundchange they often achieved quite low thresholds

It was not thought worthwhile to test separate frequencybands in this case for two reasons First body damping ismost readily altered by a physical change of material andthis is likely to have an effect over a broad frequency rangeSecond it will be shown shortly that the JND for percep-tion of a change in body damping is of the same orderas the measured standard deviation among real guitars sothat the perceptual significance is probably marginal anddigging deeper into the details is likely to be of limitedimportance

42 Physical measurements

The next step is to look at physical measurements to as-sess how these thresholds of perception compare with theactual extent of variation between guitars Of course onewould never find two real instruments which differed byhaving all the body frequencies simultaneously scaled by

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(a) 150-250 Hz

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(b) 500-1000 Hz

Figure 5 Results in the format of Figure 4 from 3AFC studies forshifting all mode frequencies of the datum guitar lying in (a) theband 150ndash250 Hz and (b) the band 500ndash1000 Hz using as inputa single note (open A string) The percentage shifts correspond-ing to the best-four averages are (a) 187 (b) 153

1 12 15 2 25 30

005

01

015

02

(a)

Scale factor

RelSTD

1 12 15 2 25 30

005

01

015

02

(b)

Scale factor

RelSTD

Figure 6 Results in the format of Figure 4 from 3AFC studiesfor scaling all modal Q factors of the datum guitar (a) upwardsand (b) downwards using as input a single note (open A string)Horizontal axis shows scale factor applied by multiplication ordivision to Q factors The scale factors corresponding to the best-four averages are (a) 119 (b) 122

exactly the same factor However it is easy to visualise achange which would in principle have this effect it wouldhappen if the dimensions of the guitar were all scaledkeeping the same materials and other details Some sim-ple deductions can then be drawn by scaling analysis Ifthe main vibrating elements of the guitar body are treatedas thin flat plates deforming in bending then the naturalfrequencies scale proportional to hd2 where d is the scalefactor for dimensions in the plane and h is the scale factorfor thickness Taking 1 as a round number for the orderof magnitude of the threshold of detection for shifting fre-quencies it follows that the JNDs for dimensional scalingare approximately 1 if d and h are both scaled together(full geometric scaling) 1 again if thickness is changedkeeping the in-plane dimensions fixed and only 05 ifin-plane dimensions are scaled keeping the thickness thesame These are all very small numbers they amount tochanging in-plane dimensions no more than a few mm and

482

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

75 80 85 90 95 1000

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(a)

160 170 180 190 200 210 2200

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(b)

Figure 7 Modal frequencies and Q factors for 15 guitars forthe highest peak in the frequency bands (a) 75ndash100 Hz theldquoHelmholtz resonancerdquo (b) 150ndash250 Hz the ldquofirst top platemoderdquo Plus and cross symbols show fits without and with com-pensation for string damping respectively Diamond symbolsmark the datum guitar for the listening tests Open squares showthe means and range bars show one standard deviation aboveand below the means for the two sets of fitted data The rangesfor the compensated results can be recognised by their higher Qvalues Dashed lines near the axes show approximately 1 JNDabove and below the uncompensated means from the results ofsection 41

to changing a typical top plate thickness by a mere 25 micromReal instruments have dimensional variations which areconsiderably bigger than this especially in thickness soon this ground alone it is no surprise that guitars can sounddifferent

The simplest approach to obtaining quantitative infor-mation from admittance measurements on a set of gui-tars is to perform explicit modal analysis This can onlybe expected to give reliable results at relatively low fre-quencies pole-residue extraction whatever the methodbecomes sensitive once the modal overlap begins to ap-proach unity To illustrate the kind of values which emergefrom such an analysis Figure 7 shows results for modalfrequency and Q factor for 15 guitars for the highest peakin each of the two low-frequency bands mentioned in thediscussion of Figure 2 75ndash100 Hz the ldquoHelmholtz reso-nancerdquo and 150ndash250 Hz the ldquofirst top plate moderdquo

A complicating factor needs to be noted The ad-mittance measurements are made with the strings welldamped because the purpose is to obtain informationabout the body behaviour without the effect of the stringsHowever the damped strings remove some energy fromthe body vibration and this will tend to reduce the mea-sured Q factors This effect can be compensated at leastapproximately The damped strings will act essentially assemi-infinite strings and will thus present an impedance tothe bridge which is real and if the strings are all assumedto be attached at the same point equal to the sum of thestring impedances given by Z =

radicTm From the data for

nylon strings in Table I the sum of these impedances is212 Nsm To compensate the inverse of Y (ω) is takenthis real quantity is subtracted then the result re-invertedto give a new admittance This approach can be expectedto work well at higher frequencies but it is hard to be sureif the strings are well enough damped at low frequenciesto justify the semi-infinite approximation

Figure 7 shows results before and after this compensa-tion For each case the mean and scatter are indicated bylines showing one standard deviation above and below themean for frequency and Q Interestingly the standard de-viations have a similar pattern for both cases The relativestandard deviations for frequency are 55 and 6 for thetwo cases and those for Q factor about 33 in both casesbefore compensation

To obtain good comparative information about modalfrequencies over a broader bandwidth is more challeng-ing It is not possible to identify the ldquosamerdquo mode in dif-ferent guitars except at very low frequencies sensitivityto small structural changes increases with frequency as isfamiliar in other vibration problems (see eg [15]) Formode frequencies the threshold for audible changes is sosmall that it is hard to obtain any measure which is goodenough scaling frequencies by 1 means that one guitarbody might have 101 modes in a frequency range whereanother guitar had 100 and there is no known analysismethod which can give robust discrimination at this levelAny measure which can be computed shows much biggervariability than this and there is always a strong suspicionthat it is revealing the vagaries of the analysis method morethan the real differences between guitars

Things are a little more promising in the case of damp-ing estimation over a wide frequency range Taking theinverse FFT of an input admittance gives an impulse re-sponse function of the guitar body and this can be pro-cessed using time-frequency analysis to give a sonogramrepresentation In each frequency band a best-fitting ex-ponential decay can then be obtained Combining resultsfrom several different choices of the timefrequency res-olution tradeoff improves the robustness of the resultsThe process is described in more detail elsewhere [16]Some results are shown in Figure 8 compensation forstring damping has been included In Figure 8a the discretepoints represent for the datum guitar frequencydecaycombinations which satisfied an assumed criterion forgood fit quality For compatibility with other plots decay

483

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70

80

90

100

Frequency

Qfactor

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100

120

Frequency

Qfactor

(b)

30 35 40 45 50 55 60 6515

20

25

30

35

40

Mean Q

Relativestandard

deviation(

)

(c)

Figure 8 Damping trends for guitar bodies determined fromtime-frequency analysis (a) individual measurements of fre-quency and equivalent Q factor (circles) and the band averageof these (solid line) for the datum guitar (b) the solid lines asin (a) for the set of 15 guitars (c) the mean and standard devia-tion of the results from (b) across the frequency range with rangebars indicating one standard deviation above and below the meanon each axis The circle marks the datum guitar for the listeningtests

rate is represented as an equivalent Q factor The continu-ous line is simply a moving band average of these discretepoints

Figure 8b then shows the band-averaged lines for the 15guitars in the test set Finally Figure 8c shows a scatterplot of the mean and standard deviation of these band-averaged lines for the 15 guitars As in Figure 7 thelarge cross represents one standard deviation above and

2 3 5 8 100

01

02

03

04 (a)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (b)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (c)

Scale factor

RelSTD

Figure 9 Results in the format of Figure 4 from 3AFC studiesfor changing the damping parameters of the 5th string based ondatum values for the new steel strings Case (a) varying ηB testnote E2 (b) varying ηB test note C3 (c) varying ηF test noteC3 The scale factors corresponding to the best-four averages are(a) 299 (b) 313 (c) 241

below the mean on each axis The main result is thatthis frequency-averaged measure of damping has a rela-tive standard deviation of the order of 20 comparable tothe JND found earlier for perception of damping changesThis suggests that the extremes of mean damping in themeasured set of guitars are sufficiently different that theeffect should be audible but the results do not lead oneto expect that this will be a very strong component of theperception of tonal differences among guitars

5 Results for properties of the strings

The final set of JND experiments relates to changes instring properties including those properties that changedas the string aged one objective was to find out whichof those changes are likely to contribute most to the per-ception of ageing Using as datum the properties of thenew steel strings listed in Table I changes to the dampingproperties of the 5th string were investigated using single-note test sounds These 3AFC tests were completed by 11musicians and 5 non-musicians One test was also con-ducted relating to changes in the bending stiffness of thisstring it gave results in line with those of Jaumlrvelaumlinen etal [7] and the details are not reproduced here The actualbending stiffness of this string was found to be close tothe threshold for detecting an increase in bending stiffnesswhen starting from a much lower level In the aged stringthe bending stiffness only increased by a factor 16 (seeTable I) If that change is audible at all the effect is likelyto be small

The results for the damping tests are shown in Figure 9in a similar format to earlier 3AFC results As noted ear-lier the perceptual effects of changing ηA for this string

484

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

were found to be so weak that it was not possible to con-duct a meaningful test However both ηB and ηF gave clearresults The parameter ηB associated with bending stiff-ness and mainly affecting higher frequencies was testedusing two different notes ldquoplayedrdquo on the same string Theresults are shown in Figures 9ab and show a good levelof consistency between the two notes The spread over-all mean and best-four mean are all very similar for bothcases The best listeners need about a factor 3 increase inthis parameter to perceive the change Note from Table Ithat in the aged string this parameter increased from 0004to 0029 that change should be very clearly audible

Figure 9c shows corresponding results for ηF All sub-jects gave a somewhat lower threshold for this parametercompared to ηB and the very best listener achieved a JNDof less than a factor 2 In the aged string ηF increased inthe ratio 1711 for this string from Table I This effect isprobably not audible to most listeners It can be concludedthat the factor most likely to account for the difference insound of the aged string is ηB

6 Conclusions

Psychoacoustical experiments have been reported to estab-lish JND levels for the perception of changes to variousparameters of a model of synthesised guitar sound [3 4]These investigations parallel earlier work exploring relatedJNDs for violin sounds [1] The parameters investigatedhere relate to the behaviour of the guitar body (modal fre-quencies and damping factors) and of the strings (bendingstiffness and damping model parameters) All tests werecarried out using a three-alternative forced-choice method-ology It is argued that for application to musical judge-ments which can be very subtle the most interesting per-ceptual thresholds are those attainable by the best listenersunder the most favourable conditions To give an estimateof these the results for the best four subjects in each testwere averaged

Results of physical measurements have also been re-ported to give an indication of the spread of each parame-ter among real guitar bodies or strings A comparison withthe psychoacoustical results gives a first impression ofwhich quantities might be responsible for the biggest per-ceptual differences between instruments By far the mostsensitive parameter of all those tested relates to shiftingthe frequencies of body modes The best four listenerscould perceive a 1 shift when all frequencies were scaledtogether a small change compared to the actual varia-tion among guitars This performance was only degradedslightly when the shifted frequencies were band-limited intwo different bands tested The results were of the sameorder as the corresponding JNDs found for violin sounds[1] if anything the JNDs seem to be somewhat lower forthe guitar sounds perhaps because the fully-coupled syn-thesis method gives the listener additional auditory cuesassociated with transient decay rates of the string over-tones Body mode damping has to be changed by about20 for perception according to the definition used hereActual variations of modal damping exceed this level but

not by a large factor so body damping is probably quite aminor contributor to tonal perception

String damping parameters are even less sensitive theyhave to be changed by a factor of the order of 3 How-ever this large factor does not mean that the effects arenever heard Measurements have been presented on stringsof different materials and on steel-cored strings in newand ldquoagedrdquo conditions The damping parameters vary be-tween these cases by factors significantly bigger than theJNDs The results strongly indicate that the most importantparameter for the perception of the degradation of soundin well-used strings is ηB the damping factor associatedwith the bending stiffness of the string This parameterincreases greatly with age of strings probably from thedamping effect of dirt and corrosion products accumulat-ing between the windings of the lower strings of the guitar[13] Strings without over-winding do not deteriorate insound to anywhere near the same extent

The correct interpretation of the high sensitivity to shift-ing body frequencies is not obvious The actual change in-vestigated here moving all frequencies in step is very arti-ficial Listeners might be responding to the actual frequen-cies or perhaps to the change in modal density (the in-verse of the mean spacing between resonant frequencies)or perhaps to the shifting amplitude envelope of the bodyresponse Modal amplitudes were not changed when thefrequencies were scaled so the amplitude pattern movesalong the frequency axis with the scaling It would re-quire further testing to establish whether one of these fac-tors dominates or whether all contribute significantly Suchtesting would be a good target for future research to bringthe results closer to the detailed concerns of instrumentmakers

AcknowledgementThe authors thank Brian Moore Ian Cross and Alan Black-well for many helpful discussions on this work and MartinWoodhouse Stewart French Ian Cross and James West-brook for providing access to guitars for testing Twoanonymous reviewers are thanked for constructive sugges-tions

References

[1] C Fritz I Cross B C J Moore J Woodhouse Perceptualthresholds for detecting modifications applied to the acous-tical properties of a violin J Acoust Soc Am 122 (2007)3640ndash3650

[2] J Woodhouse P Galluzzo The bowed string as we know ittoday Acta Acustica united with Acustica 90 (2004) 579ndash589

[3] J Woodhouse On the synthesis of guitar plucks ActaAcustica united with Acustica 90 (2004) 928ndash944

[4] J Woodhouse Plucked guitar transients comparison ofmeasurements and synthesis Acta Acustica united withAcustica 90 (2004) 945ndash965

[5] H A K Wright The acoustics and psychoacoustics of theguitar PhD Thesis Cardiff University 1996

[6] H Jaumlrvelaumlinen V Vaumllimaumlki M Karjalainen Audibility ofthe timbral effects of inharmonicity in stringed instrumenttones Acoustics Research Letters Online 2 (2001) 79ndash84

485

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

Table I Properties of guitar strings (a) DrsquoAddario Pro Arteacute ldquoComposites hard tensionrdquo nylon strings (b) Martin ldquo8020 bronzerdquo steelstrings in new condition (c) as (b) but in used condition

String 1 2 3 4 5 6

Tuning note E4 B3 G3 D3 A2 E2

Frequency fn (Hz) 3296 2469 1960 1468 1100 824

Tension (a) T (N) 703 534 583 712 739 716(b) 121 135 182 185 169 147

Massunit length (a) m (gm) 038 052 090 195 361 624(b) 057 113 242 437 713 1102

Bending stiffness (a) EI (10minus6Nm2) 130 160 310 51 40 57(b) 108 336 234 255 327 667(c) 103 394 343 418 480 719

Inharmonicity (a) B (10minus6) 43 70 124 17 13 19(b) 21 58 30 32 45 106(c) 20 68 44 53 66 114

Loss coefficients (a) ηF (10minus5) 40 40 14 5 7 2(b) 7 6 11 9 11 5(c) 6 7 17 33 17 10

(a) ηB (10minus2) 24 20 20 20 25 20(b) 07 11 18 11 04 03(c) 07 11 68 41 29 38

(a) ηA (1s) 15 12 17 12 09 12(b) 25 10 12 13 12 17(c) 15 10 06 07 06 14

The differences between the nylon and steel strings innew condition are broadly as one would expect The nylonstrings have lower tensions and a different pattern of bend-ing stiffness because of the difference in construction forthe nylon set the top three strings are monofilaments whilethe lower three have multifilament cores with metal wind-ings The steel strings have monofilament cores through-out and only the top two strings are not wrapped Themonofilament nylon strings have higher internal dampingthan the steel but the difference is much less marked forthe lower three strings

The effect of age and use on the steel strings can alsobe seen Previous investigators have reported increaseddamping between new and old guitar strings [12 13] butwithout sufficient detail to allow a fit to equation (4) Itis generally accepted that the mechanism for this ageingeffect is the accumulation of dirt and corrosion productsbetween the windings of wrapped strings [13] perhapswith some additional cumulative wear of the outer wrap-pings where the strings make contact with the frets Themeasurements reported here support this picture there isvery little change in the two highest strings which are notwrapped For the wrapped strings there is a marked in-crease in damping especially in ηB which determines thedecay at the highest frequencies

Also not noted by previous authors there is a markedincrease in the effective bending stiffness of some of thestrings This seems perfectly credible since the contactconditions between the windings in the tensioned stringwill contribute to the bending stiffness and material col-

lecting between the windings will surely change those con-ditions This is consistent with the observed increase in ηB which is the loss factor associated with the bending stiff-ness term in the governing equation It may be noted thatthe bending stiffness values given here are in good gen-eral agreement with values reported by Jaumlrvelaumlinen et al[7] noting that their steel-string results match the ldquousedrdquoresults more closely than the ldquonewrdquo results Their paperdoes not report the condition of the strings as measured

4 Results for properties of the guitar body

41 Psychoacoustical tests

The first series of psychoacoustical tests concerned theshifting of all body mode frequencies by a given factorkeeping other simulation parameters fixed This modifica-tion is of course somewhat artificial but a possible physi-cal interpretation will be discussed in the next subsectionThe datum for this test was based on the modal proper-ties of the flamenco guitar by Martin Woodhouse This testwas run three times using the three different musical in-puts described above The three tests were completed by 8musicians and 5 non-musicians (according to the criteriondescribed above) The results are summarised in Figure 4The program that runs the 3AFC protocol gives as outputthe mean and standard deviation over the range defined bythe final 6 turnpoints The plots show for each test subjectthe mean on the horizontal axis and the relative standarddeviation (standard deviation divided by mean) on the ver-tical axis The numbers on the horizontal axis indicate the

480

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

100 200 500 1000 2000 500010

-4

10-3

10-2

Frequency (Hz)

Dampingfactor

100 200 500 1000 2000 500010

-4

10-3

10-2

Frequency (Hz)

Dampingfactor

Figure 3 Damping measurements on (top) a nylon guitar string(3rd string G3) and (bottom) the corresponding string from asteel-cored set Discrete points show individual measured decayrates curves show fits to the floor level using equation (4)

percentage shift in mode frequency Musicians are indi-cated by stars non-musicians by plus symbols The aver-ages over all test subjects are shown by the open circles

The open square symbols show the average of the foursubjects who achieved the lowest mean levels As was sug-gested earlier within the scope of the present study themost interesting threshold to compare with serious musi-cal judgments is that achieved by the best listeners underthe most favourable conditions In the violin tests the bestfive results were presented for this purpose [1] In the testsreported here the number of subjects is sometimes ratherlow and it was decided to average the best four The nu-merical results of this best-four measure for all the 3AFCtests presented here are given in the figure captions

It can be seen that for each of the three musical inputsthe best subjects could discriminate a shift of 1 or lessInteresting patterns can be seen in the detailed results de-spite the small number of subjects that would make anyelaborate statistical analysis of limited value The generalspread of results and the two averages look very simi-lar for the single note and the Huntsuppe extract For thesingle note the musicians and non-musicians are intermin-gled while for Huntsuppe all but one of the non-musicianshas a worse result than all the musicians Perhaps non-musicians have their performance degraded by informa-

03 05 1 2 3 5 100

05

1

(a) Huntsuppe

Percentage shift

RelSTD

03 05 1 2 3 5 100

05

1

(b) Single note

Percentage shift

RelSTD

03 05 1 2 3 5 100

05

1

(c) Discordant chord

Percentage shiftRelSTD

Figure 4 Results from 3AFC studies for shifting all mode fre-quencies of the datum guitar using as input (a) the Huntsuppefragment (b) a single note (open A string) (c) the discordantchord Horizontal axis shows percentage frequency shift on alogarithmic scale vertical axis shows relative standard deviationPoints denote musicians points + denote non-musicians Opencircle average of all subjects open square average of best foursubjects The percentage shifts corresponding to the best-four av-erages are (a) 113 (b) 092 (c) 059

tional masking while experienced musicians do not withsuch an obviously musical sound sample as ldquoHuntsupperdquo

The chord reveals a different picture Four of the mu-sicians achieved very low thresholds with this sound in-put but all other subjects seem to have been seriouslydistracted by this rather non-musical sound and achievedonly rather high thresholds One of the musicians comesout worst of all in this test whereas for Huntsuppe all themusicians performed well This is a striking example ofinformational masking especially since the group of foursubjects who did well actually produced a lower averagethan in either of the other tests That is as one would hopethis signal contains deliberately a wider range of fre-quency components so that analytical listeners with suffi-cient skill and experience to take advantage of it can makevery subtle judgements down to 032 shift in the bestcase This corresponds to shifting the mode frequenciesabout 6 cents or 120 of a semitone an impressively smallamount that is not very much bigger than the threshold forpitch discrimination tasks (see eg [14])

The thresholds are of the same order as the ldquobest fiverdquothresholds in the corresponding violin tests but the de-tailed pattern seems different The choice of input had amore marked effect in the violin and to achieve the lowestthresholds a very short and consequently rather unmusi-cal sound sample was used

The best-performing 7 subjects of the first test serieswere asked to undertake two further tests in which fre-

481

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

quencies were shifted only in limited bands The twobands studied were shown in Figure 2 and this time theonly input sound tested was the single-note open A stringThe results of the tests are shown in Figure 5 in the sameformat as Figure 4 Rather unexpectedly the ldquobest fourrdquothreshold for the first band containing the strongest peaksof the datum guitar was much higher than for the sec-ond band Related tests for the violin using the so-calledldquoDuumlnnwald bandsrdquo gave thresholds around 2 for the firstthree bands (see Figure 6 of Fritz et al [1]) Band 1 in thepresent study gave a comparable result but band 2 gave alower result than was observed in any of the violin testsThe best-four JND in this case is virtually the same as forthe corresponding test when all frequencies were shiftedFurther studies might be worth conducting to add moredetail to this observation

The final pair of tests relating to the perception of bodymode changes investigated the effect of damping factorsThis time the mode frequencies were kept fixed but themodal Q factors were all scaled by a given factor Q fac-tor is the inverse of damping factor so high Q correspondsto low damping In two separate tests the Q factors wereincreased and decreased to look for any asymmetry in theperception Thirteen test subjects took these two tests 10musicians and 3 non-musicians Results are shown in Fig-ure 6 in the same format as Figs 4 and 5 except that thenumbers on the horizontal axis now indicate the scale fac-tor applied rather than the percentage change This meansthat ldquono changerdquo corresponds to the value 1 rather than 0as in the previous results which has the effect of makingthe relative standard deviations numerically smaller

An asymmetry is seen between the two tests but it doesnot significantly affect the threshold value for the bestfour listeners They achieved a value around 12 ie a20 change in Q factors in both cases However acrossthe whole set of test subjects the spread is much greaterwhen the Q factors were decreased compared to the in-creasing case Many participants reported initial difficultyin hearing the difference when Q factors were decreasedalthough once they had learned to recognise the soundchange they often achieved quite low thresholds

It was not thought worthwhile to test separate frequencybands in this case for two reasons First body damping ismost readily altered by a physical change of material andthis is likely to have an effect over a broad frequency rangeSecond it will be shown shortly that the JND for percep-tion of a change in body damping is of the same orderas the measured standard deviation among real guitars sothat the perceptual significance is probably marginal anddigging deeper into the details is likely to be of limitedimportance

42 Physical measurements

The next step is to look at physical measurements to as-sess how these thresholds of perception compare with theactual extent of variation between guitars Of course onewould never find two real instruments which differed byhaving all the body frequencies simultaneously scaled by

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(a) 150-250 Hz

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(b) 500-1000 Hz

Figure 5 Results in the format of Figure 4 from 3AFC studies forshifting all mode frequencies of the datum guitar lying in (a) theband 150ndash250 Hz and (b) the band 500ndash1000 Hz using as inputa single note (open A string) The percentage shifts correspond-ing to the best-four averages are (a) 187 (b) 153

1 12 15 2 25 30

005

01

015

02

(a)

Scale factor

RelSTD

1 12 15 2 25 30

005

01

015

02

(b)

Scale factor

RelSTD

Figure 6 Results in the format of Figure 4 from 3AFC studiesfor scaling all modal Q factors of the datum guitar (a) upwardsand (b) downwards using as input a single note (open A string)Horizontal axis shows scale factor applied by multiplication ordivision to Q factors The scale factors corresponding to the best-four averages are (a) 119 (b) 122

exactly the same factor However it is easy to visualise achange which would in principle have this effect it wouldhappen if the dimensions of the guitar were all scaledkeeping the same materials and other details Some sim-ple deductions can then be drawn by scaling analysis Ifthe main vibrating elements of the guitar body are treatedas thin flat plates deforming in bending then the naturalfrequencies scale proportional to hd2 where d is the scalefactor for dimensions in the plane and h is the scale factorfor thickness Taking 1 as a round number for the orderof magnitude of the threshold of detection for shifting fre-quencies it follows that the JNDs for dimensional scalingare approximately 1 if d and h are both scaled together(full geometric scaling) 1 again if thickness is changedkeeping the in-plane dimensions fixed and only 05 ifin-plane dimensions are scaled keeping the thickness thesame These are all very small numbers they amount tochanging in-plane dimensions no more than a few mm and

482

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

75 80 85 90 95 1000

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(a)

160 170 180 190 200 210 2200

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(b)

Figure 7 Modal frequencies and Q factors for 15 guitars forthe highest peak in the frequency bands (a) 75ndash100 Hz theldquoHelmholtz resonancerdquo (b) 150ndash250 Hz the ldquofirst top platemoderdquo Plus and cross symbols show fits without and with com-pensation for string damping respectively Diamond symbolsmark the datum guitar for the listening tests Open squares showthe means and range bars show one standard deviation aboveand below the means for the two sets of fitted data The rangesfor the compensated results can be recognised by their higher Qvalues Dashed lines near the axes show approximately 1 JNDabove and below the uncompensated means from the results ofsection 41

to changing a typical top plate thickness by a mere 25 micromReal instruments have dimensional variations which areconsiderably bigger than this especially in thickness soon this ground alone it is no surprise that guitars can sounddifferent

The simplest approach to obtaining quantitative infor-mation from admittance measurements on a set of gui-tars is to perform explicit modal analysis This can onlybe expected to give reliable results at relatively low fre-quencies pole-residue extraction whatever the methodbecomes sensitive once the modal overlap begins to ap-proach unity To illustrate the kind of values which emergefrom such an analysis Figure 7 shows results for modalfrequency and Q factor for 15 guitars for the highest peakin each of the two low-frequency bands mentioned in thediscussion of Figure 2 75ndash100 Hz the ldquoHelmholtz reso-nancerdquo and 150ndash250 Hz the ldquofirst top plate moderdquo

A complicating factor needs to be noted The ad-mittance measurements are made with the strings welldamped because the purpose is to obtain informationabout the body behaviour without the effect of the stringsHowever the damped strings remove some energy fromthe body vibration and this will tend to reduce the mea-sured Q factors This effect can be compensated at leastapproximately The damped strings will act essentially assemi-infinite strings and will thus present an impedance tothe bridge which is real and if the strings are all assumedto be attached at the same point equal to the sum of thestring impedances given by Z =

radicTm From the data for

nylon strings in Table I the sum of these impedances is212 Nsm To compensate the inverse of Y (ω) is takenthis real quantity is subtracted then the result re-invertedto give a new admittance This approach can be expectedto work well at higher frequencies but it is hard to be sureif the strings are well enough damped at low frequenciesto justify the semi-infinite approximation

Figure 7 shows results before and after this compensa-tion For each case the mean and scatter are indicated bylines showing one standard deviation above and below themean for frequency and Q Interestingly the standard de-viations have a similar pattern for both cases The relativestandard deviations for frequency are 55 and 6 for thetwo cases and those for Q factor about 33 in both casesbefore compensation

To obtain good comparative information about modalfrequencies over a broader bandwidth is more challeng-ing It is not possible to identify the ldquosamerdquo mode in dif-ferent guitars except at very low frequencies sensitivityto small structural changes increases with frequency as isfamiliar in other vibration problems (see eg [15]) Formode frequencies the threshold for audible changes is sosmall that it is hard to obtain any measure which is goodenough scaling frequencies by 1 means that one guitarbody might have 101 modes in a frequency range whereanother guitar had 100 and there is no known analysismethod which can give robust discrimination at this levelAny measure which can be computed shows much biggervariability than this and there is always a strong suspicionthat it is revealing the vagaries of the analysis method morethan the real differences between guitars

Things are a little more promising in the case of damp-ing estimation over a wide frequency range Taking theinverse FFT of an input admittance gives an impulse re-sponse function of the guitar body and this can be pro-cessed using time-frequency analysis to give a sonogramrepresentation In each frequency band a best-fitting ex-ponential decay can then be obtained Combining resultsfrom several different choices of the timefrequency res-olution tradeoff improves the robustness of the resultsThe process is described in more detail elsewhere [16]Some results are shown in Figure 8 compensation forstring damping has been included In Figure 8a the discretepoints represent for the datum guitar frequencydecaycombinations which satisfied an assumed criterion forgood fit quality For compatibility with other plots decay

483

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70

80

90

100

Frequency

Qfactor

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100

120

Frequency

Qfactor

(b)

30 35 40 45 50 55 60 6515

20

25

30

35

40

Mean Q

Relativestandard

deviation(

)

(c)

Figure 8 Damping trends for guitar bodies determined fromtime-frequency analysis (a) individual measurements of fre-quency and equivalent Q factor (circles) and the band averageof these (solid line) for the datum guitar (b) the solid lines asin (a) for the set of 15 guitars (c) the mean and standard devia-tion of the results from (b) across the frequency range with rangebars indicating one standard deviation above and below the meanon each axis The circle marks the datum guitar for the listeningtests

rate is represented as an equivalent Q factor The continu-ous line is simply a moving band average of these discretepoints

Figure 8b then shows the band-averaged lines for the 15guitars in the test set Finally Figure 8c shows a scatterplot of the mean and standard deviation of these band-averaged lines for the 15 guitars As in Figure 7 thelarge cross represents one standard deviation above and

2 3 5 8 100

01

02

03

04 (a)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (b)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (c)

Scale factor

RelSTD

Figure 9 Results in the format of Figure 4 from 3AFC studiesfor changing the damping parameters of the 5th string based ondatum values for the new steel strings Case (a) varying ηB testnote E2 (b) varying ηB test note C3 (c) varying ηF test noteC3 The scale factors corresponding to the best-four averages are(a) 299 (b) 313 (c) 241

below the mean on each axis The main result is thatthis frequency-averaged measure of damping has a rela-tive standard deviation of the order of 20 comparable tothe JND found earlier for perception of damping changesThis suggests that the extremes of mean damping in themeasured set of guitars are sufficiently different that theeffect should be audible but the results do not lead oneto expect that this will be a very strong component of theperception of tonal differences among guitars

5 Results for properties of the strings

The final set of JND experiments relates to changes instring properties including those properties that changedas the string aged one objective was to find out whichof those changes are likely to contribute most to the per-ception of ageing Using as datum the properties of thenew steel strings listed in Table I changes to the dampingproperties of the 5th string were investigated using single-note test sounds These 3AFC tests were completed by 11musicians and 5 non-musicians One test was also con-ducted relating to changes in the bending stiffness of thisstring it gave results in line with those of Jaumlrvelaumlinen etal [7] and the details are not reproduced here The actualbending stiffness of this string was found to be close tothe threshold for detecting an increase in bending stiffnesswhen starting from a much lower level In the aged stringthe bending stiffness only increased by a factor 16 (seeTable I) If that change is audible at all the effect is likelyto be small

The results for the damping tests are shown in Figure 9in a similar format to earlier 3AFC results As noted ear-lier the perceptual effects of changing ηA for this string

484

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

were found to be so weak that it was not possible to con-duct a meaningful test However both ηB and ηF gave clearresults The parameter ηB associated with bending stiff-ness and mainly affecting higher frequencies was testedusing two different notes ldquoplayedrdquo on the same string Theresults are shown in Figures 9ab and show a good levelof consistency between the two notes The spread over-all mean and best-four mean are all very similar for bothcases The best listeners need about a factor 3 increase inthis parameter to perceive the change Note from Table Ithat in the aged string this parameter increased from 0004to 0029 that change should be very clearly audible

Figure 9c shows corresponding results for ηF All sub-jects gave a somewhat lower threshold for this parametercompared to ηB and the very best listener achieved a JNDof less than a factor 2 In the aged string ηF increased inthe ratio 1711 for this string from Table I This effect isprobably not audible to most listeners It can be concludedthat the factor most likely to account for the difference insound of the aged string is ηB

6 Conclusions

Psychoacoustical experiments have been reported to estab-lish JND levels for the perception of changes to variousparameters of a model of synthesised guitar sound [3 4]These investigations parallel earlier work exploring relatedJNDs for violin sounds [1] The parameters investigatedhere relate to the behaviour of the guitar body (modal fre-quencies and damping factors) and of the strings (bendingstiffness and damping model parameters) All tests werecarried out using a three-alternative forced-choice method-ology It is argued that for application to musical judge-ments which can be very subtle the most interesting per-ceptual thresholds are those attainable by the best listenersunder the most favourable conditions To give an estimateof these the results for the best four subjects in each testwere averaged

Results of physical measurements have also been re-ported to give an indication of the spread of each parame-ter among real guitar bodies or strings A comparison withthe psychoacoustical results gives a first impression ofwhich quantities might be responsible for the biggest per-ceptual differences between instruments By far the mostsensitive parameter of all those tested relates to shiftingthe frequencies of body modes The best four listenerscould perceive a 1 shift when all frequencies were scaledtogether a small change compared to the actual varia-tion among guitars This performance was only degradedslightly when the shifted frequencies were band-limited intwo different bands tested The results were of the sameorder as the corresponding JNDs found for violin sounds[1] if anything the JNDs seem to be somewhat lower forthe guitar sounds perhaps because the fully-coupled syn-thesis method gives the listener additional auditory cuesassociated with transient decay rates of the string over-tones Body mode damping has to be changed by about20 for perception according to the definition used hereActual variations of modal damping exceed this level but

not by a large factor so body damping is probably quite aminor contributor to tonal perception

String damping parameters are even less sensitive theyhave to be changed by a factor of the order of 3 How-ever this large factor does not mean that the effects arenever heard Measurements have been presented on stringsof different materials and on steel-cored strings in newand ldquoagedrdquo conditions The damping parameters vary be-tween these cases by factors significantly bigger than theJNDs The results strongly indicate that the most importantparameter for the perception of the degradation of soundin well-used strings is ηB the damping factor associatedwith the bending stiffness of the string This parameterincreases greatly with age of strings probably from thedamping effect of dirt and corrosion products accumulat-ing between the windings of the lower strings of the guitar[13] Strings without over-winding do not deteriorate insound to anywhere near the same extent

The correct interpretation of the high sensitivity to shift-ing body frequencies is not obvious The actual change in-vestigated here moving all frequencies in step is very arti-ficial Listeners might be responding to the actual frequen-cies or perhaps to the change in modal density (the in-verse of the mean spacing between resonant frequencies)or perhaps to the shifting amplitude envelope of the bodyresponse Modal amplitudes were not changed when thefrequencies were scaled so the amplitude pattern movesalong the frequency axis with the scaling It would re-quire further testing to establish whether one of these fac-tors dominates or whether all contribute significantly Suchtesting would be a good target for future research to bringthe results closer to the detailed concerns of instrumentmakers

AcknowledgementThe authors thank Brian Moore Ian Cross and Alan Black-well for many helpful discussions on this work and MartinWoodhouse Stewart French Ian Cross and James West-brook for providing access to guitars for testing Twoanonymous reviewers are thanked for constructive sugges-tions

References

[1] C Fritz I Cross B C J Moore J Woodhouse Perceptualthresholds for detecting modifications applied to the acous-tical properties of a violin J Acoust Soc Am 122 (2007)3640ndash3650

[2] J Woodhouse P Galluzzo The bowed string as we know ittoday Acta Acustica united with Acustica 90 (2004) 579ndash589

[3] J Woodhouse On the synthesis of guitar plucks ActaAcustica united with Acustica 90 (2004) 928ndash944

[4] J Woodhouse Plucked guitar transients comparison ofmeasurements and synthesis Acta Acustica united withAcustica 90 (2004) 945ndash965

[5] H A K Wright The acoustics and psychoacoustics of theguitar PhD Thesis Cardiff University 1996

[6] H Jaumlrvelaumlinen V Vaumllimaumlki M Karjalainen Audibility ofthe timbral effects of inharmonicity in stringed instrumenttones Acoustics Research Letters Online 2 (2001) 79ndash84

485

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

100 200 500 1000 2000 500010

-4

10-3

10-2

Frequency (Hz)

Dampingfactor

100 200 500 1000 2000 500010

-4

10-3

10-2

Frequency (Hz)

Dampingfactor

Figure 3 Damping measurements on (top) a nylon guitar string(3rd string G3) and (bottom) the corresponding string from asteel-cored set Discrete points show individual measured decayrates curves show fits to the floor level using equation (4)

percentage shift in mode frequency Musicians are indi-cated by stars non-musicians by plus symbols The aver-ages over all test subjects are shown by the open circles

The open square symbols show the average of the foursubjects who achieved the lowest mean levels As was sug-gested earlier within the scope of the present study themost interesting threshold to compare with serious musi-cal judgments is that achieved by the best listeners underthe most favourable conditions In the violin tests the bestfive results were presented for this purpose [1] In the testsreported here the number of subjects is sometimes ratherlow and it was decided to average the best four The nu-merical results of this best-four measure for all the 3AFCtests presented here are given in the figure captions

It can be seen that for each of the three musical inputsthe best subjects could discriminate a shift of 1 or lessInteresting patterns can be seen in the detailed results de-spite the small number of subjects that would make anyelaborate statistical analysis of limited value The generalspread of results and the two averages look very simi-lar for the single note and the Huntsuppe extract For thesingle note the musicians and non-musicians are intermin-gled while for Huntsuppe all but one of the non-musicianshas a worse result than all the musicians Perhaps non-musicians have their performance degraded by informa-

03 05 1 2 3 5 100

05

1

(a) Huntsuppe

Percentage shift

RelSTD

03 05 1 2 3 5 100

05

1

(b) Single note

Percentage shift

RelSTD

03 05 1 2 3 5 100

05

1

(c) Discordant chord

Percentage shiftRelSTD

Figure 4 Results from 3AFC studies for shifting all mode fre-quencies of the datum guitar using as input (a) the Huntsuppefragment (b) a single note (open A string) (c) the discordantchord Horizontal axis shows percentage frequency shift on alogarithmic scale vertical axis shows relative standard deviationPoints denote musicians points + denote non-musicians Opencircle average of all subjects open square average of best foursubjects The percentage shifts corresponding to the best-four av-erages are (a) 113 (b) 092 (c) 059

tional masking while experienced musicians do not withsuch an obviously musical sound sample as ldquoHuntsupperdquo

The chord reveals a different picture Four of the mu-sicians achieved very low thresholds with this sound in-put but all other subjects seem to have been seriouslydistracted by this rather non-musical sound and achievedonly rather high thresholds One of the musicians comesout worst of all in this test whereas for Huntsuppe all themusicians performed well This is a striking example ofinformational masking especially since the group of foursubjects who did well actually produced a lower averagethan in either of the other tests That is as one would hopethis signal contains deliberately a wider range of fre-quency components so that analytical listeners with suffi-cient skill and experience to take advantage of it can makevery subtle judgements down to 032 shift in the bestcase This corresponds to shifting the mode frequenciesabout 6 cents or 120 of a semitone an impressively smallamount that is not very much bigger than the threshold forpitch discrimination tasks (see eg [14])

The thresholds are of the same order as the ldquobest fiverdquothresholds in the corresponding violin tests but the de-tailed pattern seems different The choice of input had amore marked effect in the violin and to achieve the lowestthresholds a very short and consequently rather unmusi-cal sound sample was used

The best-performing 7 subjects of the first test serieswere asked to undertake two further tests in which fre-

481

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

quencies were shifted only in limited bands The twobands studied were shown in Figure 2 and this time theonly input sound tested was the single-note open A stringThe results of the tests are shown in Figure 5 in the sameformat as Figure 4 Rather unexpectedly the ldquobest fourrdquothreshold for the first band containing the strongest peaksof the datum guitar was much higher than for the sec-ond band Related tests for the violin using the so-calledldquoDuumlnnwald bandsrdquo gave thresholds around 2 for the firstthree bands (see Figure 6 of Fritz et al [1]) Band 1 in thepresent study gave a comparable result but band 2 gave alower result than was observed in any of the violin testsThe best-four JND in this case is virtually the same as forthe corresponding test when all frequencies were shiftedFurther studies might be worth conducting to add moredetail to this observation

The final pair of tests relating to the perception of bodymode changes investigated the effect of damping factorsThis time the mode frequencies were kept fixed but themodal Q factors were all scaled by a given factor Q fac-tor is the inverse of damping factor so high Q correspondsto low damping In two separate tests the Q factors wereincreased and decreased to look for any asymmetry in theperception Thirteen test subjects took these two tests 10musicians and 3 non-musicians Results are shown in Fig-ure 6 in the same format as Figs 4 and 5 except that thenumbers on the horizontal axis now indicate the scale fac-tor applied rather than the percentage change This meansthat ldquono changerdquo corresponds to the value 1 rather than 0as in the previous results which has the effect of makingthe relative standard deviations numerically smaller

An asymmetry is seen between the two tests but it doesnot significantly affect the threshold value for the bestfour listeners They achieved a value around 12 ie a20 change in Q factors in both cases However acrossthe whole set of test subjects the spread is much greaterwhen the Q factors were decreased compared to the in-creasing case Many participants reported initial difficultyin hearing the difference when Q factors were decreasedalthough once they had learned to recognise the soundchange they often achieved quite low thresholds

It was not thought worthwhile to test separate frequencybands in this case for two reasons First body damping ismost readily altered by a physical change of material andthis is likely to have an effect over a broad frequency rangeSecond it will be shown shortly that the JND for percep-tion of a change in body damping is of the same orderas the measured standard deviation among real guitars sothat the perceptual significance is probably marginal anddigging deeper into the details is likely to be of limitedimportance

42 Physical measurements

The next step is to look at physical measurements to as-sess how these thresholds of perception compare with theactual extent of variation between guitars Of course onewould never find two real instruments which differed byhaving all the body frequencies simultaneously scaled by

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(a) 150-250 Hz

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(b) 500-1000 Hz

Figure 5 Results in the format of Figure 4 from 3AFC studies forshifting all mode frequencies of the datum guitar lying in (a) theband 150ndash250 Hz and (b) the band 500ndash1000 Hz using as inputa single note (open A string) The percentage shifts correspond-ing to the best-four averages are (a) 187 (b) 153

1 12 15 2 25 30

005

01

015

02

(a)

Scale factor

RelSTD

1 12 15 2 25 30

005

01

015

02

(b)

Scale factor

RelSTD

Figure 6 Results in the format of Figure 4 from 3AFC studiesfor scaling all modal Q factors of the datum guitar (a) upwardsand (b) downwards using as input a single note (open A string)Horizontal axis shows scale factor applied by multiplication ordivision to Q factors The scale factors corresponding to the best-four averages are (a) 119 (b) 122

exactly the same factor However it is easy to visualise achange which would in principle have this effect it wouldhappen if the dimensions of the guitar were all scaledkeeping the same materials and other details Some sim-ple deductions can then be drawn by scaling analysis Ifthe main vibrating elements of the guitar body are treatedas thin flat plates deforming in bending then the naturalfrequencies scale proportional to hd2 where d is the scalefactor for dimensions in the plane and h is the scale factorfor thickness Taking 1 as a round number for the orderof magnitude of the threshold of detection for shifting fre-quencies it follows that the JNDs for dimensional scalingare approximately 1 if d and h are both scaled together(full geometric scaling) 1 again if thickness is changedkeeping the in-plane dimensions fixed and only 05 ifin-plane dimensions are scaled keeping the thickness thesame These are all very small numbers they amount tochanging in-plane dimensions no more than a few mm and

482

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

75 80 85 90 95 1000

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(a)

160 170 180 190 200 210 2200

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(b)

Figure 7 Modal frequencies and Q factors for 15 guitars forthe highest peak in the frequency bands (a) 75ndash100 Hz theldquoHelmholtz resonancerdquo (b) 150ndash250 Hz the ldquofirst top platemoderdquo Plus and cross symbols show fits without and with com-pensation for string damping respectively Diamond symbolsmark the datum guitar for the listening tests Open squares showthe means and range bars show one standard deviation aboveand below the means for the two sets of fitted data The rangesfor the compensated results can be recognised by their higher Qvalues Dashed lines near the axes show approximately 1 JNDabove and below the uncompensated means from the results ofsection 41

to changing a typical top plate thickness by a mere 25 micromReal instruments have dimensional variations which areconsiderably bigger than this especially in thickness soon this ground alone it is no surprise that guitars can sounddifferent

The simplest approach to obtaining quantitative infor-mation from admittance measurements on a set of gui-tars is to perform explicit modal analysis This can onlybe expected to give reliable results at relatively low fre-quencies pole-residue extraction whatever the methodbecomes sensitive once the modal overlap begins to ap-proach unity To illustrate the kind of values which emergefrom such an analysis Figure 7 shows results for modalfrequency and Q factor for 15 guitars for the highest peakin each of the two low-frequency bands mentioned in thediscussion of Figure 2 75ndash100 Hz the ldquoHelmholtz reso-nancerdquo and 150ndash250 Hz the ldquofirst top plate moderdquo

A complicating factor needs to be noted The ad-mittance measurements are made with the strings welldamped because the purpose is to obtain informationabout the body behaviour without the effect of the stringsHowever the damped strings remove some energy fromthe body vibration and this will tend to reduce the mea-sured Q factors This effect can be compensated at leastapproximately The damped strings will act essentially assemi-infinite strings and will thus present an impedance tothe bridge which is real and if the strings are all assumedto be attached at the same point equal to the sum of thestring impedances given by Z =

radicTm From the data for

nylon strings in Table I the sum of these impedances is212 Nsm To compensate the inverse of Y (ω) is takenthis real quantity is subtracted then the result re-invertedto give a new admittance This approach can be expectedto work well at higher frequencies but it is hard to be sureif the strings are well enough damped at low frequenciesto justify the semi-infinite approximation

Figure 7 shows results before and after this compensa-tion For each case the mean and scatter are indicated bylines showing one standard deviation above and below themean for frequency and Q Interestingly the standard de-viations have a similar pattern for both cases The relativestandard deviations for frequency are 55 and 6 for thetwo cases and those for Q factor about 33 in both casesbefore compensation

To obtain good comparative information about modalfrequencies over a broader bandwidth is more challeng-ing It is not possible to identify the ldquosamerdquo mode in dif-ferent guitars except at very low frequencies sensitivityto small structural changes increases with frequency as isfamiliar in other vibration problems (see eg [15]) Formode frequencies the threshold for audible changes is sosmall that it is hard to obtain any measure which is goodenough scaling frequencies by 1 means that one guitarbody might have 101 modes in a frequency range whereanother guitar had 100 and there is no known analysismethod which can give robust discrimination at this levelAny measure which can be computed shows much biggervariability than this and there is always a strong suspicionthat it is revealing the vagaries of the analysis method morethan the real differences between guitars

Things are a little more promising in the case of damp-ing estimation over a wide frequency range Taking theinverse FFT of an input admittance gives an impulse re-sponse function of the guitar body and this can be pro-cessed using time-frequency analysis to give a sonogramrepresentation In each frequency band a best-fitting ex-ponential decay can then be obtained Combining resultsfrom several different choices of the timefrequency res-olution tradeoff improves the robustness of the resultsThe process is described in more detail elsewhere [16]Some results are shown in Figure 8 compensation forstring damping has been included In Figure 8a the discretepoints represent for the datum guitar frequencydecaycombinations which satisfied an assumed criterion forgood fit quality For compatibility with other plots decay

483

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70

80

90

100

Frequency

Qfactor

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100

120

Frequency

Qfactor

(b)

30 35 40 45 50 55 60 6515

20

25

30

35

40

Mean Q

Relativestandard

deviation(

)

(c)

Figure 8 Damping trends for guitar bodies determined fromtime-frequency analysis (a) individual measurements of fre-quency and equivalent Q factor (circles) and the band averageof these (solid line) for the datum guitar (b) the solid lines asin (a) for the set of 15 guitars (c) the mean and standard devia-tion of the results from (b) across the frequency range with rangebars indicating one standard deviation above and below the meanon each axis The circle marks the datum guitar for the listeningtests

rate is represented as an equivalent Q factor The continu-ous line is simply a moving band average of these discretepoints

Figure 8b then shows the band-averaged lines for the 15guitars in the test set Finally Figure 8c shows a scatterplot of the mean and standard deviation of these band-averaged lines for the 15 guitars As in Figure 7 thelarge cross represents one standard deviation above and

2 3 5 8 100

01

02

03

04 (a)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (b)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (c)

Scale factor

RelSTD

Figure 9 Results in the format of Figure 4 from 3AFC studiesfor changing the damping parameters of the 5th string based ondatum values for the new steel strings Case (a) varying ηB testnote E2 (b) varying ηB test note C3 (c) varying ηF test noteC3 The scale factors corresponding to the best-four averages are(a) 299 (b) 313 (c) 241

below the mean on each axis The main result is thatthis frequency-averaged measure of damping has a rela-tive standard deviation of the order of 20 comparable tothe JND found earlier for perception of damping changesThis suggests that the extremes of mean damping in themeasured set of guitars are sufficiently different that theeffect should be audible but the results do not lead oneto expect that this will be a very strong component of theperception of tonal differences among guitars

5 Results for properties of the strings

The final set of JND experiments relates to changes instring properties including those properties that changedas the string aged one objective was to find out whichof those changes are likely to contribute most to the per-ception of ageing Using as datum the properties of thenew steel strings listed in Table I changes to the dampingproperties of the 5th string were investigated using single-note test sounds These 3AFC tests were completed by 11musicians and 5 non-musicians One test was also con-ducted relating to changes in the bending stiffness of thisstring it gave results in line with those of Jaumlrvelaumlinen etal [7] and the details are not reproduced here The actualbending stiffness of this string was found to be close tothe threshold for detecting an increase in bending stiffnesswhen starting from a much lower level In the aged stringthe bending stiffness only increased by a factor 16 (seeTable I) If that change is audible at all the effect is likelyto be small

The results for the damping tests are shown in Figure 9in a similar format to earlier 3AFC results As noted ear-lier the perceptual effects of changing ηA for this string

484

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

were found to be so weak that it was not possible to con-duct a meaningful test However both ηB and ηF gave clearresults The parameter ηB associated with bending stiff-ness and mainly affecting higher frequencies was testedusing two different notes ldquoplayedrdquo on the same string Theresults are shown in Figures 9ab and show a good levelof consistency between the two notes The spread over-all mean and best-four mean are all very similar for bothcases The best listeners need about a factor 3 increase inthis parameter to perceive the change Note from Table Ithat in the aged string this parameter increased from 0004to 0029 that change should be very clearly audible

Figure 9c shows corresponding results for ηF All sub-jects gave a somewhat lower threshold for this parametercompared to ηB and the very best listener achieved a JNDof less than a factor 2 In the aged string ηF increased inthe ratio 1711 for this string from Table I This effect isprobably not audible to most listeners It can be concludedthat the factor most likely to account for the difference insound of the aged string is ηB

6 Conclusions

Psychoacoustical experiments have been reported to estab-lish JND levels for the perception of changes to variousparameters of a model of synthesised guitar sound [3 4]These investigations parallel earlier work exploring relatedJNDs for violin sounds [1] The parameters investigatedhere relate to the behaviour of the guitar body (modal fre-quencies and damping factors) and of the strings (bendingstiffness and damping model parameters) All tests werecarried out using a three-alternative forced-choice method-ology It is argued that for application to musical judge-ments which can be very subtle the most interesting per-ceptual thresholds are those attainable by the best listenersunder the most favourable conditions To give an estimateof these the results for the best four subjects in each testwere averaged

Results of physical measurements have also been re-ported to give an indication of the spread of each parame-ter among real guitar bodies or strings A comparison withthe psychoacoustical results gives a first impression ofwhich quantities might be responsible for the biggest per-ceptual differences between instruments By far the mostsensitive parameter of all those tested relates to shiftingthe frequencies of body modes The best four listenerscould perceive a 1 shift when all frequencies were scaledtogether a small change compared to the actual varia-tion among guitars This performance was only degradedslightly when the shifted frequencies were band-limited intwo different bands tested The results were of the sameorder as the corresponding JNDs found for violin sounds[1] if anything the JNDs seem to be somewhat lower forthe guitar sounds perhaps because the fully-coupled syn-thesis method gives the listener additional auditory cuesassociated with transient decay rates of the string over-tones Body mode damping has to be changed by about20 for perception according to the definition used hereActual variations of modal damping exceed this level but

not by a large factor so body damping is probably quite aminor contributor to tonal perception

String damping parameters are even less sensitive theyhave to be changed by a factor of the order of 3 How-ever this large factor does not mean that the effects arenever heard Measurements have been presented on stringsof different materials and on steel-cored strings in newand ldquoagedrdquo conditions The damping parameters vary be-tween these cases by factors significantly bigger than theJNDs The results strongly indicate that the most importantparameter for the perception of the degradation of soundin well-used strings is ηB the damping factor associatedwith the bending stiffness of the string This parameterincreases greatly with age of strings probably from thedamping effect of dirt and corrosion products accumulat-ing between the windings of the lower strings of the guitar[13] Strings without over-winding do not deteriorate insound to anywhere near the same extent

The correct interpretation of the high sensitivity to shift-ing body frequencies is not obvious The actual change in-vestigated here moving all frequencies in step is very arti-ficial Listeners might be responding to the actual frequen-cies or perhaps to the change in modal density (the in-verse of the mean spacing between resonant frequencies)or perhaps to the shifting amplitude envelope of the bodyresponse Modal amplitudes were not changed when thefrequencies were scaled so the amplitude pattern movesalong the frequency axis with the scaling It would re-quire further testing to establish whether one of these fac-tors dominates or whether all contribute significantly Suchtesting would be a good target for future research to bringthe results closer to the detailed concerns of instrumentmakers

AcknowledgementThe authors thank Brian Moore Ian Cross and Alan Black-well for many helpful discussions on this work and MartinWoodhouse Stewart French Ian Cross and James West-brook for providing access to guitars for testing Twoanonymous reviewers are thanked for constructive sugges-tions

References

[1] C Fritz I Cross B C J Moore J Woodhouse Perceptualthresholds for detecting modifications applied to the acous-tical properties of a violin J Acoust Soc Am 122 (2007)3640ndash3650

[2] J Woodhouse P Galluzzo The bowed string as we know ittoday Acta Acustica united with Acustica 90 (2004) 579ndash589

[3] J Woodhouse On the synthesis of guitar plucks ActaAcustica united with Acustica 90 (2004) 928ndash944

[4] J Woodhouse Plucked guitar transients comparison ofmeasurements and synthesis Acta Acustica united withAcustica 90 (2004) 945ndash965

[5] H A K Wright The acoustics and psychoacoustics of theguitar PhD Thesis Cardiff University 1996

[6] H Jaumlrvelaumlinen V Vaumllimaumlki M Karjalainen Audibility ofthe timbral effects of inharmonicity in stringed instrumenttones Acoustics Research Letters Online 2 (2001) 79ndash84

485

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

quencies were shifted only in limited bands The twobands studied were shown in Figure 2 and this time theonly input sound tested was the single-note open A stringThe results of the tests are shown in Figure 5 in the sameformat as Figure 4 Rather unexpectedly the ldquobest fourrdquothreshold for the first band containing the strongest peaksof the datum guitar was much higher than for the sec-ond band Related tests for the violin using the so-calledldquoDuumlnnwald bandsrdquo gave thresholds around 2 for the firstthree bands (see Figure 6 of Fritz et al [1]) Band 1 in thepresent study gave a comparable result but band 2 gave alower result than was observed in any of the violin testsThe best-four JND in this case is virtually the same as forthe corresponding test when all frequencies were shiftedFurther studies might be worth conducting to add moredetail to this observation

The final pair of tests relating to the perception of bodymode changes investigated the effect of damping factorsThis time the mode frequencies were kept fixed but themodal Q factors were all scaled by a given factor Q fac-tor is the inverse of damping factor so high Q correspondsto low damping In two separate tests the Q factors wereincreased and decreased to look for any asymmetry in theperception Thirteen test subjects took these two tests 10musicians and 3 non-musicians Results are shown in Fig-ure 6 in the same format as Figs 4 and 5 except that thenumbers on the horizontal axis now indicate the scale fac-tor applied rather than the percentage change This meansthat ldquono changerdquo corresponds to the value 1 rather than 0as in the previous results which has the effect of makingthe relative standard deviations numerically smaller

An asymmetry is seen between the two tests but it doesnot significantly affect the threshold value for the bestfour listeners They achieved a value around 12 ie a20 change in Q factors in both cases However acrossthe whole set of test subjects the spread is much greaterwhen the Q factors were decreased compared to the in-creasing case Many participants reported initial difficultyin hearing the difference when Q factors were decreasedalthough once they had learned to recognise the soundchange they often achieved quite low thresholds

It was not thought worthwhile to test separate frequencybands in this case for two reasons First body damping ismost readily altered by a physical change of material andthis is likely to have an effect over a broad frequency rangeSecond it will be shown shortly that the JND for percep-tion of a change in body damping is of the same orderas the measured standard deviation among real guitars sothat the perceptual significance is probably marginal anddigging deeper into the details is likely to be of limitedimportance

42 Physical measurements

The next step is to look at physical measurements to as-sess how these thresholds of perception compare with theactual extent of variation between guitars Of course onewould never find two real instruments which differed byhaving all the body frequencies simultaneously scaled by

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(a) 150-250 Hz

03 05 1 2 3 5 100

05

1

Percentage shift

RelSTD

(b) 500-1000 Hz

Figure 5 Results in the format of Figure 4 from 3AFC studies forshifting all mode frequencies of the datum guitar lying in (a) theband 150ndash250 Hz and (b) the band 500ndash1000 Hz using as inputa single note (open A string) The percentage shifts correspond-ing to the best-four averages are (a) 187 (b) 153

1 12 15 2 25 30

005

01

015

02

(a)

Scale factor

RelSTD

1 12 15 2 25 30

005

01

015

02

(b)

Scale factor

RelSTD

Figure 6 Results in the format of Figure 4 from 3AFC studiesfor scaling all modal Q factors of the datum guitar (a) upwardsand (b) downwards using as input a single note (open A string)Horizontal axis shows scale factor applied by multiplication ordivision to Q factors The scale factors corresponding to the best-four averages are (a) 119 (b) 122

exactly the same factor However it is easy to visualise achange which would in principle have this effect it wouldhappen if the dimensions of the guitar were all scaledkeeping the same materials and other details Some sim-ple deductions can then be drawn by scaling analysis Ifthe main vibrating elements of the guitar body are treatedas thin flat plates deforming in bending then the naturalfrequencies scale proportional to hd2 where d is the scalefactor for dimensions in the plane and h is the scale factorfor thickness Taking 1 as a round number for the orderof magnitude of the threshold of detection for shifting fre-quencies it follows that the JNDs for dimensional scalingare approximately 1 if d and h are both scaled together(full geometric scaling) 1 again if thickness is changedkeeping the in-plane dimensions fixed and only 05 ifin-plane dimensions are scaled keeping the thickness thesame These are all very small numbers they amount tochanging in-plane dimensions no more than a few mm and

482

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

75 80 85 90 95 1000

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(a)

160 170 180 190 200 210 2200

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(b)

Figure 7 Modal frequencies and Q factors for 15 guitars forthe highest peak in the frequency bands (a) 75ndash100 Hz theldquoHelmholtz resonancerdquo (b) 150ndash250 Hz the ldquofirst top platemoderdquo Plus and cross symbols show fits without and with com-pensation for string damping respectively Diamond symbolsmark the datum guitar for the listening tests Open squares showthe means and range bars show one standard deviation aboveand below the means for the two sets of fitted data The rangesfor the compensated results can be recognised by their higher Qvalues Dashed lines near the axes show approximately 1 JNDabove and below the uncompensated means from the results ofsection 41

to changing a typical top plate thickness by a mere 25 micromReal instruments have dimensional variations which areconsiderably bigger than this especially in thickness soon this ground alone it is no surprise that guitars can sounddifferent

The simplest approach to obtaining quantitative infor-mation from admittance measurements on a set of gui-tars is to perform explicit modal analysis This can onlybe expected to give reliable results at relatively low fre-quencies pole-residue extraction whatever the methodbecomes sensitive once the modal overlap begins to ap-proach unity To illustrate the kind of values which emergefrom such an analysis Figure 7 shows results for modalfrequency and Q factor for 15 guitars for the highest peakin each of the two low-frequency bands mentioned in thediscussion of Figure 2 75ndash100 Hz the ldquoHelmholtz reso-nancerdquo and 150ndash250 Hz the ldquofirst top plate moderdquo

A complicating factor needs to be noted The ad-mittance measurements are made with the strings welldamped because the purpose is to obtain informationabout the body behaviour without the effect of the stringsHowever the damped strings remove some energy fromthe body vibration and this will tend to reduce the mea-sured Q factors This effect can be compensated at leastapproximately The damped strings will act essentially assemi-infinite strings and will thus present an impedance tothe bridge which is real and if the strings are all assumedto be attached at the same point equal to the sum of thestring impedances given by Z =

radicTm From the data for

nylon strings in Table I the sum of these impedances is212 Nsm To compensate the inverse of Y (ω) is takenthis real quantity is subtracted then the result re-invertedto give a new admittance This approach can be expectedto work well at higher frequencies but it is hard to be sureif the strings are well enough damped at low frequenciesto justify the semi-infinite approximation

Figure 7 shows results before and after this compensa-tion For each case the mean and scatter are indicated bylines showing one standard deviation above and below themean for frequency and Q Interestingly the standard de-viations have a similar pattern for both cases The relativestandard deviations for frequency are 55 and 6 for thetwo cases and those for Q factor about 33 in both casesbefore compensation

To obtain good comparative information about modalfrequencies over a broader bandwidth is more challeng-ing It is not possible to identify the ldquosamerdquo mode in dif-ferent guitars except at very low frequencies sensitivityto small structural changes increases with frequency as isfamiliar in other vibration problems (see eg [15]) Formode frequencies the threshold for audible changes is sosmall that it is hard to obtain any measure which is goodenough scaling frequencies by 1 means that one guitarbody might have 101 modes in a frequency range whereanother guitar had 100 and there is no known analysismethod which can give robust discrimination at this levelAny measure which can be computed shows much biggervariability than this and there is always a strong suspicionthat it is revealing the vagaries of the analysis method morethan the real differences between guitars

Things are a little more promising in the case of damp-ing estimation over a wide frequency range Taking theinverse FFT of an input admittance gives an impulse re-sponse function of the guitar body and this can be pro-cessed using time-frequency analysis to give a sonogramrepresentation In each frequency band a best-fitting ex-ponential decay can then be obtained Combining resultsfrom several different choices of the timefrequency res-olution tradeoff improves the robustness of the resultsThe process is described in more detail elsewhere [16]Some results are shown in Figure 8 compensation forstring damping has been included In Figure 8a the discretepoints represent for the datum guitar frequencydecaycombinations which satisfied an assumed criterion forgood fit quality For compatibility with other plots decay

483

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70

80

90

100

Frequency

Qfactor

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100

120

Frequency

Qfactor

(b)

30 35 40 45 50 55 60 6515

20

25

30

35

40

Mean Q

Relativestandard

deviation(

)

(c)

Figure 8 Damping trends for guitar bodies determined fromtime-frequency analysis (a) individual measurements of fre-quency and equivalent Q factor (circles) and the band averageof these (solid line) for the datum guitar (b) the solid lines asin (a) for the set of 15 guitars (c) the mean and standard devia-tion of the results from (b) across the frequency range with rangebars indicating one standard deviation above and below the meanon each axis The circle marks the datum guitar for the listeningtests

rate is represented as an equivalent Q factor The continu-ous line is simply a moving band average of these discretepoints

Figure 8b then shows the band-averaged lines for the 15guitars in the test set Finally Figure 8c shows a scatterplot of the mean and standard deviation of these band-averaged lines for the 15 guitars As in Figure 7 thelarge cross represents one standard deviation above and

2 3 5 8 100

01

02

03

04 (a)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (b)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (c)

Scale factor

RelSTD

Figure 9 Results in the format of Figure 4 from 3AFC studiesfor changing the damping parameters of the 5th string based ondatum values for the new steel strings Case (a) varying ηB testnote E2 (b) varying ηB test note C3 (c) varying ηF test noteC3 The scale factors corresponding to the best-four averages are(a) 299 (b) 313 (c) 241

below the mean on each axis The main result is thatthis frequency-averaged measure of damping has a rela-tive standard deviation of the order of 20 comparable tothe JND found earlier for perception of damping changesThis suggests that the extremes of mean damping in themeasured set of guitars are sufficiently different that theeffect should be audible but the results do not lead oneto expect that this will be a very strong component of theperception of tonal differences among guitars

5 Results for properties of the strings

The final set of JND experiments relates to changes instring properties including those properties that changedas the string aged one objective was to find out whichof those changes are likely to contribute most to the per-ception of ageing Using as datum the properties of thenew steel strings listed in Table I changes to the dampingproperties of the 5th string were investigated using single-note test sounds These 3AFC tests were completed by 11musicians and 5 non-musicians One test was also con-ducted relating to changes in the bending stiffness of thisstring it gave results in line with those of Jaumlrvelaumlinen etal [7] and the details are not reproduced here The actualbending stiffness of this string was found to be close tothe threshold for detecting an increase in bending stiffnesswhen starting from a much lower level In the aged stringthe bending stiffness only increased by a factor 16 (seeTable I) If that change is audible at all the effect is likelyto be small

The results for the damping tests are shown in Figure 9in a similar format to earlier 3AFC results As noted ear-lier the perceptual effects of changing ηA for this string

484

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

were found to be so weak that it was not possible to con-duct a meaningful test However both ηB and ηF gave clearresults The parameter ηB associated with bending stiff-ness and mainly affecting higher frequencies was testedusing two different notes ldquoplayedrdquo on the same string Theresults are shown in Figures 9ab and show a good levelof consistency between the two notes The spread over-all mean and best-four mean are all very similar for bothcases The best listeners need about a factor 3 increase inthis parameter to perceive the change Note from Table Ithat in the aged string this parameter increased from 0004to 0029 that change should be very clearly audible

Figure 9c shows corresponding results for ηF All sub-jects gave a somewhat lower threshold for this parametercompared to ηB and the very best listener achieved a JNDof less than a factor 2 In the aged string ηF increased inthe ratio 1711 for this string from Table I This effect isprobably not audible to most listeners It can be concludedthat the factor most likely to account for the difference insound of the aged string is ηB

6 Conclusions

Psychoacoustical experiments have been reported to estab-lish JND levels for the perception of changes to variousparameters of a model of synthesised guitar sound [3 4]These investigations parallel earlier work exploring relatedJNDs for violin sounds [1] The parameters investigatedhere relate to the behaviour of the guitar body (modal fre-quencies and damping factors) and of the strings (bendingstiffness and damping model parameters) All tests werecarried out using a three-alternative forced-choice method-ology It is argued that for application to musical judge-ments which can be very subtle the most interesting per-ceptual thresholds are those attainable by the best listenersunder the most favourable conditions To give an estimateof these the results for the best four subjects in each testwere averaged

Results of physical measurements have also been re-ported to give an indication of the spread of each parame-ter among real guitar bodies or strings A comparison withthe psychoacoustical results gives a first impression ofwhich quantities might be responsible for the biggest per-ceptual differences between instruments By far the mostsensitive parameter of all those tested relates to shiftingthe frequencies of body modes The best four listenerscould perceive a 1 shift when all frequencies were scaledtogether a small change compared to the actual varia-tion among guitars This performance was only degradedslightly when the shifted frequencies were band-limited intwo different bands tested The results were of the sameorder as the corresponding JNDs found for violin sounds[1] if anything the JNDs seem to be somewhat lower forthe guitar sounds perhaps because the fully-coupled syn-thesis method gives the listener additional auditory cuesassociated with transient decay rates of the string over-tones Body mode damping has to be changed by about20 for perception according to the definition used hereActual variations of modal damping exceed this level but

not by a large factor so body damping is probably quite aminor contributor to tonal perception

String damping parameters are even less sensitive theyhave to be changed by a factor of the order of 3 How-ever this large factor does not mean that the effects arenever heard Measurements have been presented on stringsof different materials and on steel-cored strings in newand ldquoagedrdquo conditions The damping parameters vary be-tween these cases by factors significantly bigger than theJNDs The results strongly indicate that the most importantparameter for the perception of the degradation of soundin well-used strings is ηB the damping factor associatedwith the bending stiffness of the string This parameterincreases greatly with age of strings probably from thedamping effect of dirt and corrosion products accumulat-ing between the windings of the lower strings of the guitar[13] Strings without over-winding do not deteriorate insound to anywhere near the same extent

The correct interpretation of the high sensitivity to shift-ing body frequencies is not obvious The actual change in-vestigated here moving all frequencies in step is very arti-ficial Listeners might be responding to the actual frequen-cies or perhaps to the change in modal density (the in-verse of the mean spacing between resonant frequencies)or perhaps to the shifting amplitude envelope of the bodyresponse Modal amplitudes were not changed when thefrequencies were scaled so the amplitude pattern movesalong the frequency axis with the scaling It would re-quire further testing to establish whether one of these fac-tors dominates or whether all contribute significantly Suchtesting would be a good target for future research to bringthe results closer to the detailed concerns of instrumentmakers

AcknowledgementThe authors thank Brian Moore Ian Cross and Alan Black-well for many helpful discussions on this work and MartinWoodhouse Stewart French Ian Cross and James West-brook for providing access to guitars for testing Twoanonymous reviewers are thanked for constructive sugges-tions

References

[1] C Fritz I Cross B C J Moore J Woodhouse Perceptualthresholds for detecting modifications applied to the acous-tical properties of a violin J Acoust Soc Am 122 (2007)3640ndash3650

[2] J Woodhouse P Galluzzo The bowed string as we know ittoday Acta Acustica united with Acustica 90 (2004) 579ndash589

[3] J Woodhouse On the synthesis of guitar plucks ActaAcustica united with Acustica 90 (2004) 928ndash944

[4] J Woodhouse Plucked guitar transients comparison ofmeasurements and synthesis Acta Acustica united withAcustica 90 (2004) 945ndash965

[5] H A K Wright The acoustics and psychoacoustics of theguitar PhD Thesis Cardiff University 1996

[6] H Jaumlrvelaumlinen V Vaumllimaumlki M Karjalainen Audibility ofthe timbral effects of inharmonicity in stringed instrumenttones Acoustics Research Letters Online 2 (2001) 79ndash84

485

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

75 80 85 90 95 1000

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(a)

160 170 180 190 200 210 2200

10

20

30

40

50

60

70

80

Frequency (Hz)

Q

(b)

Figure 7 Modal frequencies and Q factors for 15 guitars forthe highest peak in the frequency bands (a) 75ndash100 Hz theldquoHelmholtz resonancerdquo (b) 150ndash250 Hz the ldquofirst top platemoderdquo Plus and cross symbols show fits without and with com-pensation for string damping respectively Diamond symbolsmark the datum guitar for the listening tests Open squares showthe means and range bars show one standard deviation aboveand below the means for the two sets of fitted data The rangesfor the compensated results can be recognised by their higher Qvalues Dashed lines near the axes show approximately 1 JNDabove and below the uncompensated means from the results ofsection 41

to changing a typical top plate thickness by a mere 25 micromReal instruments have dimensional variations which areconsiderably bigger than this especially in thickness soon this ground alone it is no surprise that guitars can sounddifferent

The simplest approach to obtaining quantitative infor-mation from admittance measurements on a set of gui-tars is to perform explicit modal analysis This can onlybe expected to give reliable results at relatively low fre-quencies pole-residue extraction whatever the methodbecomes sensitive once the modal overlap begins to ap-proach unity To illustrate the kind of values which emergefrom such an analysis Figure 7 shows results for modalfrequency and Q factor for 15 guitars for the highest peakin each of the two low-frequency bands mentioned in thediscussion of Figure 2 75ndash100 Hz the ldquoHelmholtz reso-nancerdquo and 150ndash250 Hz the ldquofirst top plate moderdquo

A complicating factor needs to be noted The ad-mittance measurements are made with the strings welldamped because the purpose is to obtain informationabout the body behaviour without the effect of the stringsHowever the damped strings remove some energy fromthe body vibration and this will tend to reduce the mea-sured Q factors This effect can be compensated at leastapproximately The damped strings will act essentially assemi-infinite strings and will thus present an impedance tothe bridge which is real and if the strings are all assumedto be attached at the same point equal to the sum of thestring impedances given by Z =

radicTm From the data for

nylon strings in Table I the sum of these impedances is212 Nsm To compensate the inverse of Y (ω) is takenthis real quantity is subtracted then the result re-invertedto give a new admittance This approach can be expectedto work well at higher frequencies but it is hard to be sureif the strings are well enough damped at low frequenciesto justify the semi-infinite approximation

Figure 7 shows results before and after this compensa-tion For each case the mean and scatter are indicated bylines showing one standard deviation above and below themean for frequency and Q Interestingly the standard de-viations have a similar pattern for both cases The relativestandard deviations for frequency are 55 and 6 for thetwo cases and those for Q factor about 33 in both casesbefore compensation

To obtain good comparative information about modalfrequencies over a broader bandwidth is more challeng-ing It is not possible to identify the ldquosamerdquo mode in dif-ferent guitars except at very low frequencies sensitivityto small structural changes increases with frequency as isfamiliar in other vibration problems (see eg [15]) Formode frequencies the threshold for audible changes is sosmall that it is hard to obtain any measure which is goodenough scaling frequencies by 1 means that one guitarbody might have 101 modes in a frequency range whereanother guitar had 100 and there is no known analysismethod which can give robust discrimination at this levelAny measure which can be computed shows much biggervariability than this and there is always a strong suspicionthat it is revealing the vagaries of the analysis method morethan the real differences between guitars

Things are a little more promising in the case of damp-ing estimation over a wide frequency range Taking theinverse FFT of an input admittance gives an impulse re-sponse function of the guitar body and this can be pro-cessed using time-frequency analysis to give a sonogramrepresentation In each frequency band a best-fitting ex-ponential decay can then be obtained Combining resultsfrom several different choices of the timefrequency res-olution tradeoff improves the robustness of the resultsThe process is described in more detail elsewhere [16]Some results are shown in Figure 8 compensation forstring damping has been included In Figure 8a the discretepoints represent for the datum guitar frequencydecaycombinations which satisfied an assumed criterion forgood fit quality For compatibility with other plots decay

483

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70

80

90

100

Frequency

Qfactor

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100

120

Frequency

Qfactor

(b)

30 35 40 45 50 55 60 6515

20

25

30

35

40

Mean Q

Relativestandard

deviation(

)

(c)

Figure 8 Damping trends for guitar bodies determined fromtime-frequency analysis (a) individual measurements of fre-quency and equivalent Q factor (circles) and the band averageof these (solid line) for the datum guitar (b) the solid lines asin (a) for the set of 15 guitars (c) the mean and standard devia-tion of the results from (b) across the frequency range with rangebars indicating one standard deviation above and below the meanon each axis The circle marks the datum guitar for the listeningtests

rate is represented as an equivalent Q factor The continu-ous line is simply a moving band average of these discretepoints

Figure 8b then shows the band-averaged lines for the 15guitars in the test set Finally Figure 8c shows a scatterplot of the mean and standard deviation of these band-averaged lines for the 15 guitars As in Figure 7 thelarge cross represents one standard deviation above and

2 3 5 8 100

01

02

03

04 (a)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (b)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (c)

Scale factor

RelSTD

Figure 9 Results in the format of Figure 4 from 3AFC studiesfor changing the damping parameters of the 5th string based ondatum values for the new steel strings Case (a) varying ηB testnote E2 (b) varying ηB test note C3 (c) varying ηF test noteC3 The scale factors corresponding to the best-four averages are(a) 299 (b) 313 (c) 241

below the mean on each axis The main result is thatthis frequency-averaged measure of damping has a rela-tive standard deviation of the order of 20 comparable tothe JND found earlier for perception of damping changesThis suggests that the extremes of mean damping in themeasured set of guitars are sufficiently different that theeffect should be audible but the results do not lead oneto expect that this will be a very strong component of theperception of tonal differences among guitars

5 Results for properties of the strings

The final set of JND experiments relates to changes instring properties including those properties that changedas the string aged one objective was to find out whichof those changes are likely to contribute most to the per-ception of ageing Using as datum the properties of thenew steel strings listed in Table I changes to the dampingproperties of the 5th string were investigated using single-note test sounds These 3AFC tests were completed by 11musicians and 5 non-musicians One test was also con-ducted relating to changes in the bending stiffness of thisstring it gave results in line with those of Jaumlrvelaumlinen etal [7] and the details are not reproduced here The actualbending stiffness of this string was found to be close tothe threshold for detecting an increase in bending stiffnesswhen starting from a much lower level In the aged stringthe bending stiffness only increased by a factor 16 (seeTable I) If that change is audible at all the effect is likelyto be small

The results for the damping tests are shown in Figure 9in a similar format to earlier 3AFC results As noted ear-lier the perceptual effects of changing ηA for this string

484

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

were found to be so weak that it was not possible to con-duct a meaningful test However both ηB and ηF gave clearresults The parameter ηB associated with bending stiff-ness and mainly affecting higher frequencies was testedusing two different notes ldquoplayedrdquo on the same string Theresults are shown in Figures 9ab and show a good levelof consistency between the two notes The spread over-all mean and best-four mean are all very similar for bothcases The best listeners need about a factor 3 increase inthis parameter to perceive the change Note from Table Ithat in the aged string this parameter increased from 0004to 0029 that change should be very clearly audible

Figure 9c shows corresponding results for ηF All sub-jects gave a somewhat lower threshold for this parametercompared to ηB and the very best listener achieved a JNDof less than a factor 2 In the aged string ηF increased inthe ratio 1711 for this string from Table I This effect isprobably not audible to most listeners It can be concludedthat the factor most likely to account for the difference insound of the aged string is ηB

6 Conclusions

Psychoacoustical experiments have been reported to estab-lish JND levels for the perception of changes to variousparameters of a model of synthesised guitar sound [3 4]These investigations parallel earlier work exploring relatedJNDs for violin sounds [1] The parameters investigatedhere relate to the behaviour of the guitar body (modal fre-quencies and damping factors) and of the strings (bendingstiffness and damping model parameters) All tests werecarried out using a three-alternative forced-choice method-ology It is argued that for application to musical judge-ments which can be very subtle the most interesting per-ceptual thresholds are those attainable by the best listenersunder the most favourable conditions To give an estimateof these the results for the best four subjects in each testwere averaged

Results of physical measurements have also been re-ported to give an indication of the spread of each parame-ter among real guitar bodies or strings A comparison withthe psychoacoustical results gives a first impression ofwhich quantities might be responsible for the biggest per-ceptual differences between instruments By far the mostsensitive parameter of all those tested relates to shiftingthe frequencies of body modes The best four listenerscould perceive a 1 shift when all frequencies were scaledtogether a small change compared to the actual varia-tion among guitars This performance was only degradedslightly when the shifted frequencies were band-limited intwo different bands tested The results were of the sameorder as the corresponding JNDs found for violin sounds[1] if anything the JNDs seem to be somewhat lower forthe guitar sounds perhaps because the fully-coupled syn-thesis method gives the listener additional auditory cuesassociated with transient decay rates of the string over-tones Body mode damping has to be changed by about20 for perception according to the definition used hereActual variations of modal damping exceed this level but

not by a large factor so body damping is probably quite aminor contributor to tonal perception

String damping parameters are even less sensitive theyhave to be changed by a factor of the order of 3 How-ever this large factor does not mean that the effects arenever heard Measurements have been presented on stringsof different materials and on steel-cored strings in newand ldquoagedrdquo conditions The damping parameters vary be-tween these cases by factors significantly bigger than theJNDs The results strongly indicate that the most importantparameter for the perception of the degradation of soundin well-used strings is ηB the damping factor associatedwith the bending stiffness of the string This parameterincreases greatly with age of strings probably from thedamping effect of dirt and corrosion products accumulat-ing between the windings of the lower strings of the guitar[13] Strings without over-winding do not deteriorate insound to anywhere near the same extent

The correct interpretation of the high sensitivity to shift-ing body frequencies is not obvious The actual change in-vestigated here moving all frequencies in step is very arti-ficial Listeners might be responding to the actual frequen-cies or perhaps to the change in modal density (the in-verse of the mean spacing between resonant frequencies)or perhaps to the shifting amplitude envelope of the bodyresponse Modal amplitudes were not changed when thefrequencies were scaled so the amplitude pattern movesalong the frequency axis with the scaling It would re-quire further testing to establish whether one of these fac-tors dominates or whether all contribute significantly Suchtesting would be a good target for future research to bringthe results closer to the detailed concerns of instrumentmakers

AcknowledgementThe authors thank Brian Moore Ian Cross and Alan Black-well for many helpful discussions on this work and MartinWoodhouse Stewart French Ian Cross and James West-brook for providing access to guitars for testing Twoanonymous reviewers are thanked for constructive sugges-tions

References

[1] C Fritz I Cross B C J Moore J Woodhouse Perceptualthresholds for detecting modifications applied to the acous-tical properties of a violin J Acoust Soc Am 122 (2007)3640ndash3650

[2] J Woodhouse P Galluzzo The bowed string as we know ittoday Acta Acustica united with Acustica 90 (2004) 579ndash589

[3] J Woodhouse On the synthesis of guitar plucks ActaAcustica united with Acustica 90 (2004) 928ndash944

[4] J Woodhouse Plucked guitar transients comparison ofmeasurements and synthesis Acta Acustica united withAcustica 90 (2004) 945ndash965

[5] H A K Wright The acoustics and psychoacoustics of theguitar PhD Thesis Cardiff University 1996

[6] H Jaumlrvelaumlinen V Vaumllimaumlki M Karjalainen Audibility ofthe timbral effects of inharmonicity in stringed instrumenttones Acoustics Research Letters Online 2 (2001) 79ndash84

485

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70

80

90

100

Frequency

Qfactor

(a)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100

120

Frequency

Qfactor

(b)

30 35 40 45 50 55 60 6515

20

25

30

35

40

Mean Q

Relativestandard

deviation(

)

(c)

Figure 8 Damping trends for guitar bodies determined fromtime-frequency analysis (a) individual measurements of fre-quency and equivalent Q factor (circles) and the band averageof these (solid line) for the datum guitar (b) the solid lines asin (a) for the set of 15 guitars (c) the mean and standard devia-tion of the results from (b) across the frequency range with rangebars indicating one standard deviation above and below the meanon each axis The circle marks the datum guitar for the listeningtests

rate is represented as an equivalent Q factor The continu-ous line is simply a moving band average of these discretepoints

Figure 8b then shows the band-averaged lines for the 15guitars in the test set Finally Figure 8c shows a scatterplot of the mean and standard deviation of these band-averaged lines for the 15 guitars As in Figure 7 thelarge cross represents one standard deviation above and

2 3 5 8 100

01

02

03

04 (a)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (b)

Scale factor

RelSTD

2 3 5 8 100

01

02

03

04 (c)

Scale factor

RelSTD

Figure 9 Results in the format of Figure 4 from 3AFC studiesfor changing the damping parameters of the 5th string based ondatum values for the new steel strings Case (a) varying ηB testnote E2 (b) varying ηB test note C3 (c) varying ηF test noteC3 The scale factors corresponding to the best-four averages are(a) 299 (b) 313 (c) 241

below the mean on each axis The main result is thatthis frequency-averaged measure of damping has a rela-tive standard deviation of the order of 20 comparable tothe JND found earlier for perception of damping changesThis suggests that the extremes of mean damping in themeasured set of guitars are sufficiently different that theeffect should be audible but the results do not lead oneto expect that this will be a very strong component of theperception of tonal differences among guitars

5 Results for properties of the strings

The final set of JND experiments relates to changes instring properties including those properties that changedas the string aged one objective was to find out whichof those changes are likely to contribute most to the per-ception of ageing Using as datum the properties of thenew steel strings listed in Table I changes to the dampingproperties of the 5th string were investigated using single-note test sounds These 3AFC tests were completed by 11musicians and 5 non-musicians One test was also con-ducted relating to changes in the bending stiffness of thisstring it gave results in line with those of Jaumlrvelaumlinen etal [7] and the details are not reproduced here The actualbending stiffness of this string was found to be close tothe threshold for detecting an increase in bending stiffnesswhen starting from a much lower level In the aged stringthe bending stiffness only increased by a factor 16 (seeTable I) If that change is audible at all the effect is likelyto be small

The results for the damping tests are shown in Figure 9in a similar format to earlier 3AFC results As noted ear-lier the perceptual effects of changing ηA for this string

484

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

were found to be so weak that it was not possible to con-duct a meaningful test However both ηB and ηF gave clearresults The parameter ηB associated with bending stiff-ness and mainly affecting higher frequencies was testedusing two different notes ldquoplayedrdquo on the same string Theresults are shown in Figures 9ab and show a good levelof consistency between the two notes The spread over-all mean and best-four mean are all very similar for bothcases The best listeners need about a factor 3 increase inthis parameter to perceive the change Note from Table Ithat in the aged string this parameter increased from 0004to 0029 that change should be very clearly audible

Figure 9c shows corresponding results for ηF All sub-jects gave a somewhat lower threshold for this parametercompared to ηB and the very best listener achieved a JNDof less than a factor 2 In the aged string ηF increased inthe ratio 1711 for this string from Table I This effect isprobably not audible to most listeners It can be concludedthat the factor most likely to account for the difference insound of the aged string is ηB

6 Conclusions

Psychoacoustical experiments have been reported to estab-lish JND levels for the perception of changes to variousparameters of a model of synthesised guitar sound [3 4]These investigations parallel earlier work exploring relatedJNDs for violin sounds [1] The parameters investigatedhere relate to the behaviour of the guitar body (modal fre-quencies and damping factors) and of the strings (bendingstiffness and damping model parameters) All tests werecarried out using a three-alternative forced-choice method-ology It is argued that for application to musical judge-ments which can be very subtle the most interesting per-ceptual thresholds are those attainable by the best listenersunder the most favourable conditions To give an estimateof these the results for the best four subjects in each testwere averaged

Results of physical measurements have also been re-ported to give an indication of the spread of each parame-ter among real guitar bodies or strings A comparison withthe psychoacoustical results gives a first impression ofwhich quantities might be responsible for the biggest per-ceptual differences between instruments By far the mostsensitive parameter of all those tested relates to shiftingthe frequencies of body modes The best four listenerscould perceive a 1 shift when all frequencies were scaledtogether a small change compared to the actual varia-tion among guitars This performance was only degradedslightly when the shifted frequencies were band-limited intwo different bands tested The results were of the sameorder as the corresponding JNDs found for violin sounds[1] if anything the JNDs seem to be somewhat lower forthe guitar sounds perhaps because the fully-coupled syn-thesis method gives the listener additional auditory cuesassociated with transient decay rates of the string over-tones Body mode damping has to be changed by about20 for perception according to the definition used hereActual variations of modal damping exceed this level but

not by a large factor so body damping is probably quite aminor contributor to tonal perception

String damping parameters are even less sensitive theyhave to be changed by a factor of the order of 3 How-ever this large factor does not mean that the effects arenever heard Measurements have been presented on stringsof different materials and on steel-cored strings in newand ldquoagedrdquo conditions The damping parameters vary be-tween these cases by factors significantly bigger than theJNDs The results strongly indicate that the most importantparameter for the perception of the degradation of soundin well-used strings is ηB the damping factor associatedwith the bending stiffness of the string This parameterincreases greatly with age of strings probably from thedamping effect of dirt and corrosion products accumulat-ing between the windings of the lower strings of the guitar[13] Strings without over-winding do not deteriorate insound to anywhere near the same extent

The correct interpretation of the high sensitivity to shift-ing body frequencies is not obvious The actual change in-vestigated here moving all frequencies in step is very arti-ficial Listeners might be responding to the actual frequen-cies or perhaps to the change in modal density (the in-verse of the mean spacing between resonant frequencies)or perhaps to the shifting amplitude envelope of the bodyresponse Modal amplitudes were not changed when thefrequencies were scaled so the amplitude pattern movesalong the frequency axis with the scaling It would re-quire further testing to establish whether one of these fac-tors dominates or whether all contribute significantly Suchtesting would be a good target for future research to bringthe results closer to the detailed concerns of instrumentmakers

AcknowledgementThe authors thank Brian Moore Ian Cross and Alan Black-well for many helpful discussions on this work and MartinWoodhouse Stewart French Ian Cross and James West-brook for providing access to guitars for testing Twoanonymous reviewers are thanked for constructive sugges-tions

References

[1] C Fritz I Cross B C J Moore J Woodhouse Perceptualthresholds for detecting modifications applied to the acous-tical properties of a violin J Acoust Soc Am 122 (2007)3640ndash3650

[2] J Woodhouse P Galluzzo The bowed string as we know ittoday Acta Acustica united with Acustica 90 (2004) 579ndash589

[3] J Woodhouse On the synthesis of guitar plucks ActaAcustica united with Acustica 90 (2004) 928ndash944

[4] J Woodhouse Plucked guitar transients comparison ofmeasurements and synthesis Acta Acustica united withAcustica 90 (2004) 945ndash965

[5] H A K Wright The acoustics and psychoacoustics of theguitar PhD Thesis Cardiff University 1996

[6] H Jaumlrvelaumlinen V Vaumllimaumlki M Karjalainen Audibility ofthe timbral effects of inharmonicity in stringed instrumenttones Acoustics Research Letters Online 2 (2001) 79ndash84

485

Woodhouse et al Guitar perception ACTA ACUSTICA UNITED WITH ACUSTICAVol 98 (2012)

were found to be so weak that it was not possible to con-duct a meaningful test However both ηB and ηF gave clearresults The parameter ηB associated with bending stiff-ness and mainly affecting higher frequencies was testedusing two different notes ldquoplayedrdquo on the same string Theresults are shown in Figures 9ab and show a good levelof consistency between the two notes The spread over-all mean and best-four mean are all very similar for bothcases The best listeners need about a factor 3 increase inthis parameter to perceive the change Note from Table Ithat in the aged string this parameter increased from 0004to 0029 that change should be very clearly audible

Figure 9c shows corresponding results for ηF All sub-jects gave a somewhat lower threshold for this parametercompared to ηB and the very best listener achieved a JNDof less than a factor 2 In the aged string ηF increased inthe ratio 1711 for this string from Table I This effect isprobably not audible to most listeners It can be concludedthat the factor most likely to account for the difference insound of the aged string is ηB

6 Conclusions

Psychoacoustical experiments have been reported to estab-lish JND levels for the perception of changes to variousparameters of a model of synthesised guitar sound [3 4]These investigations parallel earlier work exploring relatedJNDs for violin sounds [1] The parameters investigatedhere relate to the behaviour of the guitar body (modal fre-quencies and damping factors) and of the strings (bendingstiffness and damping model parameters) All tests werecarried out using a three-alternative forced-choice method-ology It is argued that for application to musical judge-ments which can be very subtle the most interesting per-ceptual thresholds are those attainable by the best listenersunder the most favourable conditions To give an estimateof these the results for the best four subjects in each testwere averaged

Results of physical measurements have also been re-ported to give an indication of the spread of each parame-ter among real guitar bodies or strings A comparison withthe psychoacoustical results gives a first impression ofwhich quantities might be responsible for the biggest per-ceptual differences between instruments By far the mostsensitive parameter of all those tested relates to shiftingthe frequencies of body modes The best four listenerscould perceive a 1 shift when all frequencies were scaledtogether a small change compared to the actual varia-tion among guitars This performance was only degradedslightly when the shifted frequencies were band-limited intwo different bands tested The results were of the sameorder as the corresponding JNDs found for violin sounds[1] if anything the JNDs seem to be somewhat lower forthe guitar sounds perhaps because the fully-coupled syn-thesis method gives the listener additional auditory cuesassociated with transient decay rates of the string over-tones Body mode damping has to be changed by about20 for perception according to the definition used hereActual variations of modal damping exceed this level but

not by a large factor so body damping is probably quite aminor contributor to tonal perception

String damping parameters are even less sensitive theyhave to be changed by a factor of the order of 3 How-ever this large factor does not mean that the effects arenever heard Measurements have been presented on stringsof different materials and on steel-cored strings in newand ldquoagedrdquo conditions The damping parameters vary be-tween these cases by factors significantly bigger than theJNDs The results strongly indicate that the most importantparameter for the perception of the degradation of soundin well-used strings is ηB the damping factor associatedwith the bending stiffness of the string This parameterincreases greatly with age of strings probably from thedamping effect of dirt and corrosion products accumulat-ing between the windings of the lower strings of the guitar[13] Strings without over-winding do not deteriorate insound to anywhere near the same extent

The correct interpretation of the high sensitivity to shift-ing body frequencies is not obvious The actual change in-vestigated here moving all frequencies in step is very arti-ficial Listeners might be responding to the actual frequen-cies or perhaps to the change in modal density (the in-verse of the mean spacing between resonant frequencies)or perhaps to the shifting amplitude envelope of the bodyresponse Modal amplitudes were not changed when thefrequencies were scaled so the amplitude pattern movesalong the frequency axis with the scaling It would re-quire further testing to establish whether one of these fac-tors dominates or whether all contribute significantly Suchtesting would be a good target for future research to bringthe results closer to the detailed concerns of instrumentmakers

AcknowledgementThe authors thank Brian Moore Ian Cross and Alan Black-well for many helpful discussions on this work and MartinWoodhouse Stewart French Ian Cross and James West-brook for providing access to guitars for testing Twoanonymous reviewers are thanked for constructive sugges-tions

References

[1] C Fritz I Cross B C J Moore J Woodhouse Perceptualthresholds for detecting modifications applied to the acous-tical properties of a violin J Acoust Soc Am 122 (2007)3640ndash3650

[2] J Woodhouse P Galluzzo The bowed string as we know ittoday Acta Acustica united with Acustica 90 (2004) 579ndash589

[3] J Woodhouse On the synthesis of guitar plucks ActaAcustica united with Acustica 90 (2004) 928ndash944

[4] J Woodhouse Plucked guitar transients comparison ofmeasurements and synthesis Acta Acustica united withAcustica 90 (2004) 945ndash965

[5] H A K Wright The acoustics and psychoacoustics of theguitar PhD Thesis Cardiff University 1996

[6] H Jaumlrvelaumlinen V Vaumllimaumlki M Karjalainen Audibility ofthe timbral effects of inharmonicity in stringed instrumenttones Acoustics Research Letters Online 2 (2001) 79ndash84

485

ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse et al Guitar perceptionVol 98 (2012)

[7] H Jaumlrvelaumlinen M Karjalainen Perceptibility of inhar-monicity in the acoustic guitar Acta Acustica united withAcustica 92 (2006) 842ndash847

[8] D Ewins Modal testing Theory practice and applicationResearch Studies Press 2000

[9] C Valette The mechanics of vibrating strings ndash In Me-chanics of Musical Instruments A Hirshberg J Kergo-mard G Weinreich (eds) Springer-Verlag Vienna 1995

[10] H Levitt Transformed up-down methods in psychoacous-tics J Acoust Soc Am 49 (1971) 467ndash477

[11] L Cremer The physics of the violin MIT Press Cam-bridge MA 1984

[12] J B Allen On the aging of steel guitar strings CatgutAcoust Soc Newsletter 26 (1976) 27ndash28

[13] R J Hanson Analysis of ldquoliverdquo and ldquodeadrdquo guitar stringsJ Catgut Acoust Soc 48 (1987) 10ndash16

[14] B C J Moore An introduction to the psychology of hear-ing Elsevier Academic Press London 2004

[15] C S Manohar A J Keane Statistics of energy flows inspring-coupled one-dimensional subsystems Phil TransRoyal Soc London A 346 (1994) 525ndash542

[16] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acta Acustica united withAcustica (2012) in press

486