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I recently learnt of the mathematical model behind simple forced (undamped) oscillations, and typed the equations into Desmos with some parameters here, where something subscript F means something related to the forced oscillation and something subscript N means something related to the natural oscillation.

I got some nice waveforms such as this one:graphed oscillations

and I thought I recognised this from a little research I once did on sound synthesis, where the original wave was modulated to make patterns like this one. Is this simple model a good basis for how the piano works? A hammer hits a string, which forces an oscillation, and the string also has a natural vibration, and it produces a complex sound. Is a perfectly tuned piano a masterful piece of craftmanship where each string and hammer is of a specific size/weight/tension/whatever such that each note's forced oscillation is a semitone in pitch below the next? i.e. would plugging something like this into a computer model produce authentic sound, if the parameters were appropriately tweaked (to include also damping of course)?

Thanks.

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It could be the beginning of a model for musical instruments. The forced oscillation of a hammer strike, bow movement, lip reed, etc, is not a simple sine-type wave. In the case of a hammer strike or finger pluck, the action contains very many simple frequencies due to the Fourier theorem.

If the frequencies in the strike match any of the natural frequencies of the string (they probably will!), they will persist longer than other non-matching frequencies due to the boundary conditions of the string and those are the frequencies which will compose the tone you hear (a complex sound, as you call it.) The non-matching frequencies QUICKLY damp out.

Even vibrations from other strings can cause a string to vibrate. As an experiment, gently press (without striking) and hold the middle C, then quickly strike and release the F below the C down an octave. You will hear the middle C resonating. That's because the middle C frequency is one of the frequencies in that low F string. Release the C and it will stop.

Musical physics is fun!

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  • $\begingroup$ Is it only empirically derivable which frequencies arise from which strings and hammers? $\endgroup$
    – FShrike
    Commented Apr 24, 2021 at 16:15
  • $\begingroup$ The frequency content of the strike is very dense and will contain practically all the natural frequencies of the string. How strong each frequency is depends on where the strikeoccurs on the string and how hard the hammer surface is. Near the end of the string, the higher frequencies will be stronger. The natural frequencies of the strings are derivable, to a first approximation (ideal strings). Search on natural frequencies of string fixed at both ends. $\endgroup$
    – Bill N
    Commented Apr 24, 2021 at 16:54
  • $\begingroup$ Thank you. So you're saying you can fairly well approximate the sound from just the natural frequencies of the string, as the hammer will generally reach those and not contribute much to the sound with any non-natural frequencies as those die out quickly? $\endgroup$
    – FShrike
    Commented Apr 24, 2021 at 16:57
  • $\begingroup$ I've tried looking up questions like this before on the internet with the theory of sound synthesis and how real instruments are approximated with which envelopes and waveforms etc. but this knowledge is quite hard to come by and is not easy to find $\endgroup$
    – FShrike
    Commented Apr 24, 2021 at 16:57
  • $\begingroup$ To a first approximation, yes. In reality, the strings are not ideal, so they will have some slight non-harmonics that will depend on string construction (material, solid vs wrapped, etc). That hardness of the hammer, as I said before, will affect the strengths of the string overtones, but hammer vibrations will have negligible effects. $\endgroup$
    – Bill N
    Commented Apr 24, 2021 at 18:14
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It is more complex than how the string vibrates. Strings themselves produce little sound. They set up vibrations in the body of the instrument, and the instrument sets air vibrating.

It is still forced oscillation, but calculation which frequencies are amplified how much from the shape of the instrument would be very complex. See How does a violin work?

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  • $\begingroup$ How does anyone ever make a realistic violin sound in a synth then? $\endgroup$
    – FShrike
    Commented Apr 24, 2021 at 18:27
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    $\begingroup$ You can play a violin and analyze the sound. Once you know the frequencies, you can get a violin like sound. Plus you need the right intensity over time, which is what Attack, Decay, Sustain, Release is for. But it is harder to predict those frequencies from the shape of the instrument. $\endgroup$
    – mmesser314
    Commented Apr 24, 2021 at 18:39

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