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Page 00000001 ON THE RELATIVE INFLUENCE OF EVEN AND ODD HARMONICS IN CLARINET TIMBRE Mathieu Barthet, Philippe Guillemain, Richard Kronland-Martinet, Solvi Ystad CNRS - Laboratoire de Mecanique et d'Acoustique 31, chemin Joseph Aiguier 13402 Marseille Cedex 20, France ABSTRACT The work presented here is part of a larger project directed toward a better understanding of the influence of timbre variations upon musical expressiveness. To address such an issue, we have limited our study to the relationship between the physics and playing of the clarinet, and the generated timbre. This reduction of the problem is justified by the fact that the clarinet is a physically well understood instrument, leading to an efficient synthesis model which can be run in real-time. Such a physical knowledge is of great importance since it allows the design of specific timbre descriptors adapted to this instrument and points out the most important control parameters the musician can act on. After a brief description of the physics of the clarinet, we show how specific timbre descriptors can be derived from classical ones and how the control of the instrument acts on the perceived sound. 1. INTRODUCTION Numerous authors have already studied how note durations, playing frequencies and energy variations intervene in the musical expressiveness. These studies show that such score deviations are far from being unspecified and that they follow very explicit rules . Even though several authors agree on the importance of timbre variations during the performance, few studies are dedicated to musical expressiveness and how the performer controls the timbre during the play . This is probably due to the practical difficulties related to measurements of control parameters under real playing conditions, but also to the lack of subtle timbre descriptors that take into account the specificities of the instrument. To overcome these difficulties, we based our study on the use of physical modeling, and in particular the clarinet model. This is of great interest since it allows the generation of calibrated sounds and leads to an unambiguous relationship between the control parameters and the generated timbre. Moreover, given that the model is physically relevant, it can be used to point out some specificities of the sound, directly linked to the physics of the instrument. These specificities are valuable for the design of appropriate timbre descriptors directed toward the characterization of sounds belonging to a unique category (here the clarinet sounds). In this paper, we shall first describe how a physical model of a clarinet can be used to link the control parameters to the timbre. Then, by taking into account the physical specificities of the instrument, we will show how usual timbre descriptors can be adapted to better describe the timbre space of the clarinet. Eventually, we shall show how these timbre variations can be explained from a physical point of view. 2. A SIMPLIFIED PHYSICAL MODEL OF CLARINET In this section we briefly present the physical model used for the sound synthesis made of three coupled parts. The first part is linear and represents the bore of the instrument. The second part expresses the aperture of the reed channel, linked to the reed displacement considered as a pressure driven mass-spring oscillator. The third part couples the previous ones in a nonlinear way. In what follows, we use dimensionless variables for the pressure, flow, and reed displacement according to . 2.1. Bore model The bore is considered as a perfect cylinder of length L. Under classical hypothesis, its input impedance Ze which links the acoustic pressure Pe and flow Ue in the mouthpiece in the Fourier domain is classically written as: Ze( = Pe (w) Ue (W) = ia( k(w)L) (1) Here, Ue is normalized with respect to the characteristic impedance of the resonator. The wavenumber k(w) corresponds to the classical approximation (the bore radius is large with respect to the boundary layers thicknesses). It is worth noticing that for any flow signal, the acoustic pressure mostly contains odd harmonics since the input impedance corresponds to that of a quarter-wave resonator. At high frequencies, the increase of losses taken into account in the wavenumber induces non-zero values of the impedance for even harmonic frequencies. Hence, if the flow contains high frequency even harmonics, these will also appear in the pressure.
Page 00000002 2.2. Reed model The classical single mode reed model used here describes the displacement x(t) of the reed with respect to its equilibrium point when it is submitted to an acoustic pressure Pe (t): 1 d2x(t) +q,dx() t xt)pe wr dt2 w+(dt (2) where w, = 27f, and q, are respectively the circular frequency and the quality factor of the reed. The reed displacement behaves like the pressure below the reed resonance frequency, as a pressure amplifier around the reed resonance frequency, and as a low-pass filter at higher frequencies. 2.3. Nonlinear characteristics The classical nonlinear characteristics used here is based on the stationary Bernoulli equation and links the acoustic flow (the product between the opening of the reed channel and the acoustic velocity) to the pressure difference between the bore and the mouth of the player. The opening of the reed channel S(t) is expressed from the reed displacement by: S(t) = 6(1 - -Y + X(t))((1 - -Y + X(t)) where 0 denotes the Heaviside function the role of which is to keep the opening of the reed channel positive by cancelling it when 1 - - H+ x(t) < 0. The parameter ( characterizes the whole embouchure and takes into account the lip position and the section ratio between the mouthpiece opening and the resonator. It is proportional to the square root of the reed position H at equilibrium. Common values of ( for the clarinet are between 0.2 and 0.6. The parameter -y is the ratio between the pressure Pm inside the player's mouth (assumed to be constant in the steadystate regime) and the static beating reed pressure. In a lossless bore and a massless reed model, -y evolves from I which is the oscillation step, to 2 which corresponds to the position at which the reed starts beating. Since the reed displacement corresponds to a linear filtering of the acoustic pressure, the reed opening mostly contains odd harmonics. Nevertheless, the function 0 introduces a singularity in S(t) for playing conditions (given by ( and -y) yielding a complete closing of the reed channel (dynamic beating reed case). This leads to a sudden rise of even harmonics in S(t) (saturating nonlinearity) and the generation of high frequencies. The acoustic flow is finally given by: 2.4. Coupling of the reed and the resonator Combining the impedance relation, the reed displacement and the nonlinear characteristics, the acoustic pressure, acoustic flow and reed displacement in the mouthpiece are solutions of the coupled equations (1), (2) and (3). The digital transcription of these equations and the computation scheme that explicitely solve this coupled system is achieved according to the method described in . 2.5. External pressure From the pressure and the flow inside the resonator at the mouthpiece level, the external pressure is calculated by the relation: pext (t) j(Pe (t) +H ue(t)), which corresponds to the simplest approximation of a monopolar radiation. This expression shows that in the clarinet spectrum, the odd harmonics are generated from both the flow and the pressure, while the even harmonics mostly come from the flow. The ratio between odd and even harmonics can be considered as a signature of the "strength" of the nonlinearity. 3. TIMBRE DESCRIPTORS From the previous chapter we see that the physical behavior of the clarinet leads to particular sound structures that inevitably influence the timbre of the instrument. A large number of studies have been carried out to find correlations between perceptual timbre dimensions and physical ones in order to distinguish different instruments in a socalled timbre space generally constituted by two or three dimensions . The first dimension is temporal and well represented by the attack time, the second is spectral and represented by the spectral centroid while the last dimension is usually composed by a combination of temporal and spectral variations . In our case the aim is to distinguish different sounds from the same instrument, i.e. the clarinet, meaning that it is necessary to adapt the descriptors to more subtle timbre variations. 3.1. Classical timbre descriptors In earlier studies we have tested traditional timbre descriptors to find out how they could characterize clarinet sounds . The attack time, the spectral centroid, the spectral bandwith, and irregularity were used for this purpose. The attack time is one of the most significant temporal timbre descriptor and measures how quickly the intensity of a sound rises. The spectral centroid is an important spectral timbre descriptor directly related to the brightness of a sound. It can be expressed by: te(t) = S(t Pe (t)) Y -Pc-(t) E NfiA SC - = sI (4) This nonlinear relation between pressure and opening of the reed channel explains why the flow spectrum contains all the harmonics. where fi is the frequency related to the ith spectral component, Ai its magnitude and N the total number of components of the discrete spectrum. The spectral bandwidth
Page 00000003 is linked to the frequency spreading of the spectral components, while the irregularity shows the amplitude variation between successive harmonics. As we shall see in the next section we have derived new descriptors from the spectral centroid and irregularity. 3.2. Timbre descriptors adapted to the clarinet Indeed, a specificity of the clarinet timbre can be explained by the lack of energy of its even harmonics in comparison to the odd ones. This occurence indicates that it might be interesting to split the spectrum into two parts and observe the spectral behavior of odd and even partials separately. We have defined proper clarinet timbre descriptors that give a more detailed description of the odd and even harmonics behavior than classical ones as the control parameters vary. Thus, they may reveal some fine timbre variations which we can perceive when listening to the sounds. The spectral envelope of the harmonics has been found according to the analysis method described in the next section. From a global envelope, two new envelopes are defined, one for the odd and one for the even harmonics. Hence, we can express the the even/odd harmonics spectral amplitude ratio Reo(k) as a function of the frequency by: 4.2. Loudness equalization The relative loudness of these sounds has also been equalized to Legal = 70 phones. Its calculation has been made according to the Zwicker loudness model (ISO 523B) assuming that the sounds were stationary. An optimization algorithm based on the simplex method determines the optimal a so as the loudness of aPex equals Legal4.3. Analysis parameters All the timbre descriptors - except the attack time - are calculated in the steady-state regime of the clarinet. The FFT is made on 2048 samples windowed by a gaussian. A parabolic interpolation method on the logarithm of the magnitude spectrum is used to obtain the frequencies and amplitudes of the harmonics. In the case of fr = 2500 Hz, 18 harmonics ([0-3060 Hz]) have been taken into account whereas only 12 ([0-2004 Hz]) for fr = 1100 Hz. This choice has been made relatively to the fact that the reed acts as a low-pass filter above its resonance frequency. 4.4. Global, odd and even centroids Spectral centroids (global, odd and even) for Zeta = 0.33 Derivative of spectral centroids as a function of Gamma (Zeta = 0.33) Vk\l<k<M -- -- 2,' Re (k)= oA2k-1 Reo(k)- A2k-1 (5) where the index k refers to the kth harmonic and M is assumed to be even. Its value is usually between 0 and 1. For the clarinet, this ratio is globally weak and its raise will reveal an increase of even harmonics energy in the spectrum. From the definition of the spectral centroid (4), qualified as global, two centroids are proposed: the odd spectral centroid and the even spectral centroid. The interest in studying the growth of the spectral centroids (global, odd and even) as a function of a physical control parameter p will be discussed in the next section. For this purpose we calculate their derivative: 0.44 0.46 Gamma 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 Gamma DSCglobal,odd,even (p) -- dSCglobal,odd,even (P) (6) 4. RESULTS AND DISCUSSION 4.1. Synthesis sounds In order to study the timbre variations as a function of the control parameters, a hundred sounds P,t have been generated with the model corresponding to 100 couples ((, 7) chosen in the range (0.2 < ( < 0.5 and 0.4 < 7 < 0.5). The fundamental frequency of the bore has been set to 170 Hz (note that it does not exactly correspond to the play frequency), and the time length has been fixed to 1.5 s with a sampling frequency of 44100 Hz. Two different reed resonance frequencies and quality factors have been studied (f, = 2500 Hz, q, = 0.2) and (f, = 1100 Hz, q- = 0.7). Figure 1. Global, odd and even spectral centroids as a function of 7 ((o = 0.33) Figure 1 (left) represents the evolution of the three centroids for a given (o and an increasing mouth pressure. The global spectral centroid monotonously increases up to its maximum value at 7 = 0.5. Thus, the higher the mouth pressure, the stronger the brightness of the sound. Although the evolution of odd and even SC seems to follow the same law, their derivative as a function ofy 7 (figure 1, right) brings up an interesting phenomenon. In the three cases the derivative is positive showing that the functions SCglobal,odd,even (7) are increasing. Up to a given value the even spectral centroid derivative remains the strongest, i.e. even harmonics energy moves in a faster way towards high frequencies than the odd one as the mouth pressure increases. At 7br an abrupt change occurs and the odd curve crosses the even one. Perceptively one hears a change in the sound's timbre (we invite the reader to listen to the sounds: http://www.lma.cnrs-mrs.fr/~ guillemain/ ICMC05timbre.html). This can be explained from the reed opening function S(t) which cancels at 7 = 7br
Page 00000004 meaning that the reed is beating. In a non beating-reed situation, even harmonics of the external pressure mostly come from the Bernoulli part of the nonlinear characteristics while in a beating-reed situation, the saturating nonlinearity of the reed channel opening also contributes to add even harmonics to the flow. 4.5. The even/odd harmonics relation Even/Odd ratio (Fr =1100, Qr = 0.7, Zeta = 0.33) Even/Odd ratio (Fr = 2500, Qr = 0.2, Zeta= 0.33) 1 2 3 4 5 6 Ratio index 4 5 6 Ratio index Figure 2. Even/Odd spectral amplitudes ratio for f, 1100 Hz and f, = 2500 Hz ((o = 0.33) The even/odd ratio parameter points out the frequency area where even harmonics spectral energy is maximum compared to odd ones. The mean frequency between the harmonics partial 2k - 1 and 2k is given by: f(k) (2k - ')fo. As showed in figure 2 this area is located around f(4) 1225 Hz for f, 1100 Hz (for which f0 167 Hz) and around f (8) 2635 Hz for f,= 2500 Hz (for which fo 170 Hz). The maximum of this parameter seems to be directly related to the reed resonance frequency which suggests that the energy of even harmonics is higher around the resonance frequency of the reed. This amplification observed on external pressure signals can indeed be linked to the functioning of the physical model. Up to the nonlinearity of the Bernoulli model the flow is proportional to the opening of the reed channel which amplifies the level of all the harmonics around the reed resonance frequency. Since musicians constantly change the position of their lip on the reed during the play and therefore modify its resonance frequency, the even/odd ratio may be used in further studies on the expressiveness to estimate the reed resonance frequency in playing conditions. A second relevant aspect appears when observing the maximum of Reo as a function of the physical control parameter -y of the model (figure 3). The even/odd ratio is maximum for -Ybr corresponding to the beating-reed case and decreases for higher mouth pressure values because the saturating nonlinearity adds both even and odd harmonics. Thus, we can notice that for a given (, an even harmonics energy peak emerges in the spectrum at a frequency and intensity (weighted by surrounding odd harmonics in Reo) which can be controlled both by the parameters of the reed and the mouth pressure. 5. CONCLUSION Though the results presented here have been obtained from an oversimplified model of clarinet, preliminary studies conducted on natural clarinet sounds generated with the help of an artificial mouth exhibit similar tendencies. This study represents a first step towards the characterization of the clarinet timbre as a function of its control parameters. The derived timbre descriptors proposed here should in future works be validated through perceptual tests. 6. REFERENCES  J. Sundberg, "Grouping and Differentiation. Two Main Principles in the Performance of Music.", Integrated Human Brain Science. Theory, Method, Application (Music). Elsevier Science B.V., 2000.  C. Traube, P. Depalle, M. Wanderley. "Indirect acquisition of instrumental gesture based on signal, physical and perceptual information", NIMEO3, Montreal, Canada, 2003.  J. Kergomard, "Elementary considerations on Reed-Instrument Oscillations", Mechanics of Musical Instruments. Ed. Springer-Verlag Wien New York, 1995.  Ph. Guillemain, "A Digital Synthesis Model of Double-reed Wind Instruments", Eurasup. JASPI special issue "Model based sound synthesis ", 2004.  Ph. Guillemain, R.T. Helland, R. KronlandMartinet, S. Ystad "The Clarinet Timbre as an Attribute of Expressivenes", LNCS 3310, pp.246-259, Springer Verlag, 2005.  5. McAdams, S. Winsberg, S. Donnadieu, G. De Soete, J. Krimphoff, "Perceptual scaling of synthesized musical timbres: common dimensions, specificities, and latent subject classes", Psychol. Res. 58: 177-1 92, 1995.  A. Caclin, "Interactions et independances entre dimensions du timbre des sons complexes", PhD thesis, Paris 6 University, 2004. Evn/Odd r tic maimum (Fr = 2500, C:r = 0.2,;Zeta = 0.33) O-S5 -O085 -0-8 -0-7S 07 0-65 06 - 055 - O05 - 4 -0.4 0.42 044 046 048 05 --mm Figure 3. Even/Odd ratio maximum as a function of y