# Why don't two musical instruments sometimes generate destructive interference?

I'm an electrical engineer, and I understand wave propagation, interference patterns, and so on. But I'm missing something basic, so perhaps my understanding isn't as good as I believe. I'll show my thinking; please tell me where I am mistaken.

Say that I have two guitars. Each one is precisely tuned. Each guitar has somebody play an open-string E note.

The result is a louder note, which implies to me constructive interference. But, if this is true, why is there never a destructive "noise cancelling" effect? Obviously, both guitar strings aren't always going to vibrate in the same phase.

• Unless they're playing harmonics, you're not getting anything close to pure notes. You can see this under a signal generator. – Jerry Schirmer Jun 1 '14 at 14:18
• But if you use two speakers/amps from the same guitar in different locations in a closed room, you will get all kinds of frequency-dependent cancellation without even trying. The problem then with two guitars is getting them to stay in a fixed phase-difference, which is almost impossible. – RBarryYoung Jun 1 '14 at 17:40
• I was told that the distortion effect of Mike Oldfield's "two slightly distorted electric guitars" on Tubular Bells is exactly this. – Mr Lister Jun 1 '14 at 20:29
• If they have preciesly the same frequency, regardless of their relative phases, assuming no multipath or distortion, and completely free space, there would be at least one point at which they completely destructively interfere. – wjl Jun 2 '14 at 4:17

You correctly diagnosed the problem in the last sentence. The problem is with the phase.

There is no interference happening here. The two sources do not maintain a constant phase difference. When interference occurs (with a constant phase relation between the two sources), you will have a net intensity of $(E_1 + E_2)^2$, which is four times either if they are equal. In the destructive case, the net result gives $0$ intensity (for a phase difference of $\pi$). However, when there is no constant phase relation, the phase difference would be randomly distributed between $0$ and $2\pi$, and in this case, you just have the average intensity adding up, to give $\langle E_1^2 \rangle + \langle E_2^2 \rangle$, which is 2 times either if they are equal. That's what you hear, and mistake it for constructive interference.

Sound waves do diffract just like other types of wave.

Actually it's not hard to get the two guitars playing in phase (or at constant relative phase). They just need to be in tune, and the human ear is very good at detecting even small differences in frequency. However the guitars aren't point sources, they don't produce a pure tone and in most settings there is lots of scattering and reflection of sound from nearby objects and the walls of the room. The end result is that any diffraction pattern is blurred out.

In more carefully controlled experiments you can hear the diffraction pattern. See for example this video, or Google for many similar videos.

• Some folks have been known to tune instruments by listening for beat frequencies between the strings; that's a phase-interference phenomenon. – keshlam Jun 1 '14 at 15:24
• @leftaroundabout While they do produce different spectral sounds most of the energy is contained in a pretty tight frequency spread. If they didn't then you wouldn't be able to "tune" them. Trained ears can hear within 1-2 Hz of a difference which statistically would put them in phase in many 3-d diffusion patterns. However these areas are very small so both ears won't hear canceled or reduced effects at the same time. It just makes the sound more "dynamic". – user6972 Jun 1 '14 at 20:06
• @keshlam: most musicians do this when tuning by ear. It works by far best on two strings of a bowed instrument because the interference plays into the phase-locking loop. Even on a guitar, the beat is very audible when playing a detuned unison, but here it's mostly the change in the harmonics' relative amplitudes you hear (the ears being always better at "relative" than "absolute"). There's not normally at any time destructive interference as the OP asks for, i.e. such that the total loudness (RMS peak, whatever) is less than that of either of the single strings. – leftaroundabout Jun 1 '14 at 20:09
• @user6972: "almost all of the energy in fundamental" is profound nonsense except for some special cases (e.g. glockenspiel). In a lot of instruments, in lower registers the fundamental doesn't even have the strongest amplitude. Anyway, I don't think this is the right place for that discussion... – leftaroundabout Jun 1 '14 at 20:44
• @user6972: Wait, what? How do you get "fundamental tone" from "first five harmonics bins"? Looking at the paper (e.g. Figure 2), it seems pretty clear that they're saying that most of the signal power is in the first five harmonic frequencies, i.e. from the fundamental to the fourth overtone; that's not the same as saying that the signal power would be concentrated in the fundamental frequency alone. – Ilmari Karonen Jun 2 '14 at 20:42

Your conditions are harder to achieve that you may be aware of. To achieve full destructive interference, the signals need to have exactly the same shape / phase and amplitude. This can easily be achieved with a recorded or synthesised signal, but in manually played instruments it's rather difficult to replicate all parameters to the required precision. You can do it: record an electric guitar straight into a DAW, twice the same open string struck at exactly the same place with exactly the same force, then cut the notes onto two tracks and align the transients. Flipping the phase on one of the tracks will make a quite notable difference then, but even here the out-of-phase version won't be completely silent: it will be quiet and very thin (i.e. the low harmonics cancel well), but the attack and some higher harmonics will always remain audible.

• The transient shape depends subtly on incontrollable microscopic details (exactly when the string slips off the pick/finger, and the friction pattern from there till it swings freely)
• The higher you go in the harmonics, the harder it gets to align them precisely anti-phase. This applies both to the place you strike the string (which is what determines the initial phase & amplitude relations between the string's harmonics) and to the way you superimpose the recorded samples.

Both of these points get much stronger when you're not dealing with twice the same guitar playing the same note under the same circumstances, bu merely different players playing similar guitars under similar circumstances. Firstly, they'll have a hard time playing precisely enough in time so the transients are simultaneous – that can only work in one spot in the room anyway, a couple of inches left or right will already get much of the signals partly in phase. And even with electronically triggered, syncronised attack of the strings, you still can't control the exact harmonics' interrelation of either guitar. Indeed different guitars won't even sport exactly the same frequencies of harmonics, because of slightly different inharmonicity. The best you can do is tune the fundamentals to match exactly (use an FFT scope and stay to small amplitudes). Then you can achieve exactly-cancelling fundamentals, but even for the second harmonics it will be much more difficult.

Indeed partial, even full cancellation of the fundamental or other single harmonics keeps happening all the time in ensemble playing – always just for a moment. And indeed this happens even when only a single instrument plays, because room reflections make for comb filtering effects. Because it happens so often, our ears largely compensate / ignore this kind of phenomenon. A single missing harmonic normally isn't noticable, and even for the fundamental it's not a big deal.

• This is excellent information, thank you. It's too bad I can't accept multiple answers :) – bitsmack Jun 10 '14 at 16:53
• Really is! It enhances my answer by taking it further and educated me also about issues I didn't know. +1 (deservedly). – 299792458 Jun 11 '14 at 7:39

Single instrument, fixed location

Musical instruments absolutely can and do cause destructive interference. It happens all the time. Usually, it's only noticeable if the two notes are in tune within a few Hertz, or even a few decihertz.

This causes the audience to hear the interference as "beats", where the magnitude of the sound goes up and down at a noticeable frequency. Piano tuners use this to tune pianos: this task is made easier since the speed of the beats is equal to the difference (so two strings at 100 Hz and 100.5 Hz will "beat" every two seconds), the strings for a particular note are largely identical, and they have extreme control over the pitch of the string being tuned. A slightly out of tune piano will have noticeable beats.

The more complex the tone, the harder it is to hear the beats. It is most noticeable in high-pitched instruments emitting purer waves (e.g., flutes). It can be a real problem in recorder consorts, to the point where being precisely in tune is more painful than being a few cents (hundreds of a half-step) out of tune. In electronic music, where musicians have much more control over the pitch and shape of the wave, some artists have utilized this as a specific effect.

The interference itself can also become a third voice; in classical harmony, this reinforces the chord. For example, the fifth of the scale is 50% above the fundamental; for a chord centered on 100Hz with a fifth at 150Hz, there is a 50Hz beat tone--which is the octave below the fundamental, and therefore sounds "in tune"! This is one extra reason that anharmonic chords sound "more crunchy"; the interference tone itself is dissonant as well.

Multiple instruments

If two instruments (or more--but it's most obvious with two) are very nearly in tune, the audience will hear beats as well. Typically most obvious in the upper register with a pure tone, which means it's mostly heard with flutes and piccolos, soprano singers, and the like.

(This leads to the musician's 'joke': "How do you get two sopranos to sing in tune?" "Shoot one.")

Sometimes it is best to intentionally play out of tune, 10 cents or more, so that the beats are not as obvious.

Reverberation and acoustics

In a well-prepared concert hall, the space itself is constructed in a way to provide multiple bounces for the sound to arrive at every audience member. This is on purpose, to reduce the occurrence of destructive interference. Some halls have well-known "dead spots" where performers cannot be heard, or where audience members cannot hear; modern acoustic theory uses diffusers to avoid this.

Contrariwise, in an unprepared space (e.g., a flat rectangular closet with hard walls), it is easy to hear the interference caused by reflection, and you can set up standing waves where you can increase or decrease the amplitude by moving your head around.

• This is excellent information, thank you. I can tune a guitar by listening for the beat frequencies, I just never knew what they represented... Also, thanks for the info on harmony. It's too bad I can't accept multiple answers :) – bitsmack Jun 10 '14 at 16:55

The issue is that guitars, unlike electronic oscillators which can produce nearly pure tones, are not entirely coherent. If you have two pure (coherent) tones, then if you line up the phases just right, they cancel and you get nothing.

But guitars don't produce pure tones: they produce a lot of odd harmonics and also some random noise from the string sliding off the pick. The relative phase and amplitude of these harmonics won't be the same for two guitars. They are randomized by the unique wood in each guitar and the particular way in which the strings are plucked.

Consequently, you can't get a complete cancellation as you can with say, radio transmitters. You might cancel any one of the frequency components of the sound, but the relative phase and amplitude of all the other harmonics in the sound will not be the same, so you can't simultaneously cancel those. You will never completely cancel the sound from two guitars: you can at best achieve partial cancellation.

This partial cancellation is easy to hear, and familiar to any musician: the "beat" that's audible when two instruments are very slightly out of tune is the result of alternating constructive and destructive interference.

• Thanks, @PhilFrost, this is helpful. Nice to see you on non-electrical stacks :) – bitsmack Jun 10 '14 at 16:57

Both transverse and longitudinal waves exhibit the phenomenon of reflection, refraction, diffraction and interference. Polarization is the exception, which is only exhibited by transverse waves.$_1$

Sound is transmitted through gases, plasma, and liquids as longitudinal waves, also called compression waves. Through solids, however, it can be transmitted as both longitudinal waves and transverse waves$_2$

In interference and diffraction, energy is redistributed. If you consider interference of light, if energy reduces in one region, producing a dark fringe, it increases in another region, producing a bright fringe. There is no gain or loss of energy which is consistent with the principle of conservation of energy.$_3$

As Sound is also a form of energy. Similar to light energy, Sound is going to cancel at only one region, but increases in another region. The effect might not be observed because of foreign sound. The video from John Rennie Sir has proved the diffraction of sound waves.

Credits: $_1$ Modern ABC Physics-Part 2-Polarisaction of light-23rd edition-page1021 $_2$ Wikipedia-Sound Waves $_3$ 2nd PUC-Karnataka-Physics-Part 2-Wave Optics-2013 edition- page372

• This is well sourced, but doesn't seem to answer the question fully. – jwg Jun 2 '14 at 13:39

In addition to the other answers, I'd just add that you certainly can hear phase-related effects under some circumstances, but they don't give rise to simple cancellation for all the reasons given. But you can certainly tell the difference when one speaker of your stereo is wired in reverse, or if you move around in an environment where there are multiple speakers. I used to have a home studio years ago, and one of my favourite production tricks was to use a simple op-amp inverting amplifier to split something like rhythm guitar into two antiphase signals, and pan these hard left and right on the desk. This would have the result that the guitar sat back in the mix, kind of spread out, leaving space centre-stage for lead guitar or vocals . . .very different, subjectively, from similarly splitting and panning one signal in-phase left and right.