Talk:Beat (acoustics)

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[edit] 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)

[edit] 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)

[edit] 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 2πt). 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)

[edit] 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 S1 of requency f1 and a second sound S2 of frequency f2. 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) = &\sin(a)\cos(b)+\cos(a)\sin(b)\\ +&\sin(a)\cos(b)-\cos(a)\sin(b)\\ = &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)

[edit] 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)

[edit] 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)