El Polifemo de Oro was written in 1956 and revised by the composer in 1981. This two-fold investigation clarifies the structure of El Polifemo through careful analysis of the work and its revision, and by Smith Brindle's approach to composition based on his interviews and books. The second aspect is aimed toward the pragmatic performance issues of tempo, articulation, timbre, voice leading, and the other details of execution. Although the work was written according to serial techniques, the presence, in the twelve-note row, of triadic formations (minor triad, dominant seventh, fully-diminished seventh) juxtaposed with many tritone intervals suggests the use of tonal devices, which Smith Brindle does employ to effect tension and relaxation. It is assumed that tonal devices such as leading tones, stepwise movement in the bass, fourth and fifth relationships, and triadic constructions are heard against a traditional contextual basis and are therefore ways of implying resolution. Where tonal devices are not present, other structural components, i.e., rhythm, dynamics, timbre, etc., are examined with regard to their functions in creating or dissipating tension. Following the analysis of each of the four fragments is a discussion of performance implications based on the analysis. Both the analysis and performance aspects are non-prescriptive and are presented in the spirit that there are many other valid interpretations.