Explains musical temperaments clearly and vividly, solely using basic school mathematics The new edition has been completely redesigned and features an entirely fresh approach to musical intervals Suitable for enthusiasts of both mathematics and music

Autorentext

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 Niederrhein University of Applied Sciences (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.

Klappentext

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?

• Aren't 12 perfect fifths always exactly the same as 7 octaves?

• What exactly is "consonance" or when are intervals "pure," when "impure"?

• What is meant by "temperament characteristic" in relation to whole and half tones?

• What does "old tuning" mean and is there a "new" one that differs from it, and if so, what exactly are the differences? 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 and 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 semitones and commas, as well as the combinatorial diversity of all scale-internal structures,

• 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 explanatory arguments require only basic school-level knowledge, which are then developed into a suitable algebra and analysis 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.

The Author
Prof. Dr. Karlheinz Schüffler is a mathematician, organist, and choir director. As a mathematician, he teaches at the University of Düsseldorf and formerly at the Niederrhein University of Applied Sciences in Krefeld. As a musician, he has been committed to church music from a young age; both the organ and mathematical music theory have become his primary focal points.

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

Zusammenfassung

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.

Inhalt

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.