Tubular Bells: A Physical and Algorithmic Model Rudolf Rabenstein, Member, IEEE, Tilman Koch, and Christian Popp

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1 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 18, NO. 4, MAY Tubular Bells: A Physical and Algorithmic Model Rudolf Rabenstein, Member, IEEE, Tilman Koch, and Christian Popp Abstract Tubular bells are geometrically simple representatives of three-dimensional vibrating structures. Under certain assumptions, a tubular bell can be modeled as a rectangular plate with different types of homogeneous boundary conditions. Suitable functional transformations with respect to time and space turn the corresponding initial-boundary value problem into a two-dimensional transfer function. An algorithmic model follows according to the functional transformation method in digital sound synthesis. As with simpler vibrating structures (strings, membranes) the synthesis algorithms consist of a parallel arrangement of second-order sections. Their coefficients are obtained by simple analytic expressions directly from the physical parameters of the tubular bell. Index Terms Audio signal processing, physical modeling, sound synthesis, virtual instruments. I. INTRODUCTION T UBULAR bells belong to the rich set of musical instruments of modern opera and symphony orchestras. Depending on their geometrical structure and their material, the sound of a tubular bell either has a definite pitch with the respective overtones or its sound is closer to a church bell where the pitch frequency is less dominant than the mostly inharmonic partials. In the former case, tubular bells come in chromatically tuned sets and constitute a melodic instrument in its own right. In the latter case, they are used to imitate the sound of church bells in an orchestral piece. Due to the great variety in their spectral structure, bells and bell like structures have been found hard to model in physical modeling sound synthesis. Only few and sometimes exotic cases are found in the literature. Two different methods, namely source filter models and digital waveguides have been employed in [1]. For both methods, the sound of an existing bell has been analyzed. In the source filter approach, a frequency zooming ARMA model was used for the modeling of relevant modal groups. In the waveguide approach, an inharmonic waveguide with strong dispersion has been designed by a clever combination of allpasses and comb filters. Both approaches were successful in recreating the spectrum of the recorded sound, but they did not permit a parametric Manuscript received March 15, 2009; revised August 20, Current version published April 14, The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Julius O. Smith. R. Rabenstein is with the Telecommunications Laboratory, University of Erlangen-Nuremberg, D Erlangen, Germany ( rabe@lnt.de). T. Koch is with Beyerdynamic GmbH and Co. KG, D Heilbronn, Germany, ( tilmankoch@gmx.de). C. Popp is with IR-Systeme GmbH and Co. KG, D Hassfurt, Germany, ( mail@popp-christian.de). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TASL control of the synthetic sound. These approaches have been used to build a computer carillion [2] and the waveguide approach has been extended by the inclusion of a hammer model in [3]. Waveguide models of a crystal wine glass and a Tibetanian singing bowl have been presented in [4]. Here, recorded impulse responses were analyzed to extract the main resonances for replication with the waveguide model. Related to this work, a Tibetan singing bowl controller has been presented in [5]. Physical models for flat two-dimensional structures like bars, plates, or membranes have been discussed more widely. Only a few references are given here. Waveguide and finite-difference models for bar percussion instruments were presented in [6] and in great detail in [7], [8], while functional transformation models have been described in [9]. Plate reverberation has been covered, e.g., in [10], [11]. This contribution discusses a model for a tubular bell which is based on the basic elastic and dynamic properties of a vibrating body and does not require the analysis of sound recordings. It exploits the geometry of a tube to reduce the three-dimensional structure to a two-dimensional model. The physical model presented here is a compromise between physical exactness and analytical tractability. It is complex enough to reflect physical reality and simple enough to allow an analytical derivation of the resulting algorithmic model. The purpose of this contribution is to describe the process of turning a physical description into a synthesis algorithm rather than studying all relevant physical effects. Section II gives a short account of the physical model while the synthesis paradigm is briefly introduced in Section III. The main contribution is the transfer function model in Section IV, while a real-time implementation is described in Section VI. II. PHYSICAL MODEL This section presents the geometrical and physical considerations that lead to the basic physical model in terms of a partial differential equation with initial and boundary conditions. A. Geometry The tubular bells considered here have the shape of a prism with a constant cross section between both ends. A typical geometrical body with this property is a cylinder with a circular cross section, but also elliptical or other cross sections are included. The basic assumption is that the surface of the tube can be unwound to a plate of rectangular area, as shown in Fig. 1. The resulting rectangular plate is equivalent to the previous tube only if certain assumptions hold. The tube is initially free from any forces. This means that it does not store mechanical energy like a spring. The deformation of the material during the unwinding process is negligible. This means that there is no strain or compression on either surface /$ IEEE

2 882 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 18, NO. 4, MAY 2010 Fig. 1. Converting a tube to a plate. B. Dynamics The physical analysis of the dynamical behavior of membranes and plates is usually performed in terms of the classical Euler Bernoulli beam theory. The resulting partial differential equations take different forms according to the assumed mechanical behavior (e.g., reversibility, Bernoulli s hypothesis, isotropy, etc.). One of the forms suitable for musical applications is reported in [15]. A somewhat simpler approach is taken here. Since the two-dimensional plate is already a simplification with respect to the three-dimensional tube, it is not necessary to model the plate as exact as possible. Superposing the descriptions of two elastic beams in - and -directions results in the partial differential equation for the plate deflection (1) The arguments of have been omitted for brevity. The constant depends on the material of the plate and its thickness according to (2) Fig. 2. Coordinate system for the membrane. TABLE I PHYSICAL QUANTITIES IN (1) AND (2) The physical quantities in (1) and (2) are given in Table I. The terms with fourth-order derivatives with respect to the space variables introduce dispersion into the model. In this form, the dispersion is strongly anisotropic and emphasizes the dispersive effect which is typical for metal plates, tubes, etc. The term proportional to the first time derivative of the deflection in (1) accounts for losses. The term on the right-hand side represents the excitation. The excitation functions considered here do not change their position with time, i.e., This assumption is certainly valid for short strikes with a mallet at well-defined positions. (3) The thickness of the tube wall is negligible. These assumptions are rather idealistic and for some tubes they may be only a rough approximation. Especially the assumption that bending the plate does not change its principal behavior is a simplification. Nevertheless, the above assumptions have turned out to be a good starting point for the creation of realistic tube sounds. For more elaborate models of circular, cylindrical, and spherical structures see, e.g., [12] [14]. Fig. 2 shows the coordinate system for the two-dimensional rectangular plate which is considered in lieu of the tubular structure. The two independent spatial coordinates are and, while the deflection from the rest position is denoted by. Since the deflection depends on and and on time as well, it is a dependent variable. The extension of the plate in - and -directions is given by and, respectively. With respect to the initial tube, corresponds to its circumference and to its length. C. Initial and Boundary Conditions The solution of the partial differential (1) depends on the given initial and boundary conditions. Especially the boundary conditions are important for the spectrum of the resulting vibrations. The initial conditions define the initial deflection and initial velocity. For simplicity, they are assumed to be zero in the sequel. In other words, the bell is assumed to be at rest prior to excitation. The boundary conditions are important for two reasons: First, they constitute the relation between the initial tube from Fig. 1 and the plate considered in (1). Second, they determine the eigenvalues of the spatial differential operator in the partial differential equation (in mathematical terms) or rather the character of the bell sound (in musical terms). To resemble the periodic character of the tube around its circumference, the boundary conditions of the plate in the -direction have to be periodic. To this end, the deflection and all of its relevant spatial derivatives at the boundaries at and have to be equal. Note that these boundaries have been created artificially by cutting the tube at an arbitrary line along its length

3 RABENSTEIN et al.: TUBULAR BELLS: A PHYSICAL AND ALGORITHMIC MODEL 883 (see Fig. 1). To undo this artificial separation, the boundary conditions for the plate have to ensure a smooth transition of the solution by considering it to be periodic in -direction with period. In mathematical terms, the boundary conditions at and are (4) To simplify the notation, spatial derivatives in the direction of are represented by the respective number of primes. In the -direction, the plate has free ends (in the sense of classical beam theory) at and. Since they are not connected to other structures, the bending moments and shear forces are zero, resulting in the following boundary conditions for the second and third space derivatives at and : Here, the primes denote spatial derivatives in the direction of. Equations (1) to (5) constitute the physical description of a tubular bell subject to the restrictions given in Sections II-A and II-B. Although the derivation of the partial differential equation (1) assumes a two-dimensional plate, its solution resembles a round structure through the periodic boundary conditions (4). This initial-boundary value problem is the basis for the derivation of the algorithmic model in the following sections. III. FUNCTIONAL TRANSFORMATION METHOD The functional transformation method is a mathematically rigorous way for physical modeling sound synthesis. It is described in detail, e.g., in [9], [16], and [17]. Applications for plucked, struck, bowed, and hammered strings and for membranes and plates exist [18]. The strength of this method is to turn a partial differential equation into a discrete-time model. Interconnections with models based on wave variables are possible [19]. A short description of the essential steps is given here in relation to Fig. 3. 1) The functional transformation method starts with the description of a physical model by a partial differential equation along with the required initial and boundary conditions. 2) Laplace transformation replaces the time derivatives with multiplications by the temporal frequency variable and includes the initial values as additive terms. 3) The so-called Sturm Liouville transformation replaces the remaining space derivatives with multiplications by the spatial frequency variable and includes the boundary values as additive terms. The Sturm Liouville transformation is a problem-dependent transformation (5) Fig. 3. Essential steps of the functional transformation method. and is given in terms of the set of discrete eigenvalues and eigenfunctions of the spatial derivative operator. Its determination is an important part of the presented method. 4) The remaining algebraic equation can be solved for the transform of the output signal, i.e., the solution of the partial differential equation. The result is a description of the physical problem by a multidimensional transfer function. 5) Any of the well-known continuous-to-discrete-time transformations turns the continuous-time transfer function from (4) into a discrete-time multidimensional transfer function. Choosing the impulse-invariant-transformation for this purpose preserves the modal frequencies of the continuous-time model also in the discrete-time transfer functions. 6) The discrete-time multidimensional transfer function is converted back to a discrete-time transfer function through the inverse Sturm Liouville transformation. Due to the discrete nature of its spectrum, the inverse transformation is given by an orthogonal series expansion. 7) In a final step, inverse -transformation yields a difference equation for the step-wise computation of the samples of the solution. Its algorithmic structure is given by a parallel arrangement of low-order recursive digital filters. These steps are explained in more detail in Section IV. IV. TRANSFORMATIONS FOR TIME AND SPACE The steps of the functional transformation method are now applied to the physical model of a tubular bell from Section II. The aim is to establish the continuous-time and discrete-time transfer functions from Fig. 3 as basis for the resulting algorithmic model. A. Laplace Transform The starting point denoted by (1) in Fig. 3 is the differential equation (1). Introducing the spatial differential operator as (6)

4 884 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 18, NO. 4, MAY 2010 (1) can be written as Now, turns into the sum of the products of two onedimensional integrals (7) The one-sided Laplace transformation (8) with the temporal frequency variable turns (7) into (13) Through the differentiation theorem of the Laplace transformation, the time derivatives in (7) have been replaced by the corresponding powers of. Since homogenous initial conditions have been assumed, there are no additional terms with initial values. Equation (9) is a partial differential equation for the space variables subject to the boundary conditions (4) and (5). It corresponds to (2) in Fig. 3. B. Sturm Liouville Transform The Sturm Liouville transformation shall perform the same steps for the space variables and as the Laplace transformation did for the time variable. In particular, it shall replace the spatial differentiation operator by a multiplication with a spatial frequency variable, which is yet to be defined. Further, it shall consider the boundary conditions in both directions. The derivation of the Sturm Liouville transformation is based on the theory of Sturm Liouville problems, a well-studied kind of boundary conditions with useful properties; see, e.g., [20]. However, the following description is self-contained within the scope of this contribution. Definition: The procedure starts with the definition of the Sturm Liouville transformation (10) The integration extends over the spatial definition range of (see Fig. 1). The transformation kernel contains the spatial frequency variable. The exact form of the transformation kernel still has to be found. This process will reveal the spectral structure of the tubular bell in a mathematically rigorous way. 1) Determination of the Transformation Kernel: Application of the Sturm Liouville transformation to (9) gives (9) (11) The terms without spatial derivatives are readily replaced by their transforms and. The transform of is now determined. To this end, the rectangular shape of the plate from Fig. 2 is exploited. It permits to separate the spatial functions and as (12) To remove the fourth derivatives of and inside the integrals, integration by parts has to be applied four times. This way the values of and at the boundaries appear explicitly and can be replaced by the respective boundary conditions. The result of this tedious process for is with (14) (15) Since and do not depend on [see (12)], the primes denote derivation with respect to. Now the periodic boundary conditions (4) can be used to replace by and similar for the derivatives. The result is with Integration by parts gives for (16) (17) (18) where the boundary conditions (5) have been used to set the second and third derivatives of to zero. Here, the primes denote differentiation with respect to. Inspection of the boundary terms and shows that they still contain parts that are unknown as long as the problem is not completely solved. The periodic boundary conditions (4) have been used to simplify (16), but they do not tell the actual values at the boundary. Similar the boundary values (5) helped to remove the higher derivatives in (18), but the values of and are not known. According to Sturm Liouville theory the boundary terms and can be computed without the knowledge of the unknown terms. Since all of them are multiplied by terms that depend on the transformation kernel, these products vanish whenever the

5 RABENSTEIN et al.: TUBULAR BELLS: A PHYSICAL AND ALGORITHMIC MODEL 885 transformation kernel or its derivatives are zero at the boundaries. To be specific, for with (27) and for (19) These eigenvalue problems are now solved independently. For ease of notation, two related but simpler eigenvalue problems are considered (28) (29) with (30) (20) Comparing (19) and (20) to (4) and (5) shows that the boundary terms and vanish if the transformation kernel fulfills the same boundary conditions as the initial physical problem. This parallel can be carried even further, when (14) and (17) with and are inserted into (13). By combining the separated terms it turns out that (21) Since the transformation kernel is still undetermined, an additional requirement can be formulated. When the kernel has the property then it follows from (21) and (10) that (22) (23) This result constitutes a differentiation theorem for the Sturm Liouville transformation with respect to the spatial differential operator. It holds whenever the transformation kernel fulfills the partial differential (22) with the boundary conditions (19) and (20). Since (22) constitutes an eigenvalue problem with the eigenvalue, the transformation kernel turns out to be an eigenfunction of the spatial differential operator with respect to the boundary conditions of the original problem. C. Determination of the Eigenfunctions Now the transformation kernel is determined from the eigenvalue problem (22) with the boundary conditions (19) and (20). With the separation approach (12), equation (22) becomes (24) and can be separated in two one-dimensional eigenvalue problems for and, respectively, (25) (26) The eigenfunctions and of (25), (26) and and of (28) and (29) are related by and and the eigenvalues,,, and are related by (30). The boundary conditions (19) and (20) for hold equally for. 1) Solution of the Eigenvalue Problem for : The one-dimensional eigenvalue problem for is solved by finding the general solution of (28) and then by determining the special solution with respect to the boundary conditions (19). The general solution of (25) is given by (31) where, are the four roots of and is the imaginary unit. The coefficients are determined (up to a common constant) by restricting the general solution (31) to the boundary conditions (19). The resulting set of linear equations can be written in matrix form as (32) where the vector contains the four unknown coefficients and the matrix is composed of the exponential terms in (31) for and. The vector contains four zeros. A nontrivial solution exists if the determinant of is zero. This condition yields the possible values of, where only nonzero values are of interest. Carrying out this process is rather lengthy and results in the values The corresponding solution is or (33) (34) where the remaining constant has been set to unity. The value of is arbitrary and reflects the fact that the zero position of the -axis has been chosen by cutting the tube at an arbitrary position (see Fig. 1). 2) Solution of the Eigenvalue Problem for : Due to the similarity of (25) and (26), the general solution of (26) is (35)

6 886 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 18, NO. 4, MAY 2010 With these values of, there exists a solution of (36) for the coefficients in (35). The -component of the eigenfunction is shown in (41) at the bottom of the page. 3) Complete Eigenvalue Problem: After having solved the separate eigenvalue problems for the - and the - coordinates, the solution of the two-dimensional eigenvalue problem (22) can be assembled from (12) and (27) as (42) (43) Fig. 4. Graphical representation of (38) with l =2x. The coefficients are determined by the boundary conditions (20). The corresponding matrix equation is given by (36) (37) Solutions for the coefficients exist by setting the determinant of to zero. Keeping only the terms which yield nonzero values for gives This result establishes the eigenfunctions and eigenvalues such that the differentiation theorem (23) holds. Note, that is a scalar value. It has two indexes, and, because it is composed of the two eigenvalues and. However, the values of can also be numbered consecutively as. This notation is used in the sequel for simplicity. V. ALGORITHMIC MODEL A. Continuous-Time Transfer Function Now the partial differential (9) for the space variables can be converted into an algebraic equation. To this end the term in (11) is replaced by with the help of the differentiation theorem (23). The resulting algebraic equation (44) which is equivalent to (38) (39) represents step (3) in Fig. 3. From this algebraic equation it is now easy to arrive at the continuous-time multidimensional transfer function. Combining the excitation terms to and solving (44) for gives the multidimensional transfer function Unfortunately, this equation does not have analytical solutions other than. For large arguments, approximate values for are obtained from as (40) For small arguments, the values of have to be found by numerically searching the intersections of the functions in (39). Fig. 4 shows the situation. The first intersection at zero is not of interest since it does not correspond to an oscillation. The first nonzero intersection corresponds to the fundamental frequency and the second to the first harmonic (shown by empty circles). Both are determined by a numerical solution of (39). The following intersections are approximated by (40). (45) which is completely determined by the denominator polynomial. This transfer function, which constitutes step (4) in Fig. 3, is a simple second-order transfer function with respect to in the temporal frequency variable and the spatial frequency variable. However, the spectral complexity of is hidden in the discrete values as is revealed by the solution of the eigenvalue problems in Section IV-C. The spectral structure follows from an inspection of the denominator polynomial as (46) (41)

7 RABENSTEIN et al.: TUBULAR BELLS: A PHYSICAL AND ALGORITHMIC MODEL 887 with a pair of complex conjugate poles and with real and imaginary parts ( ) (47) The real part determines the amplitude decay. In this model it is independent of frequency. More elaborate models should include also frequency-dependent damping like the string models in [16], [17]. The structure of the spectrum is given by which depends on the eigenvalues corresponding to both spatial dimensions. The sets of these eigenvalues differ, since they have been determined from different boundary conditions. In this way the eigenfrequencies of the tubular bell are preserved through a careful mathematical analysis of the physical model. B. Discrete-Time Transfer Function Several so-called analog-to-continuous transformations exist, which convert a continuous-time system into a discrete-time system with similar properties. The most popular ones are the impulse-invariant transform and the bilinear transform, both preserving the stability properties of the given system. They differ in the characteristics of the mapping between the corresponding complex planes, especially the frequency axis. The bilinear transform avoids aliasing at the expense of frequency warping. Conversely, the impulse-invariant method preserves the correct values of the eigenfrequencies, but it requires a band-limited continuous-time signal to start with. Since a high-quality sound synthesis has to cover the full audio range, preservation of the eigenfrequencies is of utmost importance. Therefore the impulse-invariant transformation is preferred over the bilinear transformation to prevent any frequency warping. Possible aliasing problems are avoided, if the number of the eigenvalues is restricted such that the range of does not extend beyond the audio bandwidth. Each eigenfrequency contributes only a narrow band to the complete spectrum since the real part is sufficiently small for long-ringing bell sounds. The application of the impulse-invariant transformation to the second-order system (45) yields the discrete-time transfer function (48) with the sampling interval and the discrete-time frequency variable. The superscript distinguishes discrete-time quantities from their continuous-time counterparts. The discrete-time damping coefficient and the angular frequency are given by (48) constitutes step (5) in Fig. 3. (49) C. Digital Filter Realization The discrete-time transfer function (48) contains the complete information on parameters and spectral behavior in the form of a set of discrete second-order sections. Each one realizes a pair of poles with the respective damping and resonant frequency of one partial of the bell sound. The remaining steps to arrive at a difference equation have been described before (see, e.g., [16] and [17]) and are only mentioned briefly here. Due to (21), the differential operator is self-adjoint with respect to the scalar product (10). Therefore its eigenfunctions are a set of orthogonal functions with (50) (51) For a deeper understanding of linear operators and their properties see, e.g., [21]. Due to the orthogonality of the eigenfunctions, the inverse Sturm Liouville transformation of is given as an orthogonal series expansion (52) (53) Note that and denote the space coordinates, while is the discrete-time frequency variable. The quantity follows from (48) and the right-hand side of (11) as (54) where. Inserting (54) into (53) and combining some terms shows that the -transform of the deflection is obtained from the -transform of the input sequence by multiplication with a transfer function as where the transfer function terms (55) is the sum of simpler (56) (57) (58)

8 888 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 18, NO. 4, MAY 2010 where and denote spatial impulse functions. The values in (60) follow then from (10) as the eigenfunctions evaluated at the point. The representation of (56) with second-order digital filters in the discrete-time domain constitutes the final step (7) in Fig. 3. It requires only the values of the physical parameters of the underlying model (1) and (2) for calculating the coefficients in Fig. 5. Therefore, this method is most suitable for parametric control of timbral spaces. On the other hand, it would also be possible to estimate the filter coefficients from recorded sounds. Then each second-order system represents an ARMA-model for one partial of the recording. Such an approach has been introduced under the name of modal synthesis by [22]. Fig. 5. Synthesis algorithm as parallel arrangement of second-order sections. are second-order sec- Due to (48) the individual terms tions of the form with the coefficients (59) (60) (61) (62) Therefore, the solution from (55) is obtained by filtering the input through a parallel arrangement of second-order systems from (59), each one weighted by, a scaled version of the respective eigenfunction according to (58). This set of one-dimensional second-order transfer functions represents step (6) in Fig. 3. The discrete-time domain version follows directly by implementing the second-order sections in (59) as difference equations. The resulting algorithmic model is shown in Fig. 5 (compare with [17] for string synthesis). The number of parallel systems for is restricted to a maximum value, corresponding to the highest spectral component. This number can be chosen such that from (47) lies near the high end of the human hearing range. The factors can be evaluated for a fixed sample point representing the position of a virtual microphone or pick up. Examining (58) and (37) shows that the eigenfunctions have to be evaluated only at a single point. Similarly, the excitation at a specific point can be described by a two-dimensional spatial impulse [see (3)] (63) VI. REAL-TIME IMPLEMENTATION The real-time capability of the presented algorithmic model has been demonstrated by an implementation as a so-called VST plugin. It is based on the Virtual Studio Technology (VST), a de facto standard supported by a number of commercial sequencer programs [23]. This standard allows to create implementations of program modules for sound synthesis and sound processing which interface to a host program, typically a powerful computer music or sequencer software. The VST plugin for the demonstration of tubular bell synthesis with the functional transformation method is called FTM TUBE. The graphical user interface is shown in Fig. 6. The plugin reads a musical score according to the Musical Instrument Digital Interface (MIDI) standard. It is provided by the host either from a file or as real-time input from a MIDI-device, typically a keyboard. According to the musical score the plugin FTM TUBE creates tubular-bell-like sounds and sends them to the host program for further processing. The graphical user interface in Fig. 6 gives access to all parameters of the physical model except for the lengths and. These are obtained first from the fundamental frequency of the requested note (coded by the MIDI key number) and second from a geometry factor for the relation between the tube length and the tube perimeter as. This parameter can be set by the element geometry from the graphical user interface (top left in Fig. 6). Thus, the bell is automatically tuned to the correct musical note. Variation of the geometry factor changes the sound but not the pitch of the tone. The other parameters of the interface correspond directly to the physical parameters from Table I. In addition the contact time of a virtual mallet (excitation time) and the decay after a MIDI note off command (release time) can be set by the user, as well as the coordinates of the excitation point of the mallet and those of the sample point, the position of a virtual pick up. The instrument allows for a number of different tubes played at the same time. Their maximum number is set by polyphony. Depending on the facilities of the host program and a possible MIDI device, these parameters can also be operated from an external fader box or the controllers of a master keyboard. Such inputs can also be recorded into a MIDI file and edited for further refinement. Parameter changes affect the coefficients of the realizing structure in Fig. 5 directly and only with minimal computational

9 RABENSTEIN et al.: TUBULAR BELLS: A PHYSICAL AND ALGORITHMIC MODEL 889 Fig. 6. Graphical user interface of the VST-Plugin FTM TUBE. expense. Most important, it is not required to recalculate any eigenvalue problems in response to a variation of a parameter value. By expressing the conditions for the solvability of the matrix (32) and (36) by the dimensionless quantities and, the eigenvalues are obtained in normalized form through (33), (39), and (40). Any readjustment of the parameters requires only a renormalization of the eigenvalues according to (2) and (30). The poles of the continuous-time model result by recalculation of (47) and those of the discrete-time model from (49). These few algebraic relations can be computed on the fly. The separation of the possibly complicated solution of eigenvalue problems from the actual calculation of the filter coefficients is one of the strengths of the functional transformation method. Changing feedback coefficients of recursive filters during operation requires some stability considerations. These are quite simple here due to the close connection between the filter coefficients and the physical nature of the original problem. Since only real values are meaningful for and and since the physical constant from (2) is always positive, also the value of in (30) is positive. The same is true for the damping constant and therefore from (46) is a Hurwitz-polynomial. Finally, since the impulse-invariant transform maps stable continuous-time systems to stable discrete-time systems, the stability of the realization from Fig. 5 is guaranteed for all physically reasonable parameter values (not considering finite-wordlength effects). The real-time implementation FTM TUBE runs at powerful personal and notebook computers without noticeable latency and is available from [24]. VII. CONCLUSION A complete physical and algorithmic model for a tubular bell has been presented. The oscillations of a tube have been described under certain reasonable assumptions which lead to a partial differential equation for the time and two space coordinates and the respective initial and boundary condition. The rigorous solution of the boundary value problems on the basis of the Sturm Liouville theory allows to derive an algorithmic model as a set of second-order digital filters. The coefficients of the algorithmic model depend directly on the parameters of the initial partial differential equation. The algorithmic model can therefore be adjusted to any set of parameters and thus allows to explore the whole timbral space of tubular bells. The generality of the approach allows to extend the method to non-tubular geometries and to space-dependent physical parameters. The presented method needs no analysis of existing bell sounds nor does it require computations on a spatial grid. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable suggestions and for pointing out reference [12]. REFERENCES [1] M. Karjalainen, V. Välimäki, and P. A. A. Esquef, Efficient modeling and synthesis of bell-like sounds, in Proc. Conf. Digital Audio Effects (DAFx-02), Hamburg, Germany, Sep. 2002, pp [2] M. Karjalainen, P. A. A. Esquef, and V. Välimäki, Making of a computer carillion, in Proc. Stockholm Music Acoust. Conf. (SMAC), Stockholm, Sweden, Aug

10 890 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 18, NO. 4, MAY 2010 [3] M. Karjalainen, J. Pakarinen, C. Erkut, P. A. A. Esquef, and V. Välimäki, Recent advances in physical modeling with K- and W-techniques, in Proc. Conf. Digital Audio Effects (DAFx-04), Naples, Italy, Oct. 2004, pp [4] S. Serafin, C. Wilkerson, and J. Smith, Modeling bowl resonators using circular waveguide networks, in Proc. Conf. Digital Audio Effects (DAFx-02), Hamburg, Germany, Sep [5] D. Young and G. Essl, Hyperpuja: A tibetan singing bowl controller, in Proc. Int. Conf. New Interfaces for Musical Expression (NIME), Montreal, QC, Canada, [6] G. Essl and P. R. Cook, Banded waveguides: Towards physical modeling of bowed bar percussion instruments, in Proc. Int. Comput. Music Conf. (ICMC), Beijing, China, Oct , 1999, pp [7] S. Bilbao, Wave and Scattering Methods for Numerical Simulation. New York: Wiley, [8] S. Bilbao, Fast modal synthesis by digital waveguide extraction, IEEE Signal Process. Lett., vol. 13, no. 1, pp. 1 4, Jan [9] S. Petrausch and R. Rabenstein, Block-based physical modeling for digital sound synthesis of membranes and plates, in Proc. Int. Comput. Music Conf. (ICMC), Barcelona, Spain, Sep [10] S. Bilbao, A physical model for plate reverberation, in Proc. Int. Conf. Acoust., Speech, Signal Process., Toulouse, France, May 2006, pp [11] C. B. Maxwell, Real-time reverb simulation using arbitrary models, in Proc. Conf. Digital Audio Effects (DAFx-07), Bordeaux, France, Sep. 2007, pp [12] S. Cheng, Accurate fourth order equation for circular cylindrical shells, J. Eng. Mech. Div., vol. 98, no. 3, pp , [13] K. Graff, Wave Motion in Elastic Solids. New York: Dover, [14] S. Bilbao, Numerical Sound Synthesis. New York: Wiley, [15] T. Rossing and N. Fletcher, The Physics of Musical Instruments. New York: Springer, [16] L. Trautmann and R. Rabenstein, Digital Sound Synthesis by Physical Modeling using the Functional Transformation Method. New York: Kluwer, [17] R. Rabenstein and L. Trautmann, Digital sound synthesis of string instruments with the functional transformation method, Signal Process., vol. 83, no. 8, pp , [18] R. Marogna and F. Avanzini, Physically-based synthesis of nonlinear circular membranes, in Proc. Conf. Digital Audio Effects (DAFx-09), Como, Italy, Sep. 2009, pp [19] S. Petrausch and R. Rabenstein, Interconnection of state space structures and wave digital filters, IEEE Trans. Circuits Syst. II, vol. 52, no. 2, pp , Feb [20] S. G. Kelly, Advanced Engineering Mathematics. Boca Raton, FL: CRC, [21] T. Kato, Perturbation Theory for Linear Operators. Berlin, Germany: Springer, [22] J. Adrien, Dynamic modeling of vibrating structures for sound synthesis, modal synthesis, in Proc. AES 7th Int. Conf., Toronto, ON, Canada, 1989, pp [23] VST3: New Standard for Virtual Studio Technology, Steinberg Media Technologies GmbH, 2009 [Online]. Available: [24] R. Rabenstein, author s home page, [Online]. Available: Rudolf Rabenstein (M 97) received the Diplom-Ing. and Doktor-Ing. degrees in electrical engineering and the Habilitation degree in signal processing, all from the University of Erlangen-Nuremberg, Erlangen, Germany, in 1981, 1991, and 1996, respectively. He worked with the Physics Department, University of Siegen, Siegen, Germany, and now is a Professor with the Telecommunications Laboratory, University of Erlangen-Nuremberg. His research interests are in the fields of multidimensional systems theory and multimedia signal processing. He is author and coauthor of many scientific publications, has contributed to various books, and holds several patents in audio engineering. Tilman Koch graduated from the University of Applied Sciences, Coburg, Germany, in Since then he has been an Acoustic Design Engineer with Beyerdynamic, Heilbronn, Germany. His interests are advanced audio systems and composing of electronic music. Christian Popp graduated from the Friedrich- Alexander-University, Erlangen, Germany, in April Since then he has been a Development and Research Engineer for automation and communication systems with IR-Systeme, Hassfurt, Germany. His interests are advanced audio systems and making music.