Beat (acoustics)

In acoustics, a beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies.

With tuning instruments that can produce sustained tones, beats can readily be recognized. Tuning two tones to a unison will present a peculiar effect: when the two tones are close in pitch but not identical, the difference in frequency generates the beating. The volume varies like in a tremolo as the sounds alternately interfere constructively and destructively. As the two tones gradually approach unison, the beating slows down and may become so slow as to be inperceptible.

Mathematics and physics of beat tones

The sum (blue) of two sine waves (red, green) is shown as one of the waves increases in frequency. The two waves are initially identical, then the frequency of the green wave is gradually increased by 25%. Constructive and destructive interference can be seen.

This phenomenon manifests acoustically. If a graph is drawn to show the function corresponding to the total sound of two strings, it can be seen that maxima and minima are no longer constant as when a pure note is played, but change over time: when the two waves are nearly 180 degrees out of phase the maxima of each cancel the minima of the other, whereas when they are nearly in phase their maxima sum up, raising the perceived volume.

It can be proven (see List of trigonometric identities) that the successive values of maxima and minima form a wave whose frequency equals the difference between the frequencies of the two starting waves. Let's demonstrate the simplest case, between two sine waves of unit amplitude:

{\cos(2\pi f_{1}t)+\cos(2\pi f_{2}t)}={2\cos \left(2\pi {\frac {f_{1}+f_{2}}{2}}t\right)\cos \left(2\pi {\frac {f_{1}-f_{2}}{2}}t\right)}

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If the two starting frequencies are quite close (for example of the order of few hertz), the frequency of the cosine of the right side of the expression above, that is (f₁−f₂)/2, is often too slow to be perceived as a pitch. Instead, it is perceived as a periodic variation of the first in the expression above (it can be said, the cosine factor is an envelope for the other faster wave), whose frequency is (f₁ + f₂)/2, that is, the average of the two frequencies. However, because the human ear is not sensitive to the phase, only the amplitude or intensity of the sound, only the absolute value of the envelope is heard. Therefore, subjectively, the frequency of the envelope seems to have twice the frequency of the cosine, which means the audible beat frequency is:

f_{beat}=f_{1}-f_{2}\,

This can be seen on the diagram on the right.

A 110 Hz A sine wave (magenta; first 2 seconds), a 104 Hz G# sine wave (cyan; following 2 seconds), their sum (blue; final 2 seconds) and the corresponding envelope (red)

A physical interpretation is that when $\cos \left(2\pi {\frac {f_{1}-f_{2}}{2}}t\right)$ equals one, the two waves are in phase and they interfere constructively. When it is zero, they are out of phase and interfere destructively. Beats occur also in more complex sounds, or in sounds of different volumes, though calculating them mathematically is not so easy.

Beating can also be heard between notes that are near to, but not exactly, a harmonic interval, due to some harmonic of the first note beating with a harmonic of the second note. For example, in the case of perfect fifth, the third harmonic (i.e. second overtone) of the bass note beats with the second harmonic (first overtone) of the other note. As well as with out-of tune notes, this can also happen with some correctly tuned equal temperament intervals, because of the differences between them and the corresponding just intonation intervals: see Harmonic series (music)#Harmonics and tuning.

Uses

Musicians commonly use interference beats to objectively check tuning at the unison, perfect fifth, or other simple harmonic intervals. Piano and organ tuners even use a method involving counting beats, aiming at a particular number for a specific interval.

The composer Alvin Lucier has written many pieces that feature interference beats as their main focus. Italian composer Giacinto Scelsi, whose mature style is grounded on microtonal oscillations of unisons, extensively explored the textural effects of interference beats, particularly in his late works such as the violin solos Xnoybis (1964) and L'âme ailée / L'âme ouverte (1973), which feature them prominently (note that Scelsi treated and notated each string of the instrument as a separate part, so that his violin solos are effectively quartets of one-strings, where different strings of the violin may be simultaneously playing the same note with microtonal shifts, so that the interference patterns are generated). Composer Phill Niblock's music is entirely based on beating caused by microtonal differences.

Binaural beats

Binaural beats are heard when the right ear listens to a slightly different tone than the left ear. Here, the tones do not interfere physically, but are summed by the brain in the olivary nucleus. This effect is related to the brain's ability to locate sounds in three dimensions.

Sample

Beating waveforms (high frequency)

220 Hz A3 (left channel) and 207.65 Hz G#3 (right channel) beating at 12.35 Hz

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Beating waveforms (low frequency)

220 Hz A3 and 222 Hz tone beating at 2.0 Hz