KEYWORDS: Tymoczko, geometry, chords, voice leading, scales, tonality, some brute facts in Tymoczko’s new book A Geometry of Music (henceforth AGOM). A Geometry of Music has 69 ratings and 6 reviews. Peter said: Tymoczko is an intriguing composer and adept mathematician, so his mix of math and (lots of. Dmitri Tymoczko: A Geometry of Music: Harmony and. Counterpoint in the Extended. Common Practice. Hardcover, , ISBN , pages.
Tymoczko, geometry, chords, voice leading, scales, tonality, chromaticism, style, common practice, jazz. Football consists of a statistically regular alternation of circular clustering, linear clustering, and random interpenetration. Clearly, though, this explanation misses the point of football in important ways. The statistical phenomena it details are nothing more than incidental byproducts of the things that football is really about: We make many observations about music, and while these observations are usually correct in some sense, they often seem to miss important points: Tymoczko is unafraid to question conventional wisdom, spurn the beaten path, and seek new viewpoints on familiar landmarks, and AGOM does not lack for bold claims with far-reaching consequences.
It promises nothing less than a rational foundation for tonality; a set of common principles that unify the music of not just one or two centuries but broad swaths of music history spanning the better part of a tymoczkk new perspectives beometry among other things scales, modulation, functional harmony, chromaticism, and the place of jazz in music history; and a powerful, overarching geometric model that ties it all together.
The geometric model that takes pride of place in the title is a monumental achievement in its own right, and is brought to bear revealingly on fundamental aspects of harmony, voice leading, consonance, tonality, and style, aptly illustrated through geomerty of figures and musical examples in a wide range of styles. The upshot of it all is a book that occasionally vexes but more often illuminates, inspires, and exhilarates: These spaces may be thought of as adding a geometric component to pitch-class set theory, situating chords of n notes in a continuous space not a discrete lattice in n dimensions, while taking into account geometrt desired combination of the OPTIC symmetries: As such, the OPTIC spaces use sophisticated techniques from contemporary mathematics to pull together many threads from musical scholarship of the last half-century, including set theory, scale theory, neo-Riemannian theory, and an assortment of work on voice leading and geometric representations.
First, AGOM offers neither the first nor the most complete presentation of musicc model. Chapter 3 limits its scope to a few of the most musically useful spaces in two, three, or four dimensions, and Appendix B provides only tantalizing glimpses of the more general theory and some of the new considerations that arise in higher dimensions. Readers looking for coverage of the OPTIC spaces in their full generality still have no alternative to the technical presentation in Science.
Secondly, A Tyymoczko of Music is not a book about geometry; it is a book about music. More exactly, it is about musicc ways in which geometry can shape and inform our understanding of music—ways that are many and varied indeed.
Anyone who finds diagrams like the pitch-class circle, the circle of fifths, or the neo-Riemannian Tonnetz useful for visualizing aspects of musical structure should, upon learning to navigate the OPTIC spaces, come to appreciate the varied perspectives and novel musical insights that these spaces provide.
A section of three-note chord space A Geometry of Musicpage 86, Figure 3. Three-dimensional chord OP space, described in Section 3. Augmented triads, equal divisions of the octave into four-semitone intervals, lie on the central axis of this prism, and major and minor triads, which divide the octave nearly equally, are grouped near this central axis see Figure 1.
In contrast, a chord that divides the octave less evenly, such as the trichord E— G —A, lies far from the central axis and far from all other chords of the same set class The repercussions of this difference for the way composers use triads and seventh chords are explored in section 3. The role of a chord in the above examples may be played instead by a scaleor more generally by what Tymoczko calls a macroharmonya collection of notes controlling the pitch content of music over a period of time.
Studying movement from one scale to another in an appropriate space provides a new way of thinking about modulation in traditional tonal music a modulation from C major to G major is a move from one seven-note collection to another nearby one5 or of thinking about relationships between different types of scales in the music of Debussy, Shostakovich, or Reich.
If you are a composer, Tymoczko says, and if you want to write music featuring a limited variety of chord types, connected by smooth voice leading, you will have more success if you choose to use nearly-even triads or seventh chords than if you try to use uneven sets like Precepts about compositional strategy permeate the book, and while it is reasonable to wonder how many composers deliberately formulate their objectives in such ways, the precepts reveal constraints that inevitably shape musical works whether composers are conscious of them or not.
It is only a small step from compositional strategies to broad pronouncements about style: These should be relatively uncontroversial; harmonic consistency and smooth voice leading are two of them.
MTO Hook, Review of Tymoczko, A Geometry of Music
Music by Stravinsky, Glass, and Bill Evans, for example, falls comfortably within this compass, thus addressing what Tymoczko perceives as a failing of his own mueic education, in which these composers were marginalized. Though AGOM is in many ways meticulously documented—Tymoczko is generous with citations of opinions that differ from his own, and the bibliography fills ten small-print pages—it occasionally gives the impression of disowning its own heritage, and therefore perhaps of being even more iconoclastic than it actually is.
The level of originality in this work is remarkable enough already, and hardly needs exaggerating. Tymoczko confronts with apparent relish the daunting challenge of selling his ideas to a broad audience of theorists, composers, musicians, and students, and his ability to capture the intricacies of complex material while presenting it clearly and comprehensibly is praiseworthy. Some parts of the book are difficult, to be sure, but the level of abstraction subsides geometrj quickly after peaking in Chapter 3.
Appendix F provides some pedagogically valuable tymczko, most of them pertaining to the most technical material in Chapters 2—4. Tymoczko responded to such objections by saying that his exposition is modeled less on a mathematics textbook than on writings in physics or other applied sciences, where authors routinely eschew the formal axiomatic style and present mysic as much mathematics as is needed for the applications at hand.
A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice
He makes available a custom-designed software tool enabling visualization of chords in several geometric spaces; one can easily display the chords shown in Figure 1, for example, and rotate geometrj picture in three dimensions to view it from different angles.
Also, various points are geoetry by statistical analyses estimating the frequencies with which certain characteristics appear in the works of certain composers or eras—a valuable if time-consuming technique that is slowly gaining a foothold in music theory scholarship.
Fortunately, most of the errors I spotted are of the sorts that many readers will likely pass over without notice or will correct with little geomstry It has also attracted the attention of at least a few composers, cognitive scientists, and psychologists, and Tymoczko has apparently taught the material with some success to Princeton undergraduates. But the substance of the book is harmony, counterpoint, tonality, scales—the stuff of music theory—and it is music theorists who stand to benefit the most from it.
Douthett, Jack, and Peter Steinbach. Theory, History, and Ideology. University of Pennsylvania Press. Parts 2 and 3. Quinn, Geomery, and Panayotis Mavromatis.
Springerpages — An Essay in the Philosophy of Language. A Comment on Hook.
A Geometry of Music – Dmitri Tymoczko – Oxford University Press
Meyer11—12, quoting Searle The theory of orbifolds was developed by William Thurston in the late s. The significant work on voice leading and musical geometry includes, among others, RoederMorrisDouthett and SteinbachStrausCohnand notable contributions by the CQ portion of the Muaic triumvirate, such as Callender and Quinn and To form the complete space, the two triangular ends of the prism in Figure 1 must be glued together with a twist, so that the two corners representing geomerry triple unison CCC are superimposed.
Of course, real music is messier than elegant theory. While it does show a statistical preference for chord connections of the sorts Tymoczko describes, the table p. In theory, diatonic scales inhabit a seven-dimensional OP space; fortunately, og spaces can geomftry the essential features when only a few scale types are involved. The middle section of the Fourth Barcarolle, op.
A passage in the enchanting Prelude, op. Spatially proximate chords often function in similar ways, and in many musical contexts a chord may often be substituted for one of its near neighbors. This failing is addressed most provocatively in the Introduction page xvii: Unfortunately, this was not the music I had come to college wanting to understand.
For or, an exercise in Appendix F asks the reader to explain why there are five efficient voice leadings that permute three closely clustered notes in a four-note chord. The intended geometric explanation, involving paths reflecting off the boundaries of four-dimensional chord space, is somewhat hard to visualize though mmusic good exercise, to be surebut algebraically the reason is obvious: Additionally, the book has a companion website http: The audio may be beneficial to those who have difficulty playing or aurally imagining the examples, but the synthesized sound will be off-putting to many listeners, and there are occasional mismatches between printed examples and their audio counterparts.
Corpus studies are widely used in some other disciplines, notably linguistics, where it is widely recognized that the results sometimes force scholars to challenge their assumptions and confront phenomena that might otherwise escape their notice. A number of corpus-based studies of music have appeared in recent years; for example, Quinn and Mavromatis use corpus-based analysis to shed light on syntactic differences between Bach chorales and a corpus of Lutheran chorales from a century earlier.
Occasional oddities in the results betray a lack of discernment in these analyses.
A Geometry of Music
Two pages later, Figure 9. Some edges in this graph lie directly atop others in this perspective, obscuring the relationships, and unlike most gymoczko such graphs, this one is drawn so that a particular voice-leading motion such as E —E is represented, at different places in the graph, by edges that are not parallel.
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