Wikipedia defines harmony as:
“Harmony is the use of simultaneous pitches (tones, notes), or chords… Harmony is often said to refer to the ‘vertical’ aspect of music…”
Timbre then, is the harmony of a tone within itself. If harmony is the “vertical” aspect of music, then timbre is the inner aspect of sound. An instrument’s overtone resonance is the timbre. Timbre is what defines the sonic characteristics of an instrument.
Timbre is the envelope that contains the fundamental tone.
Each instrument can manipulate timbre. Articulation, attack, decay, ‘brightness,’ ‘darkness,’ are all timbrel aspects of sound. It takes a deep listening practice to cultivate a personal timbre on an instrument, it’s like a fingerprint. If you’ve ever instantly recognized a musician’s sound from only hearing a single note, it’s the timbre that you’re recognizing. Seems simple enough, but the practice of timbre is far more all-encompassing than just producing an exact pitch. After all, sound is one part pitch and one part timbre.
Don Robertson also provides a very good definition of Timbre:
“It is the amplitude (loudness) and placement of the overtones that determines what the timbre, or sound quality will be. For example, clarinets sound only the odd numbered overtones, creating notes that have a purer timbre than those produced by a stringed or brass instrument. The strength the brass instrument’s higher overtones makes the sound “brassy”: a sound that is rich, but slightly dissonant. Additionally, not all musical instruments have overtones that exactly match the pure harmonic partials. The piano’s overtones, for example are increasingly sharper than perfect harmonics because stiffness of the metal strings.”
Another great resource here: Sound Perception – Timbre