John Cage's Number Pieces, a Geometric Interpretation of “Time Brackets” Notation

Conceptual musical works that lead to a multitude of realizations are of special interest. One can’t talk about a performance without considering the rules that lead to the existence of that version. After dealing with similar works of open form by Iannis Xenakis, Pierre Boulez and Karlheinz Stockhausen, the interest in John Cage’s music is evident. His works are “so free” that one can play any part of the material; even a void set is welcomed. The freedom is maximal and still there are decisions to consider in order to make the piece playable. Our research was initially intended to develop a set of conceptual and software tools that generates a representation of the work as an assistance to performance. We deal here with the Number Pieces Cage composed in the last years of his life. Over time, we realized that the shape used to represent time brackets, brought important information for the interpretation and musical analysis. In the present text, we propose a general geometric study of these time brackets representations, while trying to make the link with their musical properties to improve the performance.