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This work takes part of the ”cagima” project (supported by the ANR) which investigates the defects of the tuning of reed musical instruments as well as their homogeneity of emission and timbre. The goal consists in replacing the traditional approach adopted by instrument makers by a global and rational approach in the design of new instruments /ab initio/ (called ”logical instruments”), minimizing some identified defects. In this context, an interactive virtual model, predictive and configurable is proposed. Several approaches are available in the literature but the main difficulty is to design digital instruments that are accurate (according to measurements) and that can be implemented in real-time. In this paper, an approach based on the Extended Discrete Singular Convolution method (EDSC) is proposed. The temporal operator (including the fractional derivative term for viscothermal losses) is implemented according to the EDSC formalism. The method allows a fast, straightforward and accurate computation of the transfer functions of an axisymmetric duct with an arbitrary profile. The computation of the case where the losses are dependent from the diameter causes no noticeable difficulty. The results are compared to measurements of a trombone bell.