284
6 Shocks and Surprises: Refining the Elementary Model
losses are greater, and since they predominantly reduce the amplitudes of the highfrequency components, the timbral effect is a reduction in brightness (Chick et al.
2012).
It is not possible at this stage to quantify exactly the trade-off between B and
absolute bore size (Myers et al. 2012), but some attempts have been made to identify
a more general spectral enrichment parameter which is a function of both brassiness
potential and bore diameter (Campbell 2014b; Campbell et al. 2020)
It is possible to illustrate these brassiness properties by simulation. For example,
a finite element time domain simulation method developed by Sylvain Maugeais
has been used to compare a crescendo followed by a diminuendo on each of two
tenor brass cousins, a trombone and a Wagner tuba (Maugeais and Gilbert 2017). It
is evident from the spectrograms displayed in Fig. 6.12 that the spectral enrichment
during the crescendo is much more important in the trombone simulation than in the
Wagner tuba. Characteristic brassy effects are typical of bright brass instruments like
trombones or trumpets but are less noticeable in the mellow brass instruments like
tubas or flugelhorns.
In each of the cases shown in Fig. 6.12, the maximum pressure amplitude
simulated in the mouthpiece was 8.5 kPa, corresponding to an rms value of 6 kPa.
This is a typical level for very loud playing. The maximum pressures in the
radiated sounds are however very different: 1993 Pa for the trombone but only
243 Pa for the Wagner tuba. The reason for this gross disparity is evident from
Fig. 6.13, which shows the detail of the waveforms of the radiated sounds at the
peak of the crescendo. The Wagner tuba waveform, which started as a sine wave
at the mouthpiece, has clearly been significantly distorted, but the distortion in
the trombone is much more severe, resulting in extremely short pulses with an
amplitude of almost 2 kPa in the radiated sound.
In time domain simulations of the type described in the previous paragraph, the
input pressure signal is prescribed, and the instrument is driven in forced oscillation.
It is assumed that the nonlinear distortion which occurs in the air column of the
instrument does not influence the input signal. Msallam et al. (2002) simulated selfsustained oscillations in a trombone, but the nonlinear propagation was taken into
account only in the cylindrical section of the instrument. A simulation of vibrating
lips coupled to a cylindrical tube by Berjamin et al. (2017) confirmed the importance
of nonlinear propagation in modifying the timbre of the output signal but also noted
significant influence on mouthpiece pressure, playing frequency and time envelope.
These results point to the importance of developing simulations of the playing of
realistic brass instruments which incorporate nonlinear propagation throughout the
bore, including the flaring bell section.
6 Shocks and Surprises: Refining the Elementary Model
losses are greater, and since they predominantly reduce the amplitudes of the highfrequency components, the timbral effect is a reduction in brightness (Chick et al.
2012).
It is not possible at this stage to quantify exactly the trade-off between B and
absolute bore size (Myers et al. 2012), but some attempts have been made to identify
a more general spectral enrichment parameter which is a function of both brassiness
potential and bore diameter (Campbell 2014b; Campbell et al. 2020)
It is possible to illustrate these brassiness properties by simulation. For example,
a finite element time domain simulation method developed by Sylvain Maugeais
has been used to compare a crescendo followed by a diminuendo on each of two
tenor brass cousins, a trombone and a Wagner tuba (Maugeais and Gilbert 2017). It
is evident from the spectrograms displayed in Fig. 6.12 that the spectral enrichment
during the crescendo is much more important in the trombone simulation than in the
Wagner tuba. Characteristic brassy effects are typical of bright brass instruments like
trombones or trumpets but are less noticeable in the mellow brass instruments like
tubas or flugelhorns.
In each of the cases shown in Fig. 6.12, the maximum pressure amplitude
simulated in the mouthpiece was 8.5 kPa, corresponding to an rms value of 6 kPa.
This is a typical level for very loud playing. The maximum pressures in the
radiated sounds are however very different: 1993 Pa for the trombone but only
243 Pa for the Wagner tuba. The reason for this gross disparity is evident from
Fig. 6.13, which shows the detail of the waveforms of the radiated sounds at the
peak of the crescendo. The Wagner tuba waveform, which started as a sine wave
at the mouthpiece, has clearly been significantly distorted, but the distortion in
the trombone is much more severe, resulting in extremely short pulses with an
amplitude of almost 2 kPa in the radiated sound.
In time domain simulations of the type described in the previous paragraph, the
input pressure signal is prescribed, and the instrument is driven in forced oscillation.
It is assumed that the nonlinear distortion which occurs in the air column of the
instrument does not influence the input signal. Msallam et al. (2002) simulated selfsustained oscillations in a trombone, but the nonlinear propagation was taken into
account only in the cylindrical section of the instrument. A simulation of vibrating
lips coupled to a cylindrical tube by Berjamin et al. (2017) confirmed the importance
of nonlinear propagation in modifying the timbre of the output signal but also noted
significant influence on mouthpiece pressure, playing frequency and time envelope.
These results point to the importance of developing simulations of the playing of
realistic brass instruments which incorporate nonlinear propagation throughout the
bore, including the flaring bell section.