Geometric sensitivity of modal parameters in wind instrument models: a case study on saxophone intonation

TL;DR

This paper extends the Transfer Matrix Method to analytically assess how small geometric changes in wind instruments affect modal parameters, aiding design optimization and sound synthesis.

Contribution

It derives analytical sensitivities of modal parameters to geometric variations, enabling predictive instrument design adjustments.

Findings

Analytical gradients of modal parameters with respect to geometry are derived.

Small geometric modifications can optimize octave harmonicity in saxophones.

The method provides insights for both instrument making and sound synthesis.

Abstract

The Transfer Matrix Method is a practical approach for modeling plane wave propagation in one-dimensional waveguides. Its simplicity makes it especially attractive for accounting for viscothermal losses, enabling realistic simulations of complex waveguides such as wind instruments. Another strength of this method lies in its fully analytical formulation of wave propagation. Modal parameters naturally arise as by-products of the model, obtained by numerically solving analytical expressions. In this work, the analytical potential of the method is extended by deriving the sensitivity of modal parameters to changes in the geometry of the resonator. These analytical gradients are applied in the context of wind instrument design. A simplified model of a soprano saxophone is used to investigate how octave harmonicity can be optimized through small geometric adjustments. The proposed approach enables predictive adjustments of geometry and offers valuable insight for both sound synthesis and instrument making.