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5 Blow That Horn: An Elementary Model of Brass Playing
Fig. 5.21 Bifurcation diagrams corresponding to a model instrument with two exactly harmonic
impedance peaks at f res1 and f res2 = 2f res1 , and with Z 1 > Z 2 . All pressures are divided by the
minimum reed closing pressure p M . (a) Analytical calculation: acoustic pressure amplitude p rms
versus mouth pressure p m (Dalmont et al. 2000). (b) Numerical calculation using continuation
method. Green, standard regime; red, octave regime; blue, inverted Helmholtz regime (thick lines,
stable states; thin lines, unstable states). Upper graph, amplitude |p(t)| of oscillating pressure
versus mouth pressure p m . Lower graph: frequency of oscillation f osc (f osc /2 for octave branch).
Dashed horizontal line: reference frequency f res1 . For comparison, the black curve illustrates a
direct Hopf bifurcation branched at p m = p th1 , corresponding to an air column having only one
resonance at the frequency f res1 . Adapted from Gilbert et al. (2020) (Color figure online)