The discipline of Music Analysis consists of "the resolution of a musical structure into relatively simpler constituent elements, and the investigation of the functions of those elements within that structure". What are the 'constituent elements'? Pitch is the element most often studied when a composition is analysed. Yet pitch is only one element of music. Pitches must be sounded, so an indispensable element of music is its timbre. While analysts are able to discuss pitch structures with a greater deal of sophistication, their attempts to discuss timbre are few and often rudimentary. In this thesis, it is recognised that the discipline of Music Analysis would be enhanced if timbre could be discussed with the same degree of precision as pitch. To this end, a number of acoustic methods for analysing music timbre are proposed. The sound of a music performance is analysed through spectral analysis. This shows us what is physically present in the sound. To begin to understand timbre perception, methods of data weighting and reduction are introduced, based on psychoacoustic models. Six measures of timbre are proposed: timbral width, timbral weight, timbral pitch, roughness, sharpness and observations based on the results of cepstrum analysis. The measures of timbral width, weight and pitch are obtained from a new technique inspired by the Tristimulus Method of Pollard and Janson. The thesis reviews past attempts, both scientific and musicological, to analyse and structure music timbre. Certain pieces appear to use timbre as their principal means of organisation. The abovementioned measures of timbre are applied to two important pieces of this type: the first movement of Lutoslawski's Jeux vénitiens (1961) and Ligeti's Atmosphères (1961). These analyses, which are the focus of the thesis, incorporate an exploration of timbre with a thorough investigation of the music's other structural elements. It is in the examination of the play between timbral and other structures that we find a level of insight into these pieces which has not, until now, been possible.