Physical modeling of the guqin a Chinese string instrument - CiteSeerX

1st Nordic Music Technology Conference (NoMute 2006), Trondheim, Norway, October 12.-14., 2006.

PHYSICAL MODELING OF THE GUQIN A CHINESE STRING INSTRUMENT Henri Penttinen, Jyri Pakarinen, Vesa Välimäki

Mikael Laurson, Mika Kuuskankare

Laboratory of Acoustics and Audio Signal Processing Helsinki University of Technology, Espoo, Finland

Centre for Music and Technology Sibelius Academy, Helsinki, Finland

[email protected]

[email protected]

Henbing Li, Marc Leman Institute for Psychoacoustics and Electronic Music Department of Musicology, Ghent University, Ghent, Belgium [email protected]

ABSTRACT This paper discusses acoustical properties and proposes a sound synthesis model for the plucked stringed instrument called guqin. The guqin is an ancient Chinese musical instrument with seven strings. The strings are plucked with the right hand while the left hand pushes the strings against the top plate of the body. In the playing of the instrument, slides and flageolet tones are an important part of the expression of the instrument. The instrument disappeared almost completely during the cultural revolution, but it has began a revival and now there are already thousands of players. A database of tones was recorded in anechoic conditions. The acoustic signal analysis and the model is based on these signals. The sound synthesis model is based on the digital waveguide technique. The physical model takes into account the most vital parts of the characteristics of the instrument. 1. INTRODUCTION This paper discusses the acoustics and proposes a sound synthesis model for the plucked stringed instrument guqin. The guqin is an ancient Chinese musical instrument with seven strings. The instrument disappeared almost completely during the cultural revolution, but it has began a revival and now there are already thousands of players. It is still used in modern music. The starting point of the guqin synthesis project was the need to play guqin music with a computer. Prof. Marc Leman and one of his PhD students in systematic musicology, Ms. Henbing Li, contacted researchers of Helsinki University of Technology to collaborate with them to develop such a system. Ms. Henbing Li’s doctoral research consists of the development of performance rules for guqin music. To verify the rules it was necessary to implement them in a computer-based system, which provides accurate control of temporal events, dynamics, and special gestures used in guqin music, such as vibrato and pitch glides. The system developed by researchers at Helsinki University of Technology (TKK) in collaboration with researchers at the Sibelius Academy seems to be a suitable environment for this application. This paper reports the results of this collaboration. So, the goal of this work is the model-based sound synthesis, which gives possibilities to control a string instrument model in

various ways. It does not have to exactly reproduce an existing instrument, but the quality should be good enough to be acceptable in musical use. This implies that similar sounds should be heard from the synthesizer as from a real guqin, when traditional playing styles are used. The synthesizer does not have to be physically valid either, but the physical modeling approach is a good way to generate realistic sound in response to control events. An alternative to the model-based approach would be a sampling synthesizer, which is based on a large database of recorded samples. It appears that it may be easier to develop a model-based synthesizer that gives a sufficiently realistic and practically unlimited sound output than to sample the sounds corresponding to all possible types of gestures that guqin players can produce. The rule-based physical modeling synthesis gives the opportunity to create an incredible amount of variations based on a relatively simple algorithms described in this paper. The ENP system [1] developed at the Sibelius Academy enables the accurate control of the synthesizer from notation seen on a computer screen, which is a useful and familiar tool for musicians. The guqin synthesizer proposed in the this article is based on the commuted digital waveguide synthesis technique [2, 3]. The digital waveguide technique has been used successfully to synthesize a broad range of traditional and ethnic instruments [4, 5, 6]. Previously, a neural network based synthesis model for the guqin has been proposed [7]. To conclude, the main goal has been to develop a system, which allows a user to create Chinese guqin music without having to own the instrument or even to be able to play it. The details of the acoustic analysis and sound synthesis model will be presented in [8]. The remaining parts of this paper are organized as follows. In Sec. 2 the construction of the guqin is discussed. Section 3 shortly presents the signal analysis made and Sec. 4 proposes a synthesis model based on this. Section 5 discusses the implementation with some musical examples. Section 6 contains a short conclusion. 2. DESCRIPTION OF THE GUQIN Guqin (pronounced ku-ch’in), also called Seven-Strings-Qin, is the modern name for the fretless plucked string instrument qin, which is the oldest Chinese string instrument still used in modern


Table 1: String numbers, the note names, the measured fundamental frequencies f0 , and string diameters for the guqin used in this article. S# Tuning f0 (Hz) d (mm) 1 C2 65.4 1.38 2 D2 73.5 1.20 3 F2 87.4 1.08 4 G2 98.2 1.00 5 A2 110.5 0.90 6 C3 131.1 0.75 7 D3 147.3 0.64 Figure 1: Typical guqin playing position.

time [9]. The current structure of the instrument was formed approximately between the 5th and the 7th century, and since then there has been no major changes in the construction. Next, we briefly describe the construction and playing style of the guqin. Figure 1 displays the side of the instrument while one of the authors Henbing Li is playing the instrument in the listening room at TKK.

strings bridge

8 c m a)

end(tail)

2.1. Construction and tuning The body of the guqin is a long, narrow, hollow box made from two pieces of wooden board, and the top board is carved into an arch while the bottom is flat. For the top board soft wood is usually used (such as tung), while the wood for the bottom board is hard (catalpa or fir). The surface of the box is covered with a special layer (about 1 mm) of roughcast, which is a mixture of deer horn powder (or bone powder or tile powder) and raw lacquer, and there are several layers of raw lacquer along the top of the roughcast for polishing. The bridge is made from hard wood, and the strings are attached to it with a twisting-rope system, which allows fine-tuning of the strings in a limited range. The other ends of the strings are bent over the end (tail) and are finally tied up to the feet on the bottom. Traditionally, the strings were made of silk, but after the 1950’s they have been replaced mainly by steel-nylon strings. There are 13 marks inlayed on the roughcast at the side of the first string, which indicate the positions of the first to the fifth and the seventh overtone. These marks also function as a reference for stopped strings, i.e., when the string is pressed against the top board. Each part between two contiguous marks is divided into ten parts, for example, mark 2.5 indicates that the tone is played halfway between the second and the third mark. The seven strings are tuned basically as a pentatonic scale. The basic tuning of the open strings is C2 , D2 , F2 , G2 , A2 , C3 , and D3 from the lowest string (no. 1) to the highest (no. 7). The pitch range for so called stopped strings is from 65.2 Hz (open string no. 1) to 787.5 Hz (string no. 7, mark 2.6) which roughly correspond to notes C2 and G5 , respectively. The highest harmonic or flageolet sound is played on string no. 7 on marks no. 1 or no. 13 (f0 = 1174.7 Hz, D6 ). The guqin used in this measurement was made by Zhang Jianhua in Beijing in 1999. The boards are made of fir, and the roughcast is deer horn powder and raw lacquer. Shangyin steel-nylon strings are used. The respective string numbers, note names, fundamental frequencies and diameters of the strings are displayed in

peg

foot

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18cm b) waist

neck 120cm

Figure 2: Construction of the guqin from two angles: (a) side view and (b) top view.

Tab. 1.

2.2. How to play the instrument In modern days the guqin is usually played on a table with its two feet standing on the table and the neck laying on the right edge of the table with anti-slip mats between the contact points of the table and the instrument (see Fig. 1). The right hand plucks the strings between the bridge and the first mark, and the left hand presses the strings against the top plate of the body. The instrument is fretless, which enables smooth sliding tones. Guqin music also incorporates substantial use of harmonics or flageolet tones. The little fingers of neither hand are used. The other four fingers of the right hand pluck the string from both the fleshy and the nail side. Typically, the nail exceeding the finger is 2-3 mm long for the thumb and 1-2 mm long for the other fingers. The left thumb presses the string on the right side, where the nail and flesh joins, or at the first joint. The other three left fingers press the string with the fleshy top part of the finger or occasionally with the left side of the first ring finger joint.


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Figure 4: Initial pitch glide of a forte fortissimo tone played on the guqin (string no. 4, mark 7).

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Figure 3: Time responses of guqin tones played on string 6 mark 5, when (a) the nail of the thumb terminates the string and (b) the fingertip of the forefinger terminates the string.

3. ACOUSTIC MEASUREMENTS AND SIGNAL ANALYSIS To create a synthesizer for the instrument, an extensive set of isolated plucks was recorded. The purpose of the isolated plucks is to be able to properly analyze the characteristics of the instrument. 3.1. Measurement and recording setup Guqin tones were recorded in the small anechoic chamber of Helsinki University of Technology. The recordings were made with a microphone (AKG C 480 B, cardioid capsule) placed at a distance of about 1 m above the sound board. The signals were recorded digitally (44.1 kHz, 16 bits) with a digital mixer (Yamaha 01v) and soundcard (Digigram VX Pocket) onto the hard drive of a PC laptop. To remove infrasonic disturbances the signals were highpass filtered with a fourth-order Butterworth filter with a cutoff frequency of 52 Hz. As for the tones, four complete sets on a typical scale used in guqin music were recorded. Two different styles for terminating the string with the left hand finger were used, the nail of the thumb or the fingertip of other fingers. In the following text these styles of string termination will be referred to as nail or fingertip. Furthermore, two different plucking styles were recorded, namely plucking with the middle finger towards the player or plucking with the index finger away from the player. In addition, a complete set of harmonics or flageolet tones was recorded for all strings and marks. Moreover, three dynamic levels (pp, mf, and ff) were recorded for all open strings and marks seven. Slides, isolated vibratos, scales, and musical pieces were also included in the database. Important for this study are the basic pluck events (281 samples), sliding sounds, and the harmonic sounds (91 samples) that will be analyzed next. All in all, the database contains over 400 samples. 3.2. Analysis of guqin tones Next, two essential features in the behavior of guqin tones are investigated. The initial pitch glide and friction sound are discussed through examining them in time and frequency domains.

Initial pitch glide is a phenomenon due to tension modulation and occurs in vibrating strings [10, 11, 12]. Even a small transverse displacement of the string causes a second-order change in its length, and therefore in its tension. This causes the pitch to decay after releasing the string from its initial displacement. Hence, some initial pitch gliding occurs in guqin tones. The amount of pitch gliding for tones played as mezzoforte and forte fortissimo notes were measured. For mezzoforte tones the largest initial pitch glide value obtained was 0.075 ERB (Equivalent Rectangular Bandwidth) [13], while the mean was 0.025 ERB with a standard deviation of 0.021. Number of ERBs is defined as 21.4 log10 (4.37F + 1), where F is frequency in kHz [13]. Similarly, for forte fortissimo tones the mean value for the initial pitch glides was 0.034 ERB with a standard deviation of 0.019. The largest value was 0.096 ERB (for string 5, mark 7). According to Järveläinen [14], these initial pitch glides would remain inaudible to most listeners, since the limit for the initial pitch glide audibility is about 0.1 ERB. However, the quartile limits are quite large and hence expert listeners, such as, instrument players are able to detect smaller changes than 0.1 ERB [14]. Figure 4 shows the behavior of the fundamental frequency f0 in time for a forte fortissimo tone played on string no. 4, mark 7 (about G3 ). The x-axis displays time and the y-axis the fundamental frequency. At 0.11 seconds the fundamental frequency is 198.6 Hz and beyond 1 s it is 197 Hz. This gives a change of 1.6 Hz, which is 0.035 ERB. The fundamental frequency estimations have been calculated with the autocorrelation-based YIN algorithm [15]. The glitch in Fig. 4 is due to estimation errors during the attack, i.e., when t < 0.11 s. The largest measured initial pitch glide can be audible for some listeners, therefore the initial pitch glides are synthesized with the obtained mean value, i.e., 0.025 ERB for mf tones and 0.034 ERB for f f tones. The initial pitch glides are synthesized by changing the pitch of the string with a break point function in the PWGLSynth system [16]. Alternatively, the synthesis could use the nonlinear approach proposed by Tolonen et al. [12]. 3.2.2. Analysis of friction sounds The friction noise caused by the sliding finger-string contact was recorded using the setup described in Section 3.1. In order to record only the friction noise, the strings were not plucked. Figure 5 (a) shows the spectrogram of the friction noise when the player slides her finger from mark 7 to mark 9 on the lowest string (i.e., from 131 Hz to 99 Hz). In (a), the player was asked to perform the slide slowly. Figure 5 (b) shows the friction noise spectrogram


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Figure 6: Block diagram of a simple guqin synthesizer.

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Figure 5: (a) Spectrogram of the friction noise when sliding slowly from mark 7 to mark 9on the 1st string (131-991 Hz). (b) Spectrogram of the friction noise when sliding quickly from mark 7 to mark 9on the 7th string (297-220 Hz).

with a slide from mark 7 to mark 9 on the highest string (i.e., from 297 Hz to 220 Hz). Here, the player was asked to perform the slide quickly. Note that both sliding styles still fall under the normal playing styles of the guqin, and that they represent the two sliding velocity extremes usually played on the instrument. The spectrogram plot reveals that the friction signal is similar to lowpass-filtered noise, where the amplitude and cutoff frequency are proportional to the sliding velocity (the amplitude and cutoff frequency are highest in the middle of the slide, where also the sliding velocity is highest). In addition to this, there is a clearly observable harmonic structure in the noise (see Fig. 5 (b)), consisting of a few modes, each approximately 200-400 Hz wide. The amplitudes and frequencies of these components also seem to be proportional to the slide velocity, the lowest harmonic having its frequency near 2.7 kHz in Fig. 5 (b). 4. SOUND SYNTHESIS OF THE GUQIN 4.1. Compact synthesis algorithm and time-varying string model The structure of the synthesis model is illustrated in Fig. 6. The guqin string model constitutes essentially of a single-delay loop (SDL) [17] digital waveguide (DWG) string, S(z), and a body model filter, B(z). The length of strings is varied during the synthesis run time [17]. The SDL string model, S(z), synthesizes the transversal vibrations of the tone, with the inharmonicity coefficients of B. The input signal is read from the excitation database. The string model S(z) is illustrated in Fig. 7. The z −L1 block implements the (time varying) integer delay of the SDL. The traditional SDL blocks HLF (z), L(z), and Ad (z) [17] correspond with the figure as follows. The HLF (z) block is the loss filter implementing the frequency dependent decay due to losses in the string and L(z) is a a third-order Lagrange filter applying the fractional part of the loop delay. Ad (z) is the dispersion filter made of a chain of four allpass filters or a single second-order allpass filter. Coefficient gca is responsible for gain compensation due

+

g ca

F( z)

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1

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H LF ( z )

Figure 7: Signal flow diagram of the guqin string model. The synthesis structure differs from the traditional SDL DWG model in two ways. Firstly, the length of the delay loop changes during run time. Secondly, the signal values are scaled by gca in order to avoid the artificial changes in energy due to the pitch change.

to changing length of the string. When implementing the string model on a computer, the memory for the maximum length of the string should be allocated beforehand so that the possible elongation can be accounted for.

4.2. String model calibration To produce normal plucked tones the string model parameters must be calibrated as described below. First, the inharmonicity is determined, and then the excitation signals are obtained by canceling the partials of the guqin tone with a sinusoidal model [18]. The transversal vibrations are canceled and the parameters for the loss filter HLF (z) and ripple filter HR (z) are obtained as described in Ref. [18] and Ref. [19], respectively. In HLF (z) the parameter g controls the overall decay, and a controls the frequency dependent decay[20]. The transfer function is HLF (z) = b/(1 + az −1 ), where b = g(1 + a). Due to calibration errors and large differences in parameter values, for consecutive tones, the g and a data are smoothed in the same vein as previously proposed [18]. More specifically, the g parameters were treated with a 10th order median filter, and the a parameters were approximated by a linear regression on the logarithmic fundamental frequency scale for each string. Additionally, the excitation signals are normalized. This way a synthesis model was obtained that has natural and subtle changes from a tone to another without drastic unwanted sonic departures. The coefficients for the dispersion filter Ad (z) are obtained as described by Rauhala and Välimäki[21].

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displays the playing of the open strings, including Chinese symbols, in the ENP. Note that sliding an open string is practically a pure physical impossibility.

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Figure 8: Time responses of synthesized guqin tones played on string 6 mark 5, when (a) the nail of the thumb terminates the string and (b) the fingertip of the forefinger terminates the string.

4.3. Results Next, synthesis results produced by the model discussed in Sec. 4.1 are shown, and the measured and synthesized signals and some characteristics are compared. 4.3.1. Normal plucked tones Figure 8 depicts the time responses of synthesized signals for string 6 mark 5, for (a) nail and (b) fingertip terminations (compare with Fig. 3). The nail tone responses correspond fairly well, as do the fingertip tones, but they have a slight difference during the beginning of the decay. This can be explained by a difference in partial decay times, i.e., the higher frequencies of the real tone decay faster than the synthesized one. 5. IMPLEMENTATION AND MUSICAL EXAMPLES Figure 9 shows an excerpt from the traditional guqin repertoire notated with our music notation package ENP [1]. The musical score is used to calculate the control information for the guqin synthesizer. The guqin synthesizer is realized in our software synthesis environment PWGLSynth [16], which in turn is a part of a visual programming language called PWGL [22]. Besides basic note information the score contains several special expressions that are used to simulate typical playing techniques of the guqin. The upper part of the score shows breakpoint functions that describe how glides should be performed by the synthesizer. The user has two main options for inserting the glide information into the score. Either she/he can use existing glide gestures from a database that has been collected by Henbing Li. This database contains hundreds of glides that have been analyzed from real guqin playing. In the other option the user can define a glide from scratch by using a visual break-point function editor in ENP. The lowest row at the bottom of the score gives the JZP notation for each note (i.e. sh,7,g,2, 0,0,g,4, etc.) in Latin characters. JZP is a novel representation scheme of the ancient Chinese guqin Jian Zi Pu tabulature that allows to notate guqin music in a modern computerized system. The JZP notation is used here to find information dealing with the correct processed sample and the low-level synthesis parameters used by our guqin model. The rest of the expression markings found in the score describe vibrato gestures that in conjunction with the glides describe how pitch fluctuations should be performed by the system. Figure 10

This paper has given a short introduction to the Chinese instrument guqin and its sound synthesis. Some acoustical aspects have been discussed and a simple synthesis model has been proposed. A future publication [8] will contain essential additions to the synthesis model and acoustic analysis presented here. The additions include investigations and discussions on the effect of the termination technique, inharmonicity, and phantom partials [23, 24]. Improvements in the model include synthesis of flageolet tones, friction sound, and phantom partials. Measured and synthesized guqin tones are available on the Internet at http://www.acoustics. hut.fi/publications/papers/jasa-guqin/. 7. ACKNOWLEDGEMENTS The authors are grateful to Jussi Pekonen, Jukka Rauhala, Hanna Järveläinen, and Vesa Norilo for their help. Dr. Cumhur Erkut is thanked for his helpful comments. This work has been financially supported by COST287-ConGAS Action (Gesture Controlled Audio Systems), the Academy of Finland (project no. 104934 and no. 105557), the Pythagoras Graduate School of Music and Sound Research, the Helsinki Graduate School of Electrical and Communications Engineering, and Tekniikan edistämissäa¨ tiö. 8. REFERENCES [1] M. Kuuskankare and M. Laurson, “Expressive notation package,” Computer Music J., vol. 30, pp. 67 – 79, 2006. [2] J. O. Smith, “Efficient synthesis of stringed musical instruments,” in Proc. Int. Computer Music Conf. (ICMC’93), Tokyo, Japan, 1993, pp. 64–71. [3] M. Karjalainen, V. Välimäki, and Z. Jánozy, “Towards highquality sound synthesis of guitar and string instruments,” in Proc. Int. Computer Music Conf. (ICMC’93), Tokyo, Japan, 1993, pp. 56–63. [4] J. O. Smith, “Physical modeling synthesis update,” Computer Music J., vol. 20, no. 2, pp. 44–56, 1996, URL: http://www-ccrma.stanford.edu/˜jos/pmupd/pmupd.html. [5] C. Erkut, M. Karjalainen, P. Huang, and V. Välimäki, “Acoustical analysis and model-based sound synthesis of the kantele,” J. Acoust. Soc. Am., vol. 112, no. 4, pp. 1681–1691, 2002. [6] V. Välimäki, J. Pakarinen, C. Erkut, and M. Karjalainen, “Discrete-time modelling of musical instruments,” Reports on Progress in Physics, vol. 69, no. 1, pp. 1–78, 2006. [7] Alvin W. Y. Su, , W. C. Chang, and R. W. Wang, “An IIR synthesis method for plucked-string instruments with embedded portamento,” J. Audio Eng. Soc., vol. 50, no. 5, pp. 351–362, 2002. [8] H. Penttinen, J. Pakarinen, V. Välimäki, M. Laurson, H. Li, and M. Leman, “Model-based sound synthesis of the guqin,” J. Acoust. Soc. Am., p. Accpeted for publication, 2007.


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Figure 9: A short guqin score in the ENP enriched with the JZP notation and slides.

Figure 10: Open strings with Chinese symbols and slides in the ENP.

[9] S. Sadie, Ed., The New Grove Dictionary of Music and Musicians, vol. 5, Groves Dictionaries Inc., Oxford University Press, University of Oxford, UK, 2001. [10] G. F. Carrier, “On the non-linear vibration problem of the elastic string,” Quarterly of Appl. Math, vol. 3, 1945. [11] C. Valette, The mechanics of vibrating strings, Springer, New York, 1995. [12] T. Tolonen, V. Välimäki, and M. Karjalainen, “Modeling of tension modulation nonlinearity in plucked strings,” IEEE Transactions on Speech and Audio Processing, vol. 8, no. 3, pp. 300–310, 2000. [13] B. Glasberg and B. Moore, “Deviation of auditory filter shapes from notched noise data,” Hearing Research, vol. 47, pp. 103–138, 1990. [14] H. Järveläinen, Perception of Attributes in Real and Synthetic String Instrument Sounds, Ph.D. thesis, Helsinki University of Technology, Espoo, Finland, 2003, http://lib.tkk.fi/Diss/2003/isbn9512263149/. [15] A. de Cheveigné and H. Kawahara, “YIN, a fundamental frequency estimator for speech and music,” J. Acoust. Soc. Am., vol. 111, no. 4, pp. 1917–1930, 2002. [16] M. Laurson, V. Norilo, and M. Kuuskankare, “PWGLSynth, a visual synthesis language for virtual instrument design and control,” Computer Music J., vol. 29, no. 3, pp. 29–41, 2005. [17] M. Karjalainen, V. Välimäki, and T. Tolonen, “Pluckedstring models: From the Karplus-Strong algorithm to digital waveguides and beyond,” Computer Music J., vol. 22, no. 3, pp. 17–32, 1998. [18] V. Välimäki and T. Tolonen, “Development and calibration of a guitar synthesizer,” J. Audio Eng. Soc., vol. 46, pp. 766– 778, 1998. [19] V. Välimäki, H. Penttinen, J. Knif, M. Laurson, and C. Erkut, “Sound synthesis of the harpsichord using a computationally efficient physical model,” EURASIP J. Applied Signal Processing, vol. 2004, no. 7, pp. 934–948, 2004.

[20] V. Välimäki, J. Huopaniemi, M. Karjalainen, and Z. Jánozy, “Physical modeling of plucked string instruments with application to real-time sound synthesis,” J. Audio Eng. Soc., vol. 44, pp. 331–353, 1996. [21] J. Rauhala and V. Välimäki, “Tunable dispersion filter design for piano synthesis,” IEEE Signal Processing Letters, vol. 13, no. 5, pp. 253–256, 2006. [22] M. Kuuskankare and M. Laurson, “Recent trends in PWGL,” in Proc. Int. Computer Music Conf. (ICMC’06), New Orleans, USA, Nov. 2006, p. Accepted for publication. [23] H. A. Conklin, “Generation of partials due to nonlinear mixing in a stringed instrument,” J. Acoust. Soc. Am., vol. 105, no. 1, pp. 536–545, 1999. [24] B. Bank and L. Sujbert, “Generation of longitudinal vibrations in piano strings: From physics to sound synthesis,” J. Acoust. Soc. Am., vol. 117, no. 4, pp. 2268–2278, 2005.