Musical Scales and their Mathematics by Schüffler, Karlheinz;

Long description:

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The Musical Scale – a Triviality or a Problem?

This book examines this provocative question. It soon becomes clear that combining tones into “harmonious” tonal systems represents a challenge of unexpected complexity, encompassing a remarkably large number of interconnected problems.

• Why does a scale have precisely 12 tones? And could there be others?
• What are the roles of the famous Pythagorean comma and all its variants?
• What exactly is “consonance”or “harmony” or “purity of intervals”?
• What is meant by ""temperament characteristic"" of musical scales?
• What does ""old tuning"" mean – and is there a ""new"" one that differs from it?

and many similar ones quickly show that their answers not only require well-considered justifications but are also closely interconnected. In this examination, ""mathematics"" plays a key role. Starting with ""simple proportions and numerical relationships"", a veritable network emerges, in which both the methods of scale generation with their circle of wolf fifths an Euler grid selection procedures, as well as the models of temperament systems, can be scientifically explained. In three parts,

• a modern interval arithmetic of musical intervals and their theory of division, decomposition, and construction of musical intervals, driven by prime numbers,
• the architectural laws of musical scales with their models and patterns, their step geometries and characteristics, their semi- and quartertones and commas,
• a mathematical-methodical description of the central temperament systems with a new, consistent systematics of significant historical tunings

are presented and equipped with numerous examples. The music-mathematical calculations and explanations require only basic school-level knowledge, which are then developed and applied to musical themes. A small peculiarity: stories from the fairytale world of musical mythical creatures accompany the text and explain it in an amusing way.

This book is a translation of the original German edition Die Tonleiter und ihre Mathematik, 4th edition, by Karlheinz Schüffler, published by Springer-Verlag GmbH, DE in 2026. The translation was done with the help of an artificial intelligence machine translation tool. A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation.

Table of Contents:

Part I: Mathematical Theory of Intervals. Musical Intervals and Tones.- Commensurability of Musical Intervals.- Rational-Harmonic and Classical-Ancient Intervals. - Iterations and their Music-Mathematical Laws.- Part II: Mathematical Theory of Scales. Scales and their Models.- Combinatorial Games around the Characteristics.- Diatonic and Chromatic Structures of Wolf Fifth Circles.- Part III: Mathematical Theory of temperament. Pythagorean Interval Systems.- Mean-Tone Temperaments.- Just Intonation, Natural Harmonic Systems and Enharmonicism.- Equal step Temperaments and their Surprising World.- Historical Temperaments – Methodology and Theory.- Epilogue – Postlude.