256
5 Blow That Horn: An Elementary Model of Brass Playing
Fig. 5.19 Detail of Fig. 5.18 showing the variation of threshold pressure (upper graph) and
threshold oscillation frequency (lower graph) as a function of lip resonance frequency for the first
regime of a tenor trombone
is found at a lip frequency f
opt
l
= 50 Hz, corresponding to a threshold oscillation
frequency f
opt
thr = 56.3 Hz. This LSA prediction is very close to the frequency of
58 Hz at which the pedal note is sounded in musical performance, although it is
48% higher than the first acoustic resonance frequency f ac,1 = 38 Hz.
Comparison of the predicted optimum threshold frequencies with playing experience shows that, despite the simplifications accepted in framing the elementary
model, LSA is able to give a good account of the pitches of the natural notes of the
trombone. Threshold frequencies which correspond to minimum values of threshold
pressure are found to agree with the musically defined frequencies of the first five
natural notes to within 5%. That this level of agreement is found even for the first
natural note is perhaps surprising, given that the conventional explanation for the
existence of the pedal note in terms of nonlinear coupling seems at odds with the
linearisation of the model equations required in the LSA treatment. The linearisation
does not remove all the off-diagonal terms in the Jacobian matrix, which means
that the behaviour of the model at the destabilisation threshold of one mode is still
subject to some influence by the other modes. However a similar LSA calculation
for the case in which all the higher modes are suppressed, leaving only the first
acoustic resonance of the trombone at 38 Hz, predicted essentially the same playing
behaviour for the first regime, with an optimum threshold frequency of 61.1 Hz.
Nonlinear coupling to higher modes is not therefore essential to the pianissimo
playing of the pedal note at 58 Hz. The first acoustic mode, at 38 Hz, is too far below
to exert a significant influence on the sounding pitch, but it has the correct phase
relationship to gain energy from an outward-striking valve with a lip resonance at
50 Hz in a feedback loop (see Sect. 5.2.1). The threshold oscillation frequency at
56 Hz is 12% above the lip resonance frequency, which is broadly comparable to the
outward-striking valve behaviour when playing the higher register notes. Raising the
lip frequency by 2 Hz is enough to ‘lip’ the pitch up to the desired pitch of B 1.
5 Blow That Horn: An Elementary Model of Brass Playing
Fig. 5.19 Detail of Fig. 5.18 showing the variation of threshold pressure (upper graph) and
threshold oscillation frequency (lower graph) as a function of lip resonance frequency for the first
regime of a tenor trombone
is found at a lip frequency f
opt
l
= 50 Hz, corresponding to a threshold oscillation
frequency f
opt
thr = 56.3 Hz. This LSA prediction is very close to the frequency of
58 Hz at which the pedal note is sounded in musical performance, although it is
48% higher than the first acoustic resonance frequency f ac,1 = 38 Hz.
Comparison of the predicted optimum threshold frequencies with playing experience shows that, despite the simplifications accepted in framing the elementary
model, LSA is able to give a good account of the pitches of the natural notes of the
trombone. Threshold frequencies which correspond to minimum values of threshold
pressure are found to agree with the musically defined frequencies of the first five
natural notes to within 5%. That this level of agreement is found even for the first
natural note is perhaps surprising, given that the conventional explanation for the
existence of the pedal note in terms of nonlinear coupling seems at odds with the
linearisation of the model equations required in the LSA treatment. The linearisation
does not remove all the off-diagonal terms in the Jacobian matrix, which means
that the behaviour of the model at the destabilisation threshold of one mode is still
subject to some influence by the other modes. However a similar LSA calculation
for the case in which all the higher modes are suppressed, leaving only the first
acoustic resonance of the trombone at 38 Hz, predicted essentially the same playing
behaviour for the first regime, with an optimum threshold frequency of 61.1 Hz.
Nonlinear coupling to higher modes is not therefore essential to the pianissimo
playing of the pedal note at 58 Hz. The first acoustic mode, at 38 Hz, is too far below
to exert a significant influence on the sounding pitch, but it has the correct phase
relationship to gain energy from an outward-striking valve with a lip resonance at
50 Hz in a feedback loop (see Sect. 5.2.1). The threshold oscillation frequency at
56 Hz is 12% above the lip resonance frequency, which is broadly comparable to the
outward-striking valve behaviour when playing the higher register notes. Raising the
lip frequency by 2 Hz is enough to ‘lip’ the pitch up to the desired pitch of B 1.