254
5 Blow That Horn: An Elementary Model of Brass Playing
Figure 5.18 can be viewed as simulating a musical performance in which the
trombonist attempts to play a note with the slide in fixed position while gradually
increasing the lip frequency f l from 20 to 500 Hz, continually adjusting the mouth
pressure to the minimum value necessary to get the note to sound. The lower curve,
which effectively traces the playing frequency for a mouth pressure just above
the sounding threshold, replicates well the musician’s experience that this exercise
results in an arpeggio of notes which are close to the natural notes of the instrument
(see Sect. 1.2.2).
The span of lip frequencies displayed in Fig. 5.18 can be divided into nine distinct
ranges based on the discontinuities in the threshold frequency curve. Each range
corresponds to one regime or register of the instrument. The first regime spans the
frequency range [30 : 62 Hz], within which the pedal note B 1 is played; this
is a somewhat anomalous regime, which will be discussed later. The second to
eighth regimes correspond to the frequency ranges [72 : 123 Hz], [124 : 179 Hz],
[180 : 234 Hz], [235 : 288 Hz], [289 : 352 Hz], [353 : 404 Hz] and [408 : 454 Hz],
respectively. The ninth regime begins at 488 Hz and continues beyond the 500 Hz
limit of the calculation. No stable oscillatory state was found for lip frequencies
below 30 Hz or between 63 and 72 Hz; there are also an interval between 454 and
488 Hz in which the equilibrium state remained stable for the tested values of p m .
The shape of the lower curve in Fig. 5.18 is characterised from f l = 72 Hz
upwards by a series of plateaus of almost constant height, each plateau representing
the variation of the threshold oscillation frequency as a function of the lip frequency
within one of the regimes. In the nth regime f thr increases gently with increasing
f l from a value just above the frequency f ac,n of the nth acoustic resonance and is
also always above the lip resonance frequency f l . This behaviour is the expected
consequence of our choice of the outward-striking valve to represent the lips in
the elementary model (see Sect. 3.2.4). The plateaus correspond to the player’s
experience in finding a set of ‘slots’, the pitch being relatively insensitive to changes
in embouchure within each slot. The fact that f thr increases slightly with rising f l in
each regime also matches the experience of the brass player, who can slightly ‘bend’
the pitch up or down within a slot by adjusting f l through the muscular tension of the
lips. The maximum change in f thr obtainable within one regime, which determines
the attainable musical range within each register, has analytic limits depending on
the lip quality factor q l as detailed in Silva et al. (2007). For the model used here,
the pitch range varies from 186 cents in regime 2 to 45 cents in regime 8.
Turning to the upper curve in Fig. 5.18, it can be observed that the oscillation
threshold p thr globally increases with the rank n of the register. A greater p m
value is required to reach the higher notes of the instrument, which again agrees
with musical experience. Each regime characterised by a plateau in the curve for
f thr is represented by a U-shaped valley in the p thr curve (Silva et al. 2007). The
minimum value p
opt
thr for each register, indicated by a green circle in Fig. 5.18,
depends significantly on the losses in the resonator. Since musicians commonly
describe a playing strategy which minimises the effort to produce a sound in each
playing regime, it is a reasonable hypothesis that p
opt
thr,n and the associated lip
5 Blow That Horn: An Elementary Model of Brass Playing
Figure 5.18 can be viewed as simulating a musical performance in which the
trombonist attempts to play a note with the slide in fixed position while gradually
increasing the lip frequency f l from 20 to 500 Hz, continually adjusting the mouth
pressure to the minimum value necessary to get the note to sound. The lower curve,
which effectively traces the playing frequency for a mouth pressure just above
the sounding threshold, replicates well the musician’s experience that this exercise
results in an arpeggio of notes which are close to the natural notes of the instrument
(see Sect. 1.2.2).
The span of lip frequencies displayed in Fig. 5.18 can be divided into nine distinct
ranges based on the discontinuities in the threshold frequency curve. Each range
corresponds to one regime or register of the instrument. The first regime spans the
frequency range [30 : 62 Hz], within which the pedal note B 1 is played; this
is a somewhat anomalous regime, which will be discussed later. The second to
eighth regimes correspond to the frequency ranges [72 : 123 Hz], [124 : 179 Hz],
[180 : 234 Hz], [235 : 288 Hz], [289 : 352 Hz], [353 : 404 Hz] and [408 : 454 Hz],
respectively. The ninth regime begins at 488 Hz and continues beyond the 500 Hz
limit of the calculation. No stable oscillatory state was found for lip frequencies
below 30 Hz or between 63 and 72 Hz; there are also an interval between 454 and
488 Hz in which the equilibrium state remained stable for the tested values of p m .
The shape of the lower curve in Fig. 5.18 is characterised from f l = 72 Hz
upwards by a series of plateaus of almost constant height, each plateau representing
the variation of the threshold oscillation frequency as a function of the lip frequency
within one of the regimes. In the nth regime f thr increases gently with increasing
f l from a value just above the frequency f ac,n of the nth acoustic resonance and is
also always above the lip resonance frequency f l . This behaviour is the expected
consequence of our choice of the outward-striking valve to represent the lips in
the elementary model (see Sect. 3.2.4). The plateaus correspond to the player’s
experience in finding a set of ‘slots’, the pitch being relatively insensitive to changes
in embouchure within each slot. The fact that f thr increases slightly with rising f l in
each regime also matches the experience of the brass player, who can slightly ‘bend’
the pitch up or down within a slot by adjusting f l through the muscular tension of the
lips. The maximum change in f thr obtainable within one regime, which determines
the attainable musical range within each register, has analytic limits depending on
the lip quality factor q l as detailed in Silva et al. (2007). For the model used here,
the pitch range varies from 186 cents in regime 2 to 45 cents in regime 8.
Turning to the upper curve in Fig. 5.18, it can be observed that the oscillation
threshold p thr globally increases with the rank n of the register. A greater p m
value is required to reach the higher notes of the instrument, which again agrees
with musical experience. Each regime characterised by a plateau in the curve for
f thr is represented by a U-shaped valley in the p thr curve (Silva et al. 2007). The
minimum value p
opt
thr for each register, indicated by a green circle in Fig. 5.18,
depends significantly on the losses in the resonator. Since musicians commonly
describe a playing strategy which minimises the effort to produce a sound in each
playing regime, it is a reasonable hypothesis that p
opt
thr,n and the associated lip