How localized nonlinear losses condition the acoustical design of a self-sustained oscillator: the clarinet and its register hole

Abstract

The register tube marks the invention of the clarinet in the early eighteenth century, tripling the range of its ancestor, the chalumeau, and giving it the widest range among wind instruments. Opening this narrow tube causes the fundamental frequency of the played note to increase by a factor of three, from the first to the second register of the resonator. The geometry and location of the register hole condition not only this mode selection mechanism, but also the global tuning of the second register. However, existing self-sustained nonlinear models of reed instruments fail to predict whether a register transition can occur, limiting optimization of the register hole geometry. Here, we introduce a sparse self-oscillating clarinet model that includes localized nonlinear acoustic losses in the register hole. This nonlinear mechanism is shown to be necessary to reproduce register transitions observed experimentally. Using systematic exploration of the control and design parameter spaces, we identify combinations of register hole diameter, position, and chimney length that ensure reliable register transitions. We show that the competing demands of playability and tuning are only satisfied by a long and narrow tube. Our findings provide a predictive tool for instrument making, assisting manufacturers in refining clarinets as well as other reed instruments, including oboes, bassoons, and saxophones.