A Geometry of Music

Harmony and Counterpoint in the Extended Common Practice

Author: Dmitri Tymoczko

Publisher: Oxford University Press

ISBN: 0199887500

Category: Music

Page: 480

View: 4335

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.

Audacious Euphony

Chromatic Harmony and the Triad's Second Nature

Author: Richard Cohn

Publisher: Oxford University Press

ISBN: 0199773211

Category: Music

Page: 288

View: 3168

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.

Tonality and Transformation

Author: Steven Rings

Publisher: Oxford University Press

ISBN: 019991320X

Category: Music

Page: 272

View: 595

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.

Rameau and Musical Thought in the Enlightenment

Author: Thomas Christensen

Publisher: Cambridge University Press

ISBN: 9780521617093

Category: Music

Page: 327

View: 1251

"Ranging widely over the musical and intellectual thought of the eighteenth century, Thomas Christensen orients Rameau's accomplishments in the light of contemporaneous traditions of music theory as well as many of the scientific ideas current in the French Enlightenment. Rameau is revealed to be an unsuspectedly syncretic and sophisticated thinker, betraying influences ranging from neoplatonic thought and Cartesian mechanistic metaphysics to Locke's empirical psychology and Newtonian experimental science. Additional primary documents and manuscripts (many revealed here for the first time) help clarify Rameau's fascinating and stormy relationship with the Encyclopedists: Diderot, Rousseau, and d'Alembert." "This book will be of value to all music theorists concerned with the foundations of harmonic tonality and it should also be of interest to scholars of eighteenth-century science, the Enlightenment, and the general history of ideas."--BOOK JACKET.

Musical Form and Transformation

Four Analytic Essays

Author: David Lewin

Publisher: Oxford University Press

ISBN: 019989020X

Category: Music

Page: 192

View: 3582

Distinguished music theorist and composer David Lewin (1933-2003) applies the conceptual framework he developed in his earlier, innovative Generalized Musical Intervals and Transformations to the varied repertoire of the twentieth century in this stimulating and illustrative book. Analyzing the diverse compositions of four canonical composers--Simbolo from Dallapiccola's Quaderno musicale di Annalibera ; Stockhausen's Klavierstuck III ; Webern's Op. 10, No. 4; and Debussy's Feux d'articifice --Lewin brings forth structures which he calls "transformational networks" to reveal interesting and suggestive aspects of the music. In this complementary work, Lewin stimulates thought about the general methodology of musical analysis and issues of large-scale form as they relate to transformational analytic structuring. Musical Form and Transformation , first published in 1993 by Yale University Press, was the recipient of an ASCAP Deems Taylor Award.

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: 431

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.

Generalized Musical Intervals and Transformations

Author: David Lewin

Publisher: Oxford University Press, USA

ISBN: 0199759944

Category: Music

Page: 258

View: 7010

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

Composition, Perception, and Performance

Author: James S. Walker,Gary W. Don

Publisher: CRC Press

ISBN: 1439867097

Category: Mathematics

Page: 342

View: 9870

At first glance, mathematics and music seem to be from separate worlds—one from science, one from art. But in fact, the connections between the two go back thousands of years, such as Pythagoras’s ideas about how to quantify changes of pitch for musical tones (musical intervals). Mathematics and Music: Composition, Perception, and Performance explores the many links between mathematics and different genres of music, deepening students’ understanding of music through mathematics. In an accessible way, the text teaches the basics of reading music and explains how various patterns in music can be described with mathematics. The authors extensively use the powerful time-frequency method of spectrograms to analyze the sounds created in musical performance. Numerous examples of music notation assist students in understanding basic musical scores. The text also provides mathematical explanations for musical scales, harmony, and rhythm and includes a concise introduction to digital audio synthesis. Along with helping students master some fundamental mathematics, this book gives them a deeper appreciation of music by showing how music is informed by both its mathematical and aesthetic structures. Web Resource On the book’s CRC Press web page, students can access videos of many of the spectrograms discussed in the text as well as musical scores playable with the free music software MuseScore. An online bibliography offers many links to free downloadable articles on math and music. The web page also provides links to other websites related to math and music, including all the sites mentioned in the book.

Tuning, Timbre, Spectrum, Scale

Author: William A. Sethares

Publisher: Springer Science & Business Media

ISBN: 9781852337971

Category: Technology & Engineering

Page: 426

View: 7219

Table2. 2. Each note consists of three partials. If the sequence is played ascending, then the ?rst virtual pitch tends to be perceived, whereas if played descending, the second, lower virtual pitch tends to be heard. Only one virtual pitch is audible at a time. This can be heard in sound examples [S: 6] and [S: 7]. Note First Second Third Virtual Pitch Virtual Pitch partial partial partial ascending descending 1 600 800 1000 200. 0 158. 9 2 620 820 1020 205. 2 163. 0 3 640 840 1040 210. 4 167. 1 4 660 860 1060 215. 6 171. 2 5 680 880 1080 220. 9 175. 3 6 700 900 1100 226. 1 179. 4 7 720 920 1120 231. 3 183. 6 8 740 940 1140 236. 6 187. 7 9 760 960 1160 241. 8 191. 8 10 780 980 1180 247. 0 195. 9 11 800 1000 1200 252. 2 200. 0 Pitch and virtual pitch are properties of a single sound. For instance, a chord played by the violin, viola, and cello of a string quartet is not usually thoughtofashavingapitch;rather,pitchisassociatedwitheachinstrumental tone separately. Thus, determining the pitch or pitches of a complex sound source requires that it ?rst be partitioned into separate perceptual entities. Only when a cluster of partials fuse into a single sound can it be assigned a pitch. When listening analytically, for instance, there may be more “notes” presentthaninthesamesoundwhenlisteningholistically.

Mathematics and Music

Author: David Wright

Publisher: American Mathematical Soc.

ISBN: 0821848739

Category: Mathematics

Page: 161

View: 4981

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.


Author: Steven W. Cranford,Markus J. Buehler

Publisher: Springer Science & Business Media

ISBN: 9400716109

Category: Technology & Engineering

Page: 440

View: 5882

Biomateriomics is the holistic study of biological material systems. While such systems are undoubtedly complex, we frequently encounter similar components -- universal building blocks and hierarchical structure motifs -- which result in a diverse set of functionalities. Similar to the way music or language arises from a limited set of music notes and words, we exploit the relationships between form and function in a meaningful way by recognizing the similarities between Beethoven and bone, or Shakespeare and silk. Through the investigation of material properties, examining fundamental links between processes, structures, and properties at multiple scales and their interactions, materiomics explains system functionality from the level of building blocks. Biomateriomics specifically focuses the analysis of the role of materials in the context of biological processes, the transfer of biological material principles towards biomimetic and bioinspired applications, and the study of interfaces between living and non-living systems. The challenges of biological materials are vast, but the convergence of biology, mathematics and engineering as well as computational and experimental techniques have resulted in the toolset necessary to describe complex material systems, from nano to macro. Applying biomateriomics can unlock Nature’s secret to high performance materials such as spider silk, bone, and nacre, and elucidate the progression and diagnosis or the treatment of diseases. Similarly, it contributes to develop a de novo understanding of biological material processes and to the potential of exploiting novel concepts in innovation, material synthesis and design.


The Mathematical Foundations of Music

Author: Gareth Loy

Publisher: MIT Press

ISBN: 0262516551

Category: Education

Page: 504

View: 6619

"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 Mathematical Foundations of Music

Author: Gareth Loy,John Chowning

Publisher: MIT Press

ISBN: 0262292769

Category: Music

Page: 584

View: 3096

Volume 2 of Musimathics continues the story of music engineering begun in Volume 1, focusing on the digital and computational domain. Loy goes deeper into the mathematics of music and sound, beginning with digital audio, sampling, and binary numbers, as well as complex numbers and how they simplify representation of musical signals. Chapters cover the Fourier transform, convolution, filtering, resonance, the wave equation, acoustical systems, sound synthesis, the short-time Fourier transform, and the wavelet transform. These subjects provide the theoretical underpinnings of today's music technology. The examples given are all practical problems in music and audio. Additional material can be found at http://www.musimathics.com.

The Cambridge History of Western Music Theory

Author: Thomas Christensen

Publisher: Cambridge University Press

ISBN: 1316025489

Category: Music

Page: N.A

View: 3182

The Cambridge History of Western Music Theory is the first comprehensive history of Western music theory to be published in the English language. A collaborative project by leading music theorists and historians, the volume traces the rich panorama of music-theoretical thought from the Ancient Greeks to the present day. Recognizing the variety and complexity of music theory as an historical subject, the volume has been organized within a flexible framework. Some chapters are defined chronologically within a restricted historical domain, whilst others are defined conceptually and span longer historical periods. Together the thirty-one chapters present a synthetic overview of the fascinating and complex subject that is historical music theory. Richly enhanced with illustrations, graphics, examples and cross-citations as well as being thoroughly indexed and supplemented by comprehensive bibliographies of the most important primary and secondary literature, this book will be an invaluable resource for students and scholars alike.

Miles Davis

A Research and Information Guide

Author: Clarence Bernard Henry

Publisher: Routledge

ISBN: 1317228391

Category: Music

Page: 308

View: 6134

This research and information guide provides a wide range of scholarship on the life, career, and musical legacy of Miles Davis, and is compiled for an interdisciplinary audience of scholars in jazz and popular music, musicology, and cultural studies. It serves as an excellent tool for librarians, researchers, and scholars sorting through the massive amount of material in the field.

Music at Hand

Instruments, Bodies, and Cognition

Author: Jonathan De Souza

Publisher: Oxford University Press

ISBN: 0190271132

Category: Music

Page: 240

View: 1993

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 are grounded 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 with audiovisual 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 Topos of Music

Geometric Logic of Concepts, Theory, and Performance

Author: Guerino Mazzola

Publisher: Birkhäuser

ISBN: 303488141X

Category: Mathematics

Page: 1344

View: 2548

With contributions by numerous experts

Mahler's Symphonic Sonatas

Author: Seth Monahan

Publisher: Oxford University Press

ISBN: 0190266465

Category: Music

Page: 296

View: 3645

Why would Gustav Mahler (1860-1911), modernist titan and so-called prophet of the New Music, commit himself time and again to the venerable sonata-allegro form of Mozart and Beethoven? How could so gifted a symphonic storyteller be drawn to a framework that many have dismissed as antiquated and dramatically inert? Mahler's Symphonic Sonatas offers a striking new take on this old dilemma. Indeed, it poses these questions seriously for the first time. Rather than downplaying Mahler's sonata designs as distracting anachronisms or innocuous groundplans, author Seth Monahan argues that for much of his career, Mahler used the inner, goal-directed dynamics of sonata form as the basis for some of his most gripping symphonic stories. Laying bare the deeper narrative/processual grammar of Mahler's evolving sonata corpus, Monahan pays particular attention to its recycling of large-scale rhetorical devices and its consistent linkage of tonal plot and affect. He then sets forth an interpretive framework that combines the visionary insights of Theodor W. Adorno-whose Mahler writings are examined here lucidly and at length-with elements of Hepokoski and Darcy's renowned Sonata Theory. What emerges is a tensely dialectical image of Mahler's sonata forms, one that hears the genre's compulsion for tonal/rhetorical closure in full collision with the spontaneous narrative needs of the surrounding music and of the overarching symphonic totality. It is a practice that calls forth sonata form not as a rigid mold, but as a dynamic process-rich with historical resonances and subject to a vast range of complications, curtailments, and catastrophes. With its expert balance of riveting analytical narration and thoughtful methodological reflection, Mahler's Symphonic Sonatas promises to be a landmark text of Mahler reception, and one that will reward scholars and students of the late-Romantic symphony for years to come.