Musical Scales and their Mathematics

Are musical scales just trivial? This book explores this question, revealing the complexity of creating "harmony" in tonal systems.

Why 12 tones? Are there alternatives? Are 12 fifths equal 7 octaves? What is "consonance"? When are intervals "perfect" or "imperfect"? What is meant by "tonal characteristics", "whole tone" and "semitone"? "Ancient tuning" vs potentially new?

Answers need thoughtful explanations, revealing interconnectedness. In this context, mathematics is pivotal, explaining scale generation, temperament systems, etc.

Divided into three parts, this book covers:

• Modern interval arithmetic driven by prime numbers.


• Architectural principles of scales, with examples.


• Systematic nature of historical tunings and temperaments.

Understanding only requires school knowledge, developed into algebraic tools applied musically.

 

Prof. Dr. Karlheinz Schüffler is a mathematician, organist, and choir conductor. As a mathematician, he teaches at the University of Düsseldorf and previously at the Hochschule Niederrhein (Krefeld). As a musician, he has been dedicated to church music since his youth, with both organ and mathematical music theory being his areas of expertise.

The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content.

This book is a translation of an original German edition. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation.