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A Geometry of Music

A Geometry of Music Author Dmitri Tymoczko
ISBN-10 9780199887507
Release 2011-03-21
Pages 480
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How is the Beatles' "Help!" similar to Stravinsky's "Dance of the Adolescents?" How does Radiohead's "Just" relate to the improvisations of Bill Evans? And how do Chopin's works exploit the non-Euclidean geometry of musical chords? In this groundbreaking work, author Dmitri Tymoczko describes a new framework for thinking about music that emphasizes the commonalities among styles from medieval polyphony to contemporary rock. Tymoczko identifies five basic musical features that jointly contribute to the sense of tonality, and shows how these features recur throughout the history of Western music. In the process he sheds new light on an age-old question: what makes music sound good? A Geometry of Music provides an accessible introduction to Tymoczko's revolutionary geometrical approach to music theory. The book shows how to construct simple diagrams representing relationships among familiar chords and scales, giving readers the tools to translate between the musical and visual realms and revealing surprising degrees of structure in otherwise hard-to-understand pieces. Tymoczko uses this theoretical foundation to retell the history of Western music from the eleventh century to the present day. Arguing that traditional histories focus too narrowly on the "common practice" period from 1680-1850, he proposes instead that Western music comprises an extended common practice stretching from the late middle ages to the present. He discusses a host of familiar pieces by a wide range of composers, from Bach to the Beatles, Mozart to Miles Davis, and many in between. A Geometry of Music is accessible to a range of readers, from undergraduate music majors to scientists and mathematicians with an interest in music. Defining its terms along the way, it presupposes no special mathematical background and only a basic familiarity with Western music theory. The book also contains exercises designed to reinforce and extend readers' understanding, along with a series of appendices that explore the technical details of this exciting new theory.

The Geometry of Musical Rhythm

The Geometry of Musical Rhythm Author Godfried T. Toussaint
ISBN-10 9781466512030
Release 2016-04-19
Pages 365
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The Geometry of Musical Rhythm: What Makes a "Good" Rhythm Good? is the first book to provide a systematic and accessible computational geometric analysis of the musical rhythms of the world. It explains how the study of the mathematical properties of musical rhythm generates common mathematical problems that arise in a variety of seemingly disparate fields. For the music community, the book also introduces the distance approach to phylogenetic analysis and illustrates its application to the study of musical rhythm. Accessible to both academics and musicians, the text requires a minimal set of prerequisites. Emphasizing a visual geometric treatment of musical rhythm and its underlying structures, the author—an eminent computer scientist and music theory researcher—presents new symbolic geometric approaches and often compares them to existing methods. He shows how distance geometry and phylogenetic analysis can be used in comparative musicology, ethnomusicology, and evolutionary musicology research. The book also strengthens the bridge between these disciplines and mathematical music theory. Many concepts are illustrated with examples using a group of six distinguished rhythms that feature prominently in world music, including the clave son. Exploring the mathematical properties of good rhythms, this book offers an original computational geometric approach for analyzing musical rhythm and its underlying structures. With numerous figures to complement the explanations, it is suitable for a wide audience, from musicians, composers, and electronic music programmers to music theorists and psychologists to computer scientists and mathematicians. It can also be used in an undergraduate course on music technology, music and computers, or music and mathematics.

Audacious Euphony

Audacious Euphony Author Richard Cohn
ISBN-10 9780199773213
Release 2012-01-01
Pages 288
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Music theorists have long believed that 19th-century triadic progressions idiomatically extend the diatonic syntax of 18th-century classical tonality, and have accordingly unified the two repertories under a single mode of representation. Post-structuralist musicologists have challenged this belief, advancing the view that many romantic triadic progressions exceed the reach of classical syntax and are mobilized as the result of a transgressive, anti-syntactic impulse. In Audacious Euphony, author Richard Cohn takes both of these views to task, arguing that romantic harmony operates under syntactic principles distinct from those that underlie classical tonality, but no less susceptible to systematic definition. Charting this alternative triadic syntax, Cohn reconceives what consonant triads are, and how they relate to one another. In doing so, he shows that major and minor triads have two distinct natures: one based on their acoustic properties, and the other on their ability to voice-lead smoothly to each other in the chromatic universe. Whereas their acoustic nature underlies the diatonic tonality of the classical tradition, their voice-leading properties are optimized by the pan-triadic progressions characteristic of the 19th century. Audacious Euphony develops a set of inter-related maps that organize intuitions about triadic proximity as seen through the lens of voice-leading proximity, using various geometries related to the 19th-century Tonnetz. This model leads to cogent analyses both of particular compositions and of historical trends across the long nineteenth century. Essential reading for music theorists, Audacious Euphony is also a valuable resource for music historians, performers and composers.

A New Geometry of Musical Chords in Interval Representation Dissonance Enrichment Degeneracy and Complementation

A New Geometry of Musical Chords in Interval Representation  Dissonance  Enrichment  Degeneracy and Complementation Author Miguel Gutierrez; Makoto Taniguchi
ISBN-10 9781450227988
Release 2010-07-07
Pages 124
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This monograph covers a fresh and original look at musical chords. The idea emanates from the fact that an intervallic representation of the chord leads naturally to a discrete barycentric condition. This condition itself leads to a convenient geometric representation of the chordal space as a simplicial grid. Chords appear as points in this grid and musical inversions of the chord would generate beautiful polyhedra inscribed in concentric spheres centered at the barycenter. The radii of these spheres would effectively quantify the evenness and thus the consonance of the chord. Internal symmetries would collapse these chordal structures into polar or equatorial displays, creating a platform for a thorough degeneracy study. Appropiate morphisms would allow us to navigate through different chordal cardinalities and ultimately to characterise complementary chords.

Tonality and Transformation

Tonality and Transformation Author Steven Rings
ISBN-10 9780199913206
Release 2011-06-10
Pages 272
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Tonality and Transformation is a groundbreaking study in the analysis of tonal music. Focusing on the listener's experience, author Steven Rings employs transformational music theory to illuminate diverse aspects of tonal hearing - from the infusion of sounding pitches with familiar tonal qualities to sensations of directedness and attraction. In the process, Rings introduces a host of new analytical techniques for the study of the tonal repertory, demonstrating their application in vivid interpretive set pieces on music from Bach to Mahler. The analyses place the book's novel techniques in dialogue with existing tonal methodologies, such as Schenkerian theory, avoiding partisan debate in favor of a methodologically careful, pluralistic approach. Rings also engages neo-Riemannian theory-a popular branch of transformational thought focused on chromatic harmony-reanimating its basic operations with tonal dynamism and bringing them into closer rapprochement with traditional tonal concepts. Written in a direct and engaging style, with lively prose and plain-English descriptions of all technical ideas, Tonality and Transformation balances theoretical substance with accessibility: it will appeal to both specialists and non-specialists. It is a particularly attractive volume for those new to transformational theory: in addition to its original theoretical content, the book offers an excellent introduction to transformational thought, including a chapter that outlines the theory's conceptual foundations and formal apparatus, as well as a glossary of common technical terms. A contribution to our understanding of tonal phenomenology and a landmark in the analytical application of transformational techniques, Tonality and Transformation is an indispensible work of music theory.

Sacred Geometry

Sacred Geometry Author Stephen Skinner
ISBN-10 1402765827
Release 2009
Pages 160
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A fascinating and inspirational look at the vital link between the hidden geometrical order of the universe, geometry in nature, and the geometry of the man-made world. The Da Vinci Code has awakened the public to the powerful and very ancient idea that religious truths and mathematical principles are intimately intertwined. Sacred Geometry offers an accessible way of understanding how that connection is revealed in nature and the arts. Over the centuries, temple builders have relied on magic numbers to shape sacred spaces, astronomers have used geometry to calculate holy seasons, and philosophers have observed the harmony of the universe in the numerical properties of music. By showing how the discoveries of mathematics are manifested over and over again in biology and physics, and how they have inspired the greatest works of art, this illuminating study reveals the universal principles that link us to the infinite.

Geometry of Quantum Theory

Geometry of Quantum Theory Author Veeravalli Seshadri Varadarajan
ISBN-10 9781461577065
Release 2013-06-29
Pages 193
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The present work is the first volume of a substantially enlarged version of the mimeographed notes of a course of lectures first given by me in the Indian Statistical Institute, Calcutta, India, during 1964-65. When it was suggested that these lectures be developed into a book, I readily agreed and took the opportunity to extend the scope of the material covered. No background in physics is in principle necessary for understand ing the essential ideas in this work. However, a high degree of mathematical maturity is certainly indispensable. It is safe to say that I aim at an audience composed of professional mathematicians, advanced graduate students, and, hopefully, the rapidly increasing group of mathematical physicists who are attracted to fundamental mathematical questions. Over the years, the mathematics of quantum theory has become more abstract and, consequently, simpler. Hilbert spaces have been used from the very beginning and, after Weyl and Wigner, group representations have come in conclusively. Recent discoveries seem to indicate that the role of group representations is destined for further expansion, not to speak of the impact of the theory of several complex variables and function-space analysis. But all of this pertains to the world of interacting subatomic particles; the more modest view of the microscopic world presented in this book requires somewhat less. The reader with a knowledge of abstract integration, Hilbert space theory, and topological groups will find the going easy.

The Topos of Music

The Topos of Music Author Guerino Mazzola
ISBN-10 9783034881418
Release 2012-12-06
Pages 1344
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With contributions by numerous experts

Generalized Musical Intervals and Transformations

Generalized Musical Intervals and Transformations Author David Lewin
ISBN-10 9780199759941
Release 2010-11-04
Pages 258
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David Lewin's Generalized Musical Intervals and Transformations is recognized as the seminal work paving the way for current studies in mathematical and systematic approaches to music analysis. Lewin, one of the 20th century's most prominent figures in music theory, pushes the boundaries of the study of pitch-structure beyond its conception as a static system for classifying and inter-relating chords and sets. Known by most music theorists as "GMIT", the book is by far the most significant contribution to the field of systematic music theory in the last half-century, generating the framework for the "transformational theory" movement. Appearing almost twenty years after GMIT's initial publication, this Oxford University Press edition features a previously unpublished preface by David Lewin, as well as a foreword by Edward Gollin contextualizing the work's significance for the current field of music theory.

Mathematics and Music

Mathematics and Music Author David Wright
ISBN-10 9780821848739
Release 2009
Pages 161
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Many people intuitively sense that there is a connection between mathematics and music. If nothing else, both involve counting. There is, of course, much more to the association. David Wright's book is an investigation of the interrelationships between mathematics and music, reviewing the needed background concepts in each subject as they are encountered. Along the way, readers will augment their understanding of both mathematics and music. The text explores the common foundations of the two subjects, which are developed side by side. Musical and mathematical notions are brought together, such as scales and modular arithmetic, intervals and logarithms, tone and trigonometry, and timbre and harmonic analysis. When possible, discussions of musical and mathematical notions are directly interwoven. Occasionally the discourse dwells for a while on one subject and not the other, but eventually the connection is established, making this an integrative treatment of the two subjects. The book is a text for a freshman level college course suitable for musically inclined or mathematically inclined students, with the intent of breaking down any apprehension that either group might have for the other subject. Exercises are given at the end of each chapter. The mathematical prerequisites are a high-school level familiarity with algebra, trigonometry, functions, and graphs. Musically, the student should have had some exposure to musical staffs, standard clefs, and key signatures, though all of these are explained in the text.


Harmonograph Author Anthony Ashton
ISBN-10 9780802714091
Release 2003-04-01
Pages 58
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Ashton presents a short, illustrated introduction to the evolution of simple harmonic theory. Illustrations.

The Geometry of Desert

The Geometry of Desert Author Shelly Kagan
ISBN-10 9780190233723
Release 2014-12-04
Pages 676
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People differ in terms of how morally deserving they are. And it is a good thing if people get what they deserve. Accordingly, it is important to work out an adequate theory of moral desert. But while certain aspects of such a theory have been frequently discussed in the philosophical literature, many others have been surprisingly neglected. For example, if it is indeed true that it is morally good for people to get what they deserve, does it always do the same amount of good when someone gets what they deserve? Or does it matter how deserving the person is? If we cannot give someone exactly what they deserve, is it better to give too much-or better to give too little? Does being twice as virtuous make you twice as deserving? And how are we to take into account the thought that what you deserve depends in part on how others are doing? The Geometry of Desert explores a number of these less familiar questions, using graphs to illustrate the various possible answers. The result is a more careful investigation into the nature of moral desert than has ever previously been offered, one that reveals desert to have a hidden complexity that most of us have failed to recognize.

The Rhythmic Structure of Music

The Rhythmic Structure of Music Author Grosvenor Cooper
ISBN-10 0226115224
Release 1963-04-15
Pages 212
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In this influential book on the subject of rhythm, the authors develop a theoretical framework based essentially on a Gestalt approach, viewing rhythmic experience in terms of pattern perception or groupings. Musical examples of increasing complexity are used to provide training in the analysis, performance, and writing of rhythm, with exercises for the student's own work. "This is a path-breaking work, important alike to music students and teachers, but it will make profitable reading for performers, too."—New York Times Book Review "When at some future time theories of rhythm . . . are . . . as well understood, and as much discussed as theories of harmony and counterpoint . . . they will rest in no small measure on the foundations laid by Cooper and Meyer in this provocative dissertation on the rhythmic structure of music."—Notes ". . . . a significant, courageous and, on the whole, successful attempt to deal with a very controversial and neglected subject. Certainly no one who takes the time to read it will emerge from the experience unchanged or unmoved."—Journal of Music Theory The late GROSVENOR W. COOPER, author of Learning to Listen, was professor of music at the University of California at Santa Cruz.


Musimathics Author Gareth Loy
ISBN-10 9780262516556
Release 2011-08-19
Pages 504
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"Mathematics can be as effortless as humming a tune, if you know the tune," writes Gareth Loy. In Musimathics, Loy teaches us the tune, providing a friendly and spirited tour of the mathematics of music--a commonsense, self-contained introduction for the nonspecialist reader. It is designed for musicians who find their art increasingly mediated by technology, and for anyone who is interested in the intersection of art and science.In this volume, Loy presents the materials of music (notes, intervals, and scales); the physical properties of music (frequency, amplitude, duration, and timbre); the perception of music and sound (how we hear); and music composition. Musimathics is carefully structured so that new topics depend strictly on topics already presented, carrying the reader progressively from basic subjects to more advanced ones. Cross-references point to related topics and an extensive glossary defines commonly used terms. The book explains the mathematics and physics of music for the reader whose mathematics may not have gone beyond the early undergraduate level. Calling himself "a composer seduced into mathematics," Loy provides answers to foundational questions about the mathematics of music accessibly yet rigorously. The topics are all subjects that contemporary composers, musicians, and musical engineers have found to be important. The examples given are all practical problems in music and audio. The level of scholarship and the pedagogical approach also make Musimathics ideal for classroom use. Additional material can be found at a companion web site.

The Geometry of Love

The Geometry of Love Author Jessica Levine
ISBN-10 9781938314636
Release 2014-04-08
Pages 292
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Julia, an aspiring poet, is living with her British boyfriend, Ben, a restrained Princeton professor, when she runs into Michael, a long-lost friend. A complex and compelling composer, Michael was once a catalyzing muse for her—but his return to her life is a destabilizing influence. Julia is drawn to Michael, but feels enormous guilt at the thought of betraying Ben—not to mention fear at the idea of giving up the security of her relationship with him. So, when Michael signals that he’s too wounded to make a commitment, she turns her triangular situation into a square: she sets him up with her cousin. Why is it easier for a woman to be a muse than to have one? Are security and imagination mutually exclusive? Can one be fully creative—in art or life—without the inspiration of erotic love? These are the questions asked in The Geometry of Love, a provocative and deeply psychological tale that explores the surprising choices we make in our romantic lives.

Geometry of Design

Geometry of Design Author Kimberly Elam
ISBN-10 1568982496
Release 2001
Pages 107
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This work takes a close look at a broad range of 20th-century examples of design, architecture and illustration, revealing underlying geometric structures in their compositions.

Differential Geometry of Three Dimensions

Differential Geometry of Three Dimensions Author C. E. Weatherburn
ISBN-10 9781316606957
Release 2016-04-28
Pages 252
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Originally published in 1930, as the second of a two-part set, this informative and systematically organized textbook, primarily aimed at university students, contains a vectorial treatment of geometry, reasoning that by the use of such vector methods, geometry is able to be both simplified and condensed. Topics covered include Flexion and Applicability of Surfaces, Levi-Civita's theory of parallel displacements on a surface and the theory of Curvilinear Congruences. Diagrams are included to supplement the text. Providing a detailed overview of the subject and forming a solid foundation for study of multidimensional differential geometry and the tensor calculus, this book will prove an invaluable reference work to scholars of mathematics as well as to anyone with an interest in the history of education.