Musical Scales and their Mathematics
Series: Mathematics Study Resources; 14;
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Product details:
- Edition number 2024
- Publisher Springer
- Date of Publication 20 July 2025
- Number of Volumes 1 pieces, Book
- ISBN 9783662695401
- Binding Paperback
- No. of pages711 pages
- Size 235x155 mm
- Language English
- Illustrations 125 Illustrations, black & white; 10 Illustrations, color 700
Categories
Short description:
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. More
Long description:
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.
More
Table of Contents:
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.
More
Musical Scales and their Mathematics
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