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5 Blow That Horn: An Elementary Model of Brass Playing
where W is the state vector describing the amplitudes of the four variables and λ is
the eigenvalue of the state. The eigenvalue is in general complex; if λ has a positive
real part, the state is unstable, since its amplitude grows exponentially with time.
Linear stability analysis tracks the evolution of the eigenvalues of possible states,
while a control variable (such as the quasi-static mouth pressure p m ) increases and
identifies the point at which a given state becomes unstable.
At sufficiently low mouth pressure, the stable state of a brass instrument
corresponds to the situation in which the lips are not oscillating. At the threshold
value of the mouth pressure, this state becomes unstable, and the system switches to
a new stable state in which the lips are oscillating. This change, which is described
in the language of nonlinear dynamics as a Hopf bifurcation, is of great musical
significance, since this marks the transition from silence to sound. Linear stability
analysis allows the calculation of the pitch of the oscillating state and the threshold
pressure required to reach it. Charting the variation of these two quantities as a
function of the lip frequency provides information about the intonation and ease of
playing of an instrument.
An example of the way in which LSA can be used to compare the behaviour of
different brass instruments is shown in Fig. 5.6, which plots the threshold pressure
and playing pitch as a function of lip frequency for the tenor and bass trombones
illustrated in Fig. 4.13. The analysis is based on the elementary model, and the linear
acoustical response of each instrument is represented as a set of acoustic modes
derived from the measured impedance curves shown in Fig. 4.14. Looking first at
the tenor trombone threshold frequency curve, shown in black in Fig. 5.6b, we see
a series of almost horizontal segments separated by vertical gaps. Each line slopes
upwards from a starting point just above one of the acoustic resonance frequencies.
This is a graphical representation of a musical experience familiar to trombonists
when playing the arpeggio of natural notes shown in Fig. 5.6c: starting with the
pedal note and gradually increasing the lip frequency, the pitch stabilises at a value
close to one of the natural notes of the instrument, bends gently upwards and then
jumps to the next higher natural note. The heights, lengths and slopes of each
segment of the curve in Fig. 5.6b contain information about the overall intonation of
the instrument.
The pressure threshold curve for the tenor trombone, shown in black in Fig. 5.6a,
contains further useful information about the playing properties of the instrument.
Corresponding to each separate segment of the threshold frequency curve is a Ushaped valley in the threshold pressure curve. The bottom of the U represents the
lowest threshold pressure at which the note can be sounded; the corresponding
frequency is optimal in the sense that it requires the minimum effort to sound it.
The value of the minimum in the pressure threshold curve increases as the natural
note number rises: for the second natural note, the minimum is 0.1 Pa, while for the
eighth natural note, it is 1.2 Pa. The LSA prediction that higher pitches require more
effort to sound is borne out by musical experience.
Turning to the LSA results for the bass trombone, represented by red curves
in Fig. 5.6, the same general features can be observed in the threshold frequency
and pressure curves as for the tenor trombone. Tenor and bass trombones both
5 Blow That Horn: An Elementary Model of Brass Playing
where W is the state vector describing the amplitudes of the four variables and λ is
the eigenvalue of the state. The eigenvalue is in general complex; if λ has a positive
real part, the state is unstable, since its amplitude grows exponentially with time.
Linear stability analysis tracks the evolution of the eigenvalues of possible states,
while a control variable (such as the quasi-static mouth pressure p m ) increases and
identifies the point at which a given state becomes unstable.
At sufficiently low mouth pressure, the stable state of a brass instrument
corresponds to the situation in which the lips are not oscillating. At the threshold
value of the mouth pressure, this state becomes unstable, and the system switches to
a new stable state in which the lips are oscillating. This change, which is described
in the language of nonlinear dynamics as a Hopf bifurcation, is of great musical
significance, since this marks the transition from silence to sound. Linear stability
analysis allows the calculation of the pitch of the oscillating state and the threshold
pressure required to reach it. Charting the variation of these two quantities as a
function of the lip frequency provides information about the intonation and ease of
playing of an instrument.
An example of the way in which LSA can be used to compare the behaviour of
different brass instruments is shown in Fig. 5.6, which plots the threshold pressure
and playing pitch as a function of lip frequency for the tenor and bass trombones
illustrated in Fig. 4.13. The analysis is based on the elementary model, and the linear
acoustical response of each instrument is represented as a set of acoustic modes
derived from the measured impedance curves shown in Fig. 4.14. Looking first at
the tenor trombone threshold frequency curve, shown in black in Fig. 5.6b, we see
a series of almost horizontal segments separated by vertical gaps. Each line slopes
upwards from a starting point just above one of the acoustic resonance frequencies.
This is a graphical representation of a musical experience familiar to trombonists
when playing the arpeggio of natural notes shown in Fig. 5.6c: starting with the
pedal note and gradually increasing the lip frequency, the pitch stabilises at a value
close to one of the natural notes of the instrument, bends gently upwards and then
jumps to the next higher natural note. The heights, lengths and slopes of each
segment of the curve in Fig. 5.6b contain information about the overall intonation of
the instrument.
The pressure threshold curve for the tenor trombone, shown in black in Fig. 5.6a,
contains further useful information about the playing properties of the instrument.
Corresponding to each separate segment of the threshold frequency curve is a Ushaped valley in the threshold pressure curve. The bottom of the U represents the
lowest threshold pressure at which the note can be sounded; the corresponding
frequency is optimal in the sense that it requires the minimum effort to sound it.
The value of the minimum in the pressure threshold curve increases as the natural
note number rises: for the second natural note, the minimum is 0.1 Pa, while for the
eighth natural note, it is 1.2 Pa. The LSA prediction that higher pitches require more
effort to sound is borne out by musical experience.
Turning to the LSA results for the bass trombone, represented by red curves
in Fig. 5.6, the same general features can be observed in the threshold frequency
and pressure curves as for the tenor trombone. Tenor and bass trombones both