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6 Shocks and Surprises: Refining the Elementary Model
will have undergone the same degree of nonlinear distortion when it arrives at the
right end.
The length of a cylindrical tube of diameter D min which gives the same degree
of nonlinear distortion as a flaring tube with the same minimum diameter, sounding
length L and bore profile D(x) is
L NDeq =
L
0
D min
D(x)
dx.
(6.6)
For brass instruments that flare outwards (as virtually all do), it is clear that the
length of the equivalent cylinder will be less than the geometrical length of the
instrument.
The brassiness potential parameter B is defined as the ratio L NDeq to L ec , the
equivalent cone length of the instrument (Sect. 4.3.4). That is,
B =
1
L ec
L
0
D min
D(x)
dx.
(6.7)
The brassiness potential parameter B is thus a dimensionless number that for
normal brass instrument contours always lies between zero and unity. It is higher for
‘cylindrical’ instruments like the trumpet and trombone and lower for ‘expanding’
instruments like the flugelhorn and euphonium. For the same input signal, larger
values of B mean greater nonlinear distortion and therefore a greater tendency for
the timbre to acquire a brassy edge at louder dynamics.
For a perfect cylinder, D(x) = D min and B = L/L ec . Because of the open end
correction, the value of B for a cylinder is slightly less than 1. Brassiness potentials
for most common brass instruments fall within the range 0.25–0.85.
The brassiness potential parameter defined by Eq. 6.7 can be used as a taxonomic
tool for the classification of brass instruments. Scatter plots of brassiness parameter
against minimum bore diameter, such as those shown in Fig. 6.11a, b, clearly
separate brass instruments into identifiable subgroups. Once the areas which are
occupied by instruments of recognised design within such a scatter plot are
established, instruments of problematic identity can be measured and B values
computed. Their positions in relation to other instruments should then give an
indication of their timbral properties.
Equations 6.2 and 6.3 show that the shock length is inversely proportional to the
frequency of the played note. The normalising factor L ec is included in Eq. 6.7 to
facilitate the direct comparison of B values for instruments with different sounding
lengths (and therefore playing ranges). A modern B trumpet is half the sounding
length of a B trombone, but the frequency of the fourth natural note of the trumpet is
twice that of the trombone; if the two instruments have comparable bore shapes, they
will have similar B values, and their equivalent natural notes will have comparable
brassiness.
6 Shocks and Surprises: Refining the Elementary Model
will have undergone the same degree of nonlinear distortion when it arrives at the
right end.
The length of a cylindrical tube of diameter D min which gives the same degree
of nonlinear distortion as a flaring tube with the same minimum diameter, sounding
length L and bore profile D(x) is
L NDeq =
L
0
D min
D(x)
dx.
(6.6)
For brass instruments that flare outwards (as virtually all do), it is clear that the
length of the equivalent cylinder will be less than the geometrical length of the
instrument.
The brassiness potential parameter B is defined as the ratio L NDeq to L ec , the
equivalent cone length of the instrument (Sect. 4.3.4). That is,
B =
1
L ec
L
0
D min
D(x)
dx.
(6.7)
The brassiness potential parameter B is thus a dimensionless number that for
normal brass instrument contours always lies between zero and unity. It is higher for
‘cylindrical’ instruments like the trumpet and trombone and lower for ‘expanding’
instruments like the flugelhorn and euphonium. For the same input signal, larger
values of B mean greater nonlinear distortion and therefore a greater tendency for
the timbre to acquire a brassy edge at louder dynamics.
For a perfect cylinder, D(x) = D min and B = L/L ec . Because of the open end
correction, the value of B for a cylinder is slightly less than 1. Brassiness potentials
for most common brass instruments fall within the range 0.25–0.85.
The brassiness potential parameter defined by Eq. 6.7 can be used as a taxonomic
tool for the classification of brass instruments. Scatter plots of brassiness parameter
against minimum bore diameter, such as those shown in Fig. 6.11a, b, clearly
separate brass instruments into identifiable subgroups. Once the areas which are
occupied by instruments of recognised design within such a scatter plot are
established, instruments of problematic identity can be measured and B values
computed. Their positions in relation to other instruments should then give an
indication of their timbral properties.
Equations 6.2 and 6.3 show that the shock length is inversely proportional to the
frequency of the played note. The normalising factor L ec is included in Eq. 6.7 to
facilitate the direct comparison of B values for instruments with different sounding
lengths (and therefore playing ranges). A modern B trumpet is half the sounding
length of a B trombone, but the frequency of the fourth natural note of the trumpet is
twice that of the trombone; if the two instruments have comparable bore shapes, they
will have similar B values, and their equivalent natural notes will have comparable
brassiness.