Timbre (pronounced “tam-ber”) is the tonal character of a pitched sound given by its overtones
So even though two instruments can play the same pitch, they sound different due to the complex waveforms created by their unique combination of overtones.
For example, a piano has a different timbre than a flute. Guitar has a different timbre than a piano, and a clarinet has a different timbre from a trumpet.
While any of these instruments may play an A (440 Hz), they are producing other frequencies, some of which are higher (overtones) and some of which are lower (undertones).
Although these overtones and undertones may not be as loud as the fundamental frequency, they do affect the “timbre” or how an instrument sounds.
This is fundamentally due to the design of the instrument, the construction materials and the mechanisms used so not all musical instruments create the same strength and combination of overtones.
In fact, some instruments don’t even produce all of the overtones.
For instance, a clarinet is open at one end and closed at the other. Clarinets produce mostly odd-numbered overtones, which, along with its reed mouthpiece, creates its signature “dark” timbre.
Brass instruments have a distinctly rich and bright timbre which can become a little harsh and “unpleasant.” Brass has strong overtones that continue high up the harmonic series.
Other instruments have added complexity in their timbre because their overtones don’t precisely match the theoretical harmonic values.
When it comes to learning to engineer, we just memorize the timbre of each instrument that we deal with, rather than learning why its harmonic pattern is the way it is.
It is easy to work out why these tonal attributes majorly affect our mixing choices like equalization.