The audio samples I used for this experiment were captured using my own 1996 G&L Legacy guitar, a Fender Frontman 10 W practice amp, a MacBook Pro with GarageBand Software and the built-in microphone. In contrast, pure tones created by a single frequency tend to be grating to listen to. The harmonics created by guitar strings are what make the sound richer and more complex. It may now be obvious that harmonics take their name from the harmonic series:Įven subtle variations in these overtones can be noticeable, and ultimately determine the timbre of a sound. For each harmonic, nodes, or points of no transverse motion, exist at integer multiples of L/n along the length of the string. Therefore, the string vibrates with a sum of periodic functions with the respective harmonic frequencies. In English: not only does the string vibrate as a standing wave along its whole length (the fundamental frequency), but also along half its length, a third of its length, etc. Giving us the frequency of the nth harmonic: Observe that the angular frequency in the above solution is Where Given some initial condition and for all x at t=0, the solution of the wave equation is
In an ideal case, each individual guitar string obeys the wave equation: As a guitar player of nine years, I have always been aware of harmonics or “overtones” on guitar but did not always understand how they worked physically, which inspired this project. My goal with this project was simply to use Fourier analysis to explore the harmonics of guitar strings.
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