Beating Frequency

Diagram of beat frequency

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.

When tuning instruments that can produce sustained tones, beats can readily be recognized. In fact, organ-tuners even use a method involving counting beats, aiming at a particular number for a specific interval. Tuning two tones to a unison will present a strange effect: when the two tones are close in pitch but not yet perfectly centered, the difference in frequency generates the beating. The volume varies like in a tremolo as the sounds alternatively interfere constructively and destructively. When the two tones gradually approach fusion, the beating slows down and disappears, giving way to full-bodied unisono resonance.


A 110 Hz A sine wave (magenta), a 104 Hz G# sine wave (cyan), their sum (blue), and the corresponding envelope (red)

The reason of this phenomenon is tied to acoustics. 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.

This isn't just a curiosity: it can be proven that the successive values of maxima and minima form a wave whose frequency equals the difference between the two starting waves. Let's demonstrate the simplest case, between two sine waves of equal amplitude:

a\sin(2\pi f_1t)+a\sin(2\pi f_2t) =2a\cos\left(2\pi\frac{f_1-f_2}{2}t\right)\sin\left(2\pi\frac{f_1+f_2}{2}t\right)

If the two starting frequencies are quite close (usually differences of the order of few hertz), the frequency of the cosine of the right side of the expression above, that is (f1f2)/2 is too slow to be perceived as a pitch. Instead, it is perceived as a periodic variation of the sine in the expression above (it can be said, the cosine factor is an envelope for the sine wave), whose frequency is (f1+f2)/2, that is, the average of the two frequencies. Since the amplitude of that wave is \left|2a\cos\left(2\pi\frac{f_1-f_2}{2}t\right)\right|, which in the period of 2/(f1f2) reaches values 2a and 0 twice, there will be two beats per such period. That is, the beating frequency is f1f2, the difference between the two starting frequencies.

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.

When the two waves are in unison f = 0 and as the difference between f1 and f2 increases, the speed increases until beyond a certain proximity (usu. about 15 Hz) beating becomes undetectable and a roughness is heard instead, after which the two pitches are perceived as separate. If the beating frequency rises to the point that the envelope becomes audible (usually, much more than 20 Hz), it is called difference tone. The violinist Giuseppe Tartini was the first to describe it, dubbing it 'il Terzo Suono' (the third sound). That this comes from a violinist makes a lot of sense: playing pure harmonies (i.e., a frequency pair of a simple proportional relation, like 4/5 or 5/6, as in just intonation major and minor third respectively) on the two upper strings, such as the C above middle C against an open E-string, will produce a clearly audible C two octaves lower. Disintonation, including a major third in equal temperament, makes the sound gruffy and rough. An interesting listening experiment is to start from a perfect unison and then very slowly and regularly increase the pitch of one tone. When one tone starts to split out from his former twin-note, a slow rumbling can be heard, gradually increasing into an audible tone.

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.

'Beating' or 'beats' is directly linked to the phenomenon of the difference tone. Musicians commonly use interference beats to objectively check tuning at the unison, perfect fifth, or other simple harmonic intervals.

The composer Alvin Lucier has written many pieces which feature interference beats as their main focus.

Binaural beats Edit

Main article: 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. There are also those who believe that the beats can be used to "entrain" the brain to a desired state.

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See also Edit

de:Schwebung fr:Battementhu:Lebegésfi:Huojunta sv:Slag (musik)

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