Paper Archive

This work presents a physical model of the yaybahar, a recently invented acoustic instrument. Here, output from a bowed string is passed through a long spring, before being amplified and propagated in air via a membrane. The highly dispersive character of the spring is responsible for the typical synthetic tonal quality of this instrument. Building on previous literature, this work presents a modal discretisation of the full system, with fine control over frequency-dependent decay times, modal amplitudes and frequencies, all essential for an accurate simulation of the dispersive characteristics of reverberation. The string-bow-bridge system is also solved in the modal domain, using recently developed noniterative numerical methods allowing for efficient simulation.

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The distributed nature of coupling in helical springs presents specific challenges in obtaining efficient computational structures for accurate spring reverb simulation. For direct simulation approaches, such as finite-difference methods, this is typically manifested in significant numerical dispersion within the hearing range. Building on a recent study of a simpler spring model, this paper presents an alternative discretisation approach that employs higher-order spatial approximations and applies centred stencils at the boundaries to address the underlying linear-system eigenvalue problem. Temporal discretisation is then applied to the resultant uncoupled mode system, rendering an efficient and flexible modal reverb structure. Through dispersion analysis it is shown that numerical dispersion errors can be kept extremely small across the hearing range for a relatively low number of system nodes. Analysis of an impulse response simulated using model parameters calculated from a measured spring geometry confirms that the model captures an enhanced set of spring characteristics.

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In the design of real-time spring reverberation algorithms, a modal architecture offers several advantages, including computational efficiency and parametric control flexibility. Due to the complex, highly dispersive behavior of helical springs, computing physically accurate parameters for such a model presents specific challenges. In this paper these are addressed by applying an implicit higher-order-in-space finite difference scheme to a two-variable model of helical spring dynamics. A novel numerical boundary treatment is presented, which utilises multiple centered boundary expressions of different stencil width. The resulting scheme is unconditionally stable, and as such allows adjusting the numerical parameters independently of each other and of the physical parameters. The dispersion relation of the scheme is shown to be accurate in the audio range only for very high orders of accuracy in combination with a small temporal and spatial step. The frequency, amplitude, and decay rate of the system modes are extracted from a diagonalised form of this numerical model. After removing all modes with frequencies outside the audio range and applying a modal amplitude correction to compensate for omitting the magnetic transducers, a light-weight modal reverb algorithm is obtained. Comparison with a measured impulse response shows a reasonably good match across a wide frequency range in terms of echo density, decay characteristics, and diffusive nature.

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The virtual exploration of the domain of mechano-acoustically produced sound and music is a long-held aspiration of physical modelling. A physics-based algorithm developed for this purpose combined with an interface can be referred to as a virtual-acoustic instrument; its design, formulation, implementation, and control are subject to a mix of technical and aesthetic criteria, including sonic complexity, versatility, modal accuracy, and computational efficiency. This paper reports on the development of one such system, based on simulating the vibrations of a string and a plate coupled via a (nonlinear) bridge element. Attention is given to formulating and implementing the numerical algorithm such that any of its parameters can be adjusted in real-time, thus facilitating musician-friendly exploration of the parameter space and offering novel possibilities regarding gestural control. Simulation results are presented exemplifying the sonic potential of the string-bridgeplate model (including bridge rattling and buzzing), and details regarding efficiency, real-time implementation and control interface development are discussed.

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The construction of new virtual instruments is one long-term goal of physical modeling synthesis; a common strategy across various different physical modeling methodologies, including lumped network models, modal synthesis and scattering based methods, is to provide a canonical set of basic elements, and allow the user to build an instrument via certain specified connection rules. Such an environment may be described as modular. Percussion instruments form a good test-bed for the development of modular synthesis techniques—the basic components are bars and plates, and may be accompanied by connection elements, with a nonlinear character. Modular synthesis has been approached using all of the techniques mentioned above, but time domain finite difference schemes are an alternative, allowing many problems inherent in the above methods, including computability, large memory and precomputation requirements, and lack of extensibility to more complex systems, to be circumvented. One such network model is presented here along with the associated difference schemes, followed by a discussion of implementation details, the issues of excitation and output, and a description of various instrument configurations. The article concludes with a presentation of simulation results, generated in the Matlab prototyping language.

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Virtual analog modeling of spring reverberation presents a challenging problem to the algorithm designer, regardless of the particular strategy employed. The difficulties lie in the behaviour of the helical spring, which, due to its inherent curvature, shows characteristics of both coherent and dispersive wave propagation. Though it is possible to emulate such effects in an efficient manner using audio signal processing constructs such as delay lines (for coherent wave propagation) and chains of allpass filters (for dispersive wave propagation), another approach is to make use of direct numerical simulation techniques, such as the finite difference time domain method (FDTD) in order to solve the equations of motion directly. Such an approach, though more computationally intensive, allows a closer link with the underlying model system— and yet, there are severe numerical difficulties associated with such designs, and in particular anomalous numerical dispersion, requiring some care at the design stage. In this paper, a complete model of helical spring vibration is presented; dispersion analysis from an audio perspective allows for model simplification. A detailed description of novel FDTD designs follows, with special attention is paid to issues such as numerical stability, loss modeling, numerical boundary conditions, and computational complexity. Simulation results are presented.

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Modal methods are a long-established approach to physical modeling sound synthesis. Projecting the equation of motion of a linear, time-invariant system onto a basis of eigenfunctions yields a set of independent forced, lossy oscillators, which may be simulated efficiently and accurately by means of standard time-stepping methods. Extensions of modal techniques to nonlinear problems are possible, though often requiring the solution of densely coupled nonlinear time-dependent equations. Here, an application of recent results in numerical simulation design is employed, in which the nonlinear energy is first quadratised via a convenient auxiliary variable. The resulting equations may be updated in time explicitly, thus avoiding the need for expensive iterative solvers, dense linear system solutions, or matrix inversions. The case of a network of interconnected distributed elements is detailed, along with a real-time implementation as an audio plugin.

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In discrete-time digital models of contact of vibrating objects stability and therefore control over system energy is an important issue. While numerical approximation is problematic in this context digital algorithms may meat this challenge when based on exact mathematical solution of the underlying equation. The latter may generally be possible under certain conditions of linearity. While a system of contacting solid objects is non-linear by definition, piece-wise linear models may be used. Here however the aspect of “switching” between different linear phases is crucial. An approach is presented for exact preservation of system energy when passing between different phases of contact. One basic principle used may be pictured as inserting appropriate ideal, massless and perfectly stiff, “connection rods” at discrete moments of phase switching. Theoretic foundations are introduced and the general technique is explained and tested at two simple examples.

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This paper deals with designing material parameters for physical models. It is shown that the characteristic relation between modal frequencies and damping factors of a sound object is the acoustic invariant of the material from which the body is made. Thus, such characteristic relation can be used for designing damping models for a conservative physical model to represent a particular material.

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A combined physical model for the human vocal folds and vocal tract is presented. The vocal fold model is based on a symmetrical 16 mass model by Titze. Each vocal fold is modeled with 8 masses that represent the mucosal membrane coupled by non-linear springs to another 8 masses for the vocalis muscle together with the ligament. Iteratively, the value of the glottal flow is calculated and taken as input for calculation of the aerodynamic forces. Together with the spring forces and damping forces they yield the new positions of the masses that are then used for the calculation of a new glottal flow value. The vocal tract model consists of a number of uniform cylinders of fixed length. At each discontinuity incident, reflected and transmitted waves are calculated including damping. Assuming a linear system, the pressure signal generated by the vocal fold model is either convoluted with the Green’s function calculated by the vocal tract model or calculated interactively assuming variable reflection coefficients for the glottis and the vocal tract during phonation. The algorithms aim at real-time performance and are implemented in MATLAB.

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