Design of Timbre with Cellular Automata and B-Spline Interpolation

The origin of this paper comes from the collaboration of the authors on the UPISketch software project, which is a creation of the Iannis Xenakis Center and a descendant of the UPIC project of Xenakis. The software, created in 2017, allows the user to draw curves which can be interpreted as musical gestures, acting on elements such as pitch, time, and timbre. Two fundamental mathematical tools used in this process are splines, to model graphical gestures, and cellular automata (CA), to model musical gestures such as pitch sequences. CA are natural objects in music because of all the local computations we do when we listen to a musical sequence. In this paper we apply both of these techniques on the ``micro-level" to the timbre of waveforms, using the approach of cycle interpolation. A waveform is modeled as a sequence of cycles, and each cycle is modeled as a cubic spline curve. The B-spline coefficients of each cycle form a discrete representation, which can be manipulated through the use of CA. In this way, key cycles can be generated, and can be interpolated with B-splines to form new and interesting timbres. We illustrate this generation and design of waveforms in our own software implementation with JUCE, which in turn will become part of UPISketch.