Sign-In
Cart
Musical Scales and Their Mathematics
Paperback
Series: Mathematics Study Resources, Book 14
General MathematicsMusic ReferenceMusic Theory
ISBN13: 9783662695401
Publisher: Springer
Published: Dec 17 2026
Pages: 711
Language: English
The Musical Scale - A Triviality or a Problem? This book explores a provocative question and soon reveals that combining tones into harmonious tonal systems is a surprisingly complex challenge, involving a remarkably rich network of interconnected issues:
- Why does the familiar scale consist of exactly 12 tones? Could other scales exist?
- What roles do the famous Pythagorean comma and its various counterparts play?
- What do concepts such as consonance, harmony, and purity of intervals mean?
- What is meant by the temperament characteristic of a musical scale?
- What is old tuning, and is there a new one?
These and many other questions are closely interconnected, and mathematics plays a central role in answering them. Beginning with simple ratios and numerical relationships, a rich conceptual network unfolds, allowing for scientific explanations of methods for scale generation, the circles of wolf fifths and Euler lattice selection procedures, as well as models of temperaments. In three parts, the book presents:
- A modern arithmetic of musical intervals, together with the theory of their division, decomposition, and construction based on prime numbers;
- The manifold architecture of musical scales, including their models and patterns, step geometries and characteristics, semitones, quarter tones, and commas;
- A mathematical foundation of tempered systems, accompanied by a novel and consistent classification of historically relevant musical tunings.
Numerous examples illustrate the concepts throughout. The mathematical calculations and explanations require only elementary school mathematics, which is developed and applied to musical questions as needed. For readers less interested in the mathematics part, entertaining stories from a fairytale world of mythical musical creatures accompany the text, illustrating it in an amusing way.
The English translation was prepared with the assistance of artificial intelligence. The resulting text was subsequently reviewed and revised by the author, with particular attention to technical accuracy and content.
Also from
Schüffler, Karlheinz
Proportionen Und Ihre Musik: Was Brüche Und Tonfolgen Miteinander Zu Tun Haben
Schüffler, Karlheinz
Paperback
Die Tonleiter Und Ihre Mathematik: Mathematische Theorie Musikalischer Intervalle Und Historischer Skalen
Schüffler, Karlheinz
Paperback
Die Tonleiter Und Ihre Mathematik: Mathematische Theorie Musikalischer Intervalle Und Historischer Skalen
Schüffler, Karlheinz
Paperback
Also in
General Mathematics
The Golden Number: Pythagorean Rites and Rhythms in the Development of Western Civilization
Ghyka, Matila C.
Hardcover
How It All Works: All Scientific Laws and Phenomena Illustrated & Demonstrated
Clegg, Brian
Dant, Adam
Hardcover
Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race
Shetterly, Margot Lee
Paperback
Six Easy Pieces: Essentials of Physics Explained by Its Most Brilliant Teacher
Feynman, Richard P.
Leighton, Robert B.
Sands, Matthew
Paperback
Oliver Byrne's Elements of Euclid: The First Six Books with Coloured Diagrams and Symbols (Art Meets Science Edition)
Art Meets Science
Hardcover
Math Practice Workbook Grades 6-8: 1000+ Questions You Need to Kill in Middle School by Brain Hunter Prep
Brain Hunter Prep
Paperback
Who Gets What -- And Why: The New Economics of Matchmaking and Market Design
Roth, Alvin E.
Paperback
Tomoko Fuse's Amazing Kusudama Origami: Geometric Paper Art from Japan's Leading Expert!
Fuse, Tomoko
Paperback
The Art of Strategy: A Game Theorist's Guide to Success in Business and Life
Nalebuff, Barry J.
Dixit, Avinash K.
Paperback
Mastering Essential Math Skills Book 2, Bilingual Edition - English/Spanish
Fisher, Richard W.
Paperback
Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race
Shetterly, Margot Lee
Hardcover
The Great Math War: How Three Brilliant Minds Fought for the Foundations of Mathematics
Bardi, Jason Socrates
Hardcover
Math Games with Bad Drawings: 75 1/4 Simple, Challenging, Go-Anywhere Games--And Why They Matter
Orlin, Ben
Hardcover
Counting with - Contando Con Frida (Bilingual: English/Spanish): A Bilingual Counting Book
Stein, Ariana
Rodriguez, Patty
Hardcover
Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race
Shetterly, Margot Lee
Paperback
Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology
Lundy, Miranda
Ashton, Anthony
Sutton, Daud
Hardcover
50 Math Tricks That Will Change Your Life: Mentally Solve the Impossible in Seconds
Zakowich, Tanya
Paperback
Graph Paper Composition Notebook: Quad Ruled 5x5, Grid Paper for Students in Math and Science
Wizo, Math
Paperback
New York Post Fiendish Sudoku: The Official Utterly Addictive Number-Placing Puzzle
Gould, Wayne
Paperback
Vector: A Surprising Story of Space, Time, and Mathematical Transformation
Arianrhod, Robyn
Hardcover
Touchthinklearn: Numbers: (Board Books for Baby Learners, Touch Feel Books for Children)
Deneux, Xavier
Hardcover
ALEKS Math Placement Test Prep 2026: 1200 Verified Practice Questions with Accurate Answer Keys, Step-by-Step Worked Solutions, and Clear Concept Expl
Reid, Marcus L.
Publishing, Kettford
Paperback
A Mind for Numbers: How to Excel at Math and Science (Even If You Flunked Algebra)
Oakley, Barbara
Paperback
Sudoku for Brain Fitness: 90-Day Challenge to Sharpen the Mind and Strengthen Cognitive Skills
Clontz, Steven
Demyanovich, Julie
Paperback
WorkKeys Study Guide and Practice Test Questions: ACT WorkKeys Exam Prep and Review Book with Applied Mathematics, Locating Information, and Reading f
Trivium Exam Prep Team
Paperback
Un Elefante: Numbers / Números (Bilingual: English/Spanish): A Bilingual Counting Book
Rodriguez, Patty
Stein, Ariana
Hardcover
Quantum Mechanics Done Right: The Shortest Path from Novice to Researcher
Freericks, James K.
Hardcover
Talentos Ocultos: La Genialidad No Tiene Color. La Fuerza No Tiene Género. El Valor No Tiene Límite.
Shetterly, Margot Lee
Paperback
Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning
Liljedahl, Peter
Paperback
180 Days(tm) Math for First Grade: Practice, Assess, Diagnose
Smith, Jodene
Smith, Jodene Lynn
Paperback
Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks
Benjamin, Arthur
Shermer, Michael
Paperback
Number Systems and the Foundations of Analysis
Mathematics
Mendelson, Elliott
Dudley, Underwood
Paperback
Concrete Mathematics: A Foundation for Computer Science
Knuth, Donald
Graham, Ronald
Patashnik, Oren
Hardcover
Medical Dosage Calculations for Dummies
Snyder, Richard W.
Depuente, Vanessa
Schoenborn, Barry
Paperback
The Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science
Schneider, Michael S.
Paperback
Sciencia: Mathematics, Physics, Chemistry, Biology, and Astronomy for All
Watkins, Matthew
Betts, Moff
Tweed, Matt
Hardcover
Reaching for the Extreme: How the Quest for the Biggest, Fewest, and Weirdest Makes Math
Stewart, Ian
Hardcover
300 Math CrossWords: Addition, Subtraction, Multiplication, and Division Math CrossWords Grade 3 and Above, Age +8
Martin, Josh Ortiz
Paperback
Math Fundamentals 3 - Geometry: For Middle School - A Quickstudy Laminated Reference Guide
Warren, Peggy
Wright, Susan
Expolog
Other
Machine Learning with PyTorch and Scikit-Learn: Develop machine learning and deep learning models with Python
Raschka, Sebastian
Liu, Yuxi (Hayden)
Mirjalili, Vahid
Paperback
It's All a Game: The History of Board Games from Monopoly to Settlers of Catan
Donovan, Tristan
Paperback
30 More Math Mysteries Kids Can't Resist!: Quick & Clever Mysteries That Boost Problem-Solving Skills
Lee, Martin
Paperback