aHamburg University of Applied Sciences
bInstitute of Systematic Musicology
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.
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