6.1 Why Brass Instruments Sound Brassy
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in the wave are radiated, while the low frequencies are reflected back into the
instrument. The waveform of the radiated pressure corresponds roughly to the time
derivative of the pressure signal arriving at the pipe exit (see Fig. 6.4). The result
is a periodic emission of sharp pressure peaks, corresponding to a spectrum with
uniform harmonic amplitudes up to very high frequencies. This is the signature of
brassy sounds.
The necessity for nonlinear treatment of propagation is related to the value of
the mouthpiece pressure amplitude. For a sine wave (amplitude ˆ
p, frequency f ), a
shock wave is formed at a distance x from the mouthpiece which is larger than a
critical distance L s given for a cylindrical bore by
L s
2γp atmos c
[(γ + 1)(2πf ˆ
p]
,
(6.2)
where γ is the ratio of specific heats of air, p atmos the atmospheric pressure and c
the speed of sound (Pierce 1989). The shock length is inversely proportional to both
the mouthpiece pressure amplitude and the frequency. The larger the input pressure
amplitude, the shorter the L s is and the brassier the sound is. For a given amplitude,
increasing the frequency reduces the shock length and makes the sound brassier.
The expression for the shock length given in Eq. 6.2 is valid only if the input
wave is sinusoidal. Hirschberg et al. (1996b) pointed out that this is not the relevant
parameter for judging the severity of the nonlinear wave steepening of an arbitrary
input waveform. A more general expression for L s is
L s
2γp 0 c
[(γ + 1)(∂p m /∂t) max ]
.
(6.3)
The distance to shock formation is thus inversely proportional to the maximum rate
of change of pressure in the mouthpiece, rather than to the pressure amplitude.
For the pressure gradients measured in very loud trombone playing, L s is of the
same order as the length of the slide section of the tenor trombone, so shock wave
formation is indeed to be expected.
Brassiness does not suddenly appear at a particular dynamic level but develops
gradually once L s is of the same order as the length of the bore. Some brass players
have found that by employing slight changes in embouchure, they can exert a degree
of control over the level of brassiness at a constant dynamic level. Experimental data
from playing tests on a french horn, reported in Norman et al. (2010), suggest that
this technique is based on the player’s ability to modify the rate of change of the
input pressure wavefront as it is formed in the mouthpiece, without significantly
changing the amplitude of the mouthpiece pressure. This permits a modification of
L s and therefore the amount of distortion obtained during nonlinear propagation.
The player’s control over the mouthpiece waveform is limited and results in
a subtle change in the shock length. Its use as a musical technique is therefore
restricted to moderate dynamic levels. At low dynamic level (pp), L s is much larger
than the sounding length L, and the sound is smooth and cannot be brassy. Between
279
in the wave are radiated, while the low frequencies are reflected back into the
instrument. The waveform of the radiated pressure corresponds roughly to the time
derivative of the pressure signal arriving at the pipe exit (see Fig. 6.4). The result
is a periodic emission of sharp pressure peaks, corresponding to a spectrum with
uniform harmonic amplitudes up to very high frequencies. This is the signature of
brassy sounds.
The necessity for nonlinear treatment of propagation is related to the value of
the mouthpiece pressure amplitude. For a sine wave (amplitude ˆ
p, frequency f ), a
shock wave is formed at a distance x from the mouthpiece which is larger than a
critical distance L s given for a cylindrical bore by
L s
2γp atmos c
[(γ + 1)(2πf ˆ
p]
,
(6.2)
where γ is the ratio of specific heats of air, p atmos the atmospheric pressure and c
the speed of sound (Pierce 1989). The shock length is inversely proportional to both
the mouthpiece pressure amplitude and the frequency. The larger the input pressure
amplitude, the shorter the L s is and the brassier the sound is. For a given amplitude,
increasing the frequency reduces the shock length and makes the sound brassier.
The expression for the shock length given in Eq. 6.2 is valid only if the input
wave is sinusoidal. Hirschberg et al. (1996b) pointed out that this is not the relevant
parameter for judging the severity of the nonlinear wave steepening of an arbitrary
input waveform. A more general expression for L s is
L s
2γp 0 c
[(γ + 1)(∂p m /∂t) max ]
.
(6.3)
The distance to shock formation is thus inversely proportional to the maximum rate
of change of pressure in the mouthpiece, rather than to the pressure amplitude.
For the pressure gradients measured in very loud trombone playing, L s is of the
same order as the length of the slide section of the tenor trombone, so shock wave
formation is indeed to be expected.
Brassiness does not suddenly appear at a particular dynamic level but develops
gradually once L s is of the same order as the length of the bore. Some brass players
have found that by employing slight changes in embouchure, they can exert a degree
of control over the level of brassiness at a constant dynamic level. Experimental data
from playing tests on a french horn, reported in Norman et al. (2010), suggest that
this technique is based on the player’s ability to modify the rate of change of the
input pressure wavefront as it is formed in the mouthpiece, without significantly
changing the amplitude of the mouthpiece pressure. This permits a modification of
L s and therefore the amount of distortion obtained during nonlinear propagation.
The player’s control over the mouthpiece waveform is limited and results in
a subtle change in the shock length. Its use as a musical technique is therefore
restricted to moderate dynamic levels. At low dynamic level (pp), L s is much larger
than the sounding length L, and the sound is smooth and cannot be brassy. Between