A model of harp plucking

Transcription

1 A model o harp plucking Delphine Chadeaux, Jean-Lo ıc Le Carrou, a) and Beno^ıt Fabre LAM-d Alembert, UPMC Univ. Paris 06, UMR CNRS 7190, Paris, France (Received 26 July 2012; revised 14 January 2013; accepted 24 January 2013) In this paper, a model o the harp plucking is developed. It is split into two successive time phases, the sticking and the slipping phases, and uses a mechanical description o the human inger s behavior. The parameters o the model are identiied through measurements o the inger/string displacements during the interaction. The validity o the model is veriied using a conigurable and repeatable robotic inger, enhanced with a silicone layer. A parametric study is perormed to investigate the inluence o the model s parameters on the ree oscillations o the string. As a result, a direct implementation o the model produces an accurate simulation o a string response to a given inger motion, as compared to experimental data. The set o parameters that govern the plucking action is divided into two groups: Parameters controlled by the harpist and parameters intrinsic to the plucking. The ormer group and to a lesser extent the latter highly inluence the initial conditions o the string vibrations. The simulations o the string s ree oscillations highlight the large impact the model parameters have on the sound produced and thereore allows the understanding o how dierent players on the same instrument can produce a speciic/personal sound quality. VC 2013 Acoustical Society o America. [ PACS number(s): Gh, St [TRM] Pages: I. INTRODUCTION The question o the dierent qualities o the sound played by dierent harpists is a subject o discussion among players and acousticians. While everybody agrees that a player can easily be recognized by his/her style and technique, skilled players insist on the possibility to identiy each other rom the sound only at the individual note level. Obviously, playing at the same plucking position on the string and producing the same global sound power, players can control some other aspects o the sound quality. Earlier results 1 obtained on 10 skilled harpists indicate that each o them provide to the string a highly repeatable plucking path depending on the playing technique and the inger studied. Besides, it has been shown 1 that the plucking position is almost the same or each harpist. It can thereore not explain the plucking speciicity o the harpists. They inely control the initial shape, velocity, angle o polarization, and rotation they provide to the string beore releasing it, resulting in an accurate control o its ree oscillations and o the sound produced. However, the way the player controls the harp plucking is not yet understood on a physical basis. A better understanding o the mechanical parameters that govern the plucking would allow us to control sound synthesis o plucked string instruments in a realistic way. Indeed, although the numerous investigations o the physics o musical instruments allow the production o satisying sound synthesis, 2,3 there is a lack o realism in the control o their initial conditions, i.e., the state in which the musician sets the instrument to produce a sound. This is mostly achieved by tuning parameters until a satisactory sound is reached. Most o the studies about the plucking action and its synthesis deal with the classical guitar. 4 9 The plucking action is described as ideal (Res. 4 6), i.e., the string vibrations are a) Author to whom correspondence should be addressed. Electronic mail: jean-loic.le_carrou@upmc.r initialized only through a displacement with no velocity. 10 Furthermore, the musician s touch is reduced to that o a plectrum, corresponding to a triangular initial shape o the string. However, the presence o the musician and his control on the note produced has been investigated or the classical guitar. 7 9 In these studies, physical modeling o the inger/ string interaction has been proposed with parameters adjusted to produce the desired sound rather than physically relevant considerations. Experimentally based investigations o the concert harp plucking 11,12 has provided inger-string motion to estimate the mechanical parameters o the inger. 11,12 However, the experimental constraints do not allow to point them out in a robust manner. Thereore the estimation o relevant mechanical parameters to describe the plucking action remains a tricky issue. Besides, a study o the piano action mechanism 13 indicates that the viscoelastic behavior o the inger should be taken into account. Looking at the literature, a cautious investigation o the human inger behavior in plucking musical instruments has not yet been undertaken. The present paper aims at modeling the classical concert harp plucking action. The proposed model is based on parameters estimated using measured displacements o the inger and o the string. The latter are expected to describe both the mechanical parameters speciic to the harpist s inger morphology and the one she/he has the possibility to control during plucking. Their impact on the sound produced is also investigated to point out the set o parameters revealing the speciic sound o a musician. A modeling o the inger/string interaction is provided in Sec. II. Then an experimental procedure is described in Sec. III to capture the inger s and the string s motion during the plucking action. On one hand, these measurements help to highlight the mechanical parameters o the musician s inger and on the other hand, they validate their relevance to model the string s response under a given inger s action in Sec. IV. Section V investigates the impact o these control parameters on the 2444 J. Acoust. Soc. Am. 133 (4), April /2013/133(4)/2444/12/$30.00 VC 2013 Acoustical Society o America

2 initial conditions o the string s ree oscillations through a parametric study. Eventually, the parameters estimation is applied in Sec. VI to plucking actions in real musical context derived rom previous measurements. 1 II. HARP PLUCKING ACTION MODELING The harp plucking can be split into two successive time phases: 1 The sticking and the slipping phases. The harp plucking modeling is structured accordingly. A. Sticking phase 1. Description During the sticking phase, the inger pulls a segment o the string rom its initial position up to the point where the tangential shear orce exerted by the string on the skin reaches a threshold orce F max, controlled by the harpist. Assuming that the displacement o the inger s distal phalanx and the string displacement only take place in the plane, ixed to the harp, perpendicular to the strings, 1 we only investigate their trajectories in this plane reerred to as (x0z) in this paper. Their components are reerred to as (x s, z s ) and (x, z )infig.1, respectively. As we only deal with isolated plucking actions, the string is considered to start rom its rest position at t c, i.e., x s ðt ¼ t c Þ¼x 0 ¼ 0, z s ðt ¼ t c Þ¼z 0 ¼ 0, at the beginning o the sticking phase. The mechanical behavior o the inger has to be taken into account to describe the sticking phase because the inger is squeezed while pulling the string. This deormation depends on both the string s and the inger s mechanical properties. 2. Skin s mechanical properties Many studies have investigated the mechanical properties o human inger Whereas it is structured in three layers (the epidermis, the dermis, and the hypodermis) with various mechanical properties, it has been shown that the inger s response to an external load can be seen like a monolayer material with viscoelastic properties. Considering human tissue, Zener and Kelvin Voigt viscoelastic models 18 are commonly used. 19,20 Furthermore, the inger s response to any load depends on the dynamic properties o the stimuli, as its orce, velocity, magnitude, requency, as well as the angle between the inger and the contact surace. 21 Eventually, viscoelastic models with non-linear components depending on the inger indentation are oten used. 15,16, Modeling Figure 1 illustrates the interaction between the inger and the string during the sticking phase with its equivalent model. We model the sticking phase o the plucking action using a Kelvin Voigt model. It consists in a spring and a dash-pot connected in parallel, relecting the elastic and the viscous properties o the material. The spring s stiness and equilibrium length are denoted k and l, respectively, while the damping o the dash-pot is reerred to as c. The equilibrium length l represents the thickness o the inger at rest. On average, it is estimated at 1 cm or the oreinger. Besides, as this phase is quasi-static, we model the string as a single spring o stiness k s. Indeed, assuming that we have a lexible string o uniorm linear density q l, stretched to a tension T and ixed at its ends, its ree oscillations velocity can be easily computed. 25 For instance, regarding the 30th harp string plucked at the third o its length and released with an initial displacement o D tr ¼ 5 mm rom its rest position and an initial velocity o V tr ¼ 2m=s, the maximal string velocity during the ollowing oscillations is estimated at about 3 m/s. Because the string velocity during the sticking phase o typical duration 300 ms does not exceed 0.5 m/s, the latter is then assumed to be quasi-static. k s is estimated based on the string s tension T, its length L, the plucking position y 0, and the width o the excitation Dl as k s ¼ T y 0 Dl 1 þ L y 0 Dl! 1 : (1) 2 2 FIG. 1. Finger/string interaction during the sticking phase and its equivalent modeling. Furthermore, the string equilibrium length l s is chosen to be zero because the origin o the x and z axis is taken at the string s rest position. The harpist s inger displacement in the (x0z) plane is the input o the plucking action modeling. To model the response o the string (x s, z s ) to this excitation, we deine the rame o reerence (~u, ~v) related to the plucking action, where ~u and ~v are normal and tangential to the skin in the contact area, respectively. Figure 1 illustrates this rame o reerence at a given instant o the sticking phase. The string s and the distal phalanx s displacements are reerred to as u s and u along the u axis, respectively. The latter is deined as qiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii u ¼ l ðz z 0 Þ 2 þðx x 0 Þ 2 ; (2) and the plucking orientation a is estimated throughout the sticking phase as J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Chadeaux et al.: Model o harp plucking 2445

3 a ¼ arctan z z 0 : (3) x x 0 These parameters will convey the string s motion in the (x0z) plane: x s ¼ x 0 þ u s cos a; (4) z s ¼ z 0 u s sin a: (5) Under quasi-static hypothesis, the u-axis component o the orce balance between the inger and the spring that models the string is written as k ðu u s @u k s u s ¼ 0: (6) We deine the inger s indentation parameter du (see Fig. 1), as du ¼ l þ u s u : (7) Equation (6) then writes FIG. 2. Finger/string interaction during the slipping phase. k s u s þ k du ¼ 0: (8) The parameters k and c correspond to the elastic and the viscous characteristics o the inger. They need irst to be estimated to determine the string motion u s rom Eq. (6). Although the relation between the load applied by a probe and the indentation has been shown to be exponential by some authors, 15,16 it appears that there is no clear agreement in the literature on the inger s stiness and damping orms with respect to its indentation. Thereore parameters k and c are investigated under both the linear and exponential ollowing orms: k lin ðduþ ¼k a du; clin ðduþ ¼c a du; k exp ðduþ ¼k a ekb du ; c exp ðduþ ¼c a ecb du : (9) The estimation o the parameters k a; kb ; ca, and cb will be carried through measurements o the inger and string motion while plucking a string. This will be presented in Sec. IV. B. Slipping phase 1. Description In the inal moments o the sticking phase, the harpist s inger turns around the string. Hence she/he deines the orientation c o the slipping phase. At the beginning o the slipping phase (t ¼ t s ), the string s position in the (x0z) plane is noted (x ts ; z ts ). From the beginning o the slipping phase until the release instant (t ¼ t r ), the string slips on the inger s surace. The length d s o the slipping corresponds to the initial distance between the ingertip and the string, which is deined at t ¼ t c by the harpist. During the slipping phase, orces occurring on the string s element are the restoring orce ~F ks, the riction orce ~F t and the normal orce ~F n,see Fig Friction properties o the human inger The riction orce governing this phase is investigated in the present paragraph. Note that the string s element contacting the inger is cylindrical and can be in gut, in nylon, or in steel as we ocus on harp strings. In general terms, the riction o the human skin F t is governed by 26 F t ¼ F a þ F d þ F v þ F r ; (10) where (1) F a is the adhesive riction component related to the contacts o asperities between the inger and the contact surace, (2) F d is the riction related to the deormation o the inger, (3) F v is the riction due to capillary adhesion or viscous shearing, relecting the sel lubrication system o the inger, (4) and F r is the riction due to deormation o inger ridges. Investigations o the orearm riction indicate that Eq. (10) can be reduced 27 to the terms F a and F d. However, it has been shown 26,28 that the riction related to the deormation o the inger F d can be neglected relatively to the adhesive riction F a. Thereore to model the string slipping over the inger surace during plucking, the riction orce is assumed to be only described by the adhesive riction F a. In the literature, it is written as a unction the normal orce F n applied by the inger on the contact surace. Because o the viscoelastic properties o the human skin, the Coulomb model predicting a linear dependency o F t in F n through a riction coeicient l has been questioned. 29,30 The non-linear model F a ¼ lf k n (11) has been proposed where k is a coeicient lower than 1. However, a recent investigation 26 o the riction between human ingers and contacting suraces indicates that a 2446 J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Chadeaux et al.: Model o harp plucking

4 two-linear relationship exist between F t and F n, with the junction point at F lim n ¼ 1N: ( F t ¼ l 1 F n ; 8F n F lim n ; F t ¼ l 2 F n ; 8F n > F lim n : As the riction phenomenon in harp plucking occurs or normal orces always greater than 1 N, 1 we assume the relationship between F t and F n to be linear through a unique riction coeicient l. 3. Modeling Figure 2 illustrates the plucking action during the slipping phase. The direction o the slipping is given by the angle c, which is controlled by the harpist. According to the Fig. 2, the string s motion in the (x0z) plane writes as x s ¼ x ts u s cos c; (12) z s ¼ z ts þ u s sin c: (13) Besides, as the string velocity can reach up to 2 m/s during the slipping phase, 1,31 the quasi-static hypothesis we used during the sticking phase can not apply during slipping. Hence the application o the Newton s second law to the string s element contacting the inger surace during the slipping phase writes as q l ~u 2 ¼ ~F ks þ ~F t þ ~F n ; (14) where q l is the string mass per unit length, or linear mass density. According to the Sec. II B 2, the previous the u-axis component o the Eq. (14) is q l u 2 ¼k ~F ks k cos b l k~f n k; (15) where the amplitude o the normal orce ~F n is 32 qiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii k~f n k¼ ðx s x 0 Þ 2 þðz s z 0 Þ 2 LT ðy 0 ðl y 0 ÞÞ ; (16) the amplitude o the restoring orce ~F ks is qiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii k~f ks k¼ ðx s x 0 Þ 2 þðz s z 0 Þ 2 k s ; (17) and the angle b is written Finally, the resolution o Eqs. (12), (13), (18), and (19) is perormed through inite dierence method, conveying the string s motion during slipping (t s < t < t r ). C. Implementation o the model The sticking phase is mostly inluenced by the inger s viscoelastic compression. Because the characteristic time o the sticking phase is long compared to the time period o the string oscillation, a quasi-static description is used. Thereore the contact orce, normal to the skin in the contact area, can be deduced rom the string displacement at the contact point. The slipping phase is triggered at the time when the tangential orce exerted by the string on the inger reaches the maximum sticking orce F max, thereore when the string displacement in the tangential (skin surace) direction reaches the magnitude F max =k s. The orce F max depends on the normal contact orce applied to the skin surace, which is related to the string displacement in the normal direction. Once the slipping phase has begun, the riction orce reduces the natural string acceleration. As a consequence, by adjusting the initial contact position o the string on the inger and inger path during the sticking phase, the player can adjust the position where the slipping phase starts, the duration o the slipping phase, the position o the string release, the string velocity at release, as well as the initial polarization o the ree string oscillation. Figure 3 proposes a block diagram o harp plucking modeling, including the mechanical parameters involved. (1) The harpist s control parameters are the maximal orce F max applied by the inger on the string, as well as the length d s and the orientation c o the slipping phase. (2) The sticking and slipping parameters k a, ca, and cb describe the contact between the inger and the string or a given plucking context (angle between the inger and the string, d s, ). This model allows estimation o the inger mechanical properties rom measurements o inger and string displacements during plucking action. In the ollowing step, the model can predict the string s response to a given inger s distal phalanx motion (x ; z ). These two aspects will be discussed in Sec. V. III. EXPERIMENTAL PROCEDURE The inger/string interaction model proposed in this paper is compared to real inger and string motion during b ¼ c arctan jz s z 0 j jx s x 0 j : (18) Equation (15) then writes q l u 2 ¼ k LT s cos b l y 0 ðl y 0 Þ qiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii ðx s x 0 Þ 2 þðz s z 0 Þ 2 : (19) FIG. 3. Summary o the harp plucking modeling. J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Chadeaux et al.: Model o harp plucking 2447

5 plucking actions. Three conigurations are investigated. First, inger and string motion have been measured or plucking actions perormed by an artiicial inger shown in Fig. 4(a). It is a repeatable and conigurable tool to pluck a string as desired. To model the human plucking, it is enhanced by silicone ingertips. We use cylindrical ingertips with a rounded ending and two dierent hardnesses. Fingertips are reerred to as F1 or F2 in the ollowing with F1 soter than F2. In addition, or variability issues, measurements have been perormed three times with F2. The robot inger has been shown to reproduce accurately an input reerence displacement and to produce a sound close to that o a real harpist s. 31,33 The use o this artiicial inger is justiied by its ability to provide a repeatable plucking with a planar motion, i.e., the closest to the model analysis. Then, to gradually investigate the robustness o the model, isolated plucking actions perormed by a harpist are captured. She has been asked to pluck the 30th string eight times with the right oreinger as illustrated in Fig. 4(b). This second coniguration represents an intermediate step between the robotic inger and harpist s in a real musical context because her plucking technique is more realistic than the ormer (or instance, with an additional rotation o the inger around the string) and does not contain the transitions techniques between two succeeding notes. Eventually, inger and string motions have been measured or plucking actions perormed by 10 harpists in various musical contexts as in arpeggio or chord sequences using the oreinger as well as the annular. The database used is the same as investigated in the previous description o the plucking action. 1 These measurements will help to point out the robustness o the model and to highlight tendencies in the whole set o mechanical parameters estimated according to the musical context. The measurement protocol carried out is mostly based on capturing the motion o the inger and o the string with a high-speed camera set at rames per second. As this experimental method has already been detailed in a previous paper, 1 we summarize here the main steps. The estimation o the inger and the string trajectories is perormed by tracking markers, placed on inger and string at strategic places, through image processing. 1 More precisely, because we are interested in displacements reerred to as x s, x, z s, and z in Fig. 1, markers are positioned as close as possible to the plucking position y 0 and to the nail, respectively. The latter is assumed to be rigid and to provide a good estimation o the distal phalanx displacement. IV. PLUCKING PARAMETERS ESTIMATION A. Sticking parameters 1. Method The sticking parameters reerred to as k a; kb ; ca and c b are estimated using an experimental database o plucking actions (x s, z s )and(x, z ). Using the latter combined with Eq. (2), Eq. (8) is solved with Runge Kutta algorithm or a set o inger s stiness and damping values. Then a wide range o values are tested through the Levenberg Marquardt algorithm. 34,35 This allows determination by minimization o the best set o parameters to solve the equation. For this purpose, the experimental trajectories are previously approximated by a sixth order polynomial curve itting. The robustness o the method to input noise is investigated in Fig. 5 in the case o isolated notes played by the robotic inger. The reconstruction quadratic error is estimated as viiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii! u1 X NT s ¼ t j~u s ðtþ u N th s ðtþj2 ; (20) t¼t s where T s is the sampling period o the experimental data (10 ls) and ~u s the reconstructed string s displacement or a given inger and string (u th s ) displacements over a wide range o artiicial input noise added to the experimental data. The error is estimated or the our models investigated: (1) Model EE: Exponential stiness and damping (k exp, c exp ), (2) Model LE: Linear stiness and exponential damping (k lin, c exp ), (3) Model EL: Exponential stiness and linear damping (k exp, c lin ), (4) Model LL: Linear stiness and damping (k lin, c lin ). FIG. 4. Experimental setup using (a) robotized and (b) real plucking actions. FIG. 5. Evolution o the reconstruction error o a reerence curve versus its signal to noise ratio J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Chadeaux et al.: Model o harp plucking

6 As expected, we observe that is higher or a low signal to noise ratio (SNR). Moreover, this error tends to become stable rom a SNR above 60 db. Because the typical SNR or our measurements is about 70 db, we assume the method to be reliable. Eventually, the comparison o the our models indicate that a linear damping induces a higher reconstruction error ( 0: m) than an exponential one ( 0: m). Thereore models EE and LE appear to be the most relevant ones. Perormances o EE and LE models or several dierent plucking situations show that the latter is more accurate and thus suitable than the ormer, suggesting that a linear stiness and an exponential damping provide a better model. 2. Results The database o the robotic plucking actions and the isolated notes perormed by a harpist are used to determine the sticking parameters k a; ca and c b or model LE. The reconstruction error estimated in percent or each plucking action is reported in Table I. It is computed with respect to the distance covered by the string during the plucking. First, we observe that the percentage error is always very small and that the reconstructions are better or the robotic plucking actions than or the human ones. This result is not surprising because the artiicial inger perorms a planar motion, whereas the harpist provides additional rotation to the string. Then let us consider the robotic plucking actions in Table I. Three repetitions o the same motion have been perormed with the ingertip reerred to as F2. The estimated parameters corresponding are close; this tends to validate the parameter estimation process. Indeed, the variabilities around the mean values are, 3% and 13% or k a; ca and c b, respectively. Eventually, the sticking parameters relect the hardness character o the ingertip: The estimated values o k a and c a are greater or silicone ingertips with higher hardnesses. Thus or a given inger indentation, the inger/string interaction orce has to be higher regarding F2 than F1. Besides, c b relects the TABLE I. Stiness (k a) and damping (ca, cb ) coeicients estimated with the model LE and the percentage o reconstruction quadratic error according to the string s displacement or the whole set o robotic and human plucking actions. k a (N m 2 ) c a (N s m 1 ) c b (m 1 ) ( 10 6 %) maximal inger indentation reachable or a given ingertip. These estimations conirm that a soter material as F1 owns a larger range o possible indentations than F2. Eventually, the sticking parameters obtained or the eight plucking actions perormed by the harpist are investigated. Although the harpist used the same inger, we observe an important variability in the parameters estimations (about 40% around the mean values). It can be explained through variations in the contact surace. Indeed, as the harpist was asked to play isolated notes, she perormed less repeatable plucking actions than in a realistic musical context. Then some plucking actions were or instance perormed close to the ingertip, while others used the inger s pulp. Obviously, the mechanical properties o the inger vary along the distal phalanx, the ingertip being harder than the pulp. The estimation o d s partly supports this assumption since it is measured to be 0.2 mm or P 1;2;3;6;7, to be 0.04 mm or P 4;5, and 0.09 mm or P 8. Besides, the variability o the skin condition over the eight plucking actions and o the contact angle may explain these variations in the parameters estimations. B. Slipping parameters 1. Method The coeicient o riction l is determined or the entire set o measured plucking action. For this purpose, the orientation o the slipping phase c is irst estimated through plucking action measurements 1 to about 458. Then using measurements o the string displacements (x s ; z s ), Eq. (15) is solved using a Runge Kutta algorithm or a given value o the riction coeicient. The Levenberg Marquardt algorithm applied to this resolution with a wide range o riction coeicient values provide the more suitable one through minimization. As or the sticking phase, the experimental trajectories are approximated by a sixth order polynomial curve itting beore this estimation process. 2. Results The human and the robotic plucking action databases are used to estimate their associated inger/string riction coeicients. The reconstruction error, computed ollowing Eq. (20) or each estimation o l, presented in Table II appears to be very small compared to the amplitude o Robotic inger F F F F Harpist P P P P P P P P TABLE II. Friction coeicients (l) and percentage o reconstruction quadratic error according to the string s displacement estimated or the whole set o robotic and human plucking actions. l (10 4 %) l (10 4 %) Robotic inger Harpist F P F P F P F P P P P P J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Chadeaux et al.: Model o harp plucking 2449

7 motion. The slipping phase modeling presented in Sec. IIB3 is thereore reliable. Furthermore, as expected, the variability observed within the eight plucking actions perormed by a harpist is higher than within dierent silicone ingertips. In both cases, l is o the same order o magnitude or all the plucking actions. Hence this result indicates that the silicone ingertip o the robot inger shows slipping properties close to that o a human inger. However, the variability observed between the three estimations o l or the three plucking actions repeated by F2 is not negligible: It seems diicult to estimate accurately the riction coeicient. Indeed, it highly depends on the experimental context as the skin condition (dry, wet, clean, ) 36 and the angle between the inger and the contact surace as well as the shape and the material o the contacting object. 37 In addition, the range o riction coeicients measured or the inger in various experimental conigurations is very large, and no result about the riction o the inger with a spherical probe (the closer experimental context to ours) has been pinpointed in the tribology literature. Hence based on a recent review o experimental results or the riction coeicient o human skin, 37 results ocusing on ingers sliding on various material suraces and on spherical probe sliding on orearms indicate that the normal orce applied by the probe and its material as well as its geometry have a great inluence on the riction coeicient. C. Reconstruction o isolated plucking The parameters o the sticking and o the slipping phases estimated either or the harpist, or or the artiicial inger are used to simulate plucking actions. Figures 6(a) and 6(c) present two selected sets o inger and string displacements measured or the harpist and the artiicial inger, respectively. Taking the inger s displacement as input reerence o the simulation, the modeled string s response is also drawn in solid line. It is computed according to the process presented in Sec. II C. In both cases, the global shape o the string motion is consistent with the measurement. However, a deviation appears in the orientation o the modeled and measured string along the path or the harpist plucking action and, to a lesser extent, or the robotic plucking action in Figs. 6(a) and in 6(c). A close observation o the curves indicates that or the harpist plucking action, the measured string does not ollow the same orientation a as the inger during the plucking. This is most probably due to the rotation the inger applies to the string around its axis, inducing an erroneous estimation o a. As the robotic inger perorms a perectly planar motion, the estimation o the latter variable and the reconstruction o the string displacement are better. Moreover the slight deviation occurring Fig. 6(c) is probably due to the inger indentation, aecting the estimation o the orientation a. Thereore the latter appears to be a key variable to deduce accurately the string displacement (x s, z s ) based on Eqs. (4) and (5). Figures 6(b) and 6(d) present the same results as in Figs. 6(a) and 6(c) but with an additional adjustment o the string orientation a during the sticking phase. The string s displacement reconstruction is obviously more accurate than previously. The motion investigated or the robotic inger shows a more sinusoidal shape than that o the harpist; this helps in minimizing the error in the reconstruction at the end o the sticking phase. Eventually, due to the strong stability o the riction coeicient, the reconstruction o the slipping phase is more straightorward. Thus the simulated displacement o the string matches the measured one or both harpist and robotic plucking actions. V. INFLUENCE OF PLUCKING ON STRING OSCILLATIONS FIG. 6. Measured and simulated plucking action. (a) Plucking action perormed by a real harpist inger, (b) with an additional adjustment o the string orientation, (c) plucking action perormed by the artiicial inger, and (d) with an additional adjustment o the string orientation. A. Method The inluence o the plucking parameters on the string s oscillations are investigated in the present section. For this purpose, we input a inger s displacement (x, z ) into the model and analyze the string s displacement (x s, z s ) produced or a set o plucking characteristics (k a, ca, cb, and l), and control parameters (F max, d s, and c). At the end o the sticking phase, the string s state can be described through its position ~u ts and its velocity ~V ts. Because the string s trajectory will have a direction opposite to that o the inger during the slipping phase, its velocity is close to zero in every possible case. Hence the value o the string velocity at the end o the sticking phase is not expected to be a relevant parameter. However, as ~u ts is related to both the string s displacement relative to its rest position D ts and the slipping orientation, it is assumed to be o great importance relatively to the initial displacement D tr, velocity V tr, and angle o polarization c at the beginning o the string s ree oscillations. During the sticking phase, ~u s is governed by the inger s mechanical parameters and by the 2450 J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Chadeaux et al.: Model o harp plucking

8 threshold orce or sticking the harpist applies to the string. Based on classical string s vibration theory, Eq. (16) provides the linear dependency between F max and D ts. However, the relationship between the latter and k a, ca, cb is not straightorward. It is investigated through the path ollowed by the string or a given inger s motion. In addition, according to the previous results, F max, l, c, and d s may directly impact D tr and V tr. A parametric study is carried out to point out the inluence o this set o parameters on the initial conditions or oscillation D tr and V tr. Fixing the whole set o parameters but one to a reerence value allows to investigate variations o D tr and V tr according to the reachable range o values o the unixed parameters. Based on previous numerical or experimental estimations o the plucking parameters, the ollowing ranges o the parameters are deined as: (1) k a 2½10; 2000Š Nm 2 ; Re ¼ 326 N m 2, (2) c a 2½500; 2000Š Nsm 1 ; Re ¼ 1093 N s m 1, (3) c b 2½1; 100Š m 1 ; Re ¼ 69 m 1, (4) F max 2½1; 10Š N; Re ¼ 5N, (5) l 2½0:87; 1:0Š; Re ¼ 0.99, (6) and d s 2½0:1; 2Š mm; Re ¼ 1 mm. Note that the inluence o the slipping orientation c is not investigated here because it mostly inluences the initial angle o polarization o the string s oscillations, whereas we only consider the string s oscillation in one dimension. Finally, we evaluate the inluence o D tr and V tr on the string s vibrations through classical spectral descriptors. They are computed on the string ree oscillations simulation. The descriptors we use are oten calculated on the radiated sound rather than on the string vibration. Even i the relationship between the vibration o the string and the radiated sound is not straightorward (it actually takes into account the soundboard mobility and the radiating properties o the instrument), we expect relative values o the descriptors to give an insight on the inluence o the plucking conditions. For this purpose, as in Sec. II, the string is assumed to be lexible, o uniorm linear density q l, stretched to a tension T, ixed at its ends, and plucked at one third o its length. Hence the modal amplitudes A n and B n o the transverse vibrations are 25,38,39 A n ¼ 2D t r sinðk n y 0 Þ k 2 n y 0ðL y 0 Þ (21) X ðn 0 CGSÞ 2 ða 2 n þ B2 n Þ r 2 ¼ n X ða 2 n þ B2 n Þ ; (24) n where CGS is the spectral centroid, and r 2 is the spread o the spectrum around CGS. The ormer, CGS is expected to show a good correlation with the sensation o brightness o the sound produced, 40 while r 2 describes the spectrum s shape. B. Results The path ollowed by the string during the sticking phase is irst investigated. Figure 7 presents seven graphs. Each o them presents the inger and string motion in dashed and dotted lines, respectively. They correspond to the second and B n ¼ 2V t r sinðk n y 0 Þ k 3 n y 0ðL y 0 Þc : (22) Eventually, the ollowing set o descriptors is calculated or the dierent initial conditions o string vibration. Denoting by n ¼ n 0 the eigenrequencies and 0 the undamental requency, X n 0 ða 2 n þ B2 n Þ n CGS ¼ X ða 2 n þ B2 n Þ ; (23) n FIG. 7. Trajectories o the inger and the string estimated through measurements o a robotic plucking action with the A5-ingertip. They are associated to the modeled string trajectory in the (x0z)-plane or a large range o inger s parameters values. (a) Reerence, (b) k a: Minimum value, (c) ka : Maximum value, (d) c a : Minimum value, (e) ca : Maximum value, () cb : Minimum value, and (g) c b : Maximum value. J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Chadeaux et al.: Model o harp plucking 2451

9 measured plucking action perormed by the artiicial inger enhanced with F2. Figure 7(a) presents in solid line the simulated string s response to the inger motion with the mechanical parameters estimated previously in this paper. It is considered as a reerence in the ollowing paragraph. Figures 7(b) and 7(c) show the evolution o the simulated string s motion, while k a takes its minimum and maximum value, respectively. Similarly, Figs. 7(c) and 7(d) on one side and Figs. 7(e) and 7() on the other side report the impact o c a and c b on the plucking modeling. The entire set o inger s characteristic during the sticking phase appears to have a non negligible inluence on the string s motion. For instance in the particular case o the k a minimum value in Fig. 7(b), as the inger s stiness is rapidly compressed, the inger and the string ollow the same trajectories. Besides o the plucking s shape, the position o the string at the beginning o the slipping phase is clearly related to the mechanical parameters k a, ca, and cb, while the string s displacement D ts is mostly governed by F max. Table III reports the inluence o F max, d s, and l on the initial conditions o the string vibrations D tr and V tr. The ormer is prone to important variations (rom 0.03 to 7.4 mm) according to the maximal orce applied by the inger on the string. It represents a variability o 211% relatively to its reerence value (3.5 mm). To a lesser extent, initial conditions o the string oscillations are also impacted by the slipping distance d s with 43% o variation. As expected, the coeicient o riction does not inluence the distance o the string relatively to its rest position at the release instant. As or V tr, the three slipping parameters have almost the same impact. They imply a variability o 93%, 133%, and 80% around its reerence value (1.5 m s 1 ). In addition, the behavior o V tr according to their variations is coherent. Indeed, both a higher slipping distance and a higher riction coeicient imply a longer slipping phase and a higher velocity at the release instant. The inluence o the initial conditions o the string s oscillations D tr and V tr on the spectral descriptors are presented in Table IV. The reerence values used are 3.5 mm and 1.5 m/s or D tr and V tr, while the values investigated are in the ranges mm and m/s, respectively. The reported range o values reachable by D tr can imply a variation o 9 Hz in the spectral centroid, i.e., about 6% o the undamental requency o the studied string (D[2 at about 140 Hz). It also impacts the spectrum s spread, which can reach up to nine times its smallest value. Although the impact o V tr is clearly less important than the one o D tr on the string s oscillations, it is not negligible. Indeed, it can induce a variation o 3% o the undamental requency in the spectral centroid, and the spread o the spectrum can reach TABLE III. Inluence o the plucking parameters on the initial condition o the string vibrations. F max, d s and l are considered to vary rom 1 to 10 N, rom 0.1 to 2 mm, and rom 0.87 to 1.0, respectively. F max (N) d s (mm) l D tr (mm) V tr (m s 1 ) TABLE IV. Inluence o the initial condition o the string vibrations D tr and V tr on the spectral descriptors CGS and r 2. D tr and V tr are considered to vary rom 0.03 to 7.4 mm and rom 0.1 to 3 m/s. up to 1.7 times its smallest value. Let us remark that these results, based on signal processing attributes, are clearly conirmed by inormal listening to sound simulations o the string oscillation with the corresponding initial conditions. C. Discussion D tr (mm) V tr (m/s) CGS (Hz) r 2 (Hz) This parametric study indicates irst that the mechanical parameters governing the sticking phase in Fig. 2 have a great inluence on the string s path during this phase. Hence, they impact the position o the string at the beginning o the slipping phase and consequently at the release time. This is o great importance relative to the initial angle o polarization o the string s oscillations and thereore to the sound produced. Then as or the slipping phase, the three parameters F max, d s, and l show a strong inluence on the amplitude o the string vibration modes and the distribution o the energy on the string modes as unction o the requency. The values F max and d s, which are directly controlled by the musician, appear to have the strongest inluence. VI. APPLICATION TO A MUSICAL CONTEXT In the previous sections, we have restricted the analysis to isolated plucking actions perormed by an artiicial inger and by a harpist. The ollowing section discusses the application o the model to actions perormed in a real musical context. For this purpose, we use a inger/string motion database collected on 10 skilled harpists reerred to as H 1:::10 perorming either arpeggio or chord. 1 Only the plucking by the oreinger or the annular is analyzed. The parameters estimated or these plucking actions are reported Table V. There are no signiicant dierences between the plucking positions o the dierent players because they all pluck the string at positions between about one-third and two-iths o the distance rom the soundboard to the neck. First, the variability estimated on the mechanical parameters describing the sticking phase is globally high: About 100% or c b and and about 50% or k a and c a. Then regarding the parameters controlling the slipping phase, the variabilities are smaller but still nonnegligible (about 15%). This indicates that these mechanical parameters are highly dependent on the plucking action, i.e., the harpist s control rather than on the harpist himsel. Because o the high variabilities, no clear result can be highlighted about the parameter c a. However, ka tends to be dependent on the playing technique. Indeed, higher values are computed while playing chord than arpeggio. This is illustrated or instance by harpists H 2;3;4 and to a lesser extent by harpists H 2;8;10. Concerning c a, no rule can be extracted rom Table V. Hence this would indicate that there 2452 J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Chadeaux et al.: Model o harp plucking

10 TABLE V. Inluence o the plucking parameters on the initial conditions o the string vibrations. Note that some boxes are empty because some experimental data are missing or because o post-processing problems. Arp-Ann, Arp-For, Ch-Ann, and Ch-For reerred to the our musical context investigated, i.e., arpeggio perormed with the annular and the oreinger and chord perormed with the annular and the oreinger. Harpist Parameter Arp-Ann Arp-For Ch-Ann Ch-For k a (N m 2 ) c a (N s m 1 ) c b (m 1 ) H 2 F max (N) l d s (mm) k a (N m 2 ) c a (N s m 1 ) c b (m 1 ) H 3 F max (N) l d s (mm) k a (N m 2 ) c a (N s m 1 ) c b (m 1 ) H 4 F max (N) l d s (mm) k a (N m 2 ) c a (N s m 1 ) c b (m 1 ) H 6 F max (N) l d s (mm) k a (N m 2 ) c a (N s m 1 ) c b (m 1 ) H 7 F max (N) l d s (mm) k a (N m 2 ) c a (N s m 1 ) c b (m 1 ) H 8 F max (N) l d s (mm) k a (N m 2 ) c a (N s m 1 ) c b (m 1 ) H 9 F max (N) l d s (mm) k a (N m 2 ) c a (N s m 1 ) c b (m 1 ) H 10 F max (N) l d s (mm) is no speciic set o mechanical parameters relatively to a harpist but more probably to a plucking action. Considering the slipping phase, the maximum orce applied by the inger to the string appears to be higher while plucking with the annular than the oreinger. It is most likely explained by a compensation o the weaker control possible with the annular due to morphological reason. This result appears to be also related to the control parameter d s. For instance, considering harpists H 2 and H 4, the smaller d s, the smaller F max. Furthermore, regarding the playing technique, trends seem to be speciic to harpists. For example, harpist H 3 plays arpeggio with a smaller slipping distance than chord, independently o the playing inger, while d s is mostly speciic to the inger or harpist H 8. These results are in agreement with previous ones highlighting that each harpist produces speciic plucking actions relatively to the playing context. Furthermore, the playing context induces variations o the control that are bigger than variations amongst players, or one speciic musical task. A global survey o the six parameters o the model that describe the plucking action rom a mechanical point o view indicates that some o them are probably linked in the playing. For instance, when the player touches the string rom a longer distance d s rom the ingertip, it may be induced by the intention to play the note louder. The apparent correlation to a stronger sticking orce F max may come rom the intention to play louder rather than on mechanical constraints. Thereore global playing indicators that lumping together several parameters o the model could be developed but ranges out o the scope o the present study. VII. CONCLUSION This paper has presented a model o the plucking action in the case o the concert harp. Measurements o the inger/ string interaction have been carried out to determine the model parameters, and a parametric study provides the relevance o the model according to the string s ree oscillations. The experimental setup was mostly based on the capture o the inger and the string motion in the plane perpendicular to the string s through a high-speed camera. The validity o the model is irst discussed on ideal plucking actions perormed by a conigurable and repeatable robotic inger, enhanced with a silicone layer. Then the identiication o plucking parameters, using the model, has been carried on isolated plucking actions perormed by a real harpist and inally on plucking in real musical contexts. The model or the inger/string interaction has been split into the two plucking action phases: The sticking and the slipping phases. During the sticking phase, the viscoelastic behavior o the inger is described using the classic Kelvin Voigt model. The spring s stiness and the damping o the dash-pot have been investigated as linear and exponential parameters depending on the inger indentation. The combination o a linear stiness and an exponential damping has been shown to provide the most relevant modeling o the sticking phase. Subsequently, a consistent set o mechanical parameters can be extracted or the various silicone ingers as well as or the harpist ingers. Considering the slipping J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Chadeaux et al.: Model o harp plucking 2453

11 phase, the riction coeicient between the inger and the string is very tricky to determine accurately. This parameter depends on several variables that were not controllable in our measurements such as the contact angle between the inger and the string or the skin lubrication conditions. However, the values o the riction coeicients estimated or each experimental dataset are very similar, close to values ound in the literature. Once the dierent plucking parameters o the model have been identiied, the string and inger trajectories that are the output o the model lead to the initial conditions (displacement and velocity) o the ree oscillation o the string. The inluence o each plucking parameter has been investigated in relation to the initial conditions o the string vibrations. The mechanical parameters governing the sticking action have been shown to greatly inluence the path ollowed by the string trajectory and position. The parameters the harpist controls directly while plucking are the orce applied to the string, the slipping distance, and orientation. They highly inluence the initial displacement and velocity o the string s ree oscillations. Finally, the variations reported o the latter imply acoustically relevant dierences in the string s oscillations. Thereore the mechanical parameters intrinsic to the harpist morphology and more speciically the control parameters strongly inluence the sound produced. This justiies the musician s claim that they sound dierent at the individual note level depending on the way they put the instrument into vibrations. Dierences in plucking are expected to induce changes in the spectral content o the sound, as well as on the sound level. Moreover, results indicate that the players produce quite dierent plucking actions according to the playing context. Those dierences may be larger than the dierences observed between players or a given musical task. The rotation o the inger during the sticking phase requires more attention. Indeed, the motion o the harpist inger oten shows a rotation o the phalanx beore the beginning o the slipping phase that induces a change in the contact surace between the string and the inger as the inger gets more parallel to the string. It would then be valuable to develop a model in three phases, the sticking phase itsel being analyzed in two successive steps: During the irst one, the string being caught by the inger, controlling the string pulling, while during the second one, the string torsion and possible inger rotation control the triggering o the slipping phase. Furthermore, an experimental protocol dedicated to the contact between the inger and strings o several diameters and material, as ound through the tessiture o the harp, would be o great interest. Finally, a perceptual test would be needed to conirm the inluence o the mechanical parameters on the sound produced. Such a test, in relation to the player s technique and musical intention, may help to determine a combination o mechanical parameters that would give rise to global plucking parameters that are relevant rom the point o view o the playing technique. ACKNOWLEDGMENTS The authors thank the harpists who participated in this study: Marie Denizot, Pierrine Didier, Marie Klein, Sandie Le Conte, Camille Levecque, Caroline Lieby-Muller, Magali Monod-Cotte, Blandine Pigaglio, Ma elle Rochut, and Coralie Vincent as well as Antoine Chaigne, Laurent Daudet, and Sylvie Gibet or useul discussions. 1 D. Chadeaux, J.-L. Le Carrou, B. Fabre, and L. Daudet, Experimentally based description o harp plucking, J. Acoust. Soc. Am. 131(1), (2012). 2 H. Penttinen, J. Pakarinen, and V. V alim aki, Model-based sound synthesis o the guqin, J. Acoust. Soc. Am. 120, (2006). 3 G. Derveaux, A. Chaigne, P. Joly, and E. Becache, Time-domain simulation o a guitar: Model and method, J. Acoust. Soc. Am. 114, (2003). 4 K. 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