Cutoff frequencies of a clarinet. What is acoustical regularity?
Résumé
A characteristic of woodwind instruments is the cutoff frequency of their tone-hole lattice. Benade proposed a practical definition using the measurement of the input impedance, for which at least two frequency bands appear. The first one is a stop band, while the second is a pass band. The value of this cutoff frequency, which is a global quantity, depends on the whole geometry of the instrument, but is rather independent of the fingering. This seems to justify the consideration of a woodwind with several open holes as a periodic lattice. However the holes on a clarinet are very irregular. The paper investigates first the experimental method of determination of the cutoff frequency, then the question of the acoustical regularity: an acoustically regular lattice of tone holes is defined as a lattice built with T-shaped cells of equal eigenfrequencies. Then the paper discusses the possibility of division of a real lattice into cells of equal eigenfrequencies. It is shown that it is not straightforward but possible, explaining the apparent paradox of the Benade theory. From this division, a narrow range of possible constant cutoff frequencies is found, and it is in very good accordance with the experimental results from impedance measurements. When considering the open holes from the input of the instrument to its output, the distances between holes are enlarged together with their radii: this explains the relative constancy of the eigenfrequencies.
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