[pcc]

With acoustic instruments, sympathetic resonance can play a big role in the overall colour, because eg unplayed undamped strings also respond according to their harmonic likeness to the presence of other excited frequencies.

For example, on a piano if you undamp but don't sound one C, by setting the key down gently and holding it, then strike the C an octave below, the higher C will start ringing quite audibly and add its own partials, the same happens also of course to a lesser degree if you say undamp the G an octave + 5th above the C you're striking. This happens all the time during normal piano playing, because pianists purposefully hold long-duration keys down, and the sustain pedal keeps undamped the notes already played.

It can also work in the "downward" direction for example on the violin where the open strings are E A D G with E the highest, the well known Vivaldi A minor concerto (eg Suzuki book 4) opens with shifting on that E string to eg 3rd position to play the A which is an octave higher than the open A string. To the extent you get this perfectly in tune, presuming a properly tuned violin of sufficient quality, the open A string ie an octave lower will start "ringing" as will the D below that to a lesser degree; and if you then subsequently damp those open strings, it changes the sound significantly. This "ringing" effect is quite important to violinists in developing their sense of tuning and calibrating exactly where to set their fingers.