4.8 Going Further: The Wogram Sum Function
215
designed to ensure that there is no conflict between the tuning messages sent to
the lips by the different acoustic resonances.
In reality the acoustic resonances are never exactly harmonic, and mixed
messages can arrive at the lips. If, for example, there is a strong impedance peak
at a little more than twice the frequency of the fundamental of the played note,
this could be expected to pull the repetition frequency upwards. Such a situation
does indeed arise in the case of the pedal note, whose frequency is close to half the
frequency of the second impedance peak but much higher than the frequency of the
first peak (see Sect. 5.4.4).
The collaboration between several acoustic resonances and the mechanical
resonance of the lips cannot be properly discussed without taking account of the
nonlinear nature of the coupled system, which is the topic of Chap. 5. It is possible,
however, to develop a function based on the input impedance Z(f ) of a brass
instrument which attempts to capture the potential influence on a note played at
frequency f due to the impedance at harmonic multiples of f . This was first
proposed in the Ph.D. thesis of Klaus Wogram (1972). The Wogram sum function
is defined as
SF (f ) =
1
n
n
Re[Z(nf )].
(4.134)
The idea behind the sum function is that the real part of the input impedance is
related to the transfer of energy to the air column; the sum is designed to represent
the energy transfer to each of the standing waves present in the instrument at a given
playing frequency.
Use of the sum function has been described by various authors. Elliott and
Bowsher (1982) found that peaks in the sum function for a trombone gave a closer
estimate of the actual playing frequencies than did the peaks in the raw input
impedance curve. Caussé et al. (2013) confirmed that the sum function predicts
correctly the playing frequency of the pedal note of the trumpet (see Fig. 4.95).
They found, however, that the strict application of the sum function as defined in
Eq. 4.134 resulted in pitches which were higher than measured playing frequencies.
Benade (1976) noted that the influence of higher impedance peaks was likely to
increase with the amplitude of the played note. This can be taken into account by a
weighted sum function
SF W (f ) =
1
n
n
a(n)Re[Z(nf )],
(4.135)
with weighting factors a(n) which depend on the playing dynamic. Figure 4.96
Illustrates a practical application to the investigation of level-dependent instability in
the note E3 on a serpent in C. This note is played with the two lowest toneholes open,
and serpents with this fingering tend to have very irregular input impedance curves
(see Sect. 7.8.2). Figure 4.96a shows the detail of the impedance curve (in blue), and
the sum function weighted heavily towards the first harmonic (in red). This could
215
designed to ensure that there is no conflict between the tuning messages sent to
the lips by the different acoustic resonances.
In reality the acoustic resonances are never exactly harmonic, and mixed
messages can arrive at the lips. If, for example, there is a strong impedance peak
at a little more than twice the frequency of the fundamental of the played note,
this could be expected to pull the repetition frequency upwards. Such a situation
does indeed arise in the case of the pedal note, whose frequency is close to half the
frequency of the second impedance peak but much higher than the frequency of the
first peak (see Sect. 5.4.4).
The collaboration between several acoustic resonances and the mechanical
resonance of the lips cannot be properly discussed without taking account of the
nonlinear nature of the coupled system, which is the topic of Chap. 5. It is possible,
however, to develop a function based on the input impedance Z(f ) of a brass
instrument which attempts to capture the potential influence on a note played at
frequency f due to the impedance at harmonic multiples of f . This was first
proposed in the Ph.D. thesis of Klaus Wogram (1972). The Wogram sum function
is defined as
SF (f ) =
1
n
n
Re[Z(nf )].
(4.134)
The idea behind the sum function is that the real part of the input impedance is
related to the transfer of energy to the air column; the sum is designed to represent
the energy transfer to each of the standing waves present in the instrument at a given
playing frequency.
Use of the sum function has been described by various authors. Elliott and
Bowsher (1982) found that peaks in the sum function for a trombone gave a closer
estimate of the actual playing frequencies than did the peaks in the raw input
impedance curve. Caussé et al. (2013) confirmed that the sum function predicts
correctly the playing frequency of the pedal note of the trumpet (see Fig. 4.95).
They found, however, that the strict application of the sum function as defined in
Eq. 4.134 resulted in pitches which were higher than measured playing frequencies.
Benade (1976) noted that the influence of higher impedance peaks was likely to
increase with the amplitude of the played note. This can be taken into account by a
weighted sum function
SF W (f ) =
1
n
n
a(n)Re[Z(nf )],
(4.135)
with weighting factors a(n) which depend on the playing dynamic. Figure 4.96
Illustrates a practical application to the investigation of level-dependent instability in
the note E3 on a serpent in C. This note is played with the two lowest toneholes open,
and serpents with this fingering tend to have very irregular input impedance curves
(see Sect. 7.8.2). Figure 4.96a shows the detail of the impedance curve (in blue), and
the sum function weighted heavily towards the first harmonic (in red). This could