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Here is my problem: suppose I am a composer you have contracted to create any sheet music I dream of with the unique condition it is not pure silence and that I don't stipulate the instrument section (which means, don't specify the timbre). Suppose you are the one who is in charge of the execution of the music, and you have in your disposition any kind of instrument, room, high tech equipment you ask — and suppose also that you hate me. Could you just by changes on the non-fundamental harmonics and non-fundamental frequencies aspects of the sound wave change my original composition so that it sounds something like Baby Shark melody or whatever you want?

Well, I will try to put this question on better words.

What I learned about timbre is that it is defined by the wave format and the envelope of the sound. So the same fundamental frequency can have different wave formats, a perfect sine wave or some irregular pattern, like a saw, or other things. Others definitions seems to point that timbre is about the harmonics analysis of the sound. I think these two definition are connected, because the wave format determines the harmonics of a sound (is that right?).

But if we just think about the sound wave, it could be mathematically decomposed on a sum of sine waves (I remember people saying this is the Fourier sum). So, in thesis, you could reproduce an approximation of the timbre of an instrument on a computer by this process. I guess this is what people do when they try to imitate a violin on a computer. Equivalently, I think that in theory you could control also this wave parameters so that you could create a sound corresponding to any wave format you want. I think this is what people who synthesize sounds do.

I understand also that when you play two instruments at the same time the air molecules make some kind of sum of the waves and this is why you could have the destruction phenomena of waves that cancel each other and produce silence.

Now this is my thought process to arrive at this question: I think you could, given a sound source, to change a second sound source so that the sum of the two waves produces another fundamental frequency different of the first instrument, because you can cancel the fundamental frequency by the above process and just let the harmonics of the two sounds sources audible. I think the question is if there is some way to make a destructive interference of the fundamental frequency, and resonance of the components of the sound wave, such that in the sum of the waves you could determine the fundamental frequency of each individual instrument, even if this fundamental frequencies of the instruments were determined before they were played.

But if you could do this, perhaps with a sufficient number of sound sources (you could even repeat the same note if necessary),you could change the fundamental frequency of the first instrument when you sum up all the waves of the sound sources to some other harmonics you determine.

But this is all just second hand information I received, and this is why I am here to ask you if this makes sense. The important thing to me is not to have an answer, I just to understand if what I am thinking makes sense, what are my errors, and how can I do to make a better question. Besides, I think interesting to think what kind of parameter you could relax in this experiment to have a better result. For example, if someone have all the power of the sheet music and the timbre of the music, how much could he play with this interference so that he make a music with harmonics.

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    OK so I hate you but I hire you to write a composition for me that I then try and make sound different and worse than what you wrote even though I paid for it? That plus mentioning Baby Shark song gets a +1 from me. The question itself is very deep and almost better suited to science than music, I look forward to the responses. – John Belzaguy Jun 23 at 1:32
  • I'm pretty sure the answer to your question is yes, but I'm struggling to see what you hope to gain by going to the trouble of finding waves that would cancel the original composition out, when in most cases you could take the simpler approach of just not rendering the original composition at all. (I can guess that perhaps you might be thinking this will facilitate some interesting cross-fading out morphing? But I'm not sure that's what you mean at all...) – topo Reinstate Monica Jun 23 at 6:17
  • If I am constrained to non-fundamental frequencies, doesn't that imply that the resulting waveform must contain the fundamental frequency? Can I add non-harmonic "overtones"? Are you asking whether it's possible to invent (1) an instrument such that when you play one tune on the instrument the listener hears another tune, or (2) two instruments such that when you play tune A on one, the listener hears tune A, but when you add the other instrument, the listener hears tune B? – phoog Jun 23 at 14:53
  • Does a sampler count as an instrument? i.e. a device which plays, say, "You Spin Me Round (Like A Record)", at different pitches, every time you press a key. – user253751 Jun 23 at 15:29
  • @John Belzaguy, I think this 007-like villain plot came out of some kind of inner desire to humiliate the figure of the romantic composer who love his composition more than anything, haha. Yes, I first asked this question on mathematics stack exchange, then they transferred the question to physics community, then they close the question and said it were more adequate to here. physics.stackexchange.com/questions/560344/… – Lost definition Jun 25 at 8:49
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As John Belzaguy said, this is much about science, I'll try my best to put that in a musical point of view.

In theory, what you are are saying is yes: two waves can cancel each in the process called destructive interferences. This is the process used for tuning a guitar using harmonics. However, there are a few problems…

Coherency of the waves

In order to interferes, your (let's say two) waves need to be coherent: they should have exactly the same frequency, and a phase ratio between them which is constant. This is hardly possible when using different sources, and is why, while light is also a wave, you do not a have black spots appearing when you light an additional light bulb: you first light bulb, and your second are not coherent one with each others.

The only this could be possible is with stereo digital instrument, which can have a very good control on the emitted wave. Here comes a second problem:

Spatial interference pattern

Let's say you have your instrument (I'm using the stereo digital synth for instance) producing two waves that should interfere with each other: exact same frequency, in anti-phase (when one is at maximum of pressure, the other one is at minimum). While, outputting this in mono would actually cancel the wave, in real life your waves need to fulfill the anti-phase condition at the precise spot you are listening. Unfortunately (or fortunately I do not know), as your two waves propagate, the time it will take them two reach your ears will not be the same in the general case. So even if the waves were in anti-phase when emitted, you ones you ear have no reason to be in anti-phase. With pure waves, here is the kind of patterns you can expect, I let you imagine that, in 3D it is a nightmare… The spots of destructive interference are also dependent on the frequency of the wave… So you'll need to move according to the music (isn't that dance?!) to follow the destructive pattern to keep the effect.

To solve this at any point, you two sounds emitters need to be very close. This is why tuning a guitar using interference works, however, if you succeed in producing no sound with two strings on a guitar, please send me the video!

I typed that fast, if any point is not clear, do not hesitate and I'll edit.

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  • "if you succeed in producing no sound with two strings on a guitar": I don't suppose the vibration modes of a guitar's bridge are such that two strings could cancel each other out; even if they were 180 degrees out of phase, the bridge would probably be twisting somewhat. The wavelength of the high E string is about 104 cm ((343 m/s) / 329.63 Hz). If you put a couple of guitars a couple of meters apart, tune their E strings to the same frequency, and play both E strings, you should at least be able to find a spot somewhere between them where the amplitude of the pitch is much smaller. – phoog Jun 23 at 14:49
  • @phoog that was exactly my point, there are a lot of additional effects which makes this possible. Also, tuning two different strings to the same frequency is likely to be impossible in real life. You may have a beating sound, just like with harmonics tuning. – Tom Jun 23 at 14:57
  • Musicians regularly tune strings to avoid beats. It's not impossible at all. – phoog Jun 23 at 18:17
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    @Lostdefinition If you are in the close field of the sound emitters, those "black" spots (if they exist) will be located at different space location depending on the frequency. In that case, you need to move around to stay in a destructive interference node. – Tom Jun 25 at 9:52
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    @Lostdefinition If you are far away, and have no echo like in an anechoic chamber, situation is much more easy. If you allow yourself an electronic device in the path between the instruments and you then you possibly could interfere with them before they enter your ears. For noise cancelling headphones, this part is made easy by the fact they are very close to your ears. – Tom Jun 25 at 9:54
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Destructive interference would effect what you hear, but that doesn't change the composition.

You could do other things to make a performance or playback inaudible, like move very far away or put a jackhammer next to the listener. Interfere however you like, that won't change the composition, because the composition is purely conceptual.

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  • Yes, I think the way I made my question allows your answer if you say the noise resulting of the jackhammer is what you want to your music sound like, hehe. I guess in this conceptual view my question is even ambiguous, because timbre is part of a composition as well. I think your answer gives an afirmative answer to the question, but I want to exclude this cases to think not chaotic musics. Nice philosophical point. – Lost definition Jun 25 at 10:15
  • @Lostdefinition, how does... "and suppose also that you hate me" ...have any bearing on the question? I just ignored that part, but if you really want an unambiguous case, I think the question needs to be re-written. It seems like your question is more like: can waveform interference make a listener hear another, arbitrary composition, example, Beethoven's Fifth is playing, but waveform interference causes the listener to hear Happy Birthday played by a flute. – Michael Curtis Jun 25 at 15:00
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You can do this. All you need to do is invert the signal, i.e. generate a waveform that is the exact negative of the input waveform and add it to the input so there is no signal left. This is how noise-cancelling headphones work. Then you add in your own signal.

Why anybody would go to the trouble of doing this, is a completely different question.

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    This answer makes it clear that this is essentially a math question. – Max Jun 23 at 6:49
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    "Why anybody would go to the trouble of doing this": it's an interesting engineering exercise. But the question limits activity to "changes on the non-fundamental harmonics and non-fundamental frequencies." Would it be possible to cancel out the fundamental under that constraint? – phoog Jun 23 at 14:36
  • It seems that your answer, although not wrong, completely misses the point. Noise (all sound) cancellation is easily accomplished by 180 deg phase shift copy of the signal added to the original. This does NOT create new frequencies, or change the fundamental pitch of the song. – ggcg Jun 23 at 14:46
  • @ggcg Adding new frequencies is the second part of the operation, although the new signal could probably be mixed with the phase-shifted one – PiedPiper Jun 23 at 14:50
  • @ggcg You could noise cancel low-frequencies only, which can be easily done with a low-pass filter. If the frequency difference between the fundamental and the overtones is great enough (that might be the hard part) you could noise cancel only the fundamental. The first overtone would then, for the ear, become the fundamental. – Tom Jun 23 at 15:52
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You seem to be asking more than one question, or convoluting several ideas together to create this scenario in your mind. Almost everyone has offered an answer that addresses "cancellation", which would turn off the sound. But you have stated...

"I think you could, given a sound source, to change a second sound source so that the sum of the two waves produces another fundamental frequency different of the first instrument, because you can cancel the fundamental frequency by the above process and just let the harmonics of the two sounds sources audible."

No, by the process of linear superposition you cannot do this. You will not change the fundamental. There are several problems with this line of reasoning. Perhaps too many to address in one post. But the one that really sticks out is a process in the brain called Fundamental Tracking. By this process our brain takes the harmonic spectrum and get the fundamental from the mathematical pattern. The brain "perceives" the fundamental even when it's missing. The next thing that gets in the way of faking out a changed fundamental is that the ear is non-linear and creates Aural Harmonics. So, as long as you have enough data present the ear will feed the brain what it needs to get the "true" fundamental.

That does not mean that faking out a fundamental is completely impossible. If the listener is fed two notes that are very close together, close enough to not be distinguished, then they will hear or perceive the average of the two called a sum tone. But they will also hear a beating. This would not be perceived as a music idea and, for example, you cannot turn Einen Kleinen Nachtmusik into Whole Lotta' Love using this, or likely any, technique.

You are creating a scenario in which you generate music with NO particular timbre, either as sheet music or a a wave file. In the latter case this theoretically impossible, then asking can one alter the wave to the point where they have not theoretically altered the music but changed to tune to something unrecognizable. On its face this is not a well posed question. And it is not possible without completely rewriting the music. At that point it's not worth editing the waveform.

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  • Very interesting answer! Thanks, I will try to learn more about this fundamental tracking effect. Yes, I put my question on a abstract timbre point of view because I wanted to understand better the relation between the waveform and its sound sources interference. Yes, there seems to have a point of inflection when a wave cannot be perceived as organized and perceived as some unrecognizable sound. But I don't understand why this make the question not well posed. I want to understand the limits of this in view of sound interference. – Lost definition Jun 25 at 11:12
  • And also understand if what I am saying makes sense.. – Lost definition Jun 25 at 12:07
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Your question pertains to physics. First of all, you mentioned using destructive interference to produce silence and backed your argument by the example of noise cancellation process in headphones. True but not always. In a 3D space, the sound waves are generated in all directions and in the form of longitudinal waves(in air) so this makes perfect noise cancellation impossible. Moreover, to achieve silence by this process, one needs to replicate the original waveform with a phase difference of exactly pi radians, which is also nearly impossible while performing the music. Furthermore, in a 3D space, sound waves are generated in all directions so to experience the effect of destructive interference, one has to be at a very specific point in space given that the waveform doesn't change. If it changes with time, the person has to be at specific points in space at specific points of time. This would be a rather tedious feat.

You asked if it would be possible to produce different frequencies by playing different instruments and combining their waves. I don't actually fully understand what you mean to ask. I mean if you're gonna play different instruments and combine their sounds to generate different frequencies (which is very very very difficult), why not play a single instrument at a different frequency? Well, you mentioned that you need to change a composition in a way that the composer couldn't blame anyone but physics. It is actually difficult to do. What can be done is that one can use different instruments to change the feel of the song, intercalate beat notes in the spaces between the original beat to craft an illusion of complex odd time signatures. For example, vardavar by Tigran Hamasayan. You should listen to this song. It uses a 4/4 time signature which sounds nowhere near like a 4/4 but it is in fact a 4/4. Check out YouTube for it. By changing the instruments to change the feel of the song, manipulating rhythm section and doing stuff like that, one can easily make a composition sound entirely different than what it was intended to sound like and keep it pleasing to ears.

Manipulating complex waveforms is a lot difficult than manipulating simple sine waves. Short answer to your question, no. Long answer to your question, I think a simple no would suffice! 😄😄

All this is what I deduced and if anything I have written is wrong, I'd appreciate rectification. What you have asked is rather interesting and I'd like to see it in action if possible.

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    Thanks, I loved to know Tigran Hamasayan music. I asked this on physics but I think they didn't understand what I mean there. My original intention was to understand this concepts of timbre and wave interference better. I am an amateur musician, with very poor physics knowledge, and in reality I have a more philosophical intention to research this subject: to know how much of the material condition of instruments and physics influenced our concept of music today. The question about universality. And sometimes I get carried away by these questions. – Lost definition Jun 25 at 11:51
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    But if some musician tries to make some kind of musical experiment based on this I think this would be very interesting too, though now I understand it seems to be a very complicated to solve this air complexity problem. – Lost definition Jun 25 at 11:54
  • @Lost definition it's good that you have a philosophical approach to matters which should be actually dealt philosophically. I mean it's not fully helpful to know anything when you don't actually ask questions about its significance and influence it has. You know what I mean. The timbre has a huge huge impact on music as we know it. I mean, play für elise on a ukulele, Latin rhythm, bonga drums, shakers and see what you get. Tell a Norwegian death metal band to cover it and you get something entirely different. So yes, change the instruments, change the music(a major aspect of music). – John1085 Jun 26 at 0:57
  • I'm happy that you asked this question. These kind of questions need to be asked and people don't even think about them. – John1085 Jun 26 at 0:58
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Can you change some composition I wrote to whatever music you want just by changing the timbre of the instruments?

If the arbitrary timbre for the instruments must be fixed over the performance of the music, then no.

For instance if you make a composition that consists of nothing but a single voice playing the same note, no choice of alternative timbre will turn it into a melody.

For similar reasons, a tune that starts and ends on the same note won't be turned into one that starts and ends on a different note and such.

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You can do this on an ordinary PC or Mac these days. There is even some free software out there. You can manipulate and overlay waveforms to your heart's content. You can choose from accurate-sounding musical instruments and write a full orchestral score, then play it back - all on the computer.

You could easily make some of Mozart's early pieces sound like Baby Shark. Just pick the best part of the main melody and repeat it endlessly. Choose a ridiculous combination of instruments that sound suitably childish and, Bingo!

I'm almost tempted to do this myself now!

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The difference between rock-and-roll and country music, say, isn't just the instruments; they also tend to use different chord progressions. You can't change the chord progressions just by changing the timbre.

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