This study addresses the challenges composers and sound designers face in creating and refining tools to achieve their musical goals. Using evolutionary processes to promote diversity and foster serendipitous discoveries, we automate the search through uncharted sonic spaces for sound discovery, arguing that diversity-promoting algorithms can bridge the gap between the theoretical realisation and practical accessibility of sounds. We describe a system for generative sound synthesis combining Quality Diversity (QD) algorithms with a supervised discriminative model, inspired by the Innovation Engine algorithm, and explore different configurations and the interplay between the chosen synthesis approach and the discriminative model. We examine the interaction between Compositional Pattern Producing Networks (CPPNs) and Digital Signal Processing (DSP) graphs, introducing a novel approach that uses multiple specialised CPPNs for different frequency ranges; this yields simpler networks while maintaining performance comparable to single-CPPN setups. We also investigate evolutionary stepping stones by analysing goal switches between musical and non-musical contexts, revealing how lineages traverse unlikely paths to current elites. Expanding the behaviour space of a previous study to include various sound durations, we uncover specialisation within temporal niches. Results indicate that CPPN and DSP graphs coupled with a Multi-dimensional Archive of Phenotypic Elites (MAP-Elites) and a deep learning classifier can generate a substantial variety of synthetic sounds, diverse and innovative across temporal and contextual dimensions. We present the generated sound objects through an online explorer and as rendered sound files, and, in the context of music composition, an experimental application that showcases their creative potential across various durations and contexts.