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[Editor 2’s introduction intended to address votes to close because the question wasn’t mathematical.]

The trigonometric formula $\sin{(at)}+\sin{(bt)}=2\cos({a-b\over2}t)\sin({a+b\over2}t)$ can be interpreted musically as saying that if two notes with frequencies $a$ and $b$ are played at the same time, the result is the same as playing one note with frequency $a+b\over2$ while varying the amplitude at a frequency of $a-b\over2$. The graph of the sum has an “envelope” around the varying amplitude (see below), and the sensation of the varying amplitude is often called “beating.”

My question: When three or more sinusoids are added, the graph of the sum also has an envelope that is more or less noticeable depending on how simple the ratios of the frequencies are. Is there a straightforward way to calculate the frequency of this envelope and how prominent/audible it is?

[End Editor 2 introduction]

When playing 3 pure sine tones simultaneously as C-E-G in 12 Tone Equal Temperament (261.62 Hz, 329.63 Hz, 391.99 Hz respectively) there is audible beating (and of course the periodic amplitude fluctuation clearly visible when viewing the composite waveform). This beating does not occur if the three tones are played in integer-ratios-based Just tuning instead (where the tone frequencies are 261.62 Hz, 327.03 Hz, 392.44 Hz respectively). Question is, how to go about calculating the beating frequency(s) in the former case?

[Edit] Below you see plots of the sums of the relevant sine waves.

First is the equal temperament wave that I plotted as $$\psi_1(x):=\sin x+\sin(2^{4/12}x)+\sin(2^{7/12}x).$$

enter image description here

This is followed by the harmonic version

$$\psi_2(x):=\sin x+\sin\frac{5x}4+\sin\frac{3x}2.$$

enter image description here

In the plots $x$ ranges from $0$ to $1500$, which amounts to $1500/(2\pi)\approx239$ full wavelengths of the base note (C). That is, the shown waves would last a bit under one second. [/Edit]

Editor's note: Because the frequency ratios are irrational, there is no clear beat. I'm not sure whether the beat the OP hears will show in this short wave. The top plot does show five "somewhat repeating" regions. It may be possible to explain those by using better rational approximations of the equal temperament frequency ratios. But, as the ratios are irrational, such beats will gradually degrade and/or move around.

JL

OP's edit: I believe the following answer's the question. Beating is a result of the difference between frequencies that are close together. Beats are easily heard if the frequency difference is less than about 10 Hz and may be detected up to 15 Hz. We perceive beating even if two frequencies are nearly, but not quite, in a simple ratio (as in this case). If we consider just two tones (the first and third) playing together, the beating is a result of second and third order combination tones, and the frequency of beating is given by: ((2*261.62)-391.99)-(391.99-261.62) = 0.88 Hz. Another way of obtaining the same result: the frequency ratio is close to 3:2. Thus (3*261.62)-(2*391.99) = 0.88 Hz. If we now consider just the first and second tone playing together, their frequency ratio is close to 5:4. Thus (5*261.62)-(4*329.63) = 10.42 Hz.So, this is an interesting situation where we hear the beating between harmonics of tones, but do not hear the harmonics themselves.

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closed as off-topic by Lord_Farin, Shailesh, Namaste, Davide Giraudo, José Carlos Santos Dec 25 '18 at 21:33

This question appears to be off-topic. The users who voted to close gave these specific reasons:

  • "This question is not about mathematics, within the scope defined in the help center." – Shailesh, Namaste, José Carlos Santos
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  • $\begingroup$ We need more information. Can you estimate the beat frequency you managed to hear? Would those five "lumps" per second shown in the first wave fit? The above sample may be too short. Also, the way you produced the digital wave may have approximation errors, and what you hear may be an artefact of rounding errors in frequencies? I'm prepared to be very wrong, though. $\endgroup$ – Jyrki Lahtonen Dec 25 '18 at 16:11
  • $\begingroup$ It might be easier to first work with two equal temperament tones. Undoubtedly you tried that already. $\endgroup$ – Jyrki Lahtonen Dec 25 '18 at 16:16
  • $\begingroup$ Yes I too obtained '5 repeating lumps', and it seems it is this modulation which manifests as an audible 'beating' (the sensation is about a 5 times per second 'pulsing'). I don't know how relevant this is, but if we consider just the first two tones, the 'lumps' are not as large in amplitude but there are just over 10 per second, and we get 10 if take the frequency difference between the third harmonic of 329.63 Hz (1318.52) and fourth harmonic of 261.62 Hz (1308.1). $\endgroup$ – samiant Dec 25 '18 at 21:07
  • $\begingroup$ And if we consider all three tones, we obtain another modulation of just under a second if we take the difference between the second harmonic of 261.62 Hz (784.86) and the first harmonic of 391.99 (783.98). There is an online resource which says: "What is the beat frequency when a 220 Hz tone and 330 Hz tone are played together? Answer: 2 Hz. This is the difference between first harmonic of 331 (662) and second harmonic of 220 (660)". $\endgroup$ – samiant Dec 25 '18 at 21:21
  • $\begingroup$ I totally forgot about the harmonics creating beats also. Shows how little I know about this! Please fix any errors in my note in the main body of the question. Just click the edit button under the post to get access to the source. $\endgroup$ – Jyrki Lahtonen Dec 25 '18 at 21:31