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Page 00000001 Synthesis of Transients in Guitar Sounds Cem I. Duruoz Digital Audio Video Engineering, SONY Electronics Inc. 3300 Zanker Rd., San Jose, CA95134, U.S.A. firstname.lastname@example.org Abstract Guitar sounds are analyzed in detail to understand the origin of transients produced during the attack and damping of a note. These transients and the main note are produced on the computer using physical modeling and FM synthesis. The implementation successfully simulates the articulated multiple note sounds for a complete set of performance parameters which correspond to various playing techniques. With appropriate adjustment in the parameters, the model can easily be used for the synthesis of all plucked instrument sounds. 1 Introduction The guitar, a very popular musical instrument, is also a very interesting physical system for studying acoustics, due to the coexistance of the vibrational modes supported by its three dimensional body, its extremely non-ideal strings, and a complicated excitation mechanism. The picture becomes even more complicated when the coupling between these elements is incorporated into the mathematical model which is built to represent the instrument and synthesize its sound. Within the last few years, detailed research has been done on the topic. Karplus and Strong proposed an intuitive digital synthesis algorithm  which is very useful in making the connection between physical models and digital signal processing, in particular waveguide synthesis. Karjalainen and Smith proposed various techniques for the modeling of the body, when it is decoupled from strings . Chaigne used finite difference methods to study the two transverse vibrational modes of a string, their coupling to the bridge, and their mutual coupling . All of these models are very successful in analyzing one or few aspects of the instrumental sound and synthesizing it. It is a well known fact, however, that the perceived individuality of a musical instrument is only partly related to the main note and its harmonics. The sounds whose durations are much shorter than that of the note itself -referred to as "transients" in this paper- are crucial in a synthesis model which is designed to produce sounds to be perceived as "real". Surprizingly, they have usually been neglected in sound synthesis especially in commercial synthesizers. In this paper, the properties of these transients are explained and it is shown that they are not simply "noise" but have a harmonic or inharmonic spectrum. Then a hybrid model which applies previously successful methods to their synthesis and the synthesis of articulated note sounds is presented. 2 Qualitative Analysis of Tran sients and Articulation Fig. 1 shows the waveform for three open string G's played succesively on a classical guitar. The portion t < to in the figure represents the end portion of the first note. After to, the plucking device -the nail in this context- slides on the string, releasing it at time t = tba. The duration of this sliding motion depends on the angle of the finger from the string and its speed which is the variable determining the amplitude of the actual note. In this time interval (to < t < tba), a very soft "scratch" sound is heard. The analysis of the spectrum of this sound shows that it is harmonic with a wideband noise superimposed (Fig. 2 (c)). This is the longitudinal mode vibration of the string the details of which will be given in the next Sec. 3.1. At t = tba (body attack) the transverse vibration starts along with a percussive body sound "tap sound". In classical guitars, the amplitude of the tap sound is a function of the angle between the finger and the top plate and the overal amplitude of the note. For so called "free stroke" in which the finger moves more or less parallel to the top plate,
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Page 00000003 The synthesis model is posed by Cook  for flut noise is supplied to a looj and a onepole filter. From r found to be exponential, (or typical duration between 2C the delay line is tuned to ob from recordings. The noise 0 (a) 194Hz z -25 A -so! (b) 104Hz 25 208Hz -75 0 375 0 (c) 1546Hz -25 1U0i 1h -25 3004Hz -75 --100 /1 - 0 1875 3 freque Figure 2: Three example Fig.1 around time instants The empty and full arrowh modes of the body, and th( string modes, respectively. of the string and its ampl since as the string gets to roughness increases. 3.2 Transverse Mo Finite difference methods thesizing the transverse vi implementation, the specia to discretize the ideal strii introduces numerical dispe]  but it is satisfactory for about articulations. This used to represent the strin significant loss terms, -21 modeled by a biquad filter a is the loss per unit length string length. Fig. 3 shows an impleme modes. The four delay lin( of the rightgoing, y (x, t) a ponents of the string displa tering junction in the midd device. The "nut filter" co very similar to that pro- filters: a biquad filter to simulate the losses, and two ýes. An enveloped white first order allpass filters to tune the pitch and generp containing a delay line ate inharmonicity by introducing phase delays . 'ecordings, the envelope is At time tba, the waveguides are filled with sloped Slinear in dB scale) with a lines which simulate the initial shape of the string. ) and 50ms. The length of As the output signal, the sum of values on both sides )tain the pitches extracted of the nut filter is used. Although this is an inturepresents the roughness itive choice, the spectrum exhibits missing harmonics at the correct frequencies, as required. The scattering function parameters are adjusted according to ithe time intervals and the corresponding transients shown in Fig. 1. For tba < t < td, the transmission \ r coefficients in both directions are 1.0 and the reflection coefficients are 0.0. As soon as the string starts to be damped, the plucking device constraints the 4... displacement. This is implemented by comparing the 750 1125 1500 sum of the displacement at the junction and the constraint. If the displacement is larger, it is clipped and the leftgoing and rightgoing components are scaled to preserve their mutual ratios. The constraint is assumed to be slowly varying in time, in other words, S its characteristic time scale is larger than the maxi750 5625 7500 mum period in the string vibration. The string disny(Hz)placement becomes zero at t = tbd. It then quickly spectra from the data in becomes negative and excites the divided portions of t3 (a), t4 (b) and t2 (c). the string. Right around this moment, the reflection eads show the vibrational coefficients are increased and the transmission coeffie harmonic spectra of the cients are decreased to a value close to zero such as 0.05. Depending on the excitation, they can be made non-symmetric, for example to introduce additional itude is made adjustable losses, the sum of the reflection and transmission cobe played over time, the efficients on one side can be made less than 1.0. If the nut side is chosen, the model simulates a thumb finger, which damps the nut portion of the string. The two transverse modes are coupled to each S other at the bridge using another scattering junction. are very powerful in syn- The parameters in this junction are not modified durbrational modes. In this ing the course of a note. Moreover, in the current bl case Ax = cAt is used implementation, the parameters in this junction are ng equation. This choice kept constant in frequency. For better results, a frersion for non-ideal strings quency dependent admittance matrix should be used. demonstrating the ideas way, waveguides can be ig portions and the most Y + 2b3-  can be assuming aL <K 1, where of the string and L is the entation for the transverse es model the propagation nd leftgoing y- (x, t) comicement y(x, t). The scatle represents the plucking nsists of several cascaded 3.3 Body Vibration The body vibration, b(t), is assumed to be known -in fact estimated- and can be generated by FM synthesis or any other method. It is coupled to the transverse modes on the bridge, as a boundary condition: b(t) y+(x = bridge, t)+y-(x = bridge, t). This approach works very well if a tap sound is recorded without the transverse modes, and then synthesized with an envelope more spread in time to account for the continuous excitation due to the vibrationg string. In this paper, two sets of FM generators were used for
Page 00000004 delay line scattering junction delay line Nut Excitation Bridge Figure 3: Schematic diagram which represents the synthesis model of the transverse modes and their coupling to the tap sound. the tap sounds generated at tbd and the corresponding tba of the third note. Their carrier envelopes are added to simulate the frequent excitation of the body in fast passages. 4 Synthesis of Articulations The output sound is obtained by enveloping, timing and smoothly adding (fade in/out) all the vibrational mode sounds described in Sec. 3. W~ith the use of a more complicated timing circuit, the ideas represented here can be used for the digital synthesis of sounds produced using various playing techniques specific to any plucked string instrument. For example, in the case of a classical guitar, harmonics can be synthesized by adding another scattering junction and scaling the amplitudes of the signals. Ascending slurs which also produce a percussive body sound can simply be implemented by moving the scattering junction to the location where the finger "hammers on" and applying a constraint for the displacement. All the left hand "squeak" sounds which are a combination of longitudinal modes and noise due to string roughness, can be generated by adjusting the timing between the individual elements of the model. 5 Conclusions The sound data produced using the procedure described in this paper is perceived to be very close to the reality especially when it consists of articulated guitar notes. It is very important to notice that this result is achieved not by synthesizing the transverse or body modes very accurately, but by incorporating the correct transients into the synthesis model. 6 Acknowledgments The author would like to thank Julius 0. Smith for stimulating discussions; Chris Chafe, Jay Kadis, and Fernando Lopez-Lezcano for their support at the Center for Computer Research in Music and Acoustics (CCRMA) during his matriculation at Stanford University. Refere nces  Karplus, K. and Strong, A. 1983. "Digital synthesis of plucked-string and drum timbres", Computer Music Journal, Volume 7, Number 2, pp. 43-55.  K~arjalainen, M. and Smith, J. 1996. "Body modeling techniques for string instrument synthesis", Proc. JCMC, Hong Kong, pp. 232-239.  Chaigne, A. 1992. "On the use of finite differences for musical synthesis", Journal d'Acoustique, Volume 5, Number 2, pp. 181-211.  Chaigne, A. 1991. "Viscoelastic properties of nylon guitar strings", Catgut Acoust. Soc. J., Volume 1, Number 7, pp. 21-27.  Cook, P.R. 1992. "A Meta-wind-instrument physical model, and a meta-controller for real time performance control", Proc. JCMC, San Jose, pp. 273-276.  Jaffe, D.A. and Smith, J.O. 1983. "Extensions of the Karplus-Strong plucked-string algorithm", Computer Music Journal, Volume 7, Number 2, pp. 56-69.