## The Topos of Music

*Geometric Logic of Concepts, Theory, and Performance*

**Author**: Guerino Mazzola

**Publisher:**Birkhäuser

**ISBN:**303488141X

**Category:**Mathematics

**Page:**1344

**View:**9429

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With contributions by numerous experts

## The Topos of Music

*Geometric Logic of Concepts, Theory, and Performance*

**Author**: Guerino Mazzola

**Publisher:**Springer Science & Business Media

**ISBN:**9783764357313

**Category:**Mathematics

**Page:**1335

**View:**5793

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With contributions by numerous experts

## The Topos of Music

**Author**: Guerino Mazzola

**Publisher:**Springer

**ISBN:**9783319644332

**Category:**Mathematics

**Page:**1580

**View:**9525

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This four-volume set is the second edition of the now classic book “The Topos of Music”. In Vol. I the author explains the theory's conceptual framework of denotators and forms, the classification of local and global musical objects, the mathematical models of harmony and counterpoint, and topologies for rhythm and motives. In Vol. II the author explains his theory of musical performance, developed in the language of differential geometry, introducing performance vector fields that generalize tempo and intonation. Volume III presents gesture theory, including a gesture philosophy for music, the mathematics of gestures, concept architectures and software for musical gesture theory, the multiverse perspective which reveals the relationship between gesture theory and the string theory in theoretical physics, and applications of gesture theory to a number of musical themes, including counterpoint, modulation theory, free jazz, Hindustani music, and vocal gestures. Finally, Vol. IV contains appendices, explaining background topics in sound, mathematics, and music.

## Cool Math for Hot Music

*A First Introduction to Mathematics for Music Theorists*

**Author**: Guerino Mazzola,Maria Mannone,Yan Pang

**Publisher:**Springer

**ISBN:**331942937X

**Category:**Computers

**Page:**323

**View:**485

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This textbook is a first introduction to mathematics for music theorists, covering basic topics such as sets and functions, universal properties, numbers and recursion, graphs, groups, rings, matrices and modules, continuity, calculus, and gestures. It approaches these abstract themes in a new way: Every concept or theorem is motivated and illustrated by examples from music theory (such as harmony, counterpoint, tuning), composition (e.g., classical combinatorics, dodecaphonic composition), and gestural performance. The book includes many illustrations, and exercises with solutions.

## Music Through Fourier Space

*Discrete Fourier Transform in Music Theory*

**Author**: Emmanuel Amiot

**Publisher:**Springer

**ISBN:**3319455818

**Category:**Computers

**Page:**206

**View:**3533

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This book explains the state of the art in the use of the discrete Fourier transform (DFT) of musical structures such as rhythms or scales. In particular the author explains the DFT of pitch-class distributions, homometry and the phase retrieval problem, nil Fourier coefficients and tilings, saliency, extrapolation to the continuous Fourier transform and continuous spaces, and the meaning of the phases of Fourier coefficients. This is the first textbook dedicated to this subject, and with supporting examples and exercises this is suitable for researchers and advanced undergraduate and graduate students of music, computer science and engineering. The author has made online supplementary material available, and the book is also suitable for practitioners who want to learn about techniques for understanding musical notions and who want to gain musical insights into mathematical problems.

## Categories, Allegories

**Author**: P.J. Freyd,A. Scedrov

**Publisher:**Elsevier

**ISBN:**9780080887012

**Category:**Mathematics

**Page:**293

**View:**7145

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General concepts and methods that occur throughout mathematics – and now also in theoretical computer science – are the subject of this book. It is a thorough introduction to Categories, emphasizing the geometric nature of the subject and explaining its connections to mathematical logic. The book should appeal to the inquisitive reader who has seen some basic topology and algebra and would like to learn and explore further. The first part contains a detailed treatment of the fundamentals of Geometric Logic, which combines four central ideas: natural transformations, sheaves, adjoint functors, and topoi. A special feature of the work is a general calculus of relations presented in the second part. This calculus offers another, often more amenable framework for concepts and methods discussed in part one. Some aspects of this approach find their origin in the relational calculi of Peirce and Schroeder from the last century, and in the 1940's in the work of Tarski and others on relational algebras. The representation theorems discussed are an original feature of this approach.

## Musimathics

*The Mathematical Foundations of Music*

**Author**: Gareth Loy

**Publisher:**MIT Press

**ISBN:**0262516551

**Category:**Education

**Page:**504

**View:**472

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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.

## Mathematics and Computation in Music

*4th International Conference, MCM 2013, Montreal, Canada, June 12-14, 2013, Proceedings*

**Author**: Jason Yust,Jonathan Wild,John Ashley Burgoyne

**Publisher:**Springer

**ISBN:**3642393578

**Category:**Computers

**Page:**241

**View:**1018

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This book constitutes the thoroughly refereed proceedings of the Fourth International Conference on Mathematics and Computation in Music, MCM 2013, held in Montreal, Canada, in June 2013. The 18 papers presented were carefully reviewed and selected from numerous submissions. They are promoting the collaboration and exchange of ideas among researchers in music theory, mathematics, computer science, musicology, cognition and other related fields.

## The Geometry of Musical Rhythm

*What Makes a "Good" Rhythm Good?*

**Author**: Godfried T. Toussaint

**Publisher:**CRC Press

**ISBN:**1466512032

**Category:**Mathematics

**Page:**365

**View:**9401

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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.

## Designing the City of Reason

*Foundations and Frameworks*

**Author**: Ali Madanipour

**Publisher:**Routledge

**ISBN:**1134103980

**Category:**Architecture

**Page:**352

**View:**7171

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With a practical approach to theory, Designing the City of Reason offers new perspectives on how differing belief systems and philosophical approaches impact on city design and development, exploring how this has changed before, during and after the impact of modernism in all its rationalism. Looking at the connections between abstract ideas and material realities, this book provides a social and historical account of ideas which have emerged out of the particular concerns and cultural contexts and which inform the ways we live. By considering the changing foundations for belief and action, and their impact on urban form, it follows the history and development of city design in close conjunction with the growth of rationalist philosophy. Building on these foundations, it goes on to focus on the implications of this for urban development, exploring how public infrastructures of meaning are constructed and articulated through the dimensions of time, space, meaning, value and action. With its wide-ranging subject matter and distinctive blend of theory and practice, this book furthers the scope and range of urban design by asking new questions about the cities we live in and the values and symbols which we assign to them.

## A Generative Theory of Shape

**Author**: Michael Leyton

**Publisher:**Springer

**ISBN:**3540454888

**Category:**Computers

**Page:**549

**View:**6819

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The purpose of this book is to develop a generative theory of shape that has two properties we regard as fundamental to intelligence –(1) maximization of transfer: whenever possible, new structure should be described as the transfer of existing structure; and (2) maximization of recoverability: the generative operations in the theory must allow maximal inferentiability from data sets. We shall show that, if generativity satis?es these two basic criteria of - telligence, then it has a powerful mathematical structure and considerable applicability to the computational disciplines. The requirement of intelligence is particularly important in the gene- tion of complex shape. There are plenty of theories of shape that make the generation of complex shape unintelligible. However, our theory takes the opposite direction: we are concerned with the conversion of complexity into understandability. In this, we will develop a mathematical theory of und- standability. The issue of understandability comes down to the two basic principles of intelligence - maximization of transfer and maximization of recoverability. We shall show how to formulate these conditions group-theoretically. (1) Ma- mization of transfer will be formulated in terms of wreath products. Wreath products are groups in which there is an upper subgroup (which we will call a control group) that transfers a lower subgroup (which we will call a ?ber group) onto copies of itself. (2) maximization of recoverability is insured when the control group is symmetry-breaking with respect to the ?ber group.

## And Yet It Is Heard

*Musical, Multilingual and Multicultural History of the Mathematical Sciences -*

**Author**: Tito M. Tonietti

**Publisher:**Springer

**ISBN:**3034806752

**Category:**Mathematics

**Page:**593

**View:**7960

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We bring into full light some excerpts on musical subjects which were until now scattered throughout the most famous scientific texts. The main scientific and musical cultures outside of Europe are also taken into consideration. The first and most important property to underline in the scientific texts examined here is the language they are written in. This means that our multicultural history of the sciences necessarily also becomes a review of the various dominant languages used in the different historical contexts. In this volume, the history of the development of the sciences is told as it happened in real contexts, not in an alienated ideal world.

## Etale Cohomology (PMS-33)

**Author**: James S. Milne

**Publisher:**Princeton University Press

**ISBN:**1400883989

**Category:**Mathematics

**Page:**344

**View:**1930

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One of the most important mathematical achievements of the past several decades has been A. Grothendieck's work on algebraic geometry. In the early 1960s, he and M. Artin introduced étale cohomology in order to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry, but also in several different branches of number theory and in the representation theory of finite and p-adic groups. Yet until now, the work has been available only in the original massive and difficult papers. In order to provide an accessible introduction to étale cohomology, J. S. Milne offers this more elementary account covering the essential features of the theory. The author begins with a review of the basic properties of flat and étale morphisms and of the algebraic fundamental group. The next two chapters concern the basic theory of étale sheaves and elementary étale cohomology, and are followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Professor Milne proves the fundamental theorems in étale cohomology -- those of base change, purity, Poincaré duality, and the Lefschetz trace formula. He then applies these theorems to show the rationality of some very general L-series. Originally published in 1980. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

## Mathematics and Computation in Music

*Second International Conference, MCM 2009, New Haven, CT, USA, June 19-22, 2009. Proceedings*

**Author**: Elaine Chew,Adrian Childs,Ching-Hua Chuan

**Publisher:**Springer Science & Business Media

**ISBN:**3642023940

**Category:**Computers

**Page:**298

**View:**4996

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This book constitutes the refereed proceedings of the Second International Conference on Mathematics and Computation in Music, MCM 2009, held in New Haven, CT, USA, in June 2009. The 26 revised full papers presented were carefully reviewed and selected from 38 submissions. The MCM conference is the flagship conference of the Society for Mathematics and Computation in Music. The papers deal with topics within applied mathematics, computational models, mathematical modelling and various further aspects of the theory of music. This year’s conference is dedicated to the honor of John Clough whose research modeled the virtues of collaborative work across the disciplines.

## From Music to Mathematics

*Exploring the Connections*

**Author**: N.A

**Publisher:**JHU Press

**ISBN:**142141919X

**Category:**Mathematics

**Page:**320

**View:**4764

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Taking a "music first" approach, Gareth E. Roberts’s From Music to Mathematics will inspire students to learn important, interesting, and at times advanced mathematics. Ranging from a discussion of the geometric sequences and series found in the rhythmic structure of music to the phase-shifting techniques of composer Steve Reich, the musical concepts and examples in the book motivate a deeper study of mathematics. Comprehensive and clearly written, From Music to Mathematics is designed to appeal to readers without specialized knowledge of mathematics or music. Students are taught the relevant concepts from music theory (notation, scales, intervals, the circle of fifths, tonality, etc.), with the pertinent mathematics developed alongside the related musical topic. The mathematics advances in level of difficulty from calculating with fractions, to manipulating trigonometric formulas, to constructing group multiplication tables and proving a number is irrational. Topics discussed in the book include ? Rhythm ? Introductory music theory ? The science of sound ? Tuning and temperament? Symmetry in music ? The Bartók controversy ? Change ringing ? Twelve-tone music? Mathematical modern music ? The Hemachandra–Fibonacci numbers and the golden ratio? Magic squares ? Phase shifting Featuring numerous musical excerpts, including several from jazz and popular music, each topic is presented in a clear and in-depth fashion. Sample problems are included as part of the exposition, with carefully written solutions provided to assist the reader. The book also contains more than 200 exercises designed to help develop students’ analytical skills and reinforce the material in the text. From the first chapter through the last, readers eager to learn more about the connections between mathematics and music will find a comprehensive textbook designed to satisfy their natural curiosity.

## A Geometry of Music

*Harmony and Counterpoint in the Extended Common Practice*

**Author**: Dmitri Tymoczko

**Publisher:**OUP USA

**ISBN:**0195336674

**Category:**Mathematics

**Page:**450

**View:**8709

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In this groundbreaking book, Tymoczko uses contemporary geometry to provide a new framework for thinking about music, one that emphasizes the commonalities among styles from Medieval polyphony to contemporary jazz.

## Mind and Nature

*Selected Writings on Philosophy, Mathematics, and Physics*

**Author**: Hermann Weyl

**Publisher:**Princeton University Press

**ISBN:**9781400833320

**Category:**Mathematics

**Page:**272

**View:**9018

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Hermann Weyl (1885-1955) was one of the twentieth century's most important mathematicians, as well as a seminal figure in the development of quantum physics and general relativity. He was also an eloquent writer with a lifelong interest in the philosophical implications of the startling new scientific developments with which he was so involved. Mind and Nature is a collection of Weyl's most important general writings on philosophy, mathematics, and physics, including pieces that have never before been published in any language or translated into English, or that have long been out of print. Complete with Peter Pesic's introduction, notes, and bibliography, these writings reveal an unjustly neglected dimension of a complex and fascinating thinker. In addition, the book includes more than twenty photographs of Weyl and his family and colleagues, many of which are previously unpublished. Included here are Weyl's exposition of his important synthesis of electromagnetism and gravitation, which Einstein at first hailed as "a first-class stroke of genius"; two little-known letters by Weyl and Einstein from 1922 that give their contrasting views on the philosophical implications of modern physics; and an essay on time that contains Weyl's argument that the past is never completed and the present is not a point. Also included are two book-length series of lectures, The Open World (1932) and Mind and Nature (1934), each a masterly exposition of Weyl's views on a range of topics from modern physics and mathematics. Finally, four retrospective essays from Weyl's last decade give his final thoughts on the interrelations among mathematics, philosophy, and physics, intertwined with reflections on the course of his rich life.

## Music and Mathematics

*From Pythagoras to Fractals*

**Author**: John Fauvel,Raymond Flood,Robin J. Wilson

**Publisher:**Oxford University Press on Demand

**ISBN:**9780199298938

**Category:**Mathematics

**Page:**189

**View:**925

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From Ancient Greek times, music has been seen as a mathematical art, and the relationship between mathematics and music has fascinated generations. This collection of wide ranging, comprehensive and fully-illustrated papers, authorized by leading scholars, presents the link between these two subjects in a lucid manner that is suitable for students of both subjects, as well as the general reader with an interest in music. Physical, theoretical, physiological, acoustic, compositional and analytical relationships between mathematics and music are unfolded and explored with focus on tuning and temperament, the mathematics of sound, bell-ringing and modern compositional techniques.

## The Topos of Music III: Gestures

*Musical Multiverse Ontologies*

**Author**: Guerino Mazzola,René Guitart,Jocelyn Ho,Alex Lubet,Maria Mannone,Matt Rahaim,Florian Thalmann

**Publisher:**Springer

**ISBN:**3319644815

**Category:**Mathematics

**Page:**604

**View:**6626

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This is the third volume of the second edition of the now classic book “The Topos of Music”. The authors present gesture theory, including a gesture philosophy for music, the mathematics of gestures, concept architectures and software for musical gesture theory, the multiverse perspective which reveals the relationship between gesture theory and the string theory in theoretical physics, and applications of gesture theory to a number of musical themes, including counterpoint, modulation theory, free jazz, Hindustani music, and vocal gestures.

## Philosophy of Mathematics

*Structure and Ontology*

**Author**: Stewart Shapiro

**Publisher:**Oxford University Press

**ISBN:**9780198025450

**Category:**Philosophy

**Page:**296

**View:**3086

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Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians.