HARMONIC AND MUSICAL WAVELETS

The concept of a harmonic wavelet is generalized to describe a family of mixed wavelets with the structure w(m, n)(x) = {exp (in2pix) - exp (im2pix)}/i(n - m) 2pix. It is shown that this family provides a complete set of orthogonal basis functions for signal analysis. By choosing the (real) numbers m and n (not necessarily integers) appropriately, wavelets whose frequency content ascends according to the musical scale can be generated. These musical wavelets provide greater frequency discrimination than is possible with harmonic wavelets whose frequency interval is always an octave. An example of the wavelet analysis of music illustrates possible applications.