Understanding Beats in Music Tuning
Interesting phenomena emerge when two waves with different frequencies interfere. Beats are a result of interference and are another example of interference. So far we have focused on interference between two waves with the same frequency, and thus, wavelength. In this case, however, there are locations of destructive, constructive, and partial interference to the right of the speakers.
The diagram below is a snapshot of the two waves at some particular time. The plot above is the sum of the two waves with different frequencies. If you are standing at the most right constructive interference location marked in the figure, you hear a sound of maximum intensity. Unlike for the case of equal frequencies, we conclude that an observer at a fixed location now hears the intensity of sound changing periodically.
In other words, the combined amplitude leading to a specific type of interference is now time-dependent. We call this phenomena beats, since the sound is now literally beating from loud to silent and back to loud, like a drum being hit in a periodic manner. Since we are combining two wave equations with different frequencies resulting in a time-dependent amplitude, the combined wave can no longer be described by a simple sinusoidal wave equation. However, it is still periodic since the amplitude change is periodic.
The figure below shows an example of a combined wave from two sinusoidal waves with two different frequencies. The blue envelope wave on the combined plot outlines the change in amplitude. One can see when the envelope function is at peak, the two individual wave overlap at the same displacement (crests in this figure) resulting in constructive interference at that location.
When the envelope function is at equilibrium, the displacements of the individual waves are cancelling (one is at a crest while the other one is at a trough) resulting in destructive interference. This is a snapshot of the combined wave at a fixed time. If you are standing at a fixed position, this wave will move in time, so you will be hearing a periodic intensity of constructive and destructive interference.
The intensity of the sound is the square of the amplitude, so constructive interference (maximum intensity) occur every time the envelope wave is at a crest or a trough, or every half a cycle of the envelope function. The period of this time-dependent interference, or beats, is half the period of the envelope function drawn above.
Mathematical Representation of Beats
We can also determine the beat frequency mathematically in terms of the original frequencies of the two waves. To determine the beat frequency, we want to find the time it takes for the interference pattern to repeat itself. Let us assume that at some specific time an observer standing at a fixed location hears constructive interference (the loudest sound from the combined waves). Let us define this time as \(t=0\) sec.
At time later time which we define as the beat period, \(T_b\), the observer will again hear constructive interference. This implies that the phase difference increased from 0 to \(2\pi\). The path length difference and the phase constant difference is not time-dependent as long as the observer stays put. The absolute value in the equation above implies that the beat frequency which is the difference between two frequencies is a positive value, since frequency cannot be negative.
Thus, the absolute value assure that the the beat frequency is positive, if you happen to subtract the larger frequency from the smaller one. You can see from this results that the difference in the two frequencies needs to be small for beats to have a meaningful interpretation to the human ear.
Another important property of beats is the carrier frequency. This frequency is approximately the frequency of the "small" oscillations within the envelop function. In other words, in combined plot of Figure 8.6.2, the carrier period would be the time between two consecutive peaks (approximately). We can mathematically determine that beat and carrier frequencies by adding two wave functions of waves with different frequencies.
Recall, for sound waves we represent the displacement in terms of pressure, with equilibrium displacement at atmospheric pressure, which we set here to zero for simplicity. Let us assume that at some location and instance of time which we define as x=0 m and t=0 sec the interference is constructive, such that there is no relative phase constant difference between the two waves, \(\phi_1=\phi_2=0\). We want to see how the wave changes as a function of time at any position, so we choose x=0 m for convenience.
The above function is not the simple wave function with one sine term that we are familiar with, but represents the combination of the two waves. We can think of the second sine function as representing the carrier wave with the carrier frequency of \(f_c=(f_1+f_2)/2\). The quantity \(2P_o\) multiplied by the cosine function represents the time-dependent amplitude of the combined wave.
From the equation, we can be that frequency of this amplitude is \((f_1-f_2)/2\). The carrier frequency can be found from the period of the small oscillations inside the beat envelope. The beat period can also be obtained from by measuring the time that passed between locations of biggest amplitude.
This location does not have to be a crest, but can also be a trough, since the intensity of a wave is proportional to the square of amplitude. In this plot, we see the largest amplitude at t=5 sec. The next time that we observe the same amplitude is at t=15 sec. where we have assume that the bigger frequency is \(f_1\). Now we can solve for the individual values of frequency, \(f_1\) and \(f_2\), since we have two equations and two unknowns.
Application of Beats in Music Tuning
Musicians use the principles of beats to tune their instruments. Frequencies on strings which are confined between two ends, like a guitar string, are determined mainly by the length and the speed of the wave on the string, as we will see in a later section on standing waves. The source no longer has control of the frequencies that it generates on the string, precisely because the ends of the string are confined.
This is very different from what we have learned so far, so stay tuned! When tuning a string instrument, the musician changes the tension of the string, which in turn changes the speed and results in a different frequency. They find the desired frequency by using tuning forks (at least this was the main method before tuning apps).
A tuning fork has a predetermined frequency based on its physical properties at which it generates a sound when someone strikes it. Thus, with multiple turning forks of different frequencies a musician can tune all the strings to their desired frequencies. They do this by hitting the tuning fork and plucking or striking the string of the instrument at the same time.
If the musician hears a beat, the string is out of tune. Then, they will adjust the tension until beats are no longer heard, implying that the string is vibrating at the exact same frequency as the tuning fork.
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 as in a tremolo while 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 imperceptible.
Perception of Beats
If you're a musician, you've probably heard the sound that results when you two sounds with almost the same pitch are played together. The result is sound with a distinctive "wah-wah-wah" sound. The pitch is about the same as the two sounds that are combining, but there is a characteristic "in and out" sound as the sound cycles back and forth between being loud and being soft.
This sound is called beats. If the two notes creating the beats are really close together, the "beating" is slow- the "wahs" are long and the silences in between don't occur very often- the beat frequency is low. If the two notes are farther apart, the beat frequency is higher- the "wahs" happen more often.
As the two sounds that are being played together drift into and out of phase, the two waves shift back and forth from constructive interference to destructive interference. If the two source frequencies are nearly identical, it takes a very long time for the sounds to drift into (or out of) phase. This results in a low beat frequency. If the frequencies are farther apart, the two sounds drift in and out of phase more often, resulting in a higher beat frequency.
In this equation, [latex]f_b[/latex] represents the beat frequency; [latex]f_1[/latex] and [latex]f_1[/latex] represent the frequency of the sounds causing the beats. If a 100 Hz sound is played with a 98 Hz sound, the result is a sound with a beat frequency of 2 Hz (according to the equation). In practical terms, there will be two full "wahs" per second.
At the beginning of this section on beats, it was mentioned that the listener hears a single note that fluctuates in volume (at the beat frequency). It seems logical that the frequency of this single note would be halfway between the two frequencies being played together.
When the difference between the two sound frequencies is small (less than 10 Hz for most people), people hear a single pitch that varies in loudness. As the beat frequency increases, two things happen. First, our ears/brains have trouble keeping up with the rapidly changing amplitude and we lose the ability to hear the individual silences between the "wahs." Second, our brains begin to recognize the difference between the two pitches creating the beats.
For most people, this transition happens when the difference between the two frequencies is in the 10 Hz to 50 Hz range. If the difference between the two sound frequencies becomes large enough, difference tones (sometimes called Tartini tones, after the violinist who discovered them) occur. Listeners hear not only the two source frequencies, but also a third (much lower) tone. This tone usually corresponds to a frequency equal to the difference of the two source frequencies.
One example of this is a London police whistle. The whistle has two chambers, each of which plays a different very high note. When the whistle is sounded, listeners hear three notes together- the two high notes and a third note much lower than the other two. For musicians, beats enter into the perception of intonation.
Additional Considerations
Even if the amplitudes are slightly different, the two waves will still drift in and out of phase, and the waves will alternate between constructive and destructive interference. However, the amplitude of the combined wave will never reach zero. You will still hear beats, but the difference between the loud and soft part of the beats will not be as big.
Monaural beats are when there is only one tone that pulses on and off in a specific pattern. With only one tone (as opposed to two tones with binaural beats), your brain has a much easier time adjusting and there is no need to balance separate tones. This means that monaural beats can be used effectively via either headphones or speakers.
To experience the binaural beats perception, it is best to listen to this file with headphones. Binaural Beats Base tone 200 Hz, beat frequency from 7 Hz to 12.9 Hz. According to a 2023 systematic review, studies have investigated some of the claimed positive effects in the areas of cognitive processing.
Affective states like anxiety, mood, pain perception, meditation, relaxation, mind wandering, and creativity were studied, but it was determined that the techniques were not comparable and the results were inconclusive. Out of fourteen studies reviewed, five reported results in line with the brainwave entrainment hypothesis, eight studies reported contradictory, and one had mixed results.
The composer Alvin Lucier has written many pieces that feature interference beats as their main focus. Italian composer Giacinto Scelsi, whose 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 (Scelsi treated and notated each string of the instrument as a separate part to make his violin solos 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).