# A SIMPLE DYNAMIC TONEHOLE MODEL FOR REAL-TIME SYNTHESIS OF CLARINET-LIKE INSTRUMENTS

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Page 00000001 A SIMPLE DYNAMIC TONEHOLE MODEL FOR REAL-TIME SYNTHESIS OF CLARINET-LIKE INSTRUMENTS Jonathan TERROIR, Philippe GUILLEMAIN CNRS - Laboratoire de Mecanique et d'Acoustique 31, chemin Jospeh Aiguier. 13402, Marseille cedex 20, France ABSTRACT In this paper, we present a digital method to simulate the effect of dynamic geometry modification of a simplified clarinet bore induced by tonehole closing or opening. For this purpose, we first consider a physical tonehole model where the closing of the hole is achieved by progressively reducing its radius. In the context of real-time synthesis of woodwind instrument tones obtained from digital models such as those developped by Guillemain [5], we propose a simplified method that preserves the main characteristics of the physical model. The results of the two methods are compared. 1. INTRODUCTION This paper is devoted to the modeling of the dynamic closing of a tonehole in the context of real-time synthesis. Physical models are known to accurately reproduce the behavior of musical instruments. Consequently they are advantageous to simulate the establishment of oscillations or other states. In a musical context it is important to reproduce realistic transitions between different pitches. The aim of this study is not to accurately model the physical phenomena involved in the closing of a tonehole, but to end up with a perceptually realistic model. According to musicians there are two important aspects linked to the closing. The first one, observed by the player, is a difficulty in playing when closing the tonehole. This is induced by a loss of harmonicity of the impedance peaks and a decrease in the amplitude of the first peak. The second one is a noticeable effect of glissando during a slow closing. Consequently, we focus on the reproduction of these two aspects in a real-time context. Many studies dedicated to toneholes have been concerned with closed or open holes (see for example [6], [8] or [3]). Benade [1] showed that the equivalent length of a single hole pipe is a direct function of the geometric properties of the hole. Keefe [7] studied the influence of the size of the holes on frequency, amplitude and playing characteristics. Dalmont et al. [2] showed how the size of the hole can modify the perceived pitch, timbre, radiated energy and loudness. The knowledge of how the hole modifies the behavior of the instrument is essential, but what happens during the closing is very important too. Nederveen [9] has considered the influence of the key during the closing, as well 2i P3t3 PI P I'll 1 12 '1 "2 q / Figure 1. Cylindrical pipe with a single tonehole. L1, L2, rt and ht the geometrical properties of the single hole pipe. pi, wi pressures and flows. as Ducasse [4], who has implemented a model of progressive closing of the tonehole, and Scavone and Cook [10] who simulate the closing by varying the coefficients of the hole reflectance function between its fully open value and a value close to zero. The paper is divided in four sections. After this brief introduction, the simplified single hole pipe physical model is presented. Followed by an alternate model based on an interpolation procedure. Last section presents the conclusions and perspectives of this work. 2. SIMPLIFIED PHYSICAL MODEL 2.1. Bore model One considers a cylindrical pipe with a single hole (see figure 1). This is an oversimplified model of a clarinet body where we only consider the first open hole [1]. Localized losses and non linearities induced by sharp edges or air jets, as well as radiation impedance are ignored here. Considering electro-acoustical analogy and applying Kirchhoff's law we obtain the following pressure and flow equations. P1 = P2 = P3 Ul = U2 + U3 (1) (2) The hole input impedance Zt is defined as a function of the geometrical characteristics of the hole (equation 3) where rt is the radius, ht is the length, St wr 2is the surface and kt is the wave number (complex-valued to integrate losses). Zt(0) 0=SPC tan(ktht) (3)

Page 00000002 a) 0 500 1000 1500 2000 2500 Frequencies (Hz) Figure 2. Variation of the input impedance Z, of a single tonehole pipe as a function of the hole state. The closing is simulated by linearly reducing the hole radius. rt is 3.5mm for a totally open tonehole and zero for a totally closed tonehole. The radius of the bore is 8mm. The impedance Z, of the single hole pipe is consequently a function of the hole radius and of the impedances Z, = jZo tan(kiLl) and Z2 jZo tan(k2L2) corresponding respectively to the two bores of lengthes L1 and L2 (equation 4), which characteristic impedance is Zo. 70 u 60 E 6 40 - 30 E 20. 2000 1500 1000 10', Open Closing Closed Tonehole state 500 Open Closing Closed Tonehole state Pe (w) Ue (0j) 1 1 1+ 1 + 1 1 Zt 2 The closing of the hole is simulated by reducing its radius rt from a given value to zero. The input impedance Z, is calculated for each values of the radius. Figure 2 shows how the impedance of a single hole pipe (L1= 0.2m and L2 0.07m, corresponding to a 5 semitones variation) varies as a function of the radius of the hole. Each horizontal line corresponds to a given value of rt from 3.5mm to zero. One can observe a glissando of the impedance peaks. The figure also shows that frequencies of the initially existing peaks are sliding and that some peaks are appearing (the amplitude of the fourth peak is increasing). The variations of the amplitude of the sliding peaks is more complicated. One can observe a first decrease followed by an increasing phase (see figure 3). Moreover, although the initial and final impedances' first three peaks are nearly harmonic during the time of the closing, the inharmonicity is important and the amplitudes of the peaks are smaller (figure 2). Inharmonicity combined with lower amplitudes of the peaks cause the instrument to be more difficult to play, as shown by Dalmont and al. [2]. Despite its simplicity, this input impedance model is able to reproduce the main phenomena that appear during the tonehole closing, i.e. frequency glissando induced by the glissando of the first impedance peak and negative effects on the Figure 3. Amplitude and frequency variation of the three first peaks of the input impedance Z, of a single hole pipe as a function of the tonehole state from totally open to totally closed. The closing is simulated by reducing linearly the hole radius. Peak 1: solid line; peak 2: dotted line; peak 3: dashed-dotted line. ease of playing induced by the decrease of the amplitudes of the impedance peaks and loss of harmonicity, as shown on figure 3. 2.2. Full instrument model We consider a classical physical model of a simplified clarinet-like instrument (see e.g [13]), and its real-time oriented version as proposed by Guillemain [5]. The dimensionless physical model includes a linear part that models the reed displacement as a pressure driven mass-spring system, an other linear part that models the input impedance of the bore and a non-linear part, coupled with the linear parts, that uses the Bernoulli model to describe interactions between flow and pressure at the mouthpiece. The discrete time-domain equivalent of equation 4 is obtained according to [5]. In figure 4, where the spectrogram of the external pressure is displayed, the radius of the tonehole is linearly reduced from 0.35mm to Omm (L1 0.2m, L2 0.07m). The main effect of the closing, namely the glissando is clearly reproduced. Due to the nonlinear coupling with the reed, this glissando is different from that of the first peak of Ze (w). A drawback of this dynamic closing model is its numerical complexity. Indeed, with this model, in a real-time implementation, the coefficients of the difference equation linking the output pe (r) (the pressure in the mouthpiece) to the input.e (n) (the flow in the mouthpiece), that corresponds to the impedance part of the full model, have to be modified each time the radius of the tonehole is modified. Such a computation cost can become incompatible with a realtime implementation. This is why, in the next section, we propose a simplified method that approximates the effect of a continuous modification of the input impedance induced by a tonehole closing. 3. CROSS-FADE MODEL Though the simplified model of a dynamic closing presented in the previous section gives expected results (glis

Page 00000003 2500 2000 1500 00 Open Closing of the tonehole Closed Figure 4. Spectrogram of the external pressure signal obtained by simulating the closing of the tone hole by reducing linearly its radius from totally open to totally closed. sando, inharmonicity, loss of amplitude of the first peaks), its computational cost might be incompatible with a realtime sound synthesis. Here, we seek an alternate method which is compatible with real-time synthesis and gives perceptually relevant results. During his study dealing with trumpet modeling, Vergez [11] worked on the problem of dynamic closing of valves. The problem is similar since one aims at modelling the smooth transition from a length L1 to a length L1 + L2 with the introduction of a side-branch at intermediate states. Vergez chose to model the closing of a valve by simultaneously considering two resonators during the transition time. He took into account the change of the effective length by interpolating in the time domain between the two reflection functions corresponding to the two different lengthes of the bore. Hence, the reflected wave value is an average between the two pressures calculated considering the two reflection functions (the first one for the length L1 the second one for L1 + L2). For partially open holes, van Walstijn and Campbell [12] divided the tonehole volume into an "open part" and a "closed part", the first one behaving as an inertance and the second as a compliance. The impedance of the hole is a function of the impedances of the extreme positions and a parameter defines the ratio between opened and closed states. The discrete-time model we are interested in is based on a digital impedance filter the coefficients of which are explicitely expressed as functions of the geometrical parameters of the instrument. Following similar lines, we interpolate between two filterings corresponding to two different bore lengthes. For any bore length, equation 5 relates, at each sample n, the pressure pe(n) to the flow ue(n), where V is a function of the past (linked to (n-l, n-2,...n-D) samples, where D is the propagation delay): pe(n) = ue(n)+ V (5) V1 corresponds to the initial geometry (bore of length L1). V1+2 corresponds to the final geometry (bore of length L1 + L2). During the time of the closing, an interpolation 1500 ... 1000 U500 0 Open Cross-fade Closed Figure 5. Spectrogram of the external pressure signal of a single tonehole pipe obtained by simulating the closing of the hole by a linear cross-fade. between V1 and V1+2 is performed. We obtain a new value of V which is a direct function of the values for open and closed positions: V = VR + V1+2Rinv. R and Rin are two time varying functions that define how this crossfade is performed, satisfying the relation: R + Ri = 1. Figure 5 shows the spectrogram of the external pressure signals obtained with this interpolation procedure. Each vertical line corresponds to a fixed value of R and Rin, and the horizontal axis corresponds to a linear increase of R from 1 to 0. One can notice that the pitch changes non linearily with respect to R and that the level of the harmonics reaches a minimum value around R = 0.5. This behavior of the pressure spectrum is in good agreement with that obtained with the simplified physical model presented in figure 4. Nevertheless, one can notice that the amplitude and frequency variations obtained by reducing the hole radius and by interpolating are different. This difference makes it necessary to adjust the shape of the function R. Figure 4 shows how the amplitudes and frequencies of the first harmonics change with respect to rt. From the frequency variations of the first harmonic shown in figure 5, one can associate to any given value of rt yielding a given frequency of the first harmonic, the value of R that leads to the same frequency of the first harmonic. We focus on this criterion since we are mainly interested in reproducing the perceptive glissando effect of the closing. We found that a suitable couple R and Ri, functions of the type: R = cos(/3r), where /3 is a constant adjusted so that 0 < R < 1 when 0 < rt < rmax and Rin - 1 - R led to a comparable frequency evolution of the first two harmonics (see figure 6), and in a similar evolution of the amplitude of the first harmonic. Moreover, this shape is in accordance with the frequency evolution of the first peak of the input impedance Z, of the single hole pipe (see figure 3), which is directly related to the fundamental frequency of the sound. The difference between the level of the first impedance peaks and the am

Page 00000004 3 Harmonic 3 Harmonic 1 Open Closing Closed Tonehole state Harmonic 3 Harmonic 1 Open Closing Closed Tonehole state Z o -I 3 a, CD a o o. --h --s Figure 6. Amplitude and frequency variation of the first two harmonics of external pressure signal obtained by reducing the tonehole radius (solid line) and by cross-fade with R as a function of /cos(/3rt) (dotted line). Results are presented as a function of the tonehole state from totally open to totally closed. plitude of the first harmonics of the external pressure is due to the nonlinear generation of the sound. In normal playing conditions, transitions between notes are obtained by varying continuously rT over a few milliseconds. It is worth noting that though this interpolation model constitutes a crude approximation of the physical model of section 2, the validity of the physical model itself is questionable in transient situations from both physical and signal processing points of view, since the input impedance (or the reflection function in a wave variable model) and its time-domain equivalent are defined for fixed geometries and stationary hypotheses. 4. CONCLUSIONS The simplified model of dynamic tonehole closing obtained by interpolation is compatible with a real-time implementation. It is able to reproduce the perceptive glissando effect occuring during note transitions and can be used for bores of more complex geometries. The reduced ease of playing (e.g modification of the oscillation treshold), induced by inharmonicity and lowering of the amplitude of the first peak of the impedance has been observed on the real-time implementation of the model. But it is hard to quantify since it depends on the control parameters and requires further studies. In the same way, the dynamic behavior of the whole instrument with toneholes in intermediate states will be studied with the help of an artificial mouth and compared with that obtained with the model. Sound examples corresponding to this paper are available at: http://www.lma.cnrs-mrs.fr/~ guillemain/ICMC05hole.html 5. REFERENCES [1] Benade, A. H. "Fundamentals of musical acoustics", Oxford University Press, London, 1976. [2] Dalmont, J.P., Ducasse, E. and Ollivier, S. "Practical consequences of tone holes nonlinear behaviour", Proceedings of the ISMA 2001, Perugia, Italia, 2001. [3] Dubos, V., Kergomard, J., Khettabi, A., Dalmont, J.P., Keefe, D.H. and Nedervenn, C.J. "Theory of sound propagation in a duct with a branched tube using modal decomposition", Acta Acustica, Vol.85, p.153-169, 1999. [4] Ducasse, E. "Modelisation et simulation dans le domaine temporel d'instruments a vent a anche simple en situation de jeu: methodes et modeles", pHd Thesis, Universite du Maine, 2001. [5] Guillemain, P. "A Digital Synthesis Model of Double-Reed Wind Instruments", Eurasip Journal on Applied Signal Processing, Special Issue on Model-based Sound Synthesis, Vol.2004(7), p.990-1000, 2004. [6] Keefe, D.H. "Theory of the single woodwind tone hole", J. Acoust. Soc. Am., Vol.72(3), p.676-687, 1983. [7] Keefe, D.H. "Acoustic streaming, dimensional analysis of nonlinearities, and tone hole mutual interactions in woodwinds", J. Acoust. Soc. Am., Vol.73, p.1804-1820, 1983. [8] Nederveen, C.J., Jansen, J.K.M. and Van Hassel, R.R. "Corrections for woodwind tone-hole calculations" Acta Acustica, Vol.84, p.657 -699, 1998. [9] Nederveen, C.J. "Acoustical aspects of woodwind instruments, revised edition", Northern Illinois University Press, Illinois, 1998. [10] Scavone, G. and Cook, P., "Real-time computer modeling of woodwind instruments", Proceedings of the ISMA 1998, Leavenworth, WA, USA, 1998. [11] Vergez, C. "Trompette et trompettiste: un systeme dynamique non lineaire a analyser, modeliser et simuler dans un contexte musical", pHd Thesis, Universite Paris 6, 2000. [12] van Walstijn, M. and Campbell, M. "Discretetime modeling of woodwind instrument bores using wave variables", J. Acoust. Soc. Am., Vol.113(1), p.575-585, 2003. [13] Wilson, T. A. and Beavers, G. S. "Operating modes of the clarinet", J. Acoust. Soc. Am., Vol.56, p.653-658, 1974.