Page  00000240 PERFORMANCE CONTROL OF A FLUTE PHYSICAL MODEL USING FUZZY LOGIC Patricio de la Cuadral, Rodrigo E Cddizl Benoft Fabre2, Nicolas Montgermont2 1 Centro de Investigaci6n en Tecnologfas de Audio Pontificia Universidad Cat6lica de Chile {fpcuadraIrcadizI} 2 LAM - IJLRDA, Paris 6 University fabreb, ABSTRACT A simple model aimed to emulate the control exerted by a flautist over selected parameters while exciting the flute is proposed. Using a fuzzy logic based control system, a musically meaningful mapping between performance variables and physical model parameters is obtained. The real time implementation in Pd (Pure data) environment shows an adequate matching between the outputs of the fuzzy system and previous experimental data. 1. INTRODUCTION Digital sound synthesis algorithms, even simple ones, typically involve a large number of parameters, which makes them difficult to explore by hand. As physical models of acoustic instruments become more precise their complexity and flexibility grows enormously, together with the difficulty to produce the desired sounds out of them. Physical models normally provide control over several parameters that are not necessarily appropriate to simulate an adequate musical performance of it. While blowing pressure, friction, length and gain are examples of natural parameters of physical descriptions, dynamics, brightness and vibrato are commonly used in musical contexts. It is, therefore, necessary to establish a proper mapping between the two languages when a musical use of physical models is required. Moreover, when a musical sound is generated computationally, it is important to have a good model with parameters allowing an intuitive manipulation of the sound. The musician-user must be able to develop a musical intuition that will allow experimenting with the synthesis technique used [15]. Parameter control is an essential and very important part of the musical applicability of digital synthesis. In 1995, Jaffe proposed ten criteria for evaluating synthesis techniques [9]. Of the ten, four are directly related to the parameters: How intuitive are the parameters? How perceptible are parameter changes? How physical are the parameters? How well behaved are the parameters? Throughout this paper an intuitive mapping between selected parameters of a flute physical model and musical features is proposed. In section 2 the basic aspects of the flute physical model operation are discussed. Section 3 elaborates on the musical aspects of the flute performance that are relevant to parameters control. Section 4 contains a description of the fuzzy logic control system implemented for the mapping process and finally, section 5 presents the results, future work and main conclusions of our work. 2. CONTROL PARAMETERS OF A FLUTE PHYSICAL MODEL Instruments from the flute family share a principle of operation that consists of an unstable jet issuing from either a channel or the lips of a player. The jet flows in the direction of a sharp edge that we call the labium. The acoustic field due to the presence of the resonator triggers the jet instability and therefore, the jet oscillates at the frequency of the acoustic field. The transverse displacement of the jet is very small at the flue exit, it propagates along the jet at a certain velocity (convection velocity) and slowly grows as it travels away from it (spatial amplification). The interaction of the perturbed flow with the labium provides the necessary acoustic energy to sustain the acoustic oscillation in the pipe. Figure 1. General description of flutes operation Flute operation can be globally described as a coupling between the hydrodynamic modes of a jet with the acoustic modes of a resonator. The different elements (resonator, jet, sources) are supposed to interact locally and are therefore analyzed separately and then concatenated 240

Page  00000241 to produce a simplified caricature of the flute operation, as sketched in fig. 1. The convection velocity as well as the spatial amplification factor of the wave are functions of the dimensionless frequency, the Strouhal number (Strh fh/lU where f is the frequency, h is the height of the flute from where the jet flows and U3 is the velocity of the jet centerline as shown in fig.2). W ht Uj JLabium hl Uj Figure 2. Jet emerging from the formation channel, travelling accross the open edge of the resonator and hiting the labium Following Nolle [13] and de la Cuadra [5], the spatial amplification appears to be maximum around: most entirely to the design of the instrument builder. But in most flutes the interaction of the performer has an enormous influence on the quality of sound produced, such is the case for the European transverse flute, the South American quenas and the Japanese shakuhachi. Some of the most relevant parameters controlled by the flautist are the distance from the lips to the sharp edge, the shape of the lips hole and the speed of the jet. In [6], images of performer's lips were taken together with measurements of the blowing pressure and the sound radiated by the instrument. The players' control was analyzed in terms of the physics of the sound production. Therefore the main control parameters measured (W, h, LI) were analyzed in terms of the dimensionless numbers discussed above: Strw, Strh, Re. In fig. 3 several musical excerpts played by an expert performer have been collapsed in one plot showing the shape of Re as function of W/h. It is observed that the range of operation of W/h mainly goes from 4 to 10, which coincides with the observations made in section 2, while most of the data is below Re = 2500 where the jet is laminar. Thus for laminar jets the trajectory of data as frequency or loudness grows seems to increase linearly approximately following: 0.02 < Strh = fhlUj < 0.05 (1) depending on the jet velocity profile. The structure of the jet is related to the Reynolds number (Re=U3 h/v, where v is the kinematic viscosity of the air, v 1.5 10-5m2/s). The laminar behavior of the jet occurs approximatelly when Re < 2500 [16, 12]. When blowing harder, the jet becomes turbulent. Re up to 10000 has been measured in the higher register of the flute. Changing the blowing pressure allows the flautist to adjust the delay induced by the convection of perturbations on the jet. As shown by Coltman [4], the flute sounds at the frequency of its passive resonance for a delay equal to half of the oscillating period. This corresponds to the optimal phase relation of the sound source at the labium compared to the acoustic field as discussed by Fabre [8]. The convection velocity of perturbations on the jet is about 30% to 50% of the jet centerline velocity [14], the optimal condition corresponding to half a period delay on the jet may be expressed as: 3500 = Re+250W/h for 1000 < Re < 2500 4 < W/h < 10 (4) As the jet becomes turbulent, for Re greater than 2500, the slope of the trajectory changes in such a way that less reduction of W/h is required to follow the increment on Re. 35000 Chronatic srale Diatonic scai Minr Ord Major 3rd 4th 5th Dirninuendo 1rhnda 1,3th fW Strw = U 0.15... 0.25 (2) where W is the flue-exit to labium distance. Combining equations 1 and 2, the optimal range of the thickness ratio W/h of the jet is: 2000 1000 4 0 1 4 16 Figure 3. Collapsed data from several musical excerpts. Re vs. W/h Fig.4 shows the Strouhal numbers Strh and Strw for the same set of musical excerpts. The data is observed to collapse into a small, well-defined region, bounded by 0.07 K Strw < 0.3 and 0.005 ~ Str< < 0.035. Furthermore, both variables seem to increase as the pitch raises or the loudness is increased, roughly following the trajectory 3 < W/h = <-12 Strh 1 (3) 3. MUSICAL CONTROL OF THE FLUTE There are some instruments from the flute family such as organ pipes where the quality of the final sound is left al 241

Page  00000242 - Chromatic scale S2nd Miinor 3ird 4th D.7 + 5th S Diminuendo Crescendo 00 -i Ocltav 0.04 0.03,, *, described by: Dynamics Brightness Uj W h VL VL VH VL L L H L M M M M H H L H VH VH VL VH VL VH VH L H H M M M H L L VH VL VL Table 1. Fuzzy rules table contexts but typically subjectively defined, to the flute parameters Uj, W and h. The input variables were normalized between 0 and 1 and the ouput variables used their normal range. All variables were fuzzyfied using five triangular membership functions, labeled VL (very low), L (low), M (medium), H (high) and VH (very high). Table 1 shows all the fuzzy rules used for the system. The outputs were defuzzified using the centroid and the min/max method of implication and aggregation was used. Strh = 0.2Strw-0.01 for 0.07 < Strw < 0.3 0.005 < Strh < 0.035 (5) 4. A FUZZY LOGIC PERFORMANCE CONTROLLER FOR THE FLUTE MODEL 4.1. Fuzzy logic Fuzzy logic [1] [10] is a concept derived from the mathematical branch of fuzzy sets [18] that applies multi-valued logic to sets or groups of objects. Its flexibility, simplicity, and diversity of applicability makes it a very suitable tool for the parametric control of computer music, especially synthesis algorithms [3]. In general, when fuzzy logic is applied to a problem, it is able to emulate aspects of the human reasoning process, quantify imprecise information and make decisions based on vague and incomplete data [11]. Fuzzy systems provide several advantages for creative applications. Fuzzy systems are powerful and work in a way that resembles some characteristics of human behavior. Parallel computation of fuzzy rules usually reduces the computation time compared to a traditional mathematical approach. Fuzzy systems, due to the fuzzy approximation theorem, enable the approximation of highly non-linear systems with any degree of accuracy. Fuzzy systems are model-free estimators, in consequence, it is not necessary to know any mathematical model in advance to approximate any system. Fuzzy rules can be easily specified in the form of IF-THEN statements, allowing the building of fuzzy systems with simple linguistic terms. Fuzzy logic allows us to build systems using common sense, and the fuzzy rules can be discussed, tuned, and detuned easily. 4.2. Implementation of the fuzzy controller A fuzzy logic mamdani inference system [17] was implemented in order to map two general performance variables: Brigthness and Dynamics, widely used in musical 5. RESULTS AND CONCLUSIONS 5.1. Analysis of the results 3000 2500 # 4, 2000 ** ~* 4 4 5 6 7 8 9 10 11 12 13 W~i Figure 5. Collapsed data produced by the fuzzy system. Re vs. W/h Figures 5 and 6 show the collapsed data obtained by the fuzzy system using a series of random sequences for the input variables. It is clear that the fuzzy system accurately reproduces the findings obtained by [6] shown in figures 3 and 4, as the obtained data is bounded by the same regions described by equations 4 and 5. 5.2. Real-time implementation The fuzzy system was implemented in Pd using a modular architecture, with one block corresponding to the fuzzy logic control toolkit [2] and the second to the flute physical model developed in [7]. This allows to control the 242

Page  00000243 0.065 0.04 S"s 0.035. " f 0.025 - /t. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 6. Collapsed data produced by the fuzzy system. Strh vs. Strw underlying physical model with two simple musical variables, that can be captured in real time from a live performance using different kind of sensors or simulated inside Pd, creating a mechanism to control a complex synthesis algorithm through simple and intuitive musical parameters. 5.3. Future work The fuzzy system could be extended to include more musical features, such as fingerings, extended techniques or vibrato, and also more parameters from the physical model synthesis algorithm, such as channel length, labium position, or jet/resonator interaction. 5.4. Conclusions The proposed fuzzy logic control system is able to map flute performance variables into synthesis parameters in a musically meaningful way. By using expert knowledge of a flautist and experimental data collected from real flute performances, we were able to adjust and fine tune several fuzzy rules to specify the behavior of the mapper. Only ten simple IF-THEN rules were neccesary to specify the behavior of the control system. This is a significant advance in the process of establishing more musically appealing and intuitive manners of controlling the complex aspects of physical model synthesis algorithms and of linking performance and compositional criteria with synthesis control parameters. 6. REFERENCES [1] Hans Bandemer and Siegfried Gottwald. Fuzzy sets, fuzzy logic, fuzzy methods with applications. J. Wiley, Chichester; New York, 1995. [2] Rodrigo Cidiz and Gary S. Kendall. Fuzzy logic control toolkit: real-time fuzzy control for Max/MSP and Pd. In Proceedings of the International Computer Music Conference, New Orleans, LA, 2006. [3] Rodrigo F. Cidiz. Compositional Control of Computer Music by Fuzzy Logic. PhD thesis, Northwestern University, 2006. [4] J.W. Coltman. Sounding mechanism of the flute and organ pipe. J. Acoust. Soc. Am., 44:983-992, 1968. [5] P. de la Cuadra, B. Fabre, N. Henrich, and T. Robin. Analysis of jet instability in flute-like instruments by means of image processing: effect of the channel geometry on the jet instability. ICA, 18th International Congress on Acoustics, 2004. [6] P. de la Cuadra, B. Fabre, N. Montgermont, and C. Chafe. Analysis of flute control parameters: A comparison between a novice and and experienced flautist. Submitted to Acta Acustica, 2007. [7] Patricio de la Cuadra. The sound of oscillating air jets: Physics, modeling and simulation in flute-like instruments. PhD thesis, Stanford University, 2005. [8] B. Fabre, A. Hirschberg, and A.P.J. Wijnands. Vortex shedding in steady oscillations of a flue organ pipe. Acta Acustica, 82:811-823, 1996. [9] David A. Jaffe. Ten criteria for evaluating synthesis techniques. Computer Music Journal, 19(1):76-87, 1995. [10] George J. Klir and Bo Yuan. Fuzzy sets and fuzzy logic: theory and applications. Prentice Hall PTR, Upper Saddle River N J, 1995. [11] Bart Kosko. Fuzzy Thinking. The new science oJ fuzzy logic. Hyperion, New York, 1993. [12] R.R. Mankbadi. Transition, turbulence and noise: Theory and applications for scientists and engineers. Kluwer Academic Publishers, Boston, 2nd edition, 1994. [13] A.W. Nolle. Sinuous instability of a planar air jet: Propagation parameters and acoustic excitation. J. Acoust. Soc. Am., 103:3690-3705, 1998. [14] Lord Rayleigh. The Theory of Sound. Dover, reprint (1945), New-York, 1894. [15] Xavier Serra. Current perspectives in the digital synthesis of musical sound. Formats, 1, 1997. [16] D.J. Tritton. Physical fluid dynamics. Oxford University Press, New-York, 2nd edition, 1998. [17] John Yen and Reza Langari. Fuzzy logic: intelligence, control, and information. Prentice Hall, Upper Saddle River, N.J., 1999. [18] Lofti A. Zadeh. Fuzzy sets. Information and Control, 8:338-353, 1965. 243