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2 The Scientist’s Perspective on Brass Instrument Behaviour
laws and principles, and the predictions of these models must then be compared with
reality as revealed by experiments. A good theoretical model will not only explain
previous observations but will predict behaviours and outcomes that have not yet
been observed and can be experimentally validated. A model may also be useful to
optimise the behaviour of a system, as long as the target for optimisation can be
defined. In the case of a brass instrument, for example, a good physical model of the
instrument could help a maker to optimise the intonation of the instrument, provided
that the ideal intonation was clearly defined.
2.2.1 The Scientific Case for Simplified Models
To explain the essential features of the functioning of brass instruments, acousticians
have developed models based on a number of simplifying hypotheses. This follows
the classic approach of the physicist, which is to simplify the model of the system
being studied as much as possible while retaining the fundamental operating
principles of the system. The model can then be used to gain a deep insight into
the role of these fundamental principles and to understand how the most important
control parameters of the model affect its behaviour.
The word ‘model’ can mean a number of different things, but in the present
context, we use it to define two related approaches.
1. A theoretical model of a trumpet is a set of equations describing the ways in
which different parts of the instrument function and are related.
2. A numerical model of a trumpet is a computer program which is capable of
solving the equations numerically and predicting how the instrument will behave
under given circumstances.
Numerical models typically rely on close-grained discretisation of the object
to be studied in both time and space and therefore require the rapid solution
of a very large number of equations. The development of efficient methods for
solving such equations together with the continuing growth in the memory capacity
and calculating power of computers has meant that it is now possible to test
proposed modifications of a brass instrument by making the appropriate changes
to a numerical model rather than to an instrument in the real world. It has to be
borne in mind, however, that even with unlimited computational power, the validity
of a numerical calculation depends on the quality of the model equations.
The discussion in Sect. 2.1 of various aspects of brass instrument behaviour
suggests that a simplified model can be constructed by dividing the system into
three sub-systems (see Fig. 2.15):
1. the musician, represented by the lungs and trachea, the mouth cavity and the lips;
2. the instrument including the mouthpiece;
3. the external environment into which the sound radiates.
2 The Scientist’s Perspective on Brass Instrument Behaviour
laws and principles, and the predictions of these models must then be compared with
reality as revealed by experiments. A good theoretical model will not only explain
previous observations but will predict behaviours and outcomes that have not yet
been observed and can be experimentally validated. A model may also be useful to
optimise the behaviour of a system, as long as the target for optimisation can be
defined. In the case of a brass instrument, for example, a good physical model of the
instrument could help a maker to optimise the intonation of the instrument, provided
that the ideal intonation was clearly defined.
2.2.1 The Scientific Case for Simplified Models
To explain the essential features of the functioning of brass instruments, acousticians
have developed models based on a number of simplifying hypotheses. This follows
the classic approach of the physicist, which is to simplify the model of the system
being studied as much as possible while retaining the fundamental operating
principles of the system. The model can then be used to gain a deep insight into
the role of these fundamental principles and to understand how the most important
control parameters of the model affect its behaviour.
The word ‘model’ can mean a number of different things, but in the present
context, we use it to define two related approaches.
1. A theoretical model of a trumpet is a set of equations describing the ways in
which different parts of the instrument function and are related.
2. A numerical model of a trumpet is a computer program which is capable of
solving the equations numerically and predicting how the instrument will behave
under given circumstances.
Numerical models typically rely on close-grained discretisation of the object
to be studied in both time and space and therefore require the rapid solution
of a very large number of equations. The development of efficient methods for
solving such equations together with the continuing growth in the memory capacity
and calculating power of computers has meant that it is now possible to test
proposed modifications of a brass instrument by making the appropriate changes
to a numerical model rather than to an instrument in the real world. It has to be
borne in mind, however, that even with unlimited computational power, the validity
of a numerical calculation depends on the quality of the model equations.
The discussion in Sect. 2.1 of various aspects of brass instrument behaviour
suggests that a simplified model can be constructed by dividing the system into
three sub-systems (see Fig. 2.15):
1. the musician, represented by the lungs and trachea, the mouth cavity and the lips;
2. the instrument including the mouthpiece;
3. the external environment into which the sound radiates.