When two harmonically-related sine waves (sinusoids) are added together, the two distinct tones tend to fuse together in our auditory system to produce a sensation of a single pitch. Additionally, the timbre of the resultant tone is associated with the resultant waveform. However, when two sinusoids with nearly the same frequency (but NOT exactly the same) we perceive something quite different. Fig. 1a shows a graph of a sine wave with a frequency of f₀. Fig. 1b shows a graph of a sine wave whose frequency is 1.1 f₀. Notice that these two waveforms are just slightly out of phase with each other. Fib. 1b, because it has a higher frequency, repeats its waveform slightly ahead of the waveform in Fib. 1a.

Fig. 1a. y = sin x

Fig. 1b. y = sin 1.1 x

Fig. 2 shows the resultant waveform. Notice how its amplitude varies smoothly down and back up over the course the waveform segment shown in Fig. 2.

Fig. 3: y = sin x + sin 1.1 x. Click on the graph to see more of the waveform.

Click on Fig. 2 to reveal a more detailed graph of its waveform. Verify that this amplitude variation pattern continues over time. This pattern of periodic amplitude variation is referred to as beating, and is the physical phenomena at the heart of musical tuning.

To provide a concrete example, say the frequency of the sine wave in Fig. 1a is 60 Hz. The frequency of Fig 1b would then be 66 Hz. (60 x 1.1 = 66). When these two tones combine, the amplitude variation pattern repeats 6 times every second. Thus, we hear six beats per second. What is more, we perceive the pitch of the resultant to be the average (60 + 66 / 2 = 63 Hz.) of the two original frequencies.

Fig. 4: The combination of a 440 Hz. and 442 Hz. sinusoids produces a complex tone that beats 2 times per second.

The Beat Theorem

When two sine waves with frequencies f1 and f2 respectively are very close in frequency we experience a phenomena called beating. Beating may be described as a periodic variation in amplitude that occurs at a rate f1 - f2 with a frequency f1 - f2 / 2. For example, in Fig 4. when a 440 Hz. and 442 Hz. sinusoids are combined we hear 2 beats per second and tone whose frequency is 441 Hz. Notice that this theorem predicts that the combination of 440 Hz. and 442 Hz. sinusoids, and the combination of 440 Hz. and 438 Hz. sinusoids would both produce 2 beats per second. Thus, when tuning an instrument it is obviously imposible to tell just from the number of beats produced whether you are above or below the goal pitch.