Timbre is the amplitude of the components of the harmonic series in relation to the fundamental tone.
As you "speak" you are constantly changing the amplitude of the components of the harmonics of the fundamentals in your vocal sound, as well as adding some transients - generally consonants, to the vowel sounds. You also add "noise" - the letters C, F, S, H, X, and SH combination usually make noise (no defined frequency).
A clarinet, a trumpet, a flute, a stringed instrument all produce different harmonic series even if playing the same "note". That's what I call "timbre" in the basic case.
Bass Guitar (handy home-grown example), one note played:
The fundamental frequency is marked "1", the harmonics measured are "2" though "9". There are more, but the software (REW) chooses not to count any higher.
The harmonic series is multiples of the fundamental frequency.
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Timbre may include more information, the noise a violin bow makes scraping the string, the (often) enharmonic series of drums, cymbals...
Timbre is the characteristic "sound" something makes, defined by the amplitudes of the frequencies (or noise) that accompanies the fundamental (if there even is one).
A pure tone - sine wave - is rarely encountered. That's a Fundamental with no harmonics. Its timbre is identified by the lack of harmonics. The closest I've come to producing one occurred when blowing across the top of a beer bottle:
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If the reproduction system does not reproduce the recorded harmonic series correctly, the timbre of the sound will be changed.
I'd say the reproduction of Timbre is pretty robust - I can't remember mistaking a flute for a trumpet, on any system.