aHamburg University of Applied Sciences
bInstitute of Systematic Musicology
cHAW Hamburg
Abstract :
The Impulse Pattern Formulation (IPF) is a top-down method
proposed previously (Bader, R.: Nonlinearities and Synchronization in
Musical Acoustics and Music Psychology, 2013) which assumes
musical instruments to work with impulses which are produced at a
generator, travel through the instrument, are reflected at various
positions, are exponentially damped and finally trigger or at least
interact with succeeding impulses produced by the generator. The
underlying recursive equation relates every new system state to
previous values and their logarithm. Adding more system components
increases the number of reflection points, thus the number of terms
in the argument of the logarithmic function increases. Like other
nonlinear equations, the IPF can produce stable states but also
bifurcation and divergency and fully captures transitions between
regular periodicity at nominal pitch, bifurcation scenarios, and noise.
Applying the IPF on musical Instruments, the nonlinear behavior like
transients or multiphonics can be described, which would be very
complicated or impossible using well-established methods such as
modal analysis or finite element models. Furthermore, the IPF is used
for sound synthesis which follows the fundamental principles of real
musical instruments and, due to the simple mathematical description
of the IPF, needs a very limited amount of input parameters.