Now, this amalgamation is not perfect. Both the inevitable variations in pitch, amplitude, etc of the two separate sources, and the ability of the hearing to remember differences and maintain them even when the overlap of harmonics would tend to blend the experiences, means that we do not entirely blend the two notes. We hear them as separate notes, but also feel them as being unified, of being harmonious. Two singers or two instruments playing an octave apart tend to be regarded as harmonious.
Note that it does not take much of a shift in the pitch of one of the notes to destroy that feeling of harmony. In the example where I shifted the frequency of all of the odd modes of a guitar string by putting a tiny piece of tape at the center of the string, the change in pitch of the odd numbered modes was less than a quarter of a semi-tone. But this was enough to present a very strong impression of two different pitches to the mind, as contrasted with the situation with no tape, where the odd modes and even modes formed a harmonic series. In the latter case, it was almost impossible to hear the sound as two pitches. It was a single pitch, even though nothing had changed in the amplitudes of the various modes (which now formed a harmonic series).
This leads to the hypothesis that when notes share harmonics, the ear will tend to amalgamate the notes, and make the experience of the sound to be much more of a unity, than if they do not share harmonics.