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All the research I've been doing tells me that a guitar note is determined by the fundamental frequency played. But say you play an A on the open A string (110 Hz), and then play a higher octave A by pressing behind a certain fret on another string (220 Hz). Why is this a different A, when it is just a harmonic of the fundamental frequency? Does this mean that the fundamental frequency is now the 220 Hz, and the first harmonic of 110 Hz is no longer played? Is this correct?

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When you push down on the free you are essentially changing the length of the string. This means you are chaining the fundamental frequency.

This means that, while the $220\ \rm{Hz}$ is a harmonic of the string with the $110\ \rm{Hz}$ fundamental frequency, when you half the length of the string the fundamental frequency is now $220\ \rm{Hz}$, and the $110\ \rm{Hz}$ is no longer a harmonic of the shortened string.

Mathematically, the modes of the string are achieved when the string of length $L$ is broken up into parts such that $$\lambda_n=\frac{2L}{n}$$ Where $\lambda_n$ is the wavelength of the standing wave, and $n$ is a positive integer. Since $v=f\lambda$ is true for the waves, where $v$ is the wave velocity that depends on the string properties, we have $$f_n=\frac{nv}{2L}$$

Since the fundamental frequency is when $n=1$, if $f_1=220\ \rm{Hz}$, then this is the lowest $f_n$ can be for larger $n$.

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The question you ask is less physical and more psychological. You are correct, that there's nothing different between a 220Hz fundamental and 220Hz 1st harmonic of a 110Hz from a wave perspective. Both behave the same.

It's how the ear processes it that is different. Our brains are built with an awareness of these harmonics, and it's effective to sort of bundle the fundamental and the harmonics together into a "pitch," driven primarially by the fundamental, and the "color" of the note, which is related to the ratios of various harmonics. Our brain does this process naturally, so we often don't even notice the effects.

In your case, if you played an A110 and an A220, you would get a set of harmonics. A110 would create harmonics at 220, 330, 440, etc. A220 would create harmonics at 440, 880, etc. The brain would notice the presence of those 110Hz, 330Hz, and 550Hz signals all belong as part of one "note," processing it as an "A" (describing the pitch), with a particular color. It would then be left with a set of frequencies which fit well with an "A one octave up", with a similar color.

In effect, your brain will divide up the amplitude of the 220Hz, 440Hz, 660Hz, etc. components of the sound between the expected harmonics of a string vibrating at 110Hz, and what is left over will be treated as a second note, whose fundamental is 220Hz.

This effect can be messed with. Throat singers are famous for doing this. They sing a very low note, then alter the shape of their mouth to cause one of the overtones to be much more resonant. When our ears hear this, we get the impression that that overtone isn't part of the usual harmonic series of a human voice because its so much louder than the harmonics nearby. Thus, what we hear is something akin to someone singing two notes at the same time.

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  • $\begingroup$ It might be worth noting that you can also fool the ear into inventing a fundamental which is not there. If you have a sound which looks like $\sum_{n=2}\alpha_n \sin(n\omega t)$ (so with the fundamental missing), you will often hear this as a note an octave down. $\endgroup$ – tfb Oct 9 '18 at 6:50

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