5.4 Going Further: From Linear Stability Analysis to Oscillation Regimes
255
resonance frequency f
opt
thr,n are the optimal values for a human performer playing
the nth natural note. The musical ‘centre’ of the note (see Sect. 1.2.2) can then
be identified as the pitch whose frequency is f
opt
thr,n . The experience of musicians
broadly supports this hypothesis, although the pitch centres marked by circles in
Fig. 5.18 are a little sharper than the corresponding reference pitches of the equally
tempered scale. These pitch discrepancies, which are related to the restriction of the
lip valve to purely outward-striking behaviour (Velut et al. 2017a), are less than 50
cents for registers 3–8.
It is noticeable, particularly in regimes n = 2 − 5, that f
opt
l,n is close to the
upper frequency limit of the regime and that the pressure threshold p thr
l,n increases
faster when f l,n is above f
opt
l,n than when it is below. These results are compatible
with the brasswind playing experience that it requires less effort for a musician to
bend a note down than to bend it up it up. The downward limit of pitch bending is
lower in practice because of several factors not considered in the elementary model,
including windway resonances (Sect. 6.3) and multiple lip resonances (Sect. 6.4).
The ease with which the onset of pianissimo sounds can be controlled is an
important factor in the player’s judgement of the playability of a brass instrument.
Many wind instrumentalists describe the sensation of overcoming resistance when
starting a note, and it was suggested in Sect. 1.2.9 that this ‘sounding resistance’
might be related to the threshold pressure p thr of the note in question. If this
suggestion is adopted, the upper curve in Fig. 5.18 could be viewed as a plot of
sounding resistance, the value of p
opt
thr,n corresponding to the resistance experienced
when the nth natural note is sounded.
5.4.4 The Trombone Pedal Note Regime
We return now to consider the first regime of the trombone. The notes played in this
regime are called pedal notes; with the slide in first position, the pedal note is B 1,
which is the nominal pitch of the instrument. Pedal notes are frequently employed
in trombone playing and appear in many orchestral scores (see Sect. 7.8.5). A
remarkable feature of the pedal B 1 note is that its sounding frequency is far
from any acoustic resonance frequency. The first resonance of the trombone is at
a frequency f ac,1 = 38 Hz, while the frequency of B 1 is 58 Hz. The reason for this
discrepancy is explained in Sect. 4.3.9. The fact that the pedal note can be sounded at
the desired pitch without the support of a nearby acoustic resonance was commented
on by Bouasse (1929) and has been explained as the consequence of nonlinear
coupling of higher acoustic modes whose frequencies are close to harmonics of
58 Hz (Benade 1973).
The LSA study of the trombone reported by Velut et al. (2017a) offers a fresh
perspective on the nature of the pedal note. In the first regime, illustrated in Fig. 5.19,
oscillating states exist with threshold frequencies ranging from f thr = 47.1 Hz at
f l = 30 Hz to f thr = 65.4 Hz at f l = 62 Hz. The minimum threshold pressure p
opt
thr
255
resonance frequency f
opt
thr,n are the optimal values for a human performer playing
the nth natural note. The musical ‘centre’ of the note (see Sect. 1.2.2) can then
be identified as the pitch whose frequency is f
opt
thr,n . The experience of musicians
broadly supports this hypothesis, although the pitch centres marked by circles in
Fig. 5.18 are a little sharper than the corresponding reference pitches of the equally
tempered scale. These pitch discrepancies, which are related to the restriction of the
lip valve to purely outward-striking behaviour (Velut et al. 2017a), are less than 50
cents for registers 3–8.
It is noticeable, particularly in regimes n = 2 − 5, that f
opt
l,n is close to the
upper frequency limit of the regime and that the pressure threshold p thr
l,n increases
faster when f l,n is above f
opt
l,n than when it is below. These results are compatible
with the brasswind playing experience that it requires less effort for a musician to
bend a note down than to bend it up it up. The downward limit of pitch bending is
lower in practice because of several factors not considered in the elementary model,
including windway resonances (Sect. 6.3) and multiple lip resonances (Sect. 6.4).
The ease with which the onset of pianissimo sounds can be controlled is an
important factor in the player’s judgement of the playability of a brass instrument.
Many wind instrumentalists describe the sensation of overcoming resistance when
starting a note, and it was suggested in Sect. 1.2.9 that this ‘sounding resistance’
might be related to the threshold pressure p thr of the note in question. If this
suggestion is adopted, the upper curve in Fig. 5.18 could be viewed as a plot of
sounding resistance, the value of p
opt
thr,n corresponding to the resistance experienced
when the nth natural note is sounded.
5.4.4 The Trombone Pedal Note Regime
We return now to consider the first regime of the trombone. The notes played in this
regime are called pedal notes; with the slide in first position, the pedal note is B 1,
which is the nominal pitch of the instrument. Pedal notes are frequently employed
in trombone playing and appear in many orchestral scores (see Sect. 7.8.5). A
remarkable feature of the pedal B 1 note is that its sounding frequency is far
from any acoustic resonance frequency. The first resonance of the trombone is at
a frequency f ac,1 = 38 Hz, while the frequency of B 1 is 58 Hz. The reason for this
discrepancy is explained in Sect. 4.3.9. The fact that the pedal note can be sounded at
the desired pitch without the support of a nearby acoustic resonance was commented
on by Bouasse (1929) and has been explained as the consequence of nonlinear
coupling of higher acoustic modes whose frequencies are close to harmonics of
58 Hz (Benade 1973).
The LSA study of the trombone reported by Velut et al. (2017a) offers a fresh
perspective on the nature of the pedal note. In the first regime, illustrated in Fig. 5.19,
oscillating states exist with threshold frequencies ranging from f thr = 47.1 Hz at
f l = 30 Hz to f thr = 65.4 Hz at f l = 62 Hz. The minimum threshold pressure p
opt
thr