5.1 The Three Equations of the Brass Instrument Model
219
of a mechanical oscillator interacting with a continuous flow is described as a flowinduced vibration phenomenon.
Similar oscillations occur in reed woodwind instruments because of the destabilisation of the valve formed by the mechanical reed. There is however an important
difference between the roles played by the brass player’s lips and the clarinet reed.
The brass player is able to control and modify the lip natural resonance frequency,
which is usually very close to the playing frequency. For that reason lip dynamics
are an essential component of the brass playing model. In contrast, a cane single or
double reed has a natural resonance frequency which is much higher than the highest
note in the normal compass of the instrument and can be modified only to a limited
extent by the player’s lips. It is therefore possible to develop a two-equation model
for a woodwind instrument based on a low-frequency approximation in which the
reed is treated as a massless, lossless spring (McIntyre et al. 1983).
Normally the flow through the lips enters the mouthpiece of a brass instrument.
The lip valve and the instrument are then coupled in an acoustic feedback loop, as
shown in Fig. 5.1. The oscillations are amplified and controlled by the coupling of a
localised nonlinear sound generating element (the source resulting from the valveflow interaction) with a pipe in which acoustic energy can accumulate in resonant
modes (standing waves). A small part of the acoustic energy is not trapped inside
the pipe, but is radiated through the bell and any other open holes in the instrument.
When the mechanical system is destabilised, different permanent regimes of
oscillation can be obtained. The sounding of a note of constant pitch corresponds
to a periodic regime with a steady fundamental frequency. This frequency is often
called simply the playing frequency, although the frequency spectrum of the sound
typically contains many harmonic components. The playing frequency is usually
close to one of the resonant mode frequencies of the air column in the pipe, but
‘factitious notes’ not satisfying this criterion can also be sounded. Quasi-periodic
regimes which create multiphonics sounds, and even chaotic regimes with no
definable periodicity, are also possible outcomes.
The generation of an oscillating output from a steady input is a characteristic
feature of a nonlinear dynamical system such as that represented by Fig. 5.1. In
reality, all three components represented by the coloured boxes display nonlinear
behaviour, since neither the mechanical response of the lips (represented by the
blue box) nor the acoustical response of the air column (represented by the green
box) is strictly independent of amplitude. However the linear description of the lip
dynamics in Sect. 5.1.1 provides a good approximation for moderate playing levels,
as does the description of the acoustical response of the instrument by its linear input
impedance in Sect. 5.1.3. The strongly nonlinear behaviour of the overall dynamical
system is due to the nature of the pressure-flow relationship represented by the red
box in Fig. 5.1 and discussed in Sect. 5.1.2.
219
of a mechanical oscillator interacting with a continuous flow is described as a flowinduced vibration phenomenon.
Similar oscillations occur in reed woodwind instruments because of the destabilisation of the valve formed by the mechanical reed. There is however an important
difference between the roles played by the brass player’s lips and the clarinet reed.
The brass player is able to control and modify the lip natural resonance frequency,
which is usually very close to the playing frequency. For that reason lip dynamics
are an essential component of the brass playing model. In contrast, a cane single or
double reed has a natural resonance frequency which is much higher than the highest
note in the normal compass of the instrument and can be modified only to a limited
extent by the player’s lips. It is therefore possible to develop a two-equation model
for a woodwind instrument based on a low-frequency approximation in which the
reed is treated as a massless, lossless spring (McIntyre et al. 1983).
Normally the flow through the lips enters the mouthpiece of a brass instrument.
The lip valve and the instrument are then coupled in an acoustic feedback loop, as
shown in Fig. 5.1. The oscillations are amplified and controlled by the coupling of a
localised nonlinear sound generating element (the source resulting from the valveflow interaction) with a pipe in which acoustic energy can accumulate in resonant
modes (standing waves). A small part of the acoustic energy is not trapped inside
the pipe, but is radiated through the bell and any other open holes in the instrument.
When the mechanical system is destabilised, different permanent regimes of
oscillation can be obtained. The sounding of a note of constant pitch corresponds
to a periodic regime with a steady fundamental frequency. This frequency is often
called simply the playing frequency, although the frequency spectrum of the sound
typically contains many harmonic components. The playing frequency is usually
close to one of the resonant mode frequencies of the air column in the pipe, but
‘factitious notes’ not satisfying this criterion can also be sounded. Quasi-periodic
regimes which create multiphonics sounds, and even chaotic regimes with no
definable periodicity, are also possible outcomes.
The generation of an oscillating output from a steady input is a characteristic
feature of a nonlinear dynamical system such as that represented by Fig. 5.1. In
reality, all three components represented by the coloured boxes display nonlinear
behaviour, since neither the mechanical response of the lips (represented by the
blue box) nor the acoustical response of the air column (represented by the green
box) is strictly independent of amplitude. However the linear description of the lip
dynamics in Sect. 5.1.1 provides a good approximation for moderate playing levels,
as does the description of the acoustical response of the instrument by its linear input
impedance in Sect. 5.1.3. The strongly nonlinear behaviour of the overall dynamical
system is due to the nature of the pressure-flow relationship represented by the red
box in Fig. 5.1 and discussed in Sect. 5.1.2.