2.2 An Approach to Modelling Brass Instruments
57
instrument resonance which attempts to centre the played pitch on the corresponding
natural note.
Towards the right-hand side of the spectrogram in Fig. 2.20, other descending
stepped traces appear. These represent the higher-frequency components which
Fourier analysis tells us are also present in the sound. At t = 12 s, for example,
the natural note B 3 is played, and the fundamental f R = 233 Hz is shown as the
lowest horizontal red line. Above this line there are several more horizontal red
lines, at frequencies which are integer multiples of 233 Hz. Since these frequencies
are Fourier components of a periodic signal, they are exact members of a harmonic
series whose first member is f R .
The lowest note in the glissando, which begins around t = 15 s, is the pedal note
B 1. It is striking that the fundamental f R = 58 Hz appears much weaker than any
of the 19 higher harmonics shown in the spectrogram. Reasons for this characteristic
of the pedal note are given in Sect. 5.4.4.
2.2.4 Self-Sustained Oscillations
Making use of Approximation No. 1 in Sect. 2.2.1, we can model the musician
playing a brass instrument as a system that transforms a constant excess pressure
in the player’s mouth into an oscillating pressure in the radiated sound. A system
that transforms a constant quantity into an oscillating one in this way is described as
a self-sustained oscillator or auto-oscillator. In brass instruments the vibrating lips
of the player are the agents which transform the constant mouth pressure into the
oscillating downstream pressure through the valve effect discussed fully in Chap. 3.
Self-sustained oscillators generate different types of oscillation regime. The most
important type from the musical viewpoint is the periodic regime, an example
of which was described in Sect. 2.1.1. For a given fingering or slide position on
a brass instrument, each natural note corresponds to a different periodic regime
of oscillation. Although the regimes of oscillation are the result of a complex
interaction between the vibrating lips of the musician and the air column in the
musical instrument, the oscillation frequencies which define the pitches of the
natural notes are largely determined by the geometrical property of the instrument
tube known as the bore profile. This is a curve showing how the internal radius (or
diameter) of the tube depends on the axial length from the mouthpiece to the plane
of the bell. For purposes of illustration, the tube is usually assumed to be unwrapped,
with the axis shown as a straight line. Figure 2.21 illustrates typical bore profiles for
a trombone and a euphonium.
Other types of regimes of oscillation exist, including quasi-periodic oscillations.
These are associated with musical sounds such as multiphonics and flutter tonguing,
as well as some of the less musical sounds which are involuntarily generated by
beginners.
We saw in Sect. 2.2.3 that it is possible for an experienced brass player to
sound notes with repetition frequencies well above those of the significant input
57
instrument resonance which attempts to centre the played pitch on the corresponding
natural note.
Towards the right-hand side of the spectrogram in Fig. 2.20, other descending
stepped traces appear. These represent the higher-frequency components which
Fourier analysis tells us are also present in the sound. At t = 12 s, for example,
the natural note B 3 is played, and the fundamental f R = 233 Hz is shown as the
lowest horizontal red line. Above this line there are several more horizontal red
lines, at frequencies which are integer multiples of 233 Hz. Since these frequencies
are Fourier components of a periodic signal, they are exact members of a harmonic
series whose first member is f R .
The lowest note in the glissando, which begins around t = 15 s, is the pedal note
B 1. It is striking that the fundamental f R = 58 Hz appears much weaker than any
of the 19 higher harmonics shown in the spectrogram. Reasons for this characteristic
of the pedal note are given in Sect. 5.4.4.
2.2.4 Self-Sustained Oscillations
Making use of Approximation No. 1 in Sect. 2.2.1, we can model the musician
playing a brass instrument as a system that transforms a constant excess pressure
in the player’s mouth into an oscillating pressure in the radiated sound. A system
that transforms a constant quantity into an oscillating one in this way is described as
a self-sustained oscillator or auto-oscillator. In brass instruments the vibrating lips
of the player are the agents which transform the constant mouth pressure into the
oscillating downstream pressure through the valve effect discussed fully in Chap. 3.
Self-sustained oscillators generate different types of oscillation regime. The most
important type from the musical viewpoint is the periodic regime, an example
of which was described in Sect. 2.1.1. For a given fingering or slide position on
a brass instrument, each natural note corresponds to a different periodic regime
of oscillation. Although the regimes of oscillation are the result of a complex
interaction between the vibrating lips of the musician and the air column in the
musical instrument, the oscillation frequencies which define the pitches of the
natural notes are largely determined by the geometrical property of the instrument
tube known as the bore profile. This is a curve showing how the internal radius (or
diameter) of the tube depends on the axial length from the mouthpiece to the plane
of the bell. For purposes of illustration, the tube is usually assumed to be unwrapped,
with the axis shown as a straight line. Figure 2.21 illustrates typical bore profiles for
a trombone and a euphonium.
Other types of regimes of oscillation exist, including quasi-periodic oscillations.
These are associated with musical sounds such as multiphonics and flutter tonguing,
as well as some of the less musical sounds which are involuntarily generated by
beginners.
We saw in Sect. 2.2.3 that it is possible for an experienced brass player to
sound notes with repetition frequencies well above those of the significant input