9.2 Brass Instruments in the Digital World
409
9.2.1 Optimisation in Instrument Design
Modifications in the design of conventional brass instruments have traditionally
employed a procedure which can be described as experience-guided trial and
error. If a maker decides to change the shape of a trumpet bell, for example,
knowledge gained in previous experiments or taught by past masters might suggest
that complementary changes in other parts of the bore are necessary to preserve
good intonation. A prototype of the resulting design is made and tested. If the
intonation is not satisfactory, further small bore changes are made and another
prototype constructed. Several cycles of this expensive and time-consuming process
could be necessary before an acceptable design is achieved.
It is possible to reduce the necessity for manufacturing multiple prototype
instruments by making use of the scientific understanding of brass acoustics
reviewed in Chap. 4. The input impedance curve of the proposed instrument can
be calculated from knowledge of the bore profile; the frequencies of the peaks in
the impedance curve correspond closely to the natural notes of the instrument. If
some of the natural notes of the virtual instrument are considered to be unacceptably
sharp or flat, small changes can be made to the bore profile and the input impedance
curve recalculated. Only when a satisfactory design is achieved is it necessary to
manufacture a real instrument.
The trial and error process can also be automated using suitable computer software. Smith and Daniell (1976) proposed a method for correcting the intonation of
wind instruments using perturbation theory. This mathematical technique effectively
finds all possible modifications of the bore which give the desired changes in mode
frequencies. Many of the possible bore shapes involve large and/or abrupt changes,
and the calculation involves a minimisation process which selects the smoothest of
the calculated bore profiles.
The dramatic growth of computational speed and power in the early years of the
twenty-first century, together with refinements of the method of input impedance
calculation, has made it possible to develop efficient algorithms for optimisation of
wind instrument bore profiles using input impedance targets (Kausel 2001; Petiot
and Tavard 2008; Braden et al. 2009; Macaluso and Dalmont 2011; Noreland et
al. 2013). In these methods the continuous bore of the instrument is approximated
by a finite number of short sections, which may be cylindrical, conical or flaring;
discontinuities and toneholes can also be incorporated. The optimisation process
starts with a ‘best guess’ bore profile and systematically explores the effect on
the impedance curve of small changes in the parameters defining each section. A
version of this type of optimisation software has been successfully used by brass
manufacturers (Egger et al. 2005).
An optimisation method requires a clearly defined target. In algorithms based
on the input impedance curve, the target is usually specified as a given set of
input impedance peak frequencies (corresponding to the mode frequencies of the air
column). In their pioneering 1976 paper, Smith and Daniell pointed out that defining
an ideal set of mode frequencies for a brass instrument is not straightforward, since
409
9.2.1 Optimisation in Instrument Design
Modifications in the design of conventional brass instruments have traditionally
employed a procedure which can be described as experience-guided trial and
error. If a maker decides to change the shape of a trumpet bell, for example,
knowledge gained in previous experiments or taught by past masters might suggest
that complementary changes in other parts of the bore are necessary to preserve
good intonation. A prototype of the resulting design is made and tested. If the
intonation is not satisfactory, further small bore changes are made and another
prototype constructed. Several cycles of this expensive and time-consuming process
could be necessary before an acceptable design is achieved.
It is possible to reduce the necessity for manufacturing multiple prototype
instruments by making use of the scientific understanding of brass acoustics
reviewed in Chap. 4. The input impedance curve of the proposed instrument can
be calculated from knowledge of the bore profile; the frequencies of the peaks in
the impedance curve correspond closely to the natural notes of the instrument. If
some of the natural notes of the virtual instrument are considered to be unacceptably
sharp or flat, small changes can be made to the bore profile and the input impedance
curve recalculated. Only when a satisfactory design is achieved is it necessary to
manufacture a real instrument.
The trial and error process can also be automated using suitable computer software. Smith and Daniell (1976) proposed a method for correcting the intonation of
wind instruments using perturbation theory. This mathematical technique effectively
finds all possible modifications of the bore which give the desired changes in mode
frequencies. Many of the possible bore shapes involve large and/or abrupt changes,
and the calculation involves a minimisation process which selects the smoothest of
the calculated bore profiles.
The dramatic growth of computational speed and power in the early years of the
twenty-first century, together with refinements of the method of input impedance
calculation, has made it possible to develop efficient algorithms for optimisation of
wind instrument bore profiles using input impedance targets (Kausel 2001; Petiot
and Tavard 2008; Braden et al. 2009; Macaluso and Dalmont 2011; Noreland et
al. 2013). In these methods the continuous bore of the instrument is approximated
by a finite number of short sections, which may be cylindrical, conical or flaring;
discontinuities and toneholes can also be incorporated. The optimisation process
starts with a ‘best guess’ bore profile and systematically explores the effect on
the impedance curve of small changes in the parameters defining each section. A
version of this type of optimisation software has been successfully used by brass
manufacturers (Egger et al. 2005).
An optimisation method requires a clearly defined target. In algorithms based
on the input impedance curve, the target is usually specified as a given set of
input impedance peak frequencies (corresponding to the mode frequencies of the air
column). In their pioneering 1976 paper, Smith and Daniell pointed out that defining
an ideal set of mode frequencies for a brass instrument is not straightforward, since