While it is important for systems intended to enhance musical creativity to define and explore musical ideas conceived by individual users, many limit musical freedom by focusing on maintaining musical structure, thereby impeding the user's freedom to explore his or her individual style. This dissertation presents a comprehensive body of work that introduces a new musical representation that allows users to explore a space of musical rules that are created from their own melodies. This representation, called functional scaffolding for musical composition (FSMC), exploits a simple yet powerful property of multipart compositions: The pattern of notes and rhythms in different instrumental parts of the same song are functionally related. That is, in principle, one part can be expressed as a function of another. Music in FSMC is represented accordingly as a functional relationship between an existing human composition, or scaffold, and an additional generated voice. This relationship is encoded by a type of artificial neural network called a compositional pattern producing network (CPPN). A human user without any musical expertise can then explore how these additional generated voices should relate to the scaffold through an interactive evolutionary process akin to animal breeding. The utility of this insight is validated by two implementations of FSMC called NEAT Drummer and MaestroGenesis, that respectively help users tailor drum patterns and complete multipart arrangements from as little as a single original monophonic track. The five major contributions of this work address the overarching hypothesis in this dissertation that functional relationships alone, rather than specialized music theory, are sufficient for generating plausible additional voices. First, to validate FSMC and determine whether plausible generated voices result from the human-composed scaffold or intrinsic properties of the CPPN, drum patterns are created with NEAT Drummer to accompany several different polyphonic pieces. Extending the FSMC approach to generate pitched voices, the second contribution reinforces the importance of functional transformations through quality assessments that indicate that some partially FSMC-generated pieces are indistinguishable from those that are fully human. While the third contribution focuses on constructing and exploring a space of plausible voices with MaestroGenesis, the fourth presents results from a two-year study where students discuss their creative experience with the program. Finally, the fifth contribution is a plugin for MaestroGenesis called MaestroGenesis Voice (MG-V) that provides users a more natural way to incorporate MaestroGenesis in their creative endeavors by allowing scaffold creation through the human voice. Together, the chapters in this dissertation constitute a comprehensive approach to assisted music generation, enabling creativity without the need for musical expertise.