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I play a kalimba, and have also recently written a toy program that helps me tune it using a "naively applied FFT" without any sophisticated DSP. The Nyquist frequency is 24 kHz.

The kalimba is tuned, so the fundamental for C4 shows as very close to 261.6 Hz. The biggest surprise is that there is no significant harmonic energy, and significant, narrow, stable anharmonic energy between harmonics 6-7. The biggest non-fundamental peak shows up at about 1.714 kHz which is 6.55x the fundamental. This 6.55x component appears in almost all of the notes in the first octave (C4-C5).

spectrum

I've ruled out interaction with the other keys by manually dampening them while C4 is plucked. What physical mechanism could plausibly be creating this 1.714 kHz peak?

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    $\begingroup$ In general, there’s no reason to expect that the frequencies will be evenly spaced. That happens for some waves, like ideal waves on a string or sound in a tube, but not for others. $\endgroup$
    – knzhou
    Jan 13, 2021 at 23:33
  • $\begingroup$ The waves on a kalimba have rigidity as the restoring force, not tension. Plus they have all kinds of weird boundary conditions... $\endgroup$
    – knzhou
    Jan 13, 2021 at 23:34
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    $\begingroup$ Nice, simple harmonic series are a mathematical fiction. ;) They work ok for linear oscillators, but not so well for more complex oscillators. A reasonable introduction to this topic is via the study of bells & chimes. Eg, en.wikipedia.org/wiki/Strike_tone There are some great articles on the acoustics of bells, chimes, and idiophones by Thomas D. Rossing, but I'm not having a lot of luck finding free material... $\endgroup$
    – PM 2Ring
    Jan 14, 2021 at 0:26
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    $\begingroup$ Almost every real-world structure has anharmonic overtones. A vibrating string is a very special case. Real musical wind instruments with can also be very anharmonic, but the tone is produced by forced vibration not free vibration which hides that fact. $\endgroup$
    – alephzero
    Jan 14, 2021 at 0:41

1 Answer 1

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The kalimba tine is modelled as a cantilever which is clamped hard at its fixed end but which is elastically preloaded into hard contact upon a bridge point some distance away from the fixed end. the constraint at the bridge point prevents up-and-down movement but the tine is free to vibrate in a mode that has a displacement node at the bridge point. In this mode, the tine length between the bridge point and the clamped end can support a fundamental vibration which will show up in a spectral analysis, and that mode will have no harmonic relationship with the vibration of the free end of the tine.

In addition, an ideal cantilever, when struck, will itself not produce a harmonic series of overtones as would a plucked string because of the boundary conditions enforced by the clamped end.

Finally, because the vibrating tine is well-coupled with the sound board of the kalimba body, the width of the resonant peak for the tine fundamental will be broad which supports anharmonic modes.

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  • $\begingroup$ Fascinating! Would this 6.5x be a function of the distance from the fixed position to the bridge position? $\endgroup$
    – Reinderien
    Jan 14, 2021 at 0:35
  • $\begingroup$ probably. this can be determined experimentally by heavily clamping that portion of the tine and looking for differences in the spectrum. $\endgroup$ Jan 14, 2021 at 4:59

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