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in my attempts to synthesize a plucked electric guitar string (using supercollider's digital waveguide, which is based on the Karplus-Strong algorithm) I came to a dead end and decided to look that the Cepstrums of a sampled string and my synthesized string, to nail down the audible differences. They look like this (A-string - 110Hz)

Cepstrum of smapled and synthesized plucked A string

My questions are:

  • The sampled signal has something in the right half, which the synthesized signal does not. I cannot see how to interpret this as "excitation". So what does it stand for?

  • Is it safe to say, that the spectrum of the synthesized spectrum is more "harmonic" than the one of the sampled signal? When I look at the spectra themselves it does appear this way. The sampled spectrum (left) looks more "noisy")

Spectra

  • Could it be that the regularity (and the dull sound) of the synthesized signal is a consequence of the digital waverguide itself. The way I understand Karplus-Strong, after some time only the fundamental and its harmonics survive, which would indeed lead to a "harmonic" spectrum.

  • If that is the case, what could be the cause of the irregularties of a real plucked string. IOW: what does a real string have, which Karplus-Strong doesn't have?

  • As a side note: what does the sharp peak at the end of the Cepstrum stand for? Is it caused by the 1 second window?

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  • $\begingroup$ What was the microphone placement? Room acoustics? 6 string or 12 string guitar? How was the guitar (all stings) tuned? (equal tempered, just, etc.) Were the other strings muted, fretted, or open (thus left free to vibrate sympathetically)? $\endgroup$ – hotpaw2 Jun 1 '14 at 15:00
  • $\begingroup$ I didn't sample the guitar myself. All I know is that it was a 6 string Fender Telecaster with both pickups used. I doubt that a microphone was used, but sympathetic vibration might have occurred. $\endgroup$ – Martin Drautzburg Jun 1 '14 at 15:17
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Real strings don't have a perfect integer overtone series. Responsible for this is the stiffness of the string that gives higher frequencies (i.e. shorter wavelength) a greater force towards the equilibrium position and increases the frequency of the higher order modes.

This small inharmonicity is responsible for the general shape of your cepstrum. Our auditory system still manages to assign a single pitch sensation to the guitar string signal. This sensation does however not have to coincide with the cepstral peak you see, because that depends a lot on the duration of your signal frame and the window you apply to it. You can probably tune these parameters so that you get a reading that matches the nominal pitch of the guitar string, but it will then not give the right result for other inharmonic signals with a clear pitch sensation.

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IIRC, Karplus-Strong is usually a single linear harmonic waveguide. Plucking a stringed instrument starts several things in vibration other that the string (sound board, air in cavity, etc.), many of them producing inharmonic overtones due to material stiffness, thickness and non-uniformity, some of which exchange energy and/of change in a non-linear fashion over time. The sharp peak high in the Cepstrum could indicate some kind of tiny but sudden frequency modulation(s) of the fundamental pitch.

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