We begin by exploring why tuning is far from a trivial issue. The chapter introduces the reader to fundamental ideas in tuning theory and lays the groundwork for understanding both historical and scientific approaches to consonance.

This chapter examines the history of European tuning traditions, tracing their roots to Ancient Greece. We explore the philosophical underpinnings of these systems and analyze how they continue to influence modern thinking. By tracing their development we explain how and why modern equal temperament system emerged and what is lost by adopting it. Finally, we assess the state of empirical research on consonance and identify gaps in existing theories, setting the stage for a new approach.

Here, we delve into the physical nature of sound waves and show how they can be represented as sets of frequencies. We examine different types of sets, connect them to the properties of real-world sounds, and draw parallels with current scientific models of consonance.

In this chapter, we propose a formal definition of consonance and develop a mathematical representation based on the set-theoretic model. Two core concepts — affinity and harmonicity — are introduced to quantify consonance. We compare those measures with existing theories and demonstrate how their combination provides more general understanding of consonance.

We show how tuning systems can emerge naturally from the concepts of affinity and harmonicity. These measures allow us to quantify the consonance between any two sounds on a scale from 0 (no consonance) to 1 (perfect consonance). The chapter culminates in the presentation of the Harmonic Superset Tuning System — a naturally dynamic system with broader applicability than any other tuning system.