Musical Scales and their Mathematics

2024
ISBN: 978-3-662-69540-1
Verlag: Springer

Buch, Englisch, 711 Seiten, Format (B × H): 155 mm x 235 mm

Reihe: Mathematics Study Resources

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.

Zielgruppe

Professional/practitioner

Autoren/Hrsg.

Schüffler, Karlheinz

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Geisteswissenschaften Musikwissenschaft Musikwissenschaft Allgemein Musiktheorie, Musikästhetik, Kompositionslehre
• Geisteswissenschaften Musikwissenschaft Geschichte der Musik

Inhaltsverzeichnis

Part I: Mathematical Theory of Intervals. Musical intervals and tones.- The commensurability of musical intervals.- Harmonic-rational and classical-antique intervals. - Iterations and their music-mathematical laws.- Part II: Mathematical Theory of Scales. Scales and their models.- Combinatorial games surrounding characteristics.- Diatonic and chromatic aspects of the circle of fifths.- Part III: Mathematical Temperament Theory. The Pythagorean interval system.- Meantone temperament.- The natural-harmonic system and enharmonics.- Equal temperament and its intriguing context.- Epilogue – Postlude.- Indexes.

Über Autor(innen)

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.