4.3 Bore Profiles of Brass Instruments
157
the nominal harmonic series becomes more serious. In seventh position the nominal
pitch of the natural note series is lowered by six semitones to E1. The EFP values
in Fig. 4.47 for the trombone with the slide in seventh position show that the modes
from n = 3 to n = 12 are still very close to true harmonics of E1, but the second
mode frequency (75 Hz) is 163 cents below a true second harmonic of E1. In practice
the player can sound this note in tune by choosing the correct lip vibration frequency
of 82.4 Hz; the mechanism by which energy is supplied to sustain the air column
vibration is then primarily nonlinear coupling to the even-numbered higher modes.
The first mode is almost an octave below the nominal pitch when the trombone slide
is in seventh position; this has no musical significance, however, since the first mode
plays an insignificant role in the regime of oscillation which sustains the pedal note
(see Sect. 5.4.4).
4.3.10 Instruments with Predominantly Expanding Bore
Profiles
The trombone is not the only instrument in which the first acoustic mode frequency
is substantially lower than the nominal fundamental frequency: this behaviour is
also characteristic of trumpets and other instruments in which a large proportion
of the bore is approximately cylindrical. The impedance curves of instruments in
which most of the tubing is steadily expanding do not show this dramatic deviation
from harmonicity in the first peak. Instruments of the saxhorn family (see Chap. 7)
fall into this category, although a substantial amount of approximately cylindrical
tubing is inserted into a saxhorn when several valves are operated. The serpent
(Sect. 7.8.2) and the ophicleide (Sect. 7.8.3) are instruments with mainly conical
bore profiles, shown in Fig. 7.61. The lower resonances of such instruments can be
close to exact harmonics of the nominal pitch without the necessity for a rapidly
flaring bell (see Sect. 4.3.3). The EFP plots for the serpent in Fig. 7.63 demonstrate
that when all the toneholes are closed, the EFP for the first acoustic resonance
is almost exactly zero, meaning that the resonance frequency is very close to the
nominal fundamental frequency. The EFP plots for the ophicleide in Fig. 7.65 show
that with all holes closed, the first resonance frequency is in fact slightly above the
nominal fundamental frequency. The EFP plots for the bass tuba in Fig. 7.66 show
that the inharmonicity of the first acoustic resonance depends on the operation of
the valves, which introduce approximately cylindrical tubing.
The effect of changes in the bore profile on the pitches of the lower natural
notes can be observed in the comparison of three instruments of similar length
illustrated in Fig. 4.48; Fig. 4.49 shows their bore profiles and Fig. 4.50 their EFP
curves. The bore of the Russian bugle expands approximately exponentially and is
close to cylindrical for the first 150 mm: its resonances from the second upwards are
acceptably harmonic for an instrument pitched about 50 cents above A4 = 440 Hz.
The cavalry trumpet is cylindrical for half its length and has a flared bell: although
157
the nominal harmonic series becomes more serious. In seventh position the nominal
pitch of the natural note series is lowered by six semitones to E1. The EFP values
in Fig. 4.47 for the trombone with the slide in seventh position show that the modes
from n = 3 to n = 12 are still very close to true harmonics of E1, but the second
mode frequency (75 Hz) is 163 cents below a true second harmonic of E1. In practice
the player can sound this note in tune by choosing the correct lip vibration frequency
of 82.4 Hz; the mechanism by which energy is supplied to sustain the air column
vibration is then primarily nonlinear coupling to the even-numbered higher modes.
The first mode is almost an octave below the nominal pitch when the trombone slide
is in seventh position; this has no musical significance, however, since the first mode
plays an insignificant role in the regime of oscillation which sustains the pedal note
(see Sect. 5.4.4).
4.3.10 Instruments with Predominantly Expanding Bore
Profiles
The trombone is not the only instrument in which the first acoustic mode frequency
is substantially lower than the nominal fundamental frequency: this behaviour is
also characteristic of trumpets and other instruments in which a large proportion
of the bore is approximately cylindrical. The impedance curves of instruments in
which most of the tubing is steadily expanding do not show this dramatic deviation
from harmonicity in the first peak. Instruments of the saxhorn family (see Chap. 7)
fall into this category, although a substantial amount of approximately cylindrical
tubing is inserted into a saxhorn when several valves are operated. The serpent
(Sect. 7.8.2) and the ophicleide (Sect. 7.8.3) are instruments with mainly conical
bore profiles, shown in Fig. 7.61. The lower resonances of such instruments can be
close to exact harmonics of the nominal pitch without the necessity for a rapidly
flaring bell (see Sect. 4.3.3). The EFP plots for the serpent in Fig. 7.63 demonstrate
that when all the toneholes are closed, the EFP for the first acoustic resonance
is almost exactly zero, meaning that the resonance frequency is very close to the
nominal fundamental frequency. The EFP plots for the ophicleide in Fig. 7.65 show
that with all holes closed, the first resonance frequency is in fact slightly above the
nominal fundamental frequency. The EFP plots for the bass tuba in Fig. 7.66 show
that the inharmonicity of the first acoustic resonance depends on the operation of
the valves, which introduce approximately cylindrical tubing.
The effect of changes in the bore profile on the pitches of the lower natural
notes can be observed in the comparison of three instruments of similar length
illustrated in Fig. 4.48; Fig. 4.49 shows their bore profiles and Fig. 4.50 their EFP
curves. The bore of the Russian bugle expands approximately exponentially and is
close to cylindrical for the first 150 mm: its resonances from the second upwards are
acceptably harmonic for an instrument pitched about 50 cents above A4 = 440 Hz.
The cavalry trumpet is cylindrical for half its length and has a flared bell: although