Chapter 5
Blow That Horn: An Elementary Model
of Brass Playing
A brass instrument comes to life when a musician picks it up and plays it. In this
chapter we draw together the scientific investigations of buzzing lips discussed in
Chap. 3, and the studies of the properties of brass instrument tubes reviewed in
Chap. 4, to create an elementary model of brass instrument performance which
includes both player and instrument. The aim of this task is to provide a scientific
tool which can be used to improve our understanding of the ways in which sound is
generated and controlled in a brass performance.
The model presented in the present chapter is classed as elementary because
a number of major simplifications are made in deriving it. The vibrating lips are
modelled as a one degree of freedom oscillator, and it is assumed that the lip
vibration amplitude is small enough that the lips do not collide (or the collision
model is basic). The upstream resonances of the player’s windway are neglected,
as is nonlinear propagation of sound in the air column of the instrument. Wall
vibrations are also ignored. Despite these simplifications, the elementary model is
capable of reproducing many of the important aspects of performance by human
players which were described in Sect. 1.2. These include the musician’s ability
to select by embouchure control one of a number of pitches which form an
approximate harmonic series and to pull the pitch up or down by the subtle
embouchure modification described as ‘lipping’. The model also explains why there
is a minimum blowing pressure necessary to sound each note and why this threshold
pressure increases as the played pitch rises.
The model is based on a set of three equations, which have to be solved
simultaneously to predict the nature of the sound radiated by the instrument.
These three constituent equations of the model are reviewed in Sect. 5.1. Lowamplitude self-sustained periodic solutions of the elementary model are explored in
Sect. 5.2, which includes an introduction to the technique of linear stability analysis.
In Sect. 5.3, the restriction to small amplitudes is relaxed: numerical simulation
methods developed in physical modelling, including the use of bifurcation diagrams,
are employed to illustrate a large range of playing situations. In Sect. 5.4 the
© Springer Nature Switzerland AG 2021
M. Campbell et al., The Science of Brass Instruments, Modern Acoustics and Signal
Processing, https://doi.org/10.1007/978-3-030-55686-0_5
217
Blow That Horn: An Elementary Model
of Brass Playing
A brass instrument comes to life when a musician picks it up and plays it. In this
chapter we draw together the scientific investigations of buzzing lips discussed in
Chap. 3, and the studies of the properties of brass instrument tubes reviewed in
Chap. 4, to create an elementary model of brass instrument performance which
includes both player and instrument. The aim of this task is to provide a scientific
tool which can be used to improve our understanding of the ways in which sound is
generated and controlled in a brass performance.
The model presented in the present chapter is classed as elementary because
a number of major simplifications are made in deriving it. The vibrating lips are
modelled as a one degree of freedom oscillator, and it is assumed that the lip
vibration amplitude is small enough that the lips do not collide (or the collision
model is basic). The upstream resonances of the player’s windway are neglected,
as is nonlinear propagation of sound in the air column of the instrument. Wall
vibrations are also ignored. Despite these simplifications, the elementary model is
capable of reproducing many of the important aspects of performance by human
players which were described in Sect. 1.2. These include the musician’s ability
to select by embouchure control one of a number of pitches which form an
approximate harmonic series and to pull the pitch up or down by the subtle
embouchure modification described as ‘lipping’. The model also explains why there
is a minimum blowing pressure necessary to sound each note and why this threshold
pressure increases as the played pitch rises.
The model is based on a set of three equations, which have to be solved
simultaneously to predict the nature of the sound radiated by the instrument.
These three constituent equations of the model are reviewed in Sect. 5.1. Lowamplitude self-sustained periodic solutions of the elementary model are explored in
Sect. 5.2, which includes an introduction to the technique of linear stability analysis.
In Sect. 5.3, the restriction to small amplitudes is relaxed: numerical simulation
methods developed in physical modelling, including the use of bifurcation diagrams,
are employed to illustrate a large range of playing situations. In Sect. 5.4 the
© Springer Nature Switzerland AG 2021
M. Campbell et al., The Science of Brass Instruments, Modern Acoustics and Signal
Processing, https://doi.org/10.1007/978-3-030-55686-0_5
217