I'm starting learning music and I see that usually sounds (say from a guitar or piano) always have a lot of frequencies (usually octave aparts, like the G string will have G3, then G4, G5 over it). But the brain decide it is a G3 all the time. Why is it? Is it because the G3 is the loudest? I wonder this as I usually hear the higher notes of a chord more clearly.
Thanks a lot for your help!
It is not because the fundamental is the loudest. In fact the fundamental does not even need to be there! There is a function of the brain called fundamental tracking. We have evolved to be sensitive to the harmonic sequence, f_n = n*f_1, even though not all vibrating systems follow this sequence. Given an input of several frequencies the ear responds to them creating aural harmonics due to non-linearity and the brain looks for the harmonic sequence pattern. It identifies the fundamental of the sequence even if it is NOT there in the acoustic signal.
This leads to a lot of interesting aural illusions. The first being that if you feed someone a sequence with the fundamental missing they will claim to hear it. Another being that if one is listening to multiple sources whose fundamentals line up in a harmonic sequence some of the higher notes may not be perceived by the listener as truly individual notes but roll up into the tone of the overall sound.
I have read that John Petrucci from Dream Theater played chords in such a way that the notes imply a lower fundamental than the guitar can play to trick the ear of the listener into hearing them. I am not sure how well it works as I'm not a Dream Theater fan.
In general the amount of harmonics in any instrument depends very strongly on the attack. You can make the spectrum of your guitar change by where on the string you pluck it or if you pluck versus hammer it. The next factor is the physics of the instrument. An acoustic guitar will have one type of spectrum but for an electric you can "shape" the sound with electronic effects or even software these days. In some sense all electronic instruments are a form of synthesizer (much to the chagrin of the purists).
The harmonics are usually thought to contribute to the "tone" of the instrument. As for hearing the collection of tones as one or as individuals if they are coming from different sources we can distinguish which ones belong together from directivity (beamforming done with the bi-aural hearing). Otherwise we really don't know. In contrast to hearing higher pitched tones disappear classical instrument training usually involves learning to hear the harmonics. So it would seem that there is a precedent for being able to hear the f1, f2, and f3 etc as separate notes with sufficient training. I have had such training and I'm sure about its validity.
First, note that all overtones are not all octave apart: considering a frequency f, the first overtone (or second harmonic) will have a frequency of 2f, corresponding with an octave, the second overtone will have a frequency of 3f, corresponding to an octave plus a perfect fifth, and so on…
About your question, as it is illustrated by the mechanism of "missing fundamental":
The brain perceives the pitch of a tone not only by its fundamental frequency, but also by the periodicity implied by the relationship between the higher harmonics; we may perceive the same pitch (perhaps with a different timbre) even if the fundamental frequency is missing from a tone.
the brain actually kind of consider the fundamental and its overtones as part of a whole: this is not a G3 with a G4 (and D5 and so on), but this is a G3 with overtones.
The pitch of a tone is fact more difficult to recognize when it lacks overtones!
A tone can also be recognized if the fundamental is suppressed or missing. If you take a recording of a 440 Hz A played on a piano, and sharply roll of the lower frequencies below, say, 1000 Hz, you will still recognize the note the same way. It must be that the overtones almost certainly help the ear and brain zero in on the pitch of a note.
In fact, the brain can even "hear" a phantom fundamental anyway, when it's actually missing. Quote from the article: "It is now widely accepted that the brain processes the information present in the overtones to calculate the fundamental frequency."
Some kinds of signal processing software that recognizes pitch relies on overtones.
The overtones of a periodic signal have a spectrum that has a bunch of more or less equally spaced peaks. That spectrum can be regarded as a signal, in which those harmonic peaks look like a periodic pattern. The basic idea is that if you see multiple peaks that are 440 Hz apart, that indicates a 440 Hz A. These multiple peaks confirms the pitch better than a single peak, in the face of noise. They also deal with the suppressed fundamental. Even if the 440 Hz, or even 880 Hz components are filtered out or masked by noise, the tell-tale peaks of the other harmonics being 440 Hz apart still betray the note.
It's possible to take the spectrum of a spectrum to look for such patterns in the frequency space. This is called a "cepstrum". One of the applications of this is to determine the pitch of a voice.
