Physics of musical instruments?

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I know the maths behind how a plucked string generates frequencies based on its length. I also know that different shaped waveforms change the sound you hear, i.e. real instruments don't generate pure sine waveforms. But I've no idea about combining the two... let's consider a guitar for example. I guess I could find a sample waveform from a guitar string, but I'm wondering how I could theoretically calculate the waveform shape. Now, I'm not wanting to complicate matters considering an acoustic guitar, just the actual action of plucking the string when held across the fret. What makes this not generate a pure sine wave? Anyone know anything about this, or have links?

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A quick google for your topic title already gives you a lot of result. One of my colleagues has a book titled quite similar that explains all the maths behind instruments.

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The pages I found all just talk about ideal strings, and how the frequencies obtained are calculated. Of course acoustic instruments have massive resonance influences, but I'm thinking even an electric guitar doesn't make a nice smooth sine-wave through the pickup?

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Quote:
 Original post by d000hgThe pages I found all just talk about ideal strings, and how the frequencies obtained are calculated. Of course acoustic instruments have massive resonance influences, but I'm thinking even an electric guitar doesn't make a nice smooth sine-wave through the pickup?
I think it should produce a perfect sine-wave. Although, as you note, given the resonance of the body, sympathetic resonance in the other strings, etc. it might not appear that way.

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Check out the superposition principle and convolution, overtones, harmonics, acoustics and resonance (see also acoustic resonance and string resonance).

The quick of it is that a guitar string doesn't generate a pure sine wave because of the overtones and resonances. To calculate all that requires an understanding of differential equations. They are quite involved. Are you familiar with Fourier transforms? Signal processing?

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And I would imagine the reflection of waves off of the various areas of the body of the instrument, and their con/destructive interference with the originating waves would also affect the tone. Just think about the physics of sound waves.

FlyingIsFun1217

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Note also that the point at which you pluck the string will have an effect on the resulting waveform. Picking the string near the endpoints will favour higher-order harmonics, while picking near the center will favour lower-order harmonics. The math behind how a plucked string generates frequencies based on its length will tell you just the frequency of the fundamental harmonic, and consequently the frequency of it's other harmonics, but that doesn't tell which ones are present at which intensity. The final timbre will be a result from the combination of these simpler sounds. Plus, you can even suppress certain harmonics by lightly touching the string at special positions (the nodes for the wave), producing a sound that "doesn't have" the fundamental harmonic, and will be referred as simply harmonics by musicians.

You can get a sample spectrum for guitars here and for flutes here. If you have a guitar laying around you can record a sample and analyze it yourself using the alredy mentioned Fourier Transform. I could do it myself, but my mic is in terrible shape, it would end up with more noise than real sound.

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Quote:
 Original post by fcoelhoNote also that the point at which you pluck the string will have an effect on the resulting waveform. Picking the string near the endpoints will favour higher-order harmonics, while picking near the center will favour lower-order harmonics.
I can definitely relate that this is the case... plucking right at the end produces a much 'tinnier' sound... but why does pluck position have this impact? I'd think the energy in the string would distribute along the length really fast but evidently that's not the case.

It would be really cool to see some super slow-motion video of a plucked guitar string, anyone ever come across such a thing?

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Most enlightening, thanks.

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