From Academic Kids
In music, timbre is determined by its spectrum, which is a specific mix of keynote, overtones, noise, tune behaviour, envelope ( ... ) as well as the temporal change of the spectrum and the amplitude. Timbre is the quality of a musical note which distinguishes different types of musical instrument, speech, or singing. Though a cello, piano and saxophone may all play the same pitch at the same loudness, the instruments all have very different qualities to their sounds which make them distinctive; these qualities are often referred to as the instrument's timbre. (See also: formant) The word is pronounced as in French, "TAM-ber".
The timbre for most instruments is determined by its keynote and overtones/harmonics. The overtones are integer multiples of the keynote frequency.
Mechanical instruments have additional to the keynote always overtones. Electronical Instruments can generate tone without overtones. Music instruments do not have the same overtones for all notes. For example the Clarinet has three different registers, which have all different overtones.
Not all sounds are solely determined by the keynote and the overtones. Bell sounds are better described by keynote and undertones (integer factors of the keynote). Drum sounds are a mixture of Tones and white(?) noise. Many other instruments have also noise parts, which shape the timbre. (wind instruments and organs).
The tune behavior is crucial for the timbre. The tune is the temporal curse of the specturm and the amplitude in the first split second of the tone.
The formants, which are the minima and maxima in the spectra of a music instrument are important. They are usually independent of the played tone. They are dependent on the design of the instrument. The Formants determine also the sound of the sungen vocals.
The playing technique is another determinante of the timbre. If more energy is put into an instrument, not only the amplitude is affected, but also higher modes of vibrations are achieved. This leads to more partials in the frequency spectrum. For example a harder played piano has a brighter timpre.
American Standards Association Definition
The American Standards Association defines timbre as "Timbre is that attribute of sensation in terms of which a listener can judge that two sounds having the same loudness and pitch are dissimilar”.de:Klangfarbe ja:音色 nl:Timbre zh:音色
Stephen David Beck Designing Acoustically Viable Instruments in Csound in Richar Boulanger The Csound Book