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You asked "Why doesn't a guitar string vibrate at one frequency only?". Let's look at it from the other perspective: Every kind of musical instrument makes sounds that have overtones in them, and every pitch played on every kind of instrument has multiple frequencies in it--not just on the guitar. There is no repeating, oscillating sound made by any musical instrument that has absolutely no overtones. The only sound that can exist that has no overtones would be a completely pure sine wave. You could only create a pure sine wave with an electronic oscillator. No acoustic or electro-acoustic musical instrument can create a sound that is similar to a pure sine wave.

You asked "Why doesn't a guitar string vibrate at one frequency only?". Let's look at it from the other perspective: There is no repeating, oscillating sound made by any musical instrument that has absolutely no overtones. The only sound that can exist that has no overtones would be a completely pure sine wave. You could only create a pure sine wave with an electronic oscillator. No acoustic or electro-acoustic musical instrument can create a sound that is similar to a pure sine wave.

You asked "Why doesn't a guitar string vibrate at one frequency only?". Let's look at it from the other perspective: Every kind of musical instrument makes sounds that have overtones in them, and every pitch played on every kind of instrument has multiple frequencies in it--not just on the guitar. There is no repeating, oscillating sound made by any musical instrument that has absolutely no overtones. The only sound that can exist that has no overtones would be a completely pure sine wave. You could only create a pure sine wave with an electronic oscillator. No acoustic or electro-acoustic musical instrument can create a sound that is similar to a pure sine wave.

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You asked "Why doesn't a guitar string vibrate at one frequency only?". Let's look at it from the other perspective: There is no repeating, oscillating sound made by any musical instrument that has absolutely no overtones. The only sound that can exist that has no overtones would be a completely pure sine wave. You could only create a pure sine wave with an electronic oscillator. No acoustic or electro-acoustic musical instrument can create a sound that is similar to a pure sine wave.