We discuss the connection between free-jazz music and service-oriented computing, and advance a method for formal, algebraic analysis of improvised performances; we aim for a better understanding of both the creative process of music improvising and the complexity of service-oriented systems. We formalize free-jazz performances as complex dynamic systems of services, building on the idea that an improvisation can be seen as a collection of music phase spaces that organise themselves through concept blending, and emerge as the performed music. We first define music phase spaces as specifications written over a class of logics that satisfy a set of requirements that make them suitable for dealing with improvisations. Based on these specifications we then formalize free-jazz performances as service applications that evolve by requiring other music fragments to be added as service modules to the improvisation. Finally, we present a logic for specifying free jazz based on one of Anthony Braxton’s graphic notations for composition notes.