Chapter 10: Processes of Perception and Analysis

Section 8: Auditory Perception


The human auditory system is sensitive to sound at frequencies between about 20 Hz and 20 kHz. Middle A on a piano typically corresponds to a frequency of 440 Hz. Each octave represents a change in frequency by a factor of two. In western music there are normally 12 notes identified within an octave. These differ in frequency by successive factors of roughly 21/12—with different temperament schemes using different rational approximations to powers of this quantity.

The perceived character of a sound seems to depend most on the frequencies it contains, but also to be somewhat affected by the way its intensity ramps up with time, as well as the way frequencies change during this ramp up. Many musical instruments produce sound by vibrating strings or air in cylindrical or conical tubes, and in these cases, there is one main frequency, together with roughly equally spaced overtones. In percussion instruments, the spectrum of frequencies is usually much more complicated. In speech, vowels and voiced consonants tend to be characterized by the lowest two or three frequencies of the mouth. In nature, processes such as fluid turbulence and fracture yield a broad spectrum of frequencies. In speech, letters like "s" also yield broad spectra, presumably because they involve fluid turbulence.

Any sound can be specified by giving its amplitude or waveform as a function of time. Sin[ω t] corresponds to a pure tone. Other simple mathematical functions can also yield distinctive sounds. FM synthesis functions such as Sin[ω(t+ a Sin[b t])] can be made to sound somewhat like various musical instruments, and indeed were widely used in early synthesizers.

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From Stephen Wolfram: A New Kind of Science [citation]