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Sound is propagated by waves. Waves can interfere.

Suppose there are two tenors standing next to each other and each singing a continuous middle-C.

  1. Will it be the case that some people in the audience cannot hear them because of interference?

  2. Would it make a difference if they were two sopranos or two basses and singing a number of octaves higher or lower?

  3. How does this generalize to an array of n singers?

  4. Given a whole choir, to what extent are their voices less than simply additive because of this? Is it possible that, for some unfortunate member of the audience, the choir appears to be completely silent--if only for a moment?

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    $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – David Z Oct 26 '15 at 14:23
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The main issue in the setting of an orchestra or choir is the fact that no two voice or instruments maintain exactly the same pitch for any length of time. If you have two pure sine wave source that differ by just one Hertz, then the interference pattern between them will shift over time - in fact at any given point you will hear a cycle of constructive and destructive interference which we recognize as beats, but the exact time when each member of the audience will hear the greatest or least intensity will vary with their position.

Next let's look at the angular distribution of signal. If two tenors are singing a D3 of 147 Hz (near the bottom of their range) the wavelength of the sound is 2 m: if they stand closer together than 1 m there will be no opportunity to create a 180 degree phase shift anywhere. If they sing near the top of their range, the pitch is closer to 600 Hz and the wavelength 0.5 m. But whatever interference pattern they generate, a tiny shift in frequency would be sufficient to move the pattern - so no stationary observer would experience a "silent" interference - even of the fundamental frequency.

Enter vibrato: most singers and instruments deliberately modulate their frequency slightly - this makes the note sound more appealing and allows them to make micro corrections to the pitch. It also makes the voice stand out more against a background of instruments and tends to allow it to project better (louder for less effort on the part of the singer). This is used by soloists but more rarely by good choirs - because in the choir you want to blend voices, not have them stand out.

At any rate, the general concept here is incoherence: the different source of sound in a choir or orchestra are incoherent, meaning that they do not maintain a fixed phase relationship over time. And this means they do not produce a stationary interference pattern.

A side effect of interference is seen in the volume of a choir: if you add the amplitudes of two sound sources that are perfectly in phase, your amplitude doubles and the energy / intensity quadruples. A 32 man choir would be over 1000 times louder than a solo voice - and this would be achieved in part because the voices could only be heard "right in front" of the choir (perfectly coherent voices would act like a phased array). But since the voice are incoherent, there is no focusing, no amplification, and they can be heard everywhere.

Note that incoherence is a function of phase and frequency - every note is a mix of frequencies, and although a steady note will in principle contain just a fundamental and its harmonics, their exact relationship is very complicated. Even if you took a single singer's voice, and put it into two speakers with a delay line feeding one of the speakers, I believe you would still not find interference because of the fluctuations in pitch over even a short time. Instead, your ear would perceive this as two people singing.

And finally - because a voice (or an instrument) is such a complex mix of frequencies, there is in general no geometric arrangement of sources and receiver in which all frequencies would interfere destructively at the same time. And the ear is such a complex instrument that it will actually "synthesize" missing components in a perceived note - leading to the strange phenomenon where for certain instruments, the perceived pitch corresponds to a frequency that is not present - as is the case with a bell, for example.

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    $\begingroup$ Unless you're singing very high, one hertz is quite noticeably out of tune. See my answer. $\endgroup$ – Level River St Oct 25 '15 at 20:28
  • $\begingroup$ Why not go whole hog and mention correlation length/time? :D $\endgroup$ – DanielSank Oct 26 '15 at 0:54
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    $\begingroup$ This answer, while pieces are correct, misses some of the more important effects and emphasizes irrelevant points. $\endgroup$ – anon01 Oct 26 '15 at 7:46
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    $\begingroup$ "...if they stand closer together than 1 m there will be no destructive interference anywhere." This is incorrect, because it assumes that the people are singing in phase, which is not generally the case. $\endgroup$ – Dietrich Epp Oct 26 '15 at 8:02
  • $\begingroup$ @DietrichEpp fair point - edited $\endgroup$ – Floris Oct 26 '15 at 10:29
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Waves interference are perceptible when you have very close and pure frequencies. At guitar or flute you can hear beats when 2 strings or 2 instruments play such close and pure frequency. But for a rich timbre like human voice, forget it! They are "billions" (indeed, a continuity) of different unsynchronized frequencies (even in what you would call "C tone"), so you won't cancel all of them at the same time in a huge multi-synchrone beat.

What is more likely (for ordinary rich timbres) is the interference of your own voice travelling by 2 different paths, typically when you sing between 2 close concrete walls. Because here there is synchronism (it is 2 copies of the same signal; the canceling alignment is ruled by the distance and by the wavelength). You won't cancel all, but you will fade or amplify some range of tones, and listeners at different locations might hear different modulations.

Note also that emitters are not points and do not float in air: this blurs the paths and thus the conditions of interference (so the amount and locality of it).

Now all this is for ordinary singers or instruments with rich timbre. For opera-like voice on long standing notes, you are probably closer to the flute situation (if people play the same pure tone). Plus opera rooms take care of sound reflectors (close or far), singers are far from you, etc, so sound paths are much less blurry.

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  • $\begingroup$ So a duo or orchestra of electronic instruments that all played sine waves would suffer from the problem? Wouldn't the tenors lose the fundamental frequency of the note and just leave the overtones? That would make them sound reedy to some people and boomy to others. $\endgroup$ – chasly from UK Oct 25 '15 at 11:01
  • $\begingroup$ A thought about self-interference (maybe that is not a good phrase!!!) but even with one's own voice surely the interference pattern would be different for each frequency (?). Again it is the fundamental frequency that concerns me most. $\endgroup$ – chasly from UK Oct 25 '15 at 11:13
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    $\begingroup$ Note that with all overtones present without the fundamental, you still hear the same tone ! (so some organs can produce very low notes using this "absent fundamental" effect). Note that there also exist the traditionnal effect of "the additional singer", were the combination of voices gives you the illusion that there is one more voice. $\endgroup$ – Fabrice NEYRET Oct 25 '15 at 11:17
  • $\begingroup$ NB: I did various editings in my answer. $\endgroup$ – Fabrice NEYRET Oct 25 '15 at 11:24
  • $\begingroup$ @chaslyfromUK actually, having perfect sine waves played a little out of phase between left and right channels is a technique used by synthesizers to create some particular sounds. $\endgroup$ – Cort Ammon Oct 26 '15 at 17:21
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Total destructive interference is requires unlikely circumstances. Starting from a textbook situation that you could observe interference (two sound sources of equal frequency and amplitude), I've listed the likeliest ways these conditions are broken when moving to more realistic conditions.

Imagine two tuning forks that are struck exactly at the same time with the same impulse, in a foam covered room to prevent reflected sound waves. In this case, the (time averaged) sound intensity is given by the classic two-source interference pattern (this can be observed!). The image shows such a pattern for various frequencies and distances - more on that below.

Complication 1: Volume (amplitude) modulation

This is probably the biggest effect: if our hammer accidentally strikes one of the tuning forks harder than the other, there will be no complete cancellation anywhere, and the interference pattern will be louder/softer regions from some background volume. This is particularly important because we perceive loudness logarithmically, so the sound cancellation will have to be really good for us to notice it.

Complication 2: frequency content

Tuning forks are designed to be quite monochromatic - this is in part due to their long tine geometry. Most sounds have harmonics (multiples of that frequency) and often other frequencies too.

We could replace the tuning forks with bells, for example, which have significantly more frequency content. These additional frequencies have very different diffraction patterns, and when you stick them all together, the tend to "smooth out" the diffraction pattern. You can kind of see this by noticing how different the patterns are for different frequencies in the image, although this is a little more complicated*.

Real life gets really complicated

Now put twelve singers in a concert hall. In addition to everything mentioned above you now get sound reflections from everywhere; virtually all the structure from our original diffraction pattern has been averaged out. FWIW, the human voice is, by these measures quite unruly as an instrument too: it has greater pitch and volume variation than most instruments, and quite a complicated frequency composition.

(caveat) *Additional frequencies will make the overall pattern more complex, but you can't just "add up" interference patterns to get a total; Field amplitudes can be added, but intensity/volume (what's plotted) cannot.

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    $\begingroup$ Perhaps the most extraordinary thing is that an experienced choral/orchestral conductor can hear if someone is singing/playing slightly off key or with an unpleasant tone and pinpoint that person. How the human brain can separate out the complete mishmash that is going on and convert it back to individual melodic lines is a real mystery to me (and I am a musician). Also we can do it in stereo. $\endgroup$ – chasly from UK Oct 25 '15 at 23:16
  • $\begingroup$ Yes, that is interesting! Actually, that as much to do with the fact that we have two ears. Each ear hears a slightly different signal, and that really aids in locating a source. You can try this: plug one of your ears at a cocktail party, and it's suddenly much harder to know exactly where people are. $\endgroup$ – anon01 Oct 25 '15 at 23:19
  • $\begingroup$ @chasly I think it's because the human brain processes incoming signals using an algorithm similar to autocorrelation, and not simply listening for certain frequencies. Even with one microphone, there are computer algorithms that can detect that somebody is singing off-key. With just one ear you can't find out who it is, though. $\endgroup$ – ithisa Oct 26 '15 at 3:01
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    $\begingroup$ IT might be worth adding the scale of the patterns for sound frequencies typical in voice. 440z is about 75cm so this will define the scale of a pattern for pure tone cancellation. This implies that it is unlikely that, even for pure tones, both of your ears will be in a zone where the sound cancels completely. $\endgroup$ – matt_black Oct 26 '15 at 18:45
  • $\begingroup$ Actually, a tuning fork is a bad example of a sound generator. It vibrates symmetrically (hence the vibration energy is not transferred to the holder and does not dissipiate quickly.) Looking at the tuning fork down its long axis, sound is emitted in one phase at 0 and 180 degrees, in the opposite phase at 90 and 270 degrees, and is zero at four intermediate points. If you hold a tuning fork with its long axis perpendicular to the direction of your ear and turn it about its long axis (by rolling its stem between your fingers), you will clearly hear this. $\endgroup$ – Level River St Oct 26 '15 at 19:57
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First, I will expand a bit on what Fabrice said.

consider two singers singing an A at 220Hz (to make the maths easy.) No real world singer or instrument produces a pure sine wave, so there will be harmonics at 440, 660, 880, 1100Hz etc. The harmonics will also, for different examples of a given type of instrument/singer, tend to have the same type of phase relationship.

So, if our two singers or instruments have the 220Hz exactly out of phase, the 440Hz will be perfectly in phase! Where low frequencies are missing, your brain tends to fill them in https://en.wikipedia.org/wiki/Missing_fundamental

Regarding the idea that singers may not be singing the same frequency: the human ear is extremely good at differentiating frequencies. The difference between one note and the next on a scale (one semitone) is a ratio of 2^(1/12) or about 1.059:1. The semitone is divided into 100 cents, and 10 cents is considered seriously out of tune. For a 100Hz note that's about 0.6Hz, so for a short note, you're not going to hear singers going in and out of phase with each other. For a longer note you probably will.

If you want to hear what two sine wave sources sound like, you need a low pass filter. The silencers on large diesel engines are quite effective low pass filters (this is not intentional, it's just easier to silence high frequencies than low frequencies.) If you hear two train engines running together you will hear the sound fade in and out (I've heard this sound more in the United States where trains are commonly pulled by multiple engines than in the UK.)

Also, low frequencies are less attenuated by distance than high frequencies. You can hear the same sound from a twin engined plane (preferably propellor) when it is flying high above you. Again, sound fades in and out as the engines go in and out of phase. Note that the engines are synchronized as closely as possible, because rapid beats between the engine frequencies are annoying to the passengers: https://aviation.stackexchange.com/questions/14263/what-is-propeller-engine-sync-and-how-does-it-work

One other point: for two sine wave sources in an enclosed space, even if the direct signal from the sources is perfectly out of phase, the sound reflected off the walls will probably have a different phase relationship. So you will be able to hear the sound, it will just sound as if it is coming from somewhere else.

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  • $\begingroup$ super answer...! $\endgroup$ – Fattie Oct 26 '15 at 12:05
  • $\begingroup$ Downvoter please explain the problem. $\endgroup$ – Level River St Oct 27 '15 at 20:14
  • $\begingroup$ 2 downvotes! that's a record for me! Yet no explanation! So how am I supposed to fix the problem? I can only assume that it's because you think the part about engines is irrelevant. That's staying, though. It's important to note that the phenomenon of destructive interference can be observed audibly, though not in the context of a choir. $\endgroup$ – Level River St Oct 29 '15 at 4:50
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The key thing to keep in mind is that the phases of the different signals (including the room reflections) are effectively random. In addition to exact frequency matching, complete destructive interference requires that the the two signals be $180^\circ$ out of phase. Because of the phases of the different signals are uncorrelated with one another, the expected power at the listener's position grows with the number of sources.

Consider the signals in the Fourier domain, the phase of the signal at a particular frequency will be effectively random. As you add more of them together, the amplitude at a particular frequency will undergo a random walk in the complex plane. Thus, the intensity (amplitude squared) at a particular frequency will tend towards the sum of the intensities of the individual components.

If we have an ensemble of sources, producing an complex amplitude at a particular frequency $a_j(\omega) e^{i \phi_j(\omega)}$ Note: I'm not requiring that the sources be pure tones, only that we're just considering one frequency ($\omega$ with associated wave number $k$) at a time, thus I'll drop the explicit $\omega$ dependence. The key point is that the $\phi_i$ is uniformly distributed on $[0,2\pi)$ and independent from source to source.

The complex amplitude at an ear, in an unenclosed space will be:

$A = \sum_j a_j e^{i (k d_j +\phi_j)}$

where the phase includes both the intrinsic phase of the source, and the phase accumulated over the path from the source to the listener.

Your intuition would seem to be vindicated in that if we average over the random phases, we end up with $\langle A \rangle =0$ (I'm using angle brackets for the averaging over the random phases. But this is deceptive because it is zero due to phase symmetry, i.e. the phase of $A$ is uniformly distributed. Whats more interesting is to look at $\langle \lvert A \rvert^2 \rangle = \sum_j a_j^2$ (the cross-terms average out to zero) -- that is the expected power of the net signal grows with the number of sources. One would get a similar result if you examined the expected amplitude $\langle \lvert A \rvert \rangle$ but the math is slightly harder.

Except for special arrangements of the sources w.r.t. the walls of the room, the reflections off the wall also have, effectively, random phases due to the different path lengths from the source-wall-ear (or source-wall-wall-ear and so on), these would appear in the equation as

$$ A= \sum_j a_j e^{i (k d_j + \phi_j)} + \sum_p \sum_i a_j r_{pi} e^{ i( k d_{pj}+\phi_j)} $$ where the additional sum is over the paths, indexed by $p$, with associated reflectivity coefficients $r_{pj}$ and path lengths $d_{pj}$. Note that for typical treble sounds have wavelengths on the order of a meter (or so), and rooms (esp. concert halls) are scaled at 10s of meters, so different paths will not have similar phases as the direct path.

Disclaimer I can't absolutely, positively rule out the possibility that there might be, for particularly strong singers, some phase-locking between singers (c.f. this video for a mechanical example) and thus the assumption of independent $\phi_j$ might be invalid. Even so, for most musical situations, the path-length phases are large enough to effectively randomize the phases at the location of the listener.

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  • $\begingroup$ Spectacular! (The metronome example is fascinating.) $\endgroup$ – Fattie Oct 29 '15 at 13:52
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They do constructively interfere. Our ear/brain sorts it out. We do not 'hear' sound waves. We sense spectra.

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  • $\begingroup$ give it some thought, any audiologist will agree with me. $\endgroup$ – SkipBerne Oct 30 '15 at 17:23

protected by Qmechanic Oct 27 '15 at 18:52

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