Techniques for the simulation of the tanpura have advanced significantly in recent years allowing numerically stable inclusion of bridge contact. In this paper tension modulation is added to a tanpura model containing a stiff lossy string, distributed bridge contact and the thread. The model is proven to be unconditionally stable and the numerical solver used has a unique solution as a result of choices made in the discretisation process. Effects due to the distribution of the bridge contact forces by comparison to a single point bridge and of introducing the tension modulation are studied in simulations. This model is intended for use in furthering the understanding of the physics of the tanpura and for informing the development of algorithms for sound synthesis of the tanpura and similar stringed instruments.

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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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Slide-string instruments allow continuous control of pitch by articulation with a slide object whose position of contact with the string is time-varying. This paper presents a method for simulation of such articulation. Taking into account sensing and musical practice considerations, an appropriate physical model configuration is determined, which is then formulated in numerical form using a finite difference approach. The model simulates the attachment and detachment phases of slide articulation which generally involve rattling, while finger damping is modelled in a more phenomenological manner as a regionally induced time-varying damping. A stability bound for the numerical model is provided via energy analysis, which also reveals the driving power contributions of the separate articulatory sources. The approach is exemplified with simulations of slide articulatory gestures that involve glissando, vibrato and finger damping.

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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 physical modelling synthesis, articulation and tuning are effected via time-variation in one or more parameters. Adopting hammered strings as a test case, this paper develops extended forms of such control, proposing a numerical formulation that affords online adjustment of each of its scaled-form parameters, including those featuring in the one-sided power law for modelling hammerstring collisions. Starting from a modally-expanded representation of the string, an explicit scheme is constructed based on quadratising the contact energy. Compared to the case of time-invariant contact parameters, updating the scheme’s state variables relies on the evaluation of two additional analytic partial derivatives of the auxiliary variable. A numerical energy balance is derived and the numerical contact force is shown to be strictly non-adhesive. Example results with time-variant tension and time-variant contact stiffness are detailed, and real-time viability is demonstrated.

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