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A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice

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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 s 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.


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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 s 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.

30 review for A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice

  1. 4 out of 5

    Eric

    I noticed that at least a couple of reviewers said that they didn't really get the math. I'm almost certain that that's because the math doesn't check out, though it seem impossible to verify since Tymoczko doesn't really phrase his analysis in terms of the linear algebra that he implies. Nor am I sure that anyone else has seriously applied his analytic methodology to a range of musical pieces. Tymoczko is an excellent composer and all of this must make sense in his head to the degree that he us I noticed that at least a couple of reviewers said that they didn't really get the math. I'm almost certain that that's because the math doesn't check out, though it seem impossible to verify since Tymoczko doesn't really phrase his analysis in terms of the linear algebra that he implies. Nor am I sure that anyone else has seriously applied his analytic methodology to a range of musical pieces. Tymoczko is an excellent composer and all of this must make sense in his head to the degree that he uses his own methodology. But neither linear algebra nor topology is magic and this book seems to be 440+ pages of academically phrased "No, man, trust me, this is gonna blow your mind!" followed by feverish drawings by a sweaty man on an overused white board. Much as with Chomsky's Universal Grammar, I'm sure that this will inspire numerous PhD dissertations because it has the veneer of certainty that there is a rational method that can be applied if only one comes at it from the correct perspective. But there's not really a hypothesis to be tested here. I guess if you "get it" then you get it. I doubt that it's worth struggling to comprehend, however, and if it's composing you're looking to do, you could spend your time more productively than trying to reduce these fractions of reason to their lowest common denominator.

  2. 4 out of 5

    Peter

    Tymoczko is an intriguing composer and adept mathematician, so his mix of math and (lots of ) theory can be extremely thought-provoking for readers familiar with both. The mathematical understanding is by no means necessary, however. One can skim the terminology of the early theoretical writing and still comprehend the latter analysis and application. For those who want to dig in, Tymoczko's engaging, conversational tone keeps the reading interesting even through difficult concepts. His theories Tymoczko is an intriguing composer and adept mathematician, so his mix of math and (lots of ) theory can be extremely thought-provoking for readers familiar with both. The mathematical understanding is by no means necessary, however. One can skim the terminology of the early theoretical writing and still comprehend the latter analysis and application. For those who want to dig in, Tymoczko's engaging, conversational tone keeps the reading interesting even through difficult concepts. His theories (concerning the chord-spaces and scale-spaces that composers explore) are illuminating and have opened up all sorts of ideas for personal experimentation. Tymoczko establishes fascinating connections between early polyphony through Romantic chromaticism and into the explosion of jazz, rock, and post minimalism. An extremely stimulating read!

  3. 5 out of 5

    Eric Chapelle

    So far, it's quite amazing. I love reading the intro section, which pulled me right in when I read the part about the usual academic norm of music theory that's standardized. I felt like the author. Like the question that's usually asked "If it sounds good why can we use that chord?" I've just started reading this book, but so far I feel I will enjoy reading every page of it.

  4. 5 out of 5

    Jeffrey Martin

    This book is a fantastic introduction to ideas related to neo-Riemannian theory. While the author focuses his theory on tertian harmony, the geometrical models he presents would also be applicable to other styles. The one downside to this book is the author's emphasis on the worth of tonal music as opposed to other styles such as atonality. However, this doesn't obscure the value of the theory in the book.

  5. 5 out of 5

    Sahil Raghavan

    Took me nearly a year to get through, but finally did it. A challenge to follow for sure at times, but when it hit, it really hit.

  6. 4 out of 5

    Vladimir

    This a milestone in music theory and it constitutes an unification of many western practices in composition based in geometry. Tymoczko shows how orbifolds constitute appropriate musical spaces when each chord is mapped to a point in such space (two voice leading, for example, occurs more efficiently on certain regions of a Möbius band). Efficient voice leading techniques break some different symmetries in these musical spaces. It brings to consciousness procedures that composers of the past hav This a milestone in music theory and it constitutes an unification of many western practices in composition based in geometry. Tymoczko shows how orbifolds constitute appropriate musical spaces when each chord is mapped to a point in such space (two voice leading, for example, occurs more efficiently on certain regions of a Möbius band). Efficient voice leading techniques break some different symmetries in these musical spaces. It brings to consciousness procedures that composers of the past have used intuitively. All modern mathematical terms are very nicely clarified and illustrated with many musical examples and the book is NOT a mathematics textbook: this is a book for musicians who want to understand why music sounds "good" and what is the close relationship between counterpoint and harmony. One reads this book with the feeling of witnessing the birth of a classic in music theory. A must, for everyone interested in music!

  7. 5 out of 5

    Michael Curtis

    When considering the geometry of the title think college level math. But while the math was difficult - actually, I understood none of it - it did not matter. The five principles of the book are discussed thoroughly in other ways. Tymoczko's personal history is inspiring and is truly the basis of the book. It reads less like a dissertation and more like a life story. Covers classical, jazz, and rock styles in creating a new conception of tonality.

  8. 4 out of 5

    David

    3 1/2 stars. The geometry disappears as the book turns more to analysis. Which I think speaks to the analytical utility of the geometry. It's good for composers, though, and I enjoyed the intelligent coverage of such a broad range of music.

  9. 4 out of 5

    Steto Scopius

    Libro da corto circuito mentale, mi ha costretto a rivedere alcuni concetti matematici sepolti o non chiari e allo stesso tempo, nozioni di composizione e analisi. L'ho cominciato con un timore riverenziale pensando che coprisse solo la musica classica ma mi ha sorpreso e non poco.

  10. 4 out of 5

    Nathan

  11. 5 out of 5

    Ashley Joseph Whitney

  12. 5 out of 5

    Son Melo

  13. 4 out of 5

    Michael Brooks

  14. 5 out of 5

    Brian

  15. 4 out of 5

    Kevin Newman

  16. 5 out of 5

    Aaron

  17. 4 out of 5

    Stuart Duncan

  18. 4 out of 5

    Will Watson

  19. 5 out of 5

    Xavier

  20. 5 out of 5

    Stoica Ionut

  21. 5 out of 5

    David R. Norman

  22. 5 out of 5

    Luis Formiga

  23. 4 out of 5

    Victor Tittle

  24. 5 out of 5

    Dean Lea

  25. 5 out of 5

    Ben Harris

  26. 4 out of 5

    David Longo

  27. 5 out of 5

    Frank Lehman

  28. 4 out of 5

    Jason Quackenbush

  29. 4 out of 5

    Vedad FamourZadeh

  30. 5 out of 5

    Melissa Rinaldin

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