Any good text on
chromaticism will answer many of the questions surrounding the quality of sounds created in
jazz or by the various
intervals. On any stringed instrument (including the
piano), when a note is played, the string vibrates not only at the wavelength of the note played, but also wavelengths half as long, a third as long, a fourth as long, etc., decreasing in intensity as the wavelengths decrease in size. For example, if you play an A at 110Hz, the string will also vibrate at 220Hz, 330Hz, 440Hz, 550Hz, and possibly a few more will be discernable. The first six sounds produced are called the
senario of a tone. So, when you play the 110Hz A and the 220 Hz A together, one of the similarities comes from the fact that both are producing the exact same tone of 220Hz, and the tones in the senario of the higher A are being duplicated by the lower one as well. As a side note, the other notes produced by the senario of a note are its
perfect fifth and
major third; for example, A's 5th is the E at 330Hz, and its major 3, C#, is sounded at 550Hz (this is why the major chord has such a stable sound).
Also important in the sound of an octave interval is the idea of difference tones. Most people know that if you play 440Hz and 439Hz together, you can hear the two being out of tune as the phases will generate a tone of 1 Hz. So, when you play 220Hz and 110Hz together, a difference tone of 110Hz is produced, reinforcing the bottom octave's sound.
It gets much more complicated than all this, however; as noted above, read a book on chromaticism for more information. Most of my knowledge on the subject I got out of Chromaticism: Theory and Practice, by Howard Boatwright.