70
3 Buzzing Lips: Sound Generation in Brass Instruments
lip opening image; the height of the equivalent rectangle is described as the mean
height. The variation of maximum and mean heights with time is approximately
sinusoidal, as shown by the green and black lines, respectively, in Fig. 3.7.
The curves shown in Fig. 3.7 were obtained in experiments on a single experienced trombone player. The general features of the dependence of lip opening on
time seen in these results have been observed in studies of many different players.
It should be borne in mind, however, that every player develops an individual
embouchure, and this diversity is reflected in the detailed shapes of the open area
curves. A skilled player can also deliberately modify the nature of the lip vibration
to obtain a specific timbre or technical effect (Norman et al. 2010).
3.1.4 The Lip Opening Area-Height Function
In computer simulations of brass instrument playing, the lips are frequently
represented as an oscillating system with a single degree of freedom, representing
the mean height h(t) of the lip opening. In the co-ordinate framework defined in
Fig 3.5, h(t) is the height of the equivalent rectangle representing the open area
of the lips and is measured along the y direction in the xy plane. This approach
does not assume that the lips move only in the xy plane. The ‘swinging door’ lip
model described in Sect. 3.2.3 implies a substantial component of lip motion in the
z direction, and the experiments described in Sect. 3.1.5 confirm that the tip of a
trombone player’s lip can move almost 10 mm into the mouthpiece when playing
a fortissimo pedal note. It is assumed however that the volume flow of air into the
mouthpiece is primarily determined by the area S(t) of the lip opening projected on
to the xy plane, which is perpendicular to the flow direction.
Is the area S(t) proportional to the mean height h(t)? Such a linear relationship
will only hold if the width w of the equivalent rectangle is constant throughout
the cycle of vibration. For single-reed woodwind instruments like the clarinet and
saxophone, this is a reasonable approximation, since the reed opens and closes
uniformly across its width. In this case the open area is in fact a rectangle whose
constant width is equal to the width of the slotted opening in the mouthpiece
under the reed tip. The nature of the area variation in this case is illustrated by
the graphical construction in Fig. 3.8a. The lower rectangle is fixed, while the upper
dotted rectangle oscillates vertically. The overlapping shaded region represents the
opening, whose change in area is directly proportional to the vertical displacement.
Measurements such as those shown in Fig. 3.7 demonstrate that the assumption
of constant width is not in general valid for the lips of a brass player. The open area
is given by the product of two time-varying quantities:
S(t) = w(t)h(t).
The time-varying width of the lip opening can be incorporated into a single degree of
freedom model if an additional assumption is made about the relationship between
3 Buzzing Lips: Sound Generation in Brass Instruments
lip opening image; the height of the equivalent rectangle is described as the mean
height. The variation of maximum and mean heights with time is approximately
sinusoidal, as shown by the green and black lines, respectively, in Fig. 3.7.
The curves shown in Fig. 3.7 were obtained in experiments on a single experienced trombone player. The general features of the dependence of lip opening on
time seen in these results have been observed in studies of many different players.
It should be borne in mind, however, that every player develops an individual
embouchure, and this diversity is reflected in the detailed shapes of the open area
curves. A skilled player can also deliberately modify the nature of the lip vibration
to obtain a specific timbre or technical effect (Norman et al. 2010).
3.1.4 The Lip Opening Area-Height Function
In computer simulations of brass instrument playing, the lips are frequently
represented as an oscillating system with a single degree of freedom, representing
the mean height h(t) of the lip opening. In the co-ordinate framework defined in
Fig 3.5, h(t) is the height of the equivalent rectangle representing the open area
of the lips and is measured along the y direction in the xy plane. This approach
does not assume that the lips move only in the xy plane. The ‘swinging door’ lip
model described in Sect. 3.2.3 implies a substantial component of lip motion in the
z direction, and the experiments described in Sect. 3.1.5 confirm that the tip of a
trombone player’s lip can move almost 10 mm into the mouthpiece when playing
a fortissimo pedal note. It is assumed however that the volume flow of air into the
mouthpiece is primarily determined by the area S(t) of the lip opening projected on
to the xy plane, which is perpendicular to the flow direction.
Is the area S(t) proportional to the mean height h(t)? Such a linear relationship
will only hold if the width w of the equivalent rectangle is constant throughout
the cycle of vibration. For single-reed woodwind instruments like the clarinet and
saxophone, this is a reasonable approximation, since the reed opens and closes
uniformly across its width. In this case the open area is in fact a rectangle whose
constant width is equal to the width of the slotted opening in the mouthpiece
under the reed tip. The nature of the area variation in this case is illustrated by
the graphical construction in Fig. 3.8a. The lower rectangle is fixed, while the upper
dotted rectangle oscillates vertically. The overlapping shaded region represents the
opening, whose change in area is directly proportional to the vertical displacement.
Measurements such as those shown in Fig. 3.7 demonstrate that the assumption
of constant width is not in general valid for the lips of a brass player. The open area
is given by the product of two time-varying quantities:
S(t) = w(t)h(t).
The time-varying width of the lip opening can be incorporated into a single degree of
freedom model if an additional assumption is made about the relationship between