In-filling components between harmonics cancel wave repetition to represent aperiodic sounds
Evidence
However, some instruments, especially percus sion, do not have pitch and hence are non-periodic, or aperiodic. How can we analyze these instruments in terms of a basic waveform, such as a sine wave, which is inherently periodic? The answer is shown in Figure 1.53. Here the square wave example discussed earlier has had four more sine waves added to it. However, these sine waves are between the harmonics of the square wave and so are unrelated to the period, but they do start off in phase with the harmonics. The effect of these other components is to start canceling out the repeat periods of the square waves, because they are not related in frequency to them. By adding more components which sit in between the harmonics, this cancellation of the repeats becomes more effec tive so that when in the limit, the whole space between the harmonics is filled with sine wave components of the appropriate amplitude and phase. These extra components will add constructively only at the beginning of the waveform and will interfere with successive cycles due to their different fre quencies. Therefore, in this case, only one square wave will exist. — David Howard, Jamie Angus, coustics and Psychoacoustics, 455
Categorization
- DEFINE: Explains how Fourier analysis extends from periodic to aperiodic (single-event) sounds — the conceptual bridge that makes spectral analysis of percussion possible.
- DEEPTHINKING: A subtle conceptual leap: representation requires infinitely dense filling between harmonics, so perfect aperiodicity is a limiting case — good for grasping the Fourier modelling philosophy.
- XREF: connects to: Noise-like sounds have non-repeating waveforms and continuous, inharmonic spectra — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 822