Numerical Optimization of a Bicylindrical Resonator Impedance: Differences and Common Features Between a Saxophone Resonator and a Bicylindrical Resonator

This paper explores the analogy between a saxophone resonator and a bicylindrical resonator, sometimes called transverse saxophone or cylindrical saxophone. The dimensions of a bicylindrical resonator are optimized numerically to approximate a saxophone impedance. The target is the impedance measured on an usual saxophone. A classical gradient-based non-linear least-square fit function is used. Several cost functions corresponding to distances to the target impedance are assessed, according to their influence on the optimal geometry. Compromises appear between the frequency regions depending on the cost function. It is shown that the chosen cost functions are differentiable and locally convex. The convexity region contains the initial geometrical dimensions obtained by crude approximation of the first resonance frequency of the target. One optimal geometry is submitted to further analysis using descriptors of the impedance. Its deviations from the target saxophone are put into perspective with the discrepancies between the target saxophone and a saxophone from a different manufacture. Descriptors such as harmonicity or impedance peak ratio set the bicylindrical resonator apart from saxophone resonators, despite a good agreement of the resonance frequencies. Therefore, a reed instrument with a bicylindrical resonator could be tuned to produce the same notes as a saxophone, but due to differences in the intrinsic characteristics of the resonator, it should be considered not as a saxophone but as a distinct instrument.