Roughness and Beats

The Physics of Amplitude Modulation (Beats)

Sound is variation in air pressure relative to ambient pressure. It is measured at a point in space, usually by a microphone. The result of the measurement is a graph of air pressure (relative to ambient) against time. If the function is sinusoidal, the sound is a pure tone. The amplitude of a pure tone is the maximum deviation of air pressure from ambient pressure. That happens at the peaks and valleys of the sinusoid.

Beats occur when two pure tones of almost equal frequency and amplitude simultaneously sound. If the tones are produced by different loudspeakers, the function of air pressure against time will be the sum of two sinusoids. In other words, at any instant, the deviation of air pressure from ambient will be the sum of the deviation from one loudspeaker and the deviation from the other. This function is also like a sinusoid, except that its amplitude increases and decreases (the temporal envelope goes up and down), which is called amplitude modulation (AM), or beating.

When the two pure tones are in phase (when measured by the microphone), their contributions to air pressure add—they reinforce each other. “In phase” means that the peaks and valleys of the air pressure function for one tone coincide with the peaks and valleys of the other, so the phase difference is 0 degrees. If the peaks of one coincide with the valleys of the other and vice-versa, the tones are exactly “out of phase.” Their phase difference is 180 degrees, and they cancel each other. If two out-of-phase tones have equal amplitude at the position of the microphone, they completely cancel: the intensity of the AM tone oscillates between silence and some maximum value. In this case, the modulation depth is 100 percent. If the two amplitudes are different, the tones do not completely cancel, and modulation depth is less than 100 percent.

The sound of beating is familiar from out-of-tune pianos. In the central range, most pianos have three strings per tone. Each string produces a complex tone with several partials (pure tones). Piano tuners tune these strings to have the same frequency. As the piano goes out of tune, the frequencies become different. Audible beating is produced by the lowest partial of one string interacting with the lowest partial of its neighbor. Beating is also produced by pairs of higher partials. Perceptually, beating is most noticeable at a rate of about four beats per second—similar to the average rate of syllables in speech.

Perception of Roughness

The rate at which two tones beat is called the beat frequency or modulation frequency. For two pure tones, the beat frequency is the difference between the original frequencies. For example, a pure tone near concert A4 (440 Hz) beats with its chromatic neighbor Bb4 (466 Hz) with a beat frequency of 26 Hz (cycles per second).

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