2.2 An Approach to Modelling Brass Instruments
53
members of an exact harmonic series, whose fundamental is the repetition frequency
f R . In other words, the signal shown in Fig. 2.17 is equivalent to a simultaneous set
of sine wave signals with frequencies f R , 2f R , 3f R , . . . .
The mathematical process required to derive the amplitude and phase of the
components of a known time signal is called Fourier analysis (Hartmann 2004). A
graph showing the component amplitudes as a function of frequency is described as
the Fourier spectrum (or simply the frequency spectrum) of the signal. Figure 2.18a
shows the frequency spectrum of the trumpet sound whose waveform is shown in
Fig. 2.17. In the frequency range illustrated, only the first four harmonics appear,
but the series continues well above 10 kHz.
A tuning meter will report that the trumpet note has a pitch F4 and a frequency
349 Hz. While it is true that this is the frequency of the first harmonic in the
spectrum, the second, third and fourth harmonics have amplitudes only slightly
smaller than the first. Why do we not also hear the pitches of these components?
The reason is that the perception of pitch by the human hearing system does not
rely on either the first harmonic or the strongest harmonic but involves a complex
and not yet fully understood set of signal processing stages which uses the complete
frequency spectrum. If a harmonically related series of frequencies is detected, the
pitch is associated with the fundamental of that series, even if the first harmonic is
weak or indeed absent (Campbell and Greated 1987).
We turn now to discussing the relationship between the harmonic frequency
components of the note and the resonances of the instrument on which it is
played. The acoustic pressure variation in the mouthpiece of a brass instrument
during playing was discussed in Sect. 2.1.2, and the corresponding fluctuation of
the air flow into the mouthpiece was illustrated in Fig. 2.9. The ratio of acoustic
pressure to acoustic volume flow in the mouthpiece is called the input impedance
of the instrument (see Sect. 4.1.6). An input impedance curve, in which the input
impedance Z(f ) is plotted as a function of frequency, is a useful way of illustrating
the acoustical response of an instrument. Each peak in the input impedance curve
corresponds to one of the acoustic resonances of the air column inside the tube.
The input impedance curve for a B trumpet is shown by the blue line in
Fig. 2.18b. Frequencies corresponding to exact harmonics of the nominal fundamental pitch B 2 are marked by vertical red lines. The acoustic resonances are evidently
quite close to the nominal harmonic series, although the correspondence is not exact.
The glaring exception is the first peak, which is far below the value f R = 116.5 Hz
of the first harmonic of B 2.
The green vertical lines in Fig. 2.18b mark the frequencies of the harmonics of
F4 which were observed in Fig. 2.18a. To sound this note, the player chooses a lip
setting which favours vibration close to the third acoustic resonance; the acoustic
feedback from this resonance strengthens and stabilises the vibration. Because of
the approximately harmonic nature of the acoustic resonances, the sixth resonance
is in the right place to support the second harmonic of F4; similarly, the ninth and
twelfth resonances lend support to the third and fourth harmonics, respectively.
The collaboration of several acoustic resonances in reinforcing the sounding of a
note was recognised by Bouasse (1929) and Benade (1968), and we describe the
53
members of an exact harmonic series, whose fundamental is the repetition frequency
f R . In other words, the signal shown in Fig. 2.17 is equivalent to a simultaneous set
of sine wave signals with frequencies f R , 2f R , 3f R , . . . .
The mathematical process required to derive the amplitude and phase of the
components of a known time signal is called Fourier analysis (Hartmann 2004). A
graph showing the component amplitudes as a function of frequency is described as
the Fourier spectrum (or simply the frequency spectrum) of the signal. Figure 2.18a
shows the frequency spectrum of the trumpet sound whose waveform is shown in
Fig. 2.17. In the frequency range illustrated, only the first four harmonics appear,
but the series continues well above 10 kHz.
A tuning meter will report that the trumpet note has a pitch F4 and a frequency
349 Hz. While it is true that this is the frequency of the first harmonic in the
spectrum, the second, third and fourth harmonics have amplitudes only slightly
smaller than the first. Why do we not also hear the pitches of these components?
The reason is that the perception of pitch by the human hearing system does not
rely on either the first harmonic or the strongest harmonic but involves a complex
and not yet fully understood set of signal processing stages which uses the complete
frequency spectrum. If a harmonically related series of frequencies is detected, the
pitch is associated with the fundamental of that series, even if the first harmonic is
weak or indeed absent (Campbell and Greated 1987).
We turn now to discussing the relationship between the harmonic frequency
components of the note and the resonances of the instrument on which it is
played. The acoustic pressure variation in the mouthpiece of a brass instrument
during playing was discussed in Sect. 2.1.2, and the corresponding fluctuation of
the air flow into the mouthpiece was illustrated in Fig. 2.9. The ratio of acoustic
pressure to acoustic volume flow in the mouthpiece is called the input impedance
of the instrument (see Sect. 4.1.6). An input impedance curve, in which the input
impedance Z(f ) is plotted as a function of frequency, is a useful way of illustrating
the acoustical response of an instrument. Each peak in the input impedance curve
corresponds to one of the acoustic resonances of the air column inside the tube.
The input impedance curve for a B trumpet is shown by the blue line in
Fig. 2.18b. Frequencies corresponding to exact harmonics of the nominal fundamental pitch B 2 are marked by vertical red lines. The acoustic resonances are evidently
quite close to the nominal harmonic series, although the correspondence is not exact.
The glaring exception is the first peak, which is far below the value f R = 116.5 Hz
of the first harmonic of B 2.
The green vertical lines in Fig. 2.18b mark the frequencies of the harmonics of
F4 which were observed in Fig. 2.18a. To sound this note, the player chooses a lip
setting which favours vibration close to the third acoustic resonance; the acoustic
feedback from this resonance strengthens and stabilises the vibration. Because of
the approximately harmonic nature of the acoustic resonances, the sixth resonance
is in the right place to support the second harmonic of F4; similarly, the ninth and
twelfth resonances lend support to the third and fourth harmonics, respectively.
The collaboration of several acoustic resonances in reinforcing the sounding of a
note was recognised by Bouasse (1929) and Benade (1968), and we describe the