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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.

A New Geometry of Musical Chords in Interval Representation

A New Geometry of Musical Chords in Interval Representation Author Miguel Gutierrez
ISBN-10 145022797X
Release 2010
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.


Harmonograph Author Anthony Ashton
ISBN-10 9781912706051
Release 2018-04-01
Pages 64
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Why did Pythagoras pause outside a Blacksmith's workshop? Can the nature of Harmony really be understood visually? Why do harmonies leave gaps or 'commas' when added together? In this charming little book Anthony Ashton uses a Victorian device called a Harmonograph to tell the story of Harmony and the intervals in the scale. With useful appendices and exquisite line drawings this is a unique and original introduction to this timeless subject. WOODEN BOOKS are small but packed with information. "Fascinating" FINANCIAL TIMES. "Beautiful" LONDON REVIEW OF BOOKS. "Rich and Artful" THE LANCET. "Genuinely mind-expanding" FORTEAN TIMES. "Excellent" NEW SCIENTIST. "Stunning" NEW YORK TIMES. Small books, big ideas.

Hearing in Time

Hearing in Time Author Justin London
ISBN-10 9780199744374
Release 2012-05-24
Pages 234
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When we hear music we don't just listen; we move along with it. Hearing in Time explores our innate propensity for rhythmic synchronization, drawing on research in music psychology, neurobiology, music theory, and mathematics. It looks at music from a wide range of musical styles and cultures.

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 and the Visual Arts

Geometry and the Visual Arts Author Daniel Pedoe
ISBN-10 048624458X
Release 1976
Pages 296
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This survey traces the effects of geometry on artistic achievement and clearly discusses its importance to artists and scientists. It also surveys projective geometry, mathematical curves, theories of perspective, architectural form, and concepts of space.

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.

A Course in the Geometry of N Dimensions

A Course in the Geometry of N Dimensions Author Maurice G. Kendall
ISBN-10 9780486439273
Release 2004
Pages 63
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This text for undergraduate students provides a foundation for resolving proofs dependent on n-dimensional systems. The two-part treatment begins with simple figures in n dimensions and advances to examinations of the contents of hyperspheres, hyperellipsoids, hyperprisms, etc. The second part explores the mean in rectangular variation, the correlation coefficient in bivariate normal variation, Wishart's distribution, more. 1961 edition.

Music by the Numbers

Music by the Numbers Author Eli Maor
ISBN-10 9781400889891
Release 2018-05-15
Pages 176
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How music has influenced mathematics, physics, and astronomy from ancient Greece to the twentieth century Music is filled with mathematical elements, the works of Bach are often said to possess a math-like logic, and Igor Stravinsky said "musical form is close to mathematics," while Arnold Schoenberg, Iannis Xenakis, and Karlheinz Stockhausen went further, writing music explicitly based on mathematical principles. Yet Eli Maor argues that music has influenced math at least as much as math has influenced music. Starting with Pythagoras, proceeding through the work of Schoenberg, and ending with contemporary string theory, Music by the Numbers tells a fascinating story of composers, scientists, inventors, and eccentrics who played a role in the age-old relationship between music, mathematics, and the sciences, especially physics and astronomy. Music by the Numbers explores key moments in this history, particularly how problems originating in music have inspired mathematicians for centuries. Perhaps the most famous of these problems is the vibrating string, which pitted some of the greatest mathematicians of the eighteenth century against each other in a debate that lasted more than fifty years and that eventually led to the development of post-calculus mathematics. Other highlights in the book include a comparison between meter in music and metric in geometry, complete with examples of rhythmic patterns from Bach to Stravinsky, and an exploration of a suggestive twentieth-century development: the nearly simultaneous emergence of Einstein's theory of relativity and Schoenberg's twelve-tone system. Weaving these compelling historical episodes with Maor's personal reflections as a mathematician and lover of classical music, Music by the Numbers will delight anyone who loves mathematics and music.


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.

Topology and Geometry for Physicists

Topology and Geometry for Physicists Author Charles Nash
ISBN-10 9780486318363
Release 2013-08-16
Pages 320
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Differential geometry and topology are essential tools for many theoretical physicists, particularly in the study of condensed matter physics, gravity, and particle physics. Written by physicists for physics students, this text introduces geometrical and topological methods in theoretical physics and applied mathematics. It assumes no detailed background in topology or geometry, and it emphasizes physical motivations, enabling students to apply the techniques to their physics formulas and research. "Thoroughly recommended" by The Physics Bulletin, this volume's physics applications range from condensed matter physics and statistical mechanics to elementary particle theory. Its main mathematical topics include differential forms, homotopy, homology, cohomology, fiber bundles, connection and covariant derivatives, and Morse theory.

Music at Hand

Music at Hand Author Jonathan De Souza
ISBN-10 9780190271114
Release 2017
Pages 208
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From prehistoric bone flutes to pipe organs to digital synthesizers, instruments have been important to musical cultures around the world. Yet, how do instruments affect musical organization? And how might they influence players' bodies and minds?Music at Hand explores these questions with a distinctive blend of music theory, psychology, and philosophy. Practicing an instrument, of course, builds bodily habits and skills. But it also develops connections between auditory and motor regions in a player's brain. These multi-sensory links aregrounded in particular instrumental interfaces. They reflect the ways that an instrument converts action into sound, and the ways that it coordinates physical and tonal space. Ultimately, these connections can shape listening, improvisation, or composition. This means that pianos, guitars, horns,and bells are not simply tools for making notes. Such technologies, as creative prostheses, also open up possibilities for musical action, perception, and cognition.Throughout the book, author Jonathan De Souza examines diverse musical case studies - from Beethoven to blues harmonica, from Bach to electronic music - introducing novel methods for the analysis of body-instrument interaction. A companion website supports these analytical discussions withaudiovisual examples, including motion-capture videos and performances by the author. Written in lucid prose, Music at Hand offers substantive insights for music scholars, while remaining accessible to non-specialist readers. This wide-ranging book will engage music theorists and historians,ethnomusicologists, organologists, composers, and performers - but also psychologists, philosophers, media theorists, and anyone who is curious about how musical experience is embodied and conditioned by technology.

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 Geometry of Wealth

The Geometry of Wealth Author Brian Portnoy
ISBN-10 9780857196729
Release 2018-06-11
Pages 232
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How does money figure into a happy life? In The Geometry of Wealth, behavioral finance expert Brian Portnoy delivers an inspired answer, building on the critical distinction between being rich and being wealthy. While one is an unsatisfying treadmill, the other is the ability to underwrite a meaningful life, however one chooses to define that. Truly viewed, wealth is funded contentment. At the heart of this groundbreaking perspective, Portnoy takes readers on a journey toward wealth, informed by disciplines ranging from ancient history to modern neuroscience. He contends that tackling the big questions about a joyful life and tending to financial decisions are complementary, not separate, tasks. These big questions include: • How is the human brain wired for two distinct experiences of happiness? And why can money “buy” one but not the other? • What are the touchstones of a meaningful life, and are they affordable? • Why is market savvy among the least important sources of wealth but self-awareness is among the most? • How does one strike a balance between striving for more while being content with enough? This journey memorably contours along three basic shapes: A circle, triangle and square help us to visualize how we adapt to evolving circumstances, set clear priorities, and find empowerment in simplicity. In this accessible and entertaining book, Portnoy reveals that true wealth is achievable for many - including those who despair it is out of reach - but only in the context of a life in which purpose and practice are thoughtfully calibrated.

Music and the Making of Modern Science

Music and the Making of Modern Science Author Peter Pesic
ISBN-10 9780262027274
Release 2014-07-04
Pages 360
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In the natural science of ancient Greece, music formed the meeting place between numbers and perception; for the next two millennia, Pesic tells us in Music and the Making of Modern Science, "liberal education" connected music with arithmetic, geometry, and astronomy within a fourfold study, the quadrivium. Peter Pesic argues provocatively that music has had a formative effect on the development of modern science -- that music has been not just a charming accompaniment to thought but a conceptual force in its own right. Pesic explores a series of episodes in which music influenced science, moments in which prior developments in music arguably affected subsequent aspects of natural science. He describes encounters between harmony and fifteenth-century cosmological controversies, between musical initiatives and irrational numbers, between vibrating bodies and the emergent electromagnetism. He offers lively accounts of how Newton applied the musical scale to define the colors in the spectrum; how Euler and others applied musical ideas to develop the wave theory of light; and how a harmonium prepared Max Planck to find a quantum theory that reengaged the mathematics of vibration. Taken together, these cases document the peculiar power of music -- its autonomous force as a stream of experience, capable of stimulating insights different from those mediated by the verbal and the visual. An innovative e-book edition available for iOS devices will allow sound examples to be played by a touch and shows the score in a moving line.

The Geometry of Love

The Geometry of Love Author Margaret Visser
ISBN-10 9781504011709
Release 2015-06-23
Pages 328
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A “delightful” tour of Rome’s St. Agnes Outside the Walls, examining the stories, rituals, and architecture of this seventeen-hundred-year-old building (The Christian Science Monitor). In The Geometry of Love, acclaimed author Margaret Visser, the preeminent “anthropologist of everyday life,” takes on the living history of the ancient church of St. Agnes. Examining every facet of the building, from windows to catacombs, Visser takes readers on a mesmerizing tour of the old church, covering its social, political, religious, and architectural history. In so doing, she illuminates not only the church’s evolution but also its religious legacy in our modern lives. Written as an antidote to the usual dry and traditional studies of European churches, The Geometry of Love is infused with Visser’s unmatched warmth and wit, celebrating the remarkable ways that one building can reveal so much about our history and ourselves.