# Talk:Beat (acoustics)

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## Identity

I just changed the identity to read 4a on the right. This is correct isn't it? Also, why the amplitude in terms of 2a in the first place? Couldn't it be simplified to read just a? -- postglock 15:52, 16 August 2005 (UTC)

In origin the amplitudes were indeed a, but I got mistaken when converting the identity to TeX. Gonna fix that.--Army1987 22:18, 16 August 2005 (UTC)

## Inharmonic partials

Not sure if it's worth mentioning, but the two guitar strings "we" have tuned to the same note won't be exactly the same, since they have different stiffnesses and therefore different partials or timbres.

Also, you're not supposed to talk to the reader in encyclopedic tone. — Omegatron 02:23, 29 January 2006 (UTC)

The fondamental frequecies will be equal, and so will the frequency of each overtone. That's all what matters. Yes, the intensity and duration of each overtone may change between the two strings, but the beating will be the same.
(As for style, I'll try to rewrite that paragraph in a more impersonal manner.) --Army1987 13:33, 29 January 2006 (UTC)
I don't think so. The harmonics will beat even when the fundamentals are in tune. The overtones are slightly sharp from their mathematical relation because of the non-idealities of a real string. I assume (I could be wrong), that two strings of different diameters at different tensions with the same fundamentals will have overtones that are different in their sharpness. One of the definitions of consonance is the shared overtones, which is why lower and upper keys on the piano are tuned a little out of tune from the middle keys, but I'm not sure if that's relevant to this example. — Omegatron 14:53, 29 January 2006 (UTC)
This happens because piano strings are very stiff, near to the point of breaking. Guitar strings are much slacker. (BTW, I've changed the previous suggestion to raise B string to E to viceversa.) This way, the difference between theoretical frequencies and real ones (if any) is unaudible, or even meaningless. )See [1] and [2] to know what I mean by "meaningless", you cannot determine frequencies with perfect accuracy in a finite amount of time, assuming their sound lasts 10 s, you'll have an uncertainty of about 0.15 Hz, which is probably more than the difference between theoretical and real frequencies in this case.) --Army1987 20:58, 1 February 2006 (UTC)
It is, as far as I can assess the mathematics, true that this “should” be indiscernible in practice. However, as a musician I have seen plenty of practical evidence that differences in frequencies down to well under .2 Hz at a pitch of 415 Hz can be noticed in under a second by many. Regarding this discernibility in timbre (harmonics, overtones) ne also takes into consideration in a good ensemble that the difference of timbre in strings of potential unevenness (like gut strings) or consistency (gut vs. steel, for instance) must be counteracted with a slight adjustment of the basic pitch for the sound of the ensemble to be optimal. -- Olve 04:00, 23 March 2006 (UTC)
Yes, but in this case we are speaking of two steel strings from the same set, only with different gauge. Tune the high E string down to B, then ask someone to pick either that or the "true" B string, and see if you can tell them from each other... It is much easier to ear wheter the same string is plucked nearer to the bridge or to the fretboard (in the former case high harmonics are louder than in the latter). Also, I've noticed that, especially in a slack tuning, e.g. if you tune the bass string down to C, by plucking the string very hard (like in slapping), the extra tension at the plucking can cause the frequency to rise by almost half a semitone... Are you sure you mean 0.2 Hz not 2 Hz? --Army1987 20:39, 23 March 2006 (UTC)

## Beating Frequency

The article states that "the beating frequency is f1−f2, the difference between the two starting frequencies". However, if one takes the equation above, $\left|2a\cos\left(2\pi\frac{f_1-f_2}{2}t\right)\right|$, one can argue that the beating frequency is actually $\frac{f_1-f_2}{2}$. In the term $\left|A\cos\left(2\pi ft\right)\right|$, the frequency is $f$ (i.e. take the term inside the cosine function and divide it by $2\pi t$). This can be verified quickly using Octave or Matlab.

The frequency of $2a\cos\left(2\pi\frac{f_1-f_2}{2}t\right)$is actually $\frac{f_1-f_2}{2}$, but the frequency of its abs. value is the double of that, because every half period of the cosine is like the other half but with its sign changed. For example, in the image in the article, there is only one cycle of the cosine, but a half, a whole, and another half cycle of its magnitude. --Army1987 18:44, 11 May 2006 (UTC)
Fair enough. Maybe I'm confused between the beating frequency and the frequency of the envelope... Thanks
Lol ive spent quite a while working this out. The absolute value thing is essentially correct, but heres a better way to put it, during the negative half of $2a\cos\left(2\pi\frac{f_1-f_2}{2}t\right)$'s period, it still makes a maximum overall because the sin half of the function could also be -ve, and -ve * -ve = +ve = maximum (the sine half is actually negative many times during this half of the cosine's curve, because the frequency of the sin half is much higher). Thus though the cosine half has frequency $\frac{f_1-f_2}{2}$, the sine half multiplies with it to effectively take the absolute value of it giving an overall frequency double the cosines. Have a look at the picture in the article and see how the combined function does not move into negative and positive regions, but rather it expands and contracts for max and minima due to the 'absolute valuing'. - ATL 09 oct
I've always explained it as sin() and -sin() sound identical to the ear. - Rainwarrior 14:12, 9 October 2006 (UTC)

## Making the equation more readable

I find the justifying equation quite daunting until you realize that, for the sake of simplicity, most constants can be ignored. In addition, a variable change gets rid of the two fractions and make things clearer.

This is what I propose:

Let's consider the sum of a first sound $S_1$ of requency $f_1$ and a second sound $S_2$ of frequency $f_2$. The expression of these sounds are:
\begin{align}\\ S_1 &= \sin(2\pi f_1t)\\ S_2 &= \sin(2\pi f_2t) \end{align}
We define two variables $a$ and $b$ that represent the median and the half-difference of the two frequencies:
\begin{align}\\ a &= 2\pi\frac{f_1+f_2}{2}t\\ b &= 2\pi\frac{f_1-f_2}{2}t \end{align}
Using angle sum and difference identities, we can deduce
\begin{align} S_1+S_2 = \sin(a+b)+\sin(a-b) = 2\sin(a)\cos(b) \end{align}

Well, I think this reads better than the original:

${ a\sin(2\pi f_1t)+a\sin(2\pi f_2t) } = { 2a\cos\left(2\pi\frac{f_1-f_2}{2}t\right)\sin\left(2\pi\frac{f_1+f_2}{2}t\right) }$

We could also make the transformation explicit:

\begin{align} \sin(a+b)+\sin(a-b) &= \begin{align}\sin(a)\cos(b)&+\cos(a)\sin(b)\\ +\sin(a)\cos(b)&-\cos(a)\sin(b)\end{align}\\ &= 2\sin(a)\cos(b) \end{align}

(Here, the "+" should be indetend further than the two "=", while retaining proper alignment of the two upper terms, but I don't know how to do that) Exxos77 21:40, 25 May 2007 (UTC)

## Example in music

For an example in music, popular American punk band At the Drive-in have a song called "pickpocket" which uses notes a step apart to create beating. It's most prominent in the introduction of the song. 129.11.122.215 14:04, 22 September 2007 (UTC)

Alvin Lucier also has several pieces that explore beats. —Preceding unsigned comment added by 82.23.210.169 (talk) 16:39, 19 January 2008 (UTC)

Practically all accordion music, espacially the French (mussette) uses beats in their music, they have two sets of reeds wich are tuned slightly different to create this effect. The difference goes from 1Hz (slow mussette/beats) to 5 or 6 Hz (fast mussette/beats). —Preceding unsigned comment added by 78.29.204.69 (talk) 15:49, 12 July 2008 (UTC)

## Beating near to harmonic intervals

It is said: "This is caused by slight differences between the intervals of equal temperament and the "natural" intervals of the harmonic series". While this is, strictly speaking, possible, that's not what I had in mind when I originally wrote that sentence back in... (Who cares when?). I was thinking about out-of-tune notes. In the specific example of a fifth, using for example C4 and G4, the third harmonic of the former is 784.88 Hz and the second harmonic of the latter is 783.99 Hz. They beat at less than 1 hertz, and most times notes are played for much shorter than 1 second. Also, the interval between a just-intonation fifth and an equal-temperament fifth is less than two cents, while the just noticeable difference for the human ear is 5 cents, so it means that in practice often instrument aren't tuned to such accuracy.

On the other hand, should the G4 be 15 cents too sharp, its second harmonic would be 790.81 Hz, beating at almost 6 Hz and immediately noticeable.

So the inexactness of ET, if mentioned, should be introduced as a possible cause of beating (e.g. "This can also be caused"...), and a "worse" interval (e.g. the ET major third at −14 cents from the exact one) should be used for the example. Army1987 (talk) 11:06, 15 June 2008 (UTC)

The effect happens in both out-of-tune notes and ET notes. You are welcome to phrase it however you like, but I think ET should be mentioned as a cause for beats as well. Barak Sh (talk) 18:21, 15 June 2008 (UTC)

## No references

This material appears in many undergraduate physics texts. Just for grins, how about citing a couple...? Rb88guy (talk) 19:07, 4 September 2009 (UTC)

## Ultrasound edit

(moved from User talk:Oli Filth)

You deleted my bit on doppler-shift ultrasound in the beats section, saying that it was probably irrelevant. It's not! Beats is exactly what is used to observe and measure the small frequency shift between reference signals and those reflected. Do you want to argue this further? I'd be happy to... —Preceding unsigned comment added by Bhindibhagee (talkcontribs) 14:54, 20 April 2010 (UTC)

I didn't say it was "irrelevant". I said that this is almost certainly reliant on a heterodyning operation, which is similar, but subtly different. I don't have any sources to back this up, but this sounds like a standard mix-down operation to me. Oli Filth(talk|contribs) 15:30, 20 April 2010 (UTC)

## Difference Tones - unsourced and completely wrong!

There was a section on "difference tones" that contained the following unsourced content:

"If the beating frequency rises to the point that the envelope becomes audible (usually, much more than 20 Hz), it is called a difference tone.[citation needed] The violinist Giuseppe Tartini was the first to describe it, dubbing it il Terzo Suono (Italian for "the third sound"). Playing pure harmonies (i.e., a frequency pair of a simple proportional relation, like 4/5 or 5/6, as in just intonation major and minor third respectively) on the two upper strings of a violin, such as the C above middle C against an open E-string, will produce a clearly audible C two octaves lower. An interesting listening experiment (help·info) is to start from a perfect unison and then very slowly and regularly increase the pitch of one tone. When one tone starts to split out from the former twin-note, a slow rumbling can be heard, gradually increasing into an audible tone."

This is completely incorrect. An obvious counterexample would be to take a 100 Hz sine wave sin(100*2pi*t) and modulate it by another 100 Hz sine wave sin(100*2pi*t). Since the envelope is 100 Hz, by this (completely unsourced) theory, we should hear a 100 Hz audible tone in the result. Obviously, we don't; a 100 Hz tone is present neither mathematically nor audibly via naive experiment. sin(100*2pi*t)^2 = 1/2 - 1/2*cos(200*2pi*t), so what we end up getting is a 200 Hz tone with some DC offset.

Given a signal s(t) and an envelope e(t), it's a very, very common but *completely wrong* misconception that the spectral content of the envelope is present in the combined signal s(t)*e(t), which leads to the further misconception that if e(t) becomes fast enough to "be audible," you'll hear it. This is only true if s(t) has a nonzero frequency response at DC, and is false otherwise. You can see some computed AM spectra at [3], which clearly shows that the spectral content of the modulator isn't present in the mixed signal.

Another reason that the "envelope" might "become audible" is if some nonlinear system is present which is actually creating intermodulation distortion, e.g. literal sum and difference tones. This is a *completely* different phenomenon from beating and should never be confused with it.

I've deleted this entire section; I don't know how to salvage it.

Battaglia01 (talk) 01:27, 3 April 2013 (UTC)

I seem to recall Feynman mentioning something about difference tones due to the human hearing being non-linear. I might look at that when I have some spare time. — A. di M.  08:46, 5 April 2013 (UTC)
Correct, it is - or, if your speakers are nonlinear, or the microphone you used to record something is nonlinear, or if the instrument you're playing on exhibits nonlinear effects, all of which happen decently often. Feynman is probably talking about DPOAE's. But, generally speaking, true "combination tones" (not necessarily the simple quadratic difference tone) actually appear if the signal is being sent through a nonlinear system of some sort, which means that they have nothing to do with "beating" at all and they certainly don't have to do with "beating becoming audible." The only discussion about "difference tones" that should appear on this page is one about how it's NOT related to beating. Battaglia01 (talk) 20:08, 6 April 2013 (UTC)