How can we listen to the phenomenon of beats in a real instrument?

1. On a string instrument it is enough to play the same note simultaneously on two different strings, and slightly change the tuning of one of the two strings while keeping the note on the other fixed. On instruments with free intonation such as the violin or the cello, the effect can be obtained with the instrument in tune by bowing two strings simultaneously: one open, and the other stopped with a finger to produce the same note as the open string. Moving the finger slightly away from the position giving the unison produces beats, and their frequency can be adjusted.
2. On a fretted instrument such as the guitar, the same effect can be obtained by using the tuning key (or peg) that tunes one of two strings plucked simultaneously.
3. On the piano, beats between two strings of the same note can be heard when it is being tuned (or when it is out of tune). Indeed, in the piano two or three strings are struck simultaneously by the hammer to produce a single note, and to be in unison they must produce as few beats as possible. The tuner exploits this phenomenon, comparing the strings two at a time to obtain the unison.

A trained ear can also perceive beats between the higher harmonics of a note. Musicians exploit this too when tuning instruments, piano tuners in particular. An untrained ear perceives the beats of higher harmonics in various ways: as "a trembling sound", or a "harsh" one, or simply as "an out-of-tune interval".

See the list of links at the bottom of the page for further reading and links

Why have I never heard the phenomenon of beats between two different instruments?

Beats are a phenomenon that is well defined only between pure sounds (corresponding to sinusoidal waves) and, moreover, they show up as a complete disappearance of the sound at regular intervals only when the two waves have exactly the same intensity. This ideal condition is very hard to obtain with two different instruments. There are some exceptions, in which the instruments produce their sound in a particularly steady way, such as two organs facing each other (a very common arrangement in Italian churches). If the two organs play the same note on the same "soft" stop and are not perfectly in tune, at least partial beats can be heard. If the stop is "soft", like the one obtained with wooden pipes and without mixtures, the sound is poor in higher harmonics, and practically only the fundamental contributes to the beats.

Why does a slightly mistuned fifth or third produce beats, even though the two notes have very different frequencies? Does it also happen with pure tones?

With the sounds of instruments the beats do not occur between the fundamentals, but between the harmonics. For example, the third harmonic of an A at 220 Hz (660 Hz) coincides with the second harmonic of the E a just fifth above (330 Hz). If the E is slightly sharp, for example at 331 Hz, its second harmonic (662 Hz) produces a 2 Hz beat with the third harmonic of the A. In general, if two notes have frequencies f₁ and f₂ close to the ratio j:k, the j-th harmonic of the first and the k-th harmonic of the second almost coincide and beat with frequency

${\displaystyle b=|j{f}_1-k{f}_2|}$

(j = 3, k = 2 for the fifth; 4 and 3 for the fourth; 5 and 4 for the major third).

This formula shows why in equal temperament the fifths are "almost clean" and the thirds are not. Starting from an A at 220 Hz, the tempered fifth (329.63 Hz) gives |3·220 − 2·329.63| ≈ 0.7 Hz, a very slow beat, while the tempered major third (277.18 Hz) gives |5·220 − 4·277.18| ≈ 8.7 Hz, a fast beat that makes the third somewhat "rough". Tuners and musicians count precisely these beats to adjust intervals (see beats and tuning).

With pure tones there are no harmonics, yet a slightly mistuned consonance still produces a periodic fluctuation, weaker, at the same frequency b. These are called second-order beats. Two pure tones exactly in the ratio 3:2, added together, form a wave that always repeats identically. If the ratio is not exact, the relative phase of the two components changes slowly: the shape of the sum wave changes cyclically, returning to itself every 1/b seconds. The spectrum always contains only the two original frequencies: no new frequency arises, only the shape of the wave changes. These beats are heard mainly for simple ratios and at frequencies that are not too high. How the ear perceives them is still being studied.

The same reasoning applies to three or more sounds. For example, three pure tones at 330, 440 and 112 Hz produce a fluctuation at |(440 − 330) − 112| = 2 Hz.

Is it a physical effect or distortion? When the sum of pure tones is recorded, these slow variations can be seen in the waveform. They are not a defect of the equipment: they are simply the superposition of the waves. To check, just look at the spectrum: in the presence of distortion (see intermodulation distortion) new lines appear at the sum and difference frequencies, while if only the original frequencies appear, the fluctuation is due only to superposition.

Further reading and links

• For a physical illustration of the phenomenon of beats see interference
  • For further reading see also interference of harmonic waves, superposition principle
• For an illustration of the perceptual phenomenon see beats
• To experiment in the virtual laboratory see the guided tours
  • Interference between waves of different frequencies generated by a single source (beats)
  • Fourier analysis