11.3 Musical Uses of Warping
In this section we describe a few applications of warping in music. As already pointed out, many aspects and properties of warping musical signals are still to be explored and many results of this section may be deemed as experimental. Applications that will be discussed range from accurate pitch-shifting of inharmonic sources and inharmonization of harmonic sources, to feature and transient extraction, vibrato editing and morphing.
11.3.1 Pitch-shifting Inharmonic Sounds
The sounds from a large class of instruments are inherently inharmonic. The spacing of the frequencies of the partials is not uniform. In piano sounds, in the low register, the displacement of the partials from the harmonics becomes more and more apparent as we move towards the lower end of the keyboard. In Figure 11.14 we report data (× marks) extracted from a low-pitch piano tone ( Hz). These represent the differences between the frequency of a partial and that of the next one. If the sound were harmonic, one should observe a flat distribution of points aligned on the pitch frequency. On the contrary, one observes that the spacing between the partials increases with the order of the overtones. The distribution of the partials can be closely matched to the derivative of a Laguerre curve. This can be obtained by means of an optimization of the parameter b in (11.10). It turns out that ...