Introduction

In the early twentieth century, composers began exploring scales that repeat at intervals smaller than the octave. The result was some of the most distinctive sounds in modern music, from the shimmering whole-tone passages of Debussy to the otherworldly harmonies of Messiaen's organ works.

This book bridges two approaches to symmetrical pitch organization: Olivier Messiaen's "modes of limited transposition" and Nicolas Slonimsky's comprehensive catalog of melodic patterns based on equal divisions of the octave. While these systems emerged independently, they share a common mathematical foundation that this book will explore in depth.

What You'll Learn

• The mathematical foundations of symmetrical scales
• Messiaen's complete system of modes of limited transposition
• Slonimsky's approach to melodic pattern generation
• Practical applications in composition and improvisation
• Connections between these systems and their influence on modern music

Whether you're a composer seeking new harmonic colors, an improviser looking to expand your melodic vocabulary, or a theorist interested in the intersection of mathematics and music, these structures provide a systematic framework for exploration.