In this chapter we describe interesting audio effects that can be obtained by deforming the time and/or the frequency axis. Whilst discrete-time warping techniques were introduced in 1965 [Bro65], their interest in musical applications is fairly recent. Time warping aims at deforming the waveform or the envelope of the signal, while frequency warping modifies its spectral content, e.g., by transforming a harmonic signal into an inharmonic one or vice versa. The effects obtained by warping often increase the richness of the signal by introducing detuning or fluctuation of the waveform. The sounds from natural instruments like piano and drums already possess this property. The wave propagation in stiff strings and membranes can actually be explained in terms of frequency warping. By warping these sounds one can enhance or reduce their natural features. Even uninteresting synthetic sounds such as pulse trains may be transformed into interesting sounds by warping. Frequency warping is amenable to a time-varying version that allows us to introduce dynamic effects such as vibrato, tremolo and Flatterzunge in flute.
The quality of warping ultimately depends on the warping map, i.e., on the function describing the deformation of the time or frequency axis. Time and frequency warping are flexible techniques that give rise to a tremendous amount of possibilities, most of which are at present still unexplored from a musical point of view. By choosing the proper map one can actually morph the sound of an instrument into that produced by another instrument.
This chapter is divided into two main sections. In the first section we describe the time- and frequency-warping operations and derive algorithms for computing these effects, including recent advances in approximation. In the second section we illustrate some of their musical applications based on examples and case studies.