Page 54 - Spring 2006

Musical Acoustics
BRASS ACOUSTICS—IS PROPAGATION LINEAR OR NONLINEAR?
James W. Beauchamp
School of Music and Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign
Urbana, Illinois 61801
Since the beginning of the twentieth
century, the acoustics of brass
instruments has been and contin-
ues to be a popular topic in musical
acoustics. Of approximately 200 musi-
cal acousticians listed on the current
Technical Committee on Musical
Acoustics website, about 12% list brass
as one of their specific interests.
Actually, “brass” is not the defining
term, as instruments constructed of
brass may be considered woodwinds
(e.g., the saxophone), and instruments
constructed with other materials can be classified as “brass”
(e.g., the baroque cornett). What brass instruments have in
common is that they are excited by a vibrating lip. Based on
Martin’s seminal work on lip vibrations1, Backus and
Hundley developed a trumpet model that employed a sinu-
soidally changing lip opening and a nonlinear slit resistance
to generate harmonically rich mouthpiece pressure wave-
2
forms . More recently, Yoshikawa showed that lips under
different conditions may exhibit either “swinging door” or
“sliding door” motion3. Adachi and Sato formulated a two-
4 dimensional lip model that simulated these motions .
Meanwhile, Copley and Strong made a detailed stroboscop-
ic study of trombone lip vibrations5, and Dean Ayers has
demonstrated lip motion in real time with a stroboscope
and video camera at several ASA meetings (e.g.,
McLaughlin et al.), concluding that lips exhibit Rayleigh 6
wave motion .
While lips act as the source (or excitation) of a brass
instrument, its pipe (or horn) acts as its filter. In the late 1960s I became intrigued with the possibility of synthesizing brass sounds using a source-filter model, a model whose validity has been a prevalent assumption for acoustic signal production since the early days of acoustics. This model cer- tainly has been much used for synthesis of speech sounds, and it was quite successfully used in early analog electronic music synthesizers. Analysis by Luce and Clark showed that brass spectra follow a basic low-pass filter characteristic whose “roll off rate” varies with performance dynamic7. Thus, more intense tones are perceptually brighter while less intense tones sound darker. I also became acquainted with a Ph.D. dissertation (Univ. of Illinois, 1941) written by Daniel Martin (ASA editor-in-chief, 1985-99) which not only showed that more intense tones were brighter but also included measurements of mouthpiece-to-output transmis-
8
sion response and radiation patterns from the bell . The lat-
“Higher intensity tones appeared to contain stronger upper harmonics than would be predicted by a filter computed at lower intensities.”
ter two measurements were done with swept sine wave input. However, it seemed that simultaneous measure- ments of mouthpiece and output pres- sure waveforms, resulting from actual performance, could be used to derive a source-filter model. To do this, I arranged for four different trombonists at the University of Illinois to play a series of tones at different pitches and dynamics on the same trombone in an anechoic chamber. The mouthpiece and output pressure signals were recorded
on two tracks of a tape recorder. The mouthpiece pressures were unexpectedly high—between 150 and 170 dBC SPL, whereas output pressures varied from 67 to 100 dBC. An analog wave analyzer and, later, computer Fourier analysis were used to compute input and output spectra, and then a
9-12
transmission response had to be nonlinear.
In 1973, I visited Arthur Benade, who was posthumous-
ly awarded the Society’s Gold Medal in 1988, at his lab at Case Western Reserve University. During my visit, he made simultaneous swept-sine-wave graphs of input impedance and transmission response for my cornet. (Up until that time several researchers had published plots of input impedance response, but the only transmission response curves I had seen were Martin’s.) What was interesting about the two curves was that while they both exhibited a series of approx- imately harmonic-spaced resonances, the maxima of the input impedance corresponded to the minima of the trans- mission response, both corresponding to the playing fre- quencies of the instrument. This phenomenon is easily explained in terms of wave reflection: When a wave reflects back from the horn opening to reinforce lip motion, a maxi- mum of pressure occurs at the lips (thus, an impedance max- imum), but very little energy escapes from the horn (thus, a transmission minimum). However, in between these fre- quencies, energy radiates relatively efficiently from the bell, and very little reflects back, giving rise to pressure minima at the lips. This is at least what happens when wave lengths are longer than the bell diameter. When they are shorter, reflec- tions become weaker and weaker, transmission flattens off, and input impedance falls to a low value. The frequency
. Surprisingly, the filter was not fixed but appeared to vary with dynamic or intensity. Higher intensity tones produced stronger upper harmonics than would be predicted by a filter computed at lower intensities. Thus, it appeared that the trombone's
filter function was computed from their ratio
52 Acoustics Today, April 2006