Interference beats and Tartini tones

Beats are often observed between two vibrations with similar frequencies. The beat frequency equals the difference between frequencies of the beating signals. When sound signals interfere, the beat signal can sometimes be heard as a separate note: the Tartini tone. Both are useful and important in practice for measuring frequencies and for tuning musical instruments. (It is also worth looking at for the insight it gives to Heisenberg's Uncertainty Principle, as we shall see below.)

This background page to the multimedia chapter Interference gives sound files and derivations.

beats diagram


Let's consider two waves with the same amplitude A, and frequencies f1 and f2 that are not very different. Before we do any maths, we can see what will happen by looking at the diagram. In this plot, the wave depicted by the red graph plus that depicted by the purple one gives that represented by the blue curve. The horizontal axis is usually time. Suppose that the waves start out in phase, so that they add up, as shown at the left of the diagram. The wave depicted by the red graph has a slightly higher frequency than that depicted by the purple one, so it gradually gains on it, and eventually gets one half cycle out of phase. At this point, the two component waves cancel out, and the amplitude of the blue curve is near zero. After an equal interval of time, they get back in step again, as shown.

If the waves are sound waves, what will this sound like? Well, provided that the difference in frequency is small enough, the resultant wave will sound loud when the two components are in phase and soft or absent when they are out of phase. The frequency of the blue wave is, if you look carefully, about halfway between that of the red and the purple. So we should hear a wave of intermediate frequency, getting regularly louder and softer. This is the acoustical example of the phenomenon of interference beats.

(If the difference in frequencies is greater than a few tens of cycles per second, we won't recognise the very short period of cancellation as being soft, and in fact we'll get some interesting effects, to which we return later.)

Explanation and derivation

beats diagram Now either omit this section or let's get quantitative. For the purple and red waves respectively, let's write
    y1 = A cos (2π f1)t     and y2 = A cos (2π f2t),     so

    ytotal = y1 + y2 = A {cos (2π f1t) + cos (2π f2t)}    (1).

To get any further, we need the trigonometric identity that
    cos (a+b) = cos a * cos b - sin a * sin b,     from which it follows that
    cos (a-b) = cos a * cos b + sin a * sin b.
Adding these two equations gives us
    cos (a+b) + cos (a-b) = 2 cos a * cos b    (2).
We now use this identity by making the substitions
    a = 2π (f1t + f2t)/2   and   b = 2π (f1t - f2t)/2,     so
    a + b = 2π f1t       and       a - b = 2π f2t.
We now substitute this into equation (2) to get
    cos (2π f1t) + cos (2π f2t) = 2 cos (2π (f1t + f2t)/2) * cos (2π (f1t - f2t)/2)     (3)
Now we recognise (f1 + f2)/2 as the average frequency fav and (f1 - f2) as the frequency difference Δf.
Finally, we multiply (3) by A to get:
    ytotal = y1 + y2 = 2A {cos (2π Δf/2)t} * {cos (2π fav)t}     (4).

On the diagram below, we see that the maximum amplitude of the compound wave is twice that of the two interfering waves. The carrier wave indeed has the average frequency, as you can verify by counting cycles on the diagram.

    But how often do the beats occur? Let's write (4) this way:
      ytotal = {2A cos (2π Δf/2)} * cos (2π fav)     (4).
    The term inside the curly brackets {} can be considered as the slowly varying function that modulates the carrier wave with frequency fav. (It is indeed an example of amplitude modulation or AM.) This function--the modulation of the amplitude--is the green wave in the diagram. It has frequency Δf/2, but notice that there is a maximum in the amplitude or a beat when the green curve is either a maximum or a minimum, so beats occur at twice this frequency. (One cycle of the green curve is from time (i) to time (v). There are beats at (i), (iii) and (v), and quiet spots at (ii) and (iv).)

    So the beat frequency is simply Δf: the number of beats per second equals the difference in frequency between the two interfering waves, as you can hear for yourself in the sound files below.

    We now return to a complication raised above. If the beats occur more often than roughly 20 or 30 times per second, we no longer hear them as beats: our ears are not fast enough to respond to events that quickly. (Nor are our eyes: we cannot recognise a light that is flashing 30 times per second.)

    Consider, for example, what happens when we play two tones with frequencies 400 Hz (approximately the note G4) and 500 Hz (approximately the note B4). The resultant waveform will look rather like a wave of 450 Hz whose amplitude varies at a rate of 100 times per second. But that is not what we hear: we hear the chord G4 plus B4 (and perhaps also the note G2, which is an auditory illusion: see below).

    Sound files of beating sine waves

    The two signals (courtesy of John Tann) have the same amplitude. The lower
    frequency one is 400 Hz, which is between G4 and G#4. In most recordings, the starting and ending transients are removed by attenuating the amplitude at the beginning and end. That has not been done here, so that the uncompressed (au and wav) wave files really do follow the equations given above. Consequently, there are perceptible clicks at the beginning and end. The mp3 files have been compressed according to the mpeg alogorithm, i.e. distorted in such a way that they sound the same but require less memory. They do not look like the equations given above.

    1. 400 & 400.5 Hz    1 beat every 2 seconds
    2. 400 & 401 Hz      1 beat per second
    3. 400 & 403 Hz      3 beats per second
    4. 400 & 410 Hz      10 beats per second
    5. 400 & 420 Hz      can you...
    6. 400 & 430 Hz      still hear...
    7. 400 & 440 Hz      interference beats?
    8. 400 & 450 Hz      Frequency ratio 9:8 is a Pythagorean major second or a just major tone.
    9. 400 & 480 Hz      Frequency ratio 6:5 is a just minor third
    10 400 & 500 Hz      Frequency ratio 5:4 is a just major third
    11 400 & 533 Hz      Frequency ratio 4:3 is a Pythagorean or just perfect fourth
    12 400 & 600 Hz      Frequency ratio 3:2 is a Pythagorean or just perfect fifth
    13 400 & 667 Hz      Frequency ratio 5:3 is a just major sixth
    14 400 & 800 Hz      Frequency ratio 2:1 is an octave.


Tartini Tones

    If you have listened to the sound samples above at a reasonably high sound level, you may have heard Tartini tones or difference tones, particularly on numbers 10, 11 and 12. Tartini tones sound like a low pitched buzzing note with a frequency equal to the difference between the frequencies of the two interfering tones. If you play these sound files (repeated here below) moderately loudly, you will hear Tartini tones with frequencies 100, 133 and 200 Hz, corresponding approximately to the notes G2, C3, G3, as shown in the illustration at right. (See note names to convert among notes, notations and frequencies.) There is more about this on the FAQ in music acoustics.

    ( )
    400 & 500 Hz      Frequency ratio 5:4 is a just major third
    ( )
    400 & 533 Hz      Frequency ratio 4:3 is a Pythagorean or just perfect fourth
    ( )
    400 & 600 Hz      Frequency ratio 3:2 is a Pythagorean or just perfect fifth

    As well as the sound samples above, there are some more samples showing Tartini tones below. We also have a page on Tartini tones and their relation to temperament, which has more examples.

    Musical notation of the Tartini tones illustrated

Varying the beat frequency

Here are some film clips made with oscilloscopes to illustrate the beats and Tartini tones. The laboratory set-up is described here. The first shows signals that differ in frequency by 1 Hz.

Next, we increase the higher frequency, so that the difference increases smoothly from 1 to 10 Hz. How does this sound to you?

Most people agree that the sound in the clip above is a single pure tone, with amplitude varying as beats, from slow to fast. (By the way, the window on the spectrum analyser (screen bottom right) is not long enough to resolve the two frequencies.)

In the clip below, the lower frequency is still 400 Hz, but the higher is increased smoothly from 410 to 435 Hz. What do you hear?

You may have heard rapid beats initially, increasing at first. What after that? For some listeners, the end of the clip is distinctly two different notes. (The spectrum analyser also resolves them.) For others, it just sounds like an unclear sound. Some people refer to a range lying between fast beats and separate notes as a region of roughness. You can investigate this further by listening again, by using the sample sound files above, or by downloading the two channel sine wave generator from our lab site (Mac only at this stage).

Interference and consonance

The ratios 3:2 and 5:4 are called (by many Western people, at least), musical consonances (in just intonation). In this example, one tone remains constant at 400 Hz. The other is varied rapidly from 400 to 500 Hz, where it pauses briefly, before increasing to 600 Hz. You may hear a descending Tartini tone during the first of the variations. If you do not hear it at first, you might try increasing the volume and using headphones. Do you hear a Tartini tone during the second?

Here are some more consonances in just intonation.

and here a scale constructed using the notes used the consonances given above, plus two consonances at 5:4 and 3:2 based on the fifth note in the scale.

However, this has taken us some distance from beats and Tartini tones. For more detail and explanation, go to our page on Tartini tones and their relation to temperament.

Tuning a guitar with beats and harmonics

harmonics of E and A strings Here I use the fourth 'harmonic' on the low E string and the third 'harmonic' on the A string. The E string plays the note E2 so its fourth harmonic is E4, two octaves above. The A string plays A2, so its third harmonic is also E4, a (just) twelfth above, as shown in the diagram at right. In this video clip, I have already tuned the low E string, so here I adjust the tuning of the A string so its third 'harmonic' stops beating with the fourth 'harmonic' of the E string.

Look at the soundtrack. From 0.6 to 1.0 s, only the E string sounds, so there are no beats. From 1.0 to 2.4 s, there are beats at about 8 Hz, so the third 'harmonic' of the A string is about 8 Hz flat, so the A string is 8/3 Hz ~ 3 Hz flat. From 2.5 to about 5 s, as I increase the tension in the A string, we see and hear that the beat frequency decreases. Listening and looking closely after about 5 s, we can still hear beats, but their frequency is less than half a Hz. This can be reduced a little in a second tuning step, not shown here. (Using this method to tune all strings, and its limitations, are discussed on Strings and harmonics.)

In the clip above, we notice that, because of the finite ring time of the strings, there is a limit to the precision of the tuning. Which is a good time to introduce

What beats have to do with Heisenberg's Uncertainty Principle

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