On the Mathematical Theory of Woodwind Finger Holes
A. H. Benade · The Journal of the Acoustical Society of America · 1960
The acoustical effects of open and closed finger holes on woodwind bores in the lower two playing registers are investigated in a mathematical formulation which permits a coherent and comprehensive understanding of the interaction of holes with the bore of a woodwind. Results are expressed in a way which permits accurate engineering calculation of all effects which are discussed. It is shown that when the holes are closed at their outer ends, the system is simply and accurately representable by an adaptation of standard transmission line theory for a tube with side branches. Interestingly, this representation is only possible for musically usable hole sizes and spacings. A related formulation is also possible for a sequence of open finger holes: once again the accuracy of the formulation is greatest for musically usable holes. The part of a woodwind bore that is provided with closed side holes functions as a low-pass filter. Similarly the open holes lower down on the bore function as a high-pass filter. The positions of both cutoff frequencies depend end critically upon the hole sizes and spacings. Both fall at frequencies which allow them to play a role in the tune production. Methods are given for calculating “end corrections” for bores with some open and some closed holes as well as for bores with perturbations to the bore cross section. The effects of “misplaced” or “mis-sized” holes are investigated by these methods, and the position of the lowest open hole calculated. An estimate of the errors in these calculations shows them to be essentially exact for musical purposes. The radiation behavior of a row of open finger holes is analyzed. Frequencies below the “cutoff” of the open hole system are radiated essentially isotropically, while each of the higher components is emitted with its own pattern, all of which are roughly conical, in analogy with the shock wave produced by a supersonic projectile. The musical implications of this are discussed briefly. Light is shed on the function of the bell on woodwind instruments, and on the reason why a bell is not needed on certain of them. The dominant rule of the “closed hole” properties of a bore with finger holes is stressed throughout the paper.