264
5 Blow That Horn: An Elementary Model of Brass Playing
The bifurcation diagrams in Figs. 5.21 and 5.22 show that continuation methods
are able to find branches of periodic solutions and also to detect bifurcation
points from them. For example, the red line in Fig. 5.22a represents an octave
regime with oscillation frequency equal to that of the higher of the two quasiharmonic acoustic resonances. The blue line shows a period doubling bifurcation
to an inverted Helmholtz motion state. When the inharmonicity between the two
resonances is sufficiently large, quasi-periodic regimes can be observed (see a study
of saxophones in Dalmont et al. (1995) and Doc et al. (2014)). Continuation methods
are in principle able to predict the occurrence of quasi-periodic regimes by detecting
a Neimark-Sacker bifurcation point localised on the branch of a periodic solution.
Nonlinear dynamical systems typically display a large variety of oscillation
behaviours (see, e.g. Nayfeh (1995) and Manneville (2010)). Several different
types of oscillation state which can occur in brass instruments are illustrated by
simulations in Vergez and Rodet (2001b)). The ‘route to chaos’ through a succession
of period doubling bifurcations (known as the Feigenbaum process) has been
simulated in reed instruments using elementary models (Maganza et al. 1986).
Measured non-periodic regimes in wind instruments have been extensively analysed
using the phase space representation (Gibiat 1988; Gibiat and Castellengo 2000).
Quasi-periodic regimes are of great interest to musicians, since they correspond to
the special sounds known as multiphonics. In Sect. 5.4.6 some of the multiphonics
available on the trombone are described and simulated.
5.4.6 Multiphonics
Brass instruments are sometimes described as ‘monodic’ or ‘monophonic’. Is it true,
as the etymology of these terms implies, that a brass instrument can only generate
one sound at a time? The answer to this question depends on the precise meaning
attached to the word ‘sound’. Every note played on a wind instrument has a frequency spectrum with many different components, but in standard performance, the
steady-state oscillation is periodic and the frequency components are harmonically
related. Such a note is normally perceived as a single sound, with a unique pitch
closely related to the fundamental frequency of the harmonic series (Sect. 2.2.3). In
this case the behaviour of the instrument can indeed be described as monophonic.
The fusion of frequency components which results in the perception of a single
pitch is ineffective if the components are not perceived as members of the same
harmonic series. Two or more pitches may then be heard. Although the result is
still strictly speaking one sound, the impression is of several separate sound sources
acting simultaneously, and such sounds are described as multiphonics.
Many wind instruments are capable of generating multiphonics (Castellengo
1981). Two different categories of multiphonic can be distinguished, based on
the acoustic phenomena involved in their production. One category involves the
generation of multiple simultaneous pitches through an extension of conventional
playing techniques. Woodwind multiphonics come into this category, since they rely
5 Blow That Horn: An Elementary Model of Brass Playing
The bifurcation diagrams in Figs. 5.21 and 5.22 show that continuation methods
are able to find branches of periodic solutions and also to detect bifurcation
points from them. For example, the red line in Fig. 5.22a represents an octave
regime with oscillation frequency equal to that of the higher of the two quasiharmonic acoustic resonances. The blue line shows a period doubling bifurcation
to an inverted Helmholtz motion state. When the inharmonicity between the two
resonances is sufficiently large, quasi-periodic regimes can be observed (see a study
of saxophones in Dalmont et al. (1995) and Doc et al. (2014)). Continuation methods
are in principle able to predict the occurrence of quasi-periodic regimes by detecting
a Neimark-Sacker bifurcation point localised on the branch of a periodic solution.
Nonlinear dynamical systems typically display a large variety of oscillation
behaviours (see, e.g. Nayfeh (1995) and Manneville (2010)). Several different
types of oscillation state which can occur in brass instruments are illustrated by
simulations in Vergez and Rodet (2001b)). The ‘route to chaos’ through a succession
of period doubling bifurcations (known as the Feigenbaum process) has been
simulated in reed instruments using elementary models (Maganza et al. 1986).
Measured non-periodic regimes in wind instruments have been extensively analysed
using the phase space representation (Gibiat 1988; Gibiat and Castellengo 2000).
Quasi-periodic regimes are of great interest to musicians, since they correspond to
the special sounds known as multiphonics. In Sect. 5.4.6 some of the multiphonics
available on the trombone are described and simulated.
5.4.6 Multiphonics
Brass instruments are sometimes described as ‘monodic’ or ‘monophonic’. Is it true,
as the etymology of these terms implies, that a brass instrument can only generate
one sound at a time? The answer to this question depends on the precise meaning
attached to the word ‘sound’. Every note played on a wind instrument has a frequency spectrum with many different components, but in standard performance, the
steady-state oscillation is periodic and the frequency components are harmonically
related. Such a note is normally perceived as a single sound, with a unique pitch
closely related to the fundamental frequency of the harmonic series (Sect. 2.2.3). In
this case the behaviour of the instrument can indeed be described as monophonic.
The fusion of frequency components which results in the perception of a single
pitch is ineffective if the components are not perceived as members of the same
harmonic series. Two or more pitches may then be heard. Although the result is
still strictly speaking one sound, the impression is of several separate sound sources
acting simultaneously, and such sounds are described as multiphonics.
Many wind instruments are capable of generating multiphonics (Castellengo
1981). Two different categories of multiphonic can be distinguished, based on
the acoustic phenomena involved in their production. One category involves the
generation of multiple simultaneous pitches through an extension of conventional
playing techniques. Woodwind multiphonics come into this category, since they rely