Mathematical Theory of Woodwind Fingerholes

Abstract

It has been shown on general grounds that there are only two types of bore that are suitable for use in woodwinds: the cylinder and the cone. Elementary consideration of the effect of closed side holes showed that their sizes must be graduated along the bore if its special properties are to be preserved [A. H. Benade, J. Acoust. Soc. Am. 31, 137–146 (1959)]. Further work will be described showing that although woodwind side holes are not evenly sized or spaced, lumped-constant transmission line theory can be applied. Equations for the frequency dependent “length corrections” for a bore with some open and some closed holes are obtained. The formulation implies several ways of dealing with the intonation problems of woodwinds, and explains the effect of side holes on timbre. The function of the bell is clarified, and the radiation pattern from the holes is deduced. It is shown that there is a unified design procedure for bores, holes, and bells, which can be based if desired on the “best” notes of an existing instrument. Experimental evidence will be presented in support of the theory.