From beats, to roughness, to a smooth sensation

The animation in Fig. 1 is designed to demonstrate two concepts related to beats: 1) roughness, and 2) critical bandwidth. Two sinusoids are set against one another. Sine 2 remains a constant 96 Hz. throughout the animiation while Sine 1 glissandos up an octave to 192 Hz. The exact pattern is shown below in Fig. 2. When the two sinusoids are no longer the same frequency beats emerge.

Fig. 2. Musical staff notation for the audio track in Fig. 1

As the frequency distance between the two sinusoids continues to increase the number of beats per second corrospondingly increase. However, at about 137 Hz., there are so many beats per second (41) we can no longer can hear individual beats. What we do hear is usually described as a certain roughness (R) in tone quality. As the frequency continues to increase, at some point our perception of roughness gives way to a smooth sensation. View the Fig. 1 animation a number of times noting each time where you can no longer hear beating. Did it occur before or after the 137 Hz. marker? Try to do the same for the point at which you perceive the smooth sensation to begin.

Critical Bandwidth

The critical bandwidth is a window around (above and below) a reference pitch f₀ where beating and roughness are perceived. In Fig. 1, the reference pitch is 96 Hz. The critical bandwidth varies considerably throughout the frequency spectrum. In his 1937 book Science & Music, Sir James Jeans reports some observations about beats made by Mayer and Stumpf using 2 tuning forks. Based on these observations, note how the critical band varies for different regions of the frequency spectrum.

Fig. 3. Some observations about the nature of beats made by Mayer and Stumpf.
as cited in James Jeans, Science & Music (New York: Dover, 1968), p. 50.

Frequency (in Hz.) of tuning fork I	No. of beats per second at which		Interval (in semitones) until beats dissappear
	beats are most unpleasant	beats can no longer be heard
96	16	41	6
256	23	58	4
575	43	107	3
1707	84	210	2
2800	106	265	1.5
4000	---	400	1.6