Periodicity series of initial transients of musical instruments for real-time implementation.
Rolf Bader · The Journal of the Acoustical Society of America · 2010
A real-time algorithm for initial transients of musical instruments displaying the periodicity changes of the system during the transient is presented as shown with blown instruments. When the system consists of a reed and an air column, the standpoint of the reed is taken and only one equation formulates the action of the reed upon itself as delayed and damped via the air column. A general system variable is used which includes both the periodicity and the amplitude of the system. This is possible as a back impulse acting in-phase with the reed will not only cause an enlargement of the reed’s amplitude but will in accordance with this enlargement also lead to an enlargement of the reed’s periodicity. When using an exponential damping and solving the equation for the system parameter of the next time step, an equation similar to that of the logarithmic map appears. According to the system parameters, for the steady state, a whole bifurcation scenario occurs with stable, bi-stable, and chaotic motion representing a stable tone, a multiphonic, or a blowing noise. When formulated for instruments with multiple reflection points, such as violins or guitars, depending on the delays and dampings, the system shows an astonishing stability over the whole playing range because of these multiple reflections.