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Is a pure sine wave equivalent to the sum of its every harmonic (up to infinity and without the fundamental lets assume)?

Moreover, if it is so, is this the reason why all the harmonics are present in an open string played or other parts of the instrument played have a role?

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    $\begingroup$ A pure sine wave doesn't have any harmonics, by definition. That's why it's called pure. $\endgroup$ – knzhou Jul 7 '18 at 16:13
  • $\begingroup$ This is a very confusing question. The "harmonics" are all pure sine (or cosine, or exp(i*something)) wave. No mixing required to produce a sine. I think you are getting terms mixed up. Consider editing this or deleting it. Even though it has answers they can't possible the answer to the question. $\endgroup$ – ggcg Jul 11 '18 at 3:03
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A pure sinusoidal wave does not contain overtones. Overtones are present when a periodic wave is non-sinusoidal: triangular, square, or any arbitrary shape other than purely sinusoid.

There are several reasons a musical instrument does not provide a pure, single-frequency, tone. Here are two of the reasons:

  • The stimulation of the vibration (e.g., plucking the string, blowing air past a reed, or buzzing of the lips) usually is not in the form of a pure sinusoid. A plucked string is typically initially distorted into a triangular, rather than sinusoid shape; and the point along the string where it is plucked affects the shape of the triangle and thus the initial mixture of tones. Buzzing lips or a vibrating reed provide a series of pressure pulses rather than sinusoid pressure waves.

  • the vibrating system (e.g., string plus other strings plus guitar body, air in the chamber comprising a wind instrument or brass instrument) usually does not behave linearly. This can result in an initially pure tone (corresponding to a single vibrational mode) evolving into a mixture of tones (corresponding to multiple vibrational modes which are not necessarily harmonics of the initial tone). Examples of nonlinearities would be interactions between strings, different decay rates of different vibration modes, leaky pads in a flute, old strings that have lost some of their springiness, and very large vibration amplitudes.

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Since $e^{ik\omega t}$ and $e^{im\omega t}$ are orthogonal over any interval of length $2\pi/\omega$ once cannot decompose $e^{ik\omega t}$ into the linear sum of $e^{im\omega t}$ if $k\ne m$ for any $k$, specifically the decomposition $\sum_{m,m\ne 1} A_m e^{im\omega t}$ cannot equal $e^{i \omega t}$.

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