On designing stacked canons with relative chord tones
Jurjen L. van Geenen · Journal of Mathematics and Music · 2012
The established counterpointing and intervallic approaches to the construction of stacked canons provide limited harmonic or structural control and are computationally complex. While efficient and in complete control of harmony, Morris’ Tonnetz approach targets serial stacked canons ad minimum and does not encompass voice-leading constraints. A style-independent, constructive approach using relative chord tones, chord sequences and chord-sequence modulations is presented along with its connections to graph-theory in address of these issues. The problem of obtaining complete chords is connected to the Hamiltonian cycle problem, the problem of avoiding parallel intervals to the Eulerian cycle problem. Several efficient canon-generation algorithms are presented, along with an analysis of their computational complexity. The maximum number of distinct sub-canons of stacked canon is counted in demonstration of the connection between stacked canons and the design of larger scale imitative polyphonic structures.