Effect of Dispersion and Scattering on the Startup of Brass Instrument Tones

Abstract

A clean attack is favored if the round-trip time Tr for the initial sound from the lips sent down the bore and back is an integral multiple of the playing frequency period Pn. Consider a Bessel horn of length L, radius r, with r(y)=B/yα. For a pulse with spectral center ωn, the group velocity vg=c[1−(c/ωn)2α(α+1)]12, whence Tr=(2/c) [L2−ln2]12, where ln2=(c/ωn)2α(α+1). The oscillation period of the nth mode of this horn is approximately Pn=(4/c) (L−ln)/(2n−1) implying that a Bessel horn does not give a clean attack. A mouthpiece, mouthpipe, cylindrical tube, and Bessel bell can be designed to give good attack plus correct tuning. Short constrictions in a bore give early partial reflections which also damage tone onset. A 10% diam reduction 2 cm long (as in some horn valves) gives a 7% reflected amplitude at 330 Hz. Valves in combination can give considerably larger reflections. Missed notes and burbles are less common in instruments with slight constrictions.