144
4 After the Lips: Acoustic Resonances and Radiation
mouthpiece is slapped are relaxed or fully extended, since this alters the extent to
which the palm surface protrudes into the mouthpiece.
In assessing the influence of the geometrical parameters and resonance frequency
of a mouthpiece on the intonation and timbre of the instrument of which it forms
part, there are two useful general rules:
1. At frequencies well below f R , a cup mouthpiece behaves like a cylindrical
continuation of the entrance tube with a length which gives it the same internal
volume as the actual mouthpiece (Bouasse 1929; Benade 1976):
L eff =
V int
πr 2 [f f R ],
(4.54)
where r is the radius at the tube entrance and V int is the internal volume of the
mouthpiece;
2. At the frequency f R , a cup mouthpiece behaves like a cylindrical tube whose
length is a quarter wavelength (Pyle 1975):
L eff =
c
4f R
[f = f R ].
(4.55)
Taking the Denis Wick trombone mouthpiece as an example, the internal volume
of the mouthpiece was measured by blocking the exit from the backbore and filling
the mouthpiece to the rim with water using a calibrated syringe. This took 12 ml
of water, so the internal volume was V = 12 × 10 3 mm 3 . The radius of the exit
is 6.9 mm, so L eff (f f R ) = 80 mm. This is just the external length of the
mouthpiece, so in this case, the actual length and the low-frequency effective length
coincide.
Assuming f R = 535 Hz, L eff (f = f R ) = 161 mm, suggesting that the effective
length of the mouthpiece increases by 81 mm as the frequency rises from a low value
to the resonance frequency. This is around 3% of the sounding length of a trombone,
implying that the pitch difference between the first and tenth impedance peaks is
about half a semitone lower than it would be if the mouthpiece were replaced by a
cylindrical tube of the same length.
It is important to remember that the model illustrated in Fig. 4.33 is a very crude
approximation to a realistic trombone mouthpiece with a tapering backbore, and
its acoustical behaviour when attached to an instrument will be more complicated
than the simple lumped impedance we assumed in deriving Eq. 4.53. Figure
4.37 shows the predicted effect of adding a mouthpiece to the cylindrical tube
discussed in Sect. 4.3.2. In this case the complete bore including the mouthpiece
was approximated by a series of cones and cylinders, and the wave equation was
solved by a method which took into account the fact that the wavefronts in the
mouthpiece are not plane (Braden 2006).
The blue circles in Fig. 4.37 show the calculated EFP values for a simple cylinder
of length 2834 mm, while the red squares show the EFP values when the first
261 mm of the cylinder are replaced by the Denis Wick 5AL mouthpiece and a
4 After the Lips: Acoustic Resonances and Radiation
mouthpiece is slapped are relaxed or fully extended, since this alters the extent to
which the palm surface protrudes into the mouthpiece.
In assessing the influence of the geometrical parameters and resonance frequency
of a mouthpiece on the intonation and timbre of the instrument of which it forms
part, there are two useful general rules:
1. At frequencies well below f R , a cup mouthpiece behaves like a cylindrical
continuation of the entrance tube with a length which gives it the same internal
volume as the actual mouthpiece (Bouasse 1929; Benade 1976):
L eff =
V int
πr 2 [f f R ],
(4.54)
where r is the radius at the tube entrance and V int is the internal volume of the
mouthpiece;
2. At the frequency f R , a cup mouthpiece behaves like a cylindrical tube whose
length is a quarter wavelength (Pyle 1975):
L eff =
c
4f R
[f = f R ].
(4.55)
Taking the Denis Wick trombone mouthpiece as an example, the internal volume
of the mouthpiece was measured by blocking the exit from the backbore and filling
the mouthpiece to the rim with water using a calibrated syringe. This took 12 ml
of water, so the internal volume was V = 12 × 10 3 mm 3 . The radius of the exit
is 6.9 mm, so L eff (f f R ) = 80 mm. This is just the external length of the
mouthpiece, so in this case, the actual length and the low-frequency effective length
coincide.
Assuming f R = 535 Hz, L eff (f = f R ) = 161 mm, suggesting that the effective
length of the mouthpiece increases by 81 mm as the frequency rises from a low value
to the resonance frequency. This is around 3% of the sounding length of a trombone,
implying that the pitch difference between the first and tenth impedance peaks is
about half a semitone lower than it would be if the mouthpiece were replaced by a
cylindrical tube of the same length.
It is important to remember that the model illustrated in Fig. 4.33 is a very crude
approximation to a realistic trombone mouthpiece with a tapering backbore, and
its acoustical behaviour when attached to an instrument will be more complicated
than the simple lumped impedance we assumed in deriving Eq. 4.53. Figure
4.37 shows the predicted effect of adding a mouthpiece to the cylindrical tube
discussed in Sect. 4.3.2. In this case the complete bore including the mouthpiece
was approximated by a series of cones and cylinders, and the wave equation was
solved by a method which took into account the fact that the wavefronts in the
mouthpiece are not plane (Braden 2006).
The blue circles in Fig. 4.37 show the calculated EFP values for a simple cylinder
of length 2834 mm, while the red squares show the EFP values when the first
261 mm of the cylinder are replaced by the Denis Wick 5AL mouthpiece and a