# For creating beats how small the difference should be between the two frequencies

It is said that to create beats we need two "slightly" different frequencies, and subtract it.

1- My question is why do we need slightly different frequencies? Why not large difference?

2- Also how slightly different? what is the limit on maximum difference?

• The answer depends entirely on your application. Humans can hear beat frequencies to maybe 10Hz or so, but in electronics beat frequencies of tens of GHz are not unusual and in optics frequency differences of 1e15Hz are commonly used. Dec 16, 2014 at 6:30

By superposition principle we will arrive at, $$y_{total} = {[ 2Acos(2\pi \Delta f/2) ]cos(2\pi f_{av})}$$

(source: unsw.edu.au)

The term inside the [] brackets can be considered as the slowly varying function that modulates the carrier wave with frequency $$f_{av}$$. (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.

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 .

Reference here 1 and 2:

The beat frequency is very simply:

$$f_{beat}=|f_1-f_2|$$

So there is no limit on how far apart they can be. In demonstrating the beat frequency one frequently uses frequencies that are slightly apart because it produces the typical "beating."

If for example you were using the frequencies $561.6$ Hz and $300$ Hz you would get a resulting frequency of $261.6$ Hz which is in fact middle C.

In this described case you wouldn't actually notice a beating, but rather you would just hear the pure note C. It wouldn't sound like two frequencies at all, it would just sound like a single crisp note.

This is why in a lab demonstration you use frequencies that are 10 - 15 Hz apart. It produces a beat noticeable to the human ear. Using large differences produces "musical" (for lack of a better word) notes.

You can create beats with "slightly" different frequencies. But we do definitely perceive great differences between random and deterministic components. They depend on the ‘composition’ of the contributing sinusoids. But also on the length of the period of listening. And in such compositions both the choices of frequencies and phases have strong influence.

Reference from here: ( Direct copy & paste from here: ) 1

For example:

Please calculate with high resolution the following three compositions, using five sinusoids:

1. 10,000 / 10,002 / 10,004 / 10,006 / 10,008 Hz. All sine contributions.

In that case you will hear the high tone that corresponds with 10,004 Hz but with a strong beat of 2 Hz.

1. 10,000 / 10,004 / 10,008 Hz. All three sine contributions. 10,002 / 10,006 Hz. Both cosine contributions. So a 90 degree phase shift.

In that case you will hear the high tone that corresponds again with 10,004 Hz but now with a strong 4 Hz beat.

1. 10,000 / 10,002.0333 / 10,004 / 10,006.0333 / 10,008 Hz. All sine contributions.

In that case you will hear the high tone of 10,004 Hz again,

but within a period of 30 seconds

and starting with a 2 Hz beat

after 7.5 seconds the beat will gradually change into a 4 Hz beat.

After 15 seconds the beat is back again at 2 Hz.

At 22.5 seconds again at 4 Hz

and after 30 seconds the composition ends with a 2 Hz beat in the 10,004 Hz tone.

If you change the sine contributions of 10,002.0333 and 10,006.0333 Hz into cosine the composition

starts with a beat of 4 Hz,

2 Hz at 7.5 sec,

4 Hz at 15 sec,

2 Hz at 22.5 sec

and finally 4 Hz at 30 sec.

In all these proposed experiments the calculation of the different contributions to the sound energy frequency spectrum resulted per experiment in exact predictions of the final beat rhythm.

Reference from here: 2

Based on the concept that our hearing sense is differentiating and squaring the incoming sound pressure stimulus, this mechanism evokes in front of the basilar membrane the sound energy frequency spectrum.

So it is the sound energy signal, present inside the perilymph fluid, as a uniform pressure stimulus all over the volume. And therefore all the existing Fourier frequency components in the sound energy signal are present inside the perilymph to stimulate the basilar membrane including their relative amplitudes and their relative, but extremely precise, phase relations.

Direct copy & paste from here: www.a3ccm-apmas-eakoh.be/pcbwp/experiments.htm

About 10 kHz experiments (frequencies together with some prescribed phase relations in a standard summation procedure to compose a Fourier series) with a 0.0625 Hz detuning, there exists an accuracy in the periodicity pattern of 6.25 parts per million.