How do natural harmonics and also ‘pinch’ harmonics work on a guitar?
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The fundamental frequency of a guitar string is the time it takes for the guitar string to vibrate back and forth based on the tension, length, and mass of the string. Like the way a slinky or spring will bounce at a certain rate.
The harmonics are the time it takes for a fractional piece of the guitar string to vibrate back and forth. For example, 1/2, 1/3, 1/4, etc. When you strike an open string, all the harmonics vibrate together, but our ear translates this to just the fundamental frequency. Sort of how we see combinations of colors as just one color.
When you place your finger gently on the string at the twelfth fret, you are preventing the entire string from vibrating, but you still allow each half string to vibrate. Thus, each half string becomes the new fundamental and your ear hears that instead (at twice the frequency of the fundamental, or one octave). There are no new frequencies here, just the lack of the fundamental frequency and some other intermediate frequencies that you are now blocking.
Pinch harmonics are caused by unusual ratios when you mute the string at a funny place. For example maybe you pinched at 4/13ths of the string’s length, so the string vibrates in thirteenths. The reason pinch harmonics work on electric guitar and not acoustic guitar is because the signal is heavily amplified.
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