Continuous Spaces, Continuous FT

Abstract

The formula for the Fourier Transform can be extended to continuous settings in several ways: transforming the discrete into a continuous with some appropriate (usually Lebesgue) measure, i.e. summing on an infinite set; or having the variable t move on the real line instead of a cyclic group; or having the frequency become infinitesimal, perhaps keeping value constant while both k and n grow to infinite. A last variant considers ordered collections of pcs with fixed cardinality, leading unexpectedly to a good measure of quality for temperaments such as might have been used by J.S. Bach. All these changes, advertised by various researchers [25, 88], require however precise definitions of their contexts and limitations, which will be scrupulously enunciated hereafter. Several practical situations of music devised by playing directly in some continuous Fourier space have occurred in recent years, and are reviewed in the last section.