262
5 Blow That Horn: An Elementary Model of Brass Playing
p th1 , the branch turns to the left, and the solution becomes unstable. The branch then
displays the character of a branch just after an inverse Hopf bifurcation: periodic
oscillations are accessible from above for mouth pressures between p m = p th1 and
a lower limit p m = p subth which is higher than the subthreshold pressure in the case
= 0.
The lower graph in Fig. 5.22a shows that the oscillation frequency of the
fundamental regime (green curve) is no longer locked at the value f res1 , but is
attracted upwards towards the value f res2 /2. If the inharmonicity is negative, the
oscillation frequency is deviated downwards towards f res2 /2 < f res1 .
The effect of a further increase in inharmonicity on the threshold behaviour is
shown in Fig. 5.22b, which is the bifurcation diagram for = 0.04. The pitch
interval between the two resonances is now 68 cents greater than an octave. The
branch coming from this threshold p m = p th1 , corresponding to the fundamental
regime, has the appearance near threshold of a classic direct Hopf bifurcation (see
Fig. 5.12 in Sect. 5.4.1). The green curve representing the fundamental regime never
turns backwards, so p th1 remains the threshold of oscillation. The frequency of the
fundamental regime has the threshold value f th1 at the direct Hopf bifurcation point
and for higher values of p m is attracted upwards towards the value f res2 /2.
In Figs. 5.21b, 5.22a and b a branch corresponding to a direct Hopf bifurcation
beginning at p m = p th1 is indicated by a black line. This branch is the bifurcation
diagram for an air column having only one resonance at the frequency f res1 . It is
interesting to note that the air column with two quasi-harmonic resonances behaves
more and more like a one resonance air column as the inharmonicity increases,
although the inharmonic upper resonance continues to exert an influence on the
oscillation frequency.
The bifurcation diagrams shown above relate to the case Z 2 < Z 1 . Similar
diagrams are obtained for the case Z 2 > Z 1 (see Fig. 10 of Dalmont et al. (2000),
or bifurcation diagrams shown in Gilbert et al. (2020)).
Some general conclusions can be drawn from the bifurcation diagrams for
the two resonance quasi-harmonic model, which are relevant to discussions of
playability in brass instruments as well as reed woodwinds. If we accept that ease
of playing is determined by the minimum mouth pressure needed to sustain a
pianissimo sound, the instrument will be easiest to play when the resonances are
perfectly harmonic ( = 0). The inverse Hopf bifurcation then offers a playing
threshold p subth significantly below p th1 . This can be viewed as a theoretical support
for the Bouasse-Benade prescription. On the other hand, this very low threshold can
only be reached by first sounding the finite amplitude note at p th and then quickly
reducing the mouth pressure, requiring a high degree of expertise from the player.
If the criterion is the ability to start a steady note at an arbitrary low oscillation
amplitude, the direct Hopf bifurcation offered by the inharmonic = 0.04 case
might be preferred.
Questions of intonation must also be considered. For = 0 the oscillation
frequency is independent of the mouth pressure and therefore the oscillation
amplitude. The oscillation frequency is also largely independent of amplitude
for = 0.02, although this small degree of inharmonicity does result in an
5 Blow That Horn: An Elementary Model of Brass Playing
p th1 , the branch turns to the left, and the solution becomes unstable. The branch then
displays the character of a branch just after an inverse Hopf bifurcation: periodic
oscillations are accessible from above for mouth pressures between p m = p th1 and
a lower limit p m = p subth which is higher than the subthreshold pressure in the case
= 0.
The lower graph in Fig. 5.22a shows that the oscillation frequency of the
fundamental regime (green curve) is no longer locked at the value f res1 , but is
attracted upwards towards the value f res2 /2. If the inharmonicity is negative, the
oscillation frequency is deviated downwards towards f res2 /2 < f res1 .
The effect of a further increase in inharmonicity on the threshold behaviour is
shown in Fig. 5.22b, which is the bifurcation diagram for = 0.04. The pitch
interval between the two resonances is now 68 cents greater than an octave. The
branch coming from this threshold p m = p th1 , corresponding to the fundamental
regime, has the appearance near threshold of a classic direct Hopf bifurcation (see
Fig. 5.12 in Sect. 5.4.1). The green curve representing the fundamental regime never
turns backwards, so p th1 remains the threshold of oscillation. The frequency of the
fundamental regime has the threshold value f th1 at the direct Hopf bifurcation point
and for higher values of p m is attracted upwards towards the value f res2 /2.
In Figs. 5.21b, 5.22a and b a branch corresponding to a direct Hopf bifurcation
beginning at p m = p th1 is indicated by a black line. This branch is the bifurcation
diagram for an air column having only one resonance at the frequency f res1 . It is
interesting to note that the air column with two quasi-harmonic resonances behaves
more and more like a one resonance air column as the inharmonicity increases,
although the inharmonic upper resonance continues to exert an influence on the
oscillation frequency.
The bifurcation diagrams shown above relate to the case Z 2 < Z 1 . Similar
diagrams are obtained for the case Z 2 > Z 1 (see Fig. 10 of Dalmont et al. (2000),
or bifurcation diagrams shown in Gilbert et al. (2020)).
Some general conclusions can be drawn from the bifurcation diagrams for
the two resonance quasi-harmonic model, which are relevant to discussions of
playability in brass instruments as well as reed woodwinds. If we accept that ease
of playing is determined by the minimum mouth pressure needed to sustain a
pianissimo sound, the instrument will be easiest to play when the resonances are
perfectly harmonic ( = 0). The inverse Hopf bifurcation then offers a playing
threshold p subth significantly below p th1 . This can be viewed as a theoretical support
for the Bouasse-Benade prescription. On the other hand, this very low threshold can
only be reached by first sounding the finite amplitude note at p th and then quickly
reducing the mouth pressure, requiring a high degree of expertise from the player.
If the criterion is the ability to start a steady note at an arbitrary low oscillation
amplitude, the direct Hopf bifurcation offered by the inharmonic = 0.04 case
might be preferred.
Questions of intonation must also be considered. For = 0 the oscillation
frequency is independent of the mouth pressure and therefore the oscillation
amplitude. The oscillation frequency is also largely independent of amplitude
for = 0.02, although this small degree of inharmonicity does result in an