410
9 Looking Back and Looking Forward
a specific mode can play a number of different roles. Each mode can support the
fundamental frequency of a played note: it could then be expected that the ideal
mode frequencies should be equal to the frequencies of the corresponding notes
in an equally tempered scale. This requirement is however incompatible with the
Bouasse-Benade prescription that a strong and well-centred played note requires
a harmonically related set of modes: apart from the octave, none of the intervals
in the harmonic series correspond exactly to intervals in the equally tempered
scale. In addition, the assumption that brass instrumentalists always play in equal
temperament is questionable (Farkas 1956; Kopiez 2003; Bronk 2010).
The ultimate judge of good intonation is of course the musician who has to play
the instrument. From the performer’s point of view, the optimum target should be
specified in terms of the playing frequencies of the notes rather than the air column
resonances. The frequency of a played note is the result of the nonlinear coupling
of the lips and the multiple resonances of the air column, and depending on the
player’s choice of embouchure, it can differ significantly from the frequency of
the lowest supporting air column mode (Eveno et al. 2014). Poirson et al. (2007)
included playing tests by musicians in the definition of an impedance-based target
for trumpet optimisation. A standard orchestral B trumpet was provided with an
experimental leadpipe made up by coupling together four short conical sections,
each of which could be chosen from a set with several slightly different values
of the input and output radii. By selecting suitable combinations of the sections,
twelve test trumpets with subtly different input impedance curves were created, and
ten professional trumpet players were asked to evaluate the intonation of each test
instrument. Using sophisticated statistical analysis, it was possible to find a model
equation relating the intonation scores of the experts and the frequencies of the
instrument air column; from this relationship, it was possible to define a target set
of frequency ratios corresponding to the maximum intonation score. Interestingly,
these ratios were close to but not identical with the integer ratios of the perfect
harmonic series. An optimisation calculation was finally carried out to calculate the
bore of the ideal leadpipe based on the target frequencies.
While the involvement of professional performers in the setting of the optimisation target helps to ensure that the results are musically meaningful, the use of
human subjects in sensory evaluation is not straightforward and is impracticable for
systematic large-scale testing of new instrument designs. An alternative method for
defining the optimisation target in terms of playing frequency, using the physical
modelling approach described in Chap. 5, has been described by Tournemenne et
al. (2019). The system studied was a trumpet played by a ‘virtual musician’, whose
embouchure was modelled as an outward-striking lip valve. In this case, the target
was defined as a set of equally tempered playing frequencies. The procedure started
with the initial bore profile of a trumpet, whose objective properties were defined by
the impedance peak frequencies. Three of the parameters defining the vibrational
behaviour of the lip valve (the mouth pressure p m , the mass per unit area μ l and
the lip resonance frequency f l ) were treated as variables in a numerical simulation
which searched for stable periodic solutions, using the harmonic balance method
(Gilbert et al. 1989). Many embouchures satisfying this criterion were found; the
9 Looking Back and Looking Forward
a specific mode can play a number of different roles. Each mode can support the
fundamental frequency of a played note: it could then be expected that the ideal
mode frequencies should be equal to the frequencies of the corresponding notes
in an equally tempered scale. This requirement is however incompatible with the
Bouasse-Benade prescription that a strong and well-centred played note requires
a harmonically related set of modes: apart from the octave, none of the intervals
in the harmonic series correspond exactly to intervals in the equally tempered
scale. In addition, the assumption that brass instrumentalists always play in equal
temperament is questionable (Farkas 1956; Kopiez 2003; Bronk 2010).
The ultimate judge of good intonation is of course the musician who has to play
the instrument. From the performer’s point of view, the optimum target should be
specified in terms of the playing frequencies of the notes rather than the air column
resonances. The frequency of a played note is the result of the nonlinear coupling
of the lips and the multiple resonances of the air column, and depending on the
player’s choice of embouchure, it can differ significantly from the frequency of
the lowest supporting air column mode (Eveno et al. 2014). Poirson et al. (2007)
included playing tests by musicians in the definition of an impedance-based target
for trumpet optimisation. A standard orchestral B trumpet was provided with an
experimental leadpipe made up by coupling together four short conical sections,
each of which could be chosen from a set with several slightly different values
of the input and output radii. By selecting suitable combinations of the sections,
twelve test trumpets with subtly different input impedance curves were created, and
ten professional trumpet players were asked to evaluate the intonation of each test
instrument. Using sophisticated statistical analysis, it was possible to find a model
equation relating the intonation scores of the experts and the frequencies of the
instrument air column; from this relationship, it was possible to define a target set
of frequency ratios corresponding to the maximum intonation score. Interestingly,
these ratios were close to but not identical with the integer ratios of the perfect
harmonic series. An optimisation calculation was finally carried out to calculate the
bore of the ideal leadpipe based on the target frequencies.
While the involvement of professional performers in the setting of the optimisation target helps to ensure that the results are musically meaningful, the use of
human subjects in sensory evaluation is not straightforward and is impracticable for
systematic large-scale testing of new instrument designs. An alternative method for
defining the optimisation target in terms of playing frequency, using the physical
modelling approach described in Chap. 5, has been described by Tournemenne et
al. (2019). The system studied was a trumpet played by a ‘virtual musician’, whose
embouchure was modelled as an outward-striking lip valve. In this case, the target
was defined as a set of equally tempered playing frequencies. The procedure started
with the initial bore profile of a trumpet, whose objective properties were defined by
the impedance peak frequencies. Three of the parameters defining the vibrational
behaviour of the lip valve (the mouth pressure p m , the mass per unit area μ l and
the lip resonance frequency f l ) were treated as variables in a numerical simulation
which searched for stable periodic solutions, using the harmonic balance method
(Gilbert et al. 1989). Many embouchures satisfying this criterion were found; the