Here's an example of a bass note on a piano:
And here are its harmonic ingredients or partials, electronically isolated and played in order, and played in sequence, up to the 17th partial:
And here are the rest of the partials, isolated from the first 17, and played all together:
If you listen to that sound, you might now be able to hear it happening in the first recording of the bass note. Similarly, you may be able to hear some individual partials in that recording, from the first 17.
Because the harmonic series is related to our pitch perception, which is a perception, depending on our brain and ear, there is an element of psychoacoustics involved. Psychoacoustics is a crossover between psychology and acoustics. Here, in the 2nd recording of the electronically isolated partials, all the partials you can hear are the actual partials of the bass note, extracted from it electronically. Except, that is, for the first, lowest one.
That one has been artificially created and added in, just to complete the series, because it doesn't exist in the original sound of the bass note. At least not to any significant amount. Our brain, so to speak, "expects" it to be there, because it is "pre-wired" to unconsciously recognise the pattern of the harmonic series, in sounds.
The musical pitch of that first partial is the same as the musical pitch of the bass note, although of course the tone of the isolated partial is quite different to the tone of the bass note as a whole. It's an A1 on the piano.
That's the way the harmonic series works in relation to pitch perception. As far as our brain is concerned, the whole series "points" psychologically, as it were, to the pitch of the first partial or harmonic, as being the musical pitch, or thereabouts, of the musical tone.
In the lowest notes of many pianos this partial is missing, due to string stiffness, but we still hear the musical pitch as what it would have been, had the partial been there, because there is enough of a harmonic series present, for our brain to recognise the pattern, and hence, we perceive the definite musical pitch as being there. It is almost as if our brain unconsciously presumes that missing partial to be there. So in the second recording we’ve put it back, so to speak, where it should be.
Actually, regarding this whole business, in many smaller pianos, the musical pitch of the lower bass notes does become ambiguous, due to this missing partial. It also becomes ambiguous when partials deviate too much for a true harmonic series.
This is why the bass in many smaller pianos will often sound a little too sharp in pitch, and if we simply lowered the string tensions to lower the musical pitch of individual notes, the instrument would begin to sound out of tune due to mismatches in the partial structures of different notes when played together as chords or intervals.
Even more extreme, it's perfectly possible and even quite common in older pianos, that if you lower the tension in a bass string, which would normally be expected to lower the pitch, there is a jump upwards in pitch. This is because musical pitch perception really is psychoacoustic, and not really an objective property of sound. If the partials in the sound only approximate roughly to a harmonic series, and are missing the lowest partials, then as you change their arrangement the apparent musical pitch can jump around.
Sounds whose ingredients or partials are not anything like the harmonic series, especially if they are closely packed, or even a continuum, or are rapidly changing, don't have a definite musical pitch.
An example of musical sound with a small amount of pitch ambiguity is these Tubular Bells.
There's more on this piano tone in the article on the harmonic series.