On the scaling of free-reed organ pipes: A comparison of 19th-century theories to modern nonlinear models

Abstract

After their introduction in the 19th century, the scaling of free reed organ pipes was often done on the basis of investigations by W. Weber, J. G. Töpfer, and F. Haas. Especially, Töpfer is often named as the founder of mathematical/physical oriented scaling. Unfortunately, it was impossible in the 19th century to derive a number of measures (e.g., optimal lengths of a conical pipe and the boot) theoretically. In this presentation it is demonstrated how recent theories can be used to derive some of those measures. For this purpose, a one-dimensional physical model was developed. The model uses the reed generator developed by Tarnopolsky et al. [J. Acoust. Soc. Am. 108, 400–406 (2001)] and a wave-guide model for the resonator so as to determine the optimal resonator and boot dimensions. Wrong assumptions that were made in the 19th century will be also addressed. For example, organ builders frequently used the solution of Weber to determine the optimal resonator length. The solution of Weber was aimed to make the reed pipe’s frequency independent from temperature and blowing pressure, but what the organ builders would have needed is a pipe whose frequency is mainly determined by the resonator length, so that the pipes would detune with the flue pipes, when the temperature changes.