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

Dmitri Tymoczko
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Book Description

March 4 2011 0195336674 978-0195336672
Most listeners prefer tonal music to atonal music, but what exactly is the difference between them? In this groundbreaking work, author Dmitri 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. Tymoczko creates for the reader a new framework for thinking about music, one that emphasizes the commonalities among styles from Medieval polyphony to contemporary jazz. 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 the relationships among familiar chords and scales. This gives readers the tools to translate between the musical and visual realms, revealing surprising degrees of structure in otherwise hard-to-understand pieces. Tymoczko uses these theoretical ideas 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. Using analysis to make his argument clear, he discusses a host of familiar pieces by Bach, Mozart, Chopin, Debussy, Stravinsky, Shostakovich, Miles Davis, Bill Evans, and others. 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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"As far as I know, the intersection of those who are distinguished composers and those who have published in Science contains one member: the author of this book. If you are interested in tonality in music, you must read it, because it describes by far the most comprehensive theory of what makes tonal music work." --Philip Johnson-Laird, Stuart Professor of Psychology, Princeton University


"A Geometry of Music is an epoch-making publication in music theory and will certainly stimulate other new and innovative work in the field. Tymoczko has produced an outstandingly original synthesis of new music theory that unifies quite a large number of separate subfields and realizes the theorist's dream of finding the rational basis for tonality and tonal-compositional practices in music." --Daniel Harrison, Allen Forte Professor of Music Theory and Chair, Yale University Department of Music


"A provocative and ingenious melding of music, geometry, and history that promises to change the way that composers, music theorists, and cognitive scientists view music." --
Gary Marcus, Professor of Psychology, New York University and author of Kluge: The Haphazard Evolution of The Human Mind


"Tymoczko's A Geometry of Music is an appealingly written, substantial treatise on tonal harmony. The author introduces his original concepts with clarity and fearlessness. Musicologists, musicians, and listeners with an analytical bent will find plenty of ideas to chew on in this intriguing, rewarding book." --Vijay Iyer, musician


"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...If the author's way of doing music theory or promulgating his results is not quite like most of the music theory that we have learned and taught, that is hardly a sufficient reason why we should not give his powerful ideas the attention they deserve." --Music Theory Online


"A tour de force, a rich and suggestive summation of an exciting new perspective, -a jumping-off point for further explorations. His geometric diagrams provide new kinds of spatialized representations of the aural facts of tonal experience. They may help composers and musicians to 'see' new possibilities within that intricate labyrinth, as well as to bring the old ones to life anew." --Times Literary Supplement


"Formidable...The strongest aspect of Tymoczko's book is the case that he gives for voice-leading in the common practice." --Reason Papers


About the Author

Dmitri Tymoczko is a composer and music theorist who teaches at Princeton University. His 2006 article "The Geometry of Musical Chords" was the first music theory article published in the 127-year history of Science magazine, and was widely covered in the popular press. His music has been performed by ensembles throughout the country, and he has received a Rhodes scholarship, a Guggenheim fellowship, and numerous other awards.

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0 of 2 people found the following review helpful
1.0 out of 5 stars What's all the fuss about? Jan. 3 2014
Format:Hardcover|Verified Purchase
I wish somebody would explain what all the fuss is about. As pointed out elsewhere, what's with an endorsement from the Psychology Department at Princeton? Couldn't the author get anybody from his own department to say something nice?
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Amazon.com: 3.6 out of 5 stars  19 reviews
196 of 240 people found the following review helpful
2.0 out of 5 stars Some good insights, but the fundamental premise is flawed May 22 2011
By Athanasius - Published on Amazon.com
Format:Hardcover|Verified Purchase
For a work purporting to be an "epoch-making publication in music theory," one thing immediately stands out before the book is even opened. Why are 2 of the 4 supportive quotes on the dust jacket from psychologists, instead of music theorists? The first informs us that the author of this book had publications in the journal Science; therefore we should pay attention to his work in music theory. But the journal Science is not generally concerned with fields like music theory, and the two articles in question (the genesis of this book) were not really music theory publications, but rather a mathematical description of a type of n-dimensional space which the author claimed could encompass all previous geometrical models for music. I humbly submit that I can easily make the same claim, by pointing out that the n-dimensional space of real numbers also could encompass all previous geometrical models for music, with suitable transformations introduced as necessary. Defining an all-encompassing numerical or spatial model is easy (and, honestly, trivial); claiming that it is specific enough to model music and interesting enough to provide analytical insight is a different thing entirely.

The geometrical space defined (not "discovered," as the author claims in this book) in the journal Science is the underlying rationale for the "Geometry of Music" mentioned in the title. This "first music theory article in the history of the journal Science" could best be seen as an attention-getting stunt, and attention it got: magazines in many disciplines having little in common with music theory decided that the author's claims to have discovered the true underlying order of music -- "why music sounds good" in popular accounts and in this book -- must be true, since he was the only music theorist they had ever heard of. This book is the fleshing out of this theory, and it is obviously marketed to take advantage of the exposure in other academic disciplines.

But does it live up to its grandiose claims? There is actually quite a bit to be said for some of the research done in this book, which illuminates a number of fundamental questions in music theory. In particular, the statistical analyses of various trends in musical structure shed a great deal of light on the history of music, challenging some assumptions (or, rather, stereotyped straw-man assumptions that no actual music historian believes) about the breaks in musical style occurring before and after the so-called Common Practice Period (roughly 1700 to 1900). Similarly, statistical analysis of progressions in standard repertoires such as Bach chorales and Mozart piano sonatas provide some new insight into how this music works (and how it may differ in some intriguing ways from the way traditional music theory says it does). Unfortunately, whenever statistics are cited, the methodology is often not explained in detail or the highlighted parameters presuppose the existence of a structure similar to that which is already presumed to be there; nevertheless, there are certainly a number of provocative and novel claims to be found in this evidence. The author is to be commended for his initiative in doing this sort of brute-force stylistic analysis, which is incredibly time-consuming.

However, the two central claims of the book need to be evaluated separately. (1) Music of the past millennium all fits into an "extended common practice" of tonal music that can be effectively described by five constraints (conjunct melodic motion, acoustic consonance, harmonic consistency, limited macroharmony, and [tonal] centricity). (2) This extended common practice is effectively modeled within a complicated geometrical space (described in the aforementioned Science articles) that often extends into n dimensions and has interesting topological properties.

First is the claim of the "extended common practice." Without even getting into the details of the chapter surveying the history, one thing is readily apparent: only a single example dates from before 1500. The author claims that we should expand our tonal common practice by 800 years, changing it from the period 1700-1900 to the period from the year 1000 to the present. Yet he provides just one example from the 11th century and then skips to the 16th, omitting the details of 500 out of the 800 years he wishes to add. Such a move should be immediately suspect to a reader with any knowledge of music history. Moreover, the example from the 11th century fails to interact with the author's five constraints on tonal music in any meaningful way, particularly once we consider the contemporary standards of tuning and the constraints already placed on the scale -- not because of the author's theory, but because of a rather arbitrary superimposition of ancient Greek scale systems onto Western chant that occurred in the late first millennium.

If anything, the author perhaps demonstrates some sense of an extended common practice from about 1500 to present -- which is not new. Scholars for the past century have pointed out a number of developments in the late 15th and early 16th centuries that led to a use of sonorities and progressions which begin to sound much more like Western tonal music, though still constrained within the earlier modal system. And no music historian would dispute that there is some continuity to the tonal progressions of late Romanticism when compared with the developments in 20th-century jazz, as the author claims. At best, then, these historical claims are well-known, rather than the groundbreaking revisionist history the author imagines himself writing.

Within the author's 5-constraint definition of tonality, however, these claims are quite trivial. Of the five constraints, three are inherited from the structure of Greek scales as reinterpreted and used for polyphony in the medieval period -- namely acoustic consonance, limited macroharmony, and centricity. Harmonic consistency is obtained by the desire to make use of acoustic consonance within these scales (though the emphasis on consonance is a bit odd, given that the true arrival of the proto-tonal common practice music in the late 15th century came about at a time when dissonance, not consonance, was first being regulated by detailed principles). Conjunct melodic motion at first seems to be the only mildly interesting claim here, except as the author himself shows, if we use his definition of "voice crossing," such a claim is trivially part of most voice-leading within the diatonic and then chromatic scale in use since the predominantly triadic sonorities became common and began to resemble modern progressions around 1500. (Despite the supposed fascination with voice-leading, the author rarely engages with the specific types of voice-leading that actually drive progressions in chords before 1800, like tendency tones and stereotypical uses of dissonance that are outside the bounds of his theory, which focuses on the "chord" as fundamental, a view more indebted to Rameau and the history of music theory than to the way most standard tonal progressions are actually driven by counterpoint.)

I say such claims are trivial because, although they may describe some aspects of tonal music, they are largely well-known, and even if they are accurate, they are far from sufficient to describe the syntax of tonal music. If anything, they seem to be tailored to support the author's own compositional style, which makes use of a kind of tonality described by these constraints, rather than features that are important to define historical styles. Moreover, they derive not from the author's geometrical theories nor any topological properties thereof, but mostly from acoustical properties defined in scalar systems by the ancient Greeks, who did not even consider vertical harmony to be important, even though vertical harmony ("chords") is the primary component of the author's geometrical model. Thus, from a historical standpoint, it is impossible for the author's model to be "why tonal music sounds good," since the scales and properties that created the author's defining constraints came from an ancient system developed by Greeks (who thought it sounded good) but which has no relationship to the author's model.

(As an aside, many of the author's claims about chordal structure in his geometrical model depend fundamentally on the existence of 12-tone equal temperament or some approximation thereof. The chapter on the construction of scales in the book, while intriguing, says little about why a 12-note chromatic scale came into existence in the first place. Nor does it acknowledge the significant role played by this historical development and how geometrical models of tonal space such as the author's only became possible once composers forgot about tuning their own instruments and embraced an enharmonically equivalent 12-tone ET in the 19th century.)

This brings us finally to the question of the geometrical model that is the basis for the title of this book. In simple terms, it is completely unnecessary for almost all analytical claims made. For example, the author often points to the E minor prelude of Chopin as a quintessential application for his theory. The author claims that a 4-dimensional representation of the chords in this piece is the best way to understand it, and, oddly, that Chopin therefore must have implicitly understood higher dimensional spaces better than mathematicians in the early 19th century. (For an author with training in philosophy, this requires a very strange idea of epistemology and causality -- just because a model might describe something doesn't make understanding of that model a necessary condition for that thing's creation, particularly when the model is deliberately, according to the author's own claims, much more broad and complex than any previous theory of music.)

In Chopin's case, the author himself gives a reasonable, simple algorithm involving changing certain notes in the chord that adequately and accurately describes exactly what the voice-leading does. It involves only concepts like "within the 4 notes of the chord, move one at a time by half-step." Why the heck do we need a 4-dimensional hyperspace with wacky topology if we can describe this piece so simply? The answer, quite simply, is that we don't, anymore than we need the apparatus of the real number system as defined in college analysis classes to explain how a 4-year-old counts from 1 to 10. The question is not whether we *could* model the child's simple counting within college level math, but whether we gain any insight at all into how the child thinks or what the child is doing when counting by invoking the complexity of such a system. In that case, as in the case of the author's geometrical space, we gain absolutely no new insights. In the case of Chopin, there is something interesting going on in the voice-leading, but we don't need 4 dimensions to describe it.

In sum, we don't need the author's geometry of music at all. While there are many analytical insights offered in the book, none of them really benefit from a projection into a complicated geometrical space, and the few that seem to could often be improved by much simpler and straightforward geometries that the author seems to overlook or discount. There are some theoretical insights about the most consonant chords which supposedly relate to this system, but the properties within the system are again not sufficient to uniquely define the consonant chords or to throw out less consonant ones -- despite the author's bizarre parable about God handing off a "suitcase of chords" with the author's properties (a story whose implicit deification would highlight the author's arrogance if it didn't sound more like a drug deal in a bathroom than a rationale for why we should accept the truth of a multidimensional chordal space).

On a final note, the sheer egomania of the author's claims are apparent in the grandiosity he attributes to them himself, but it is unfortunate that he also feels the need to denigrate his colleagues (and entire styles of music that he doesn't like and clearly doesn't understand) in sometimes subtle and sometimes outrageous ways throughout the book. Once the author has claimed that the faculty who taught at his undergraduate institution, Harvard, "knew nothing about, and cared little for" music outside of a few avant-garde modernist composers, there is no amount of back-pedaling and claims later in the book that he is not judgmental that will make up for it. The author here impugns the names of some of the greatest scholars in music theory and musicology, whose breadth of knowledge speaks for itself in their publications. If this book indeed begins a new epoch in music theory, I hope that this unprofessional and unscholarly rudeness and arrogance is not part of it.

EDIT -- At some point after the appearance of my review, the author and his editor posted the Amazon "exclusive" interview which you can read above. The author addresses a few of my criticisms in a very broad and general way (e.g., the huge historical gap between his examples of early music from 1000-1500). I still stand by my original assessment.
23 of 30 people found the following review helpful
5.0 out of 5 stars Music Theory for the 21st Century Nov. 13 2011
By Mark D. LaDue - Published on Amazon.com
Format:Hardcover|Verified Purchase
Several years ago I happened to see one of Dmitri Tymoczko's Science articles. I was hoping to find an introduction to that paper's ideas that would be suitable for a course aimed at students with an undergraduate background in both mathematics and music theory, but none existed at that time. When OUP published A Geometry of Music, I was excited to see it; however, when I saw that half of the original six reviews here were strongly negative, I hesitated to buy it.

Unfortunately, some previous reviewers seem to feel an animus toward the author and insist on using the book's dust jacket and introduction against him. But plainly OUP selected the blurbs it did to emphasize the book's potentially broader appeal, and it's hard to see how the author's account of his undergraduate music theory education in the introduction can be regarded as an affront to his alma mater. One has only to turn to the author's acknowledgments and read his copious footnotes to see that he has given ample credit where credit is due. The homoousian claims that most of the book is trivial, that much of the work has been purlioned, and that the remainder is useless are easily seen to be ludicrous. Reviewers who claim to have written books on "Algorithmic Computer Music" which don't seem to be available or claim that "[d]efining an all-encompassing numerical or spatial model is easy (and, honestly, trivial)" should make their works available for all to see. Perhaps their papers will appear in the journal Science - right after the next article on crafting a perpetuum mobile.

After using A Geometry of Music as the basis for teaching a course, I find myself in agreement with those previous reviewers who praised its strengths; however, I do have two criticisms to offer. The book's most significant shortcoming from my perspective is its paucity of exercises. Unfortunately, there are only 38 of them, and those are relegated to Appendix F. Of those exercises only eight relate to the entire second half of the book, and four of those concern one of the topics of Chapter 7. While I understand that the book is intended to serve as an introduction for music theorists, composers, and amateurs, it seems likely that its main audience is going to be music students. In order to better serve students future editions of this book should include many more exercises, and the exercises should appear as an integral part of the book, not in an appendix. More casual readers can skip the exercises as they see fit. Kostka's text on twentieth-century music seems like a good model for the breadth and depth of musical analysis exercises that I would like to see, and the workbook for Cadwallader and Gagne's text on Schenkerian analysis could serve as a model for the level of difficulty that would be appropriate for this book.

One other shortcoming of A Geometry of Music is its general avoidance of the underlying mathematics, even in the more technical appendices. I think many more computational examples and exercises should be included in a book at this level, and the general level of computational difficulty I would like to see in future editions would meet or exceed those of Straus's text on post-tonal theory. The material currently in Appendices A through E, together with additional examples and exercises, would enhance the body of the book. More technical appendices could then explore topics along the lines of the author's on-line supplements to his Science papers. This additional material would also demonstrate the falsehood of homoousian claims that the book is "trivial," that it lacks original ideas, and that topology adds nothing to our understanding of music.

All in all, I highly recommend this book to anyone interested in seeing where music theory is going in the twenty-first century.
39 of 54 people found the following review helpful
5.0 out of 5 stars Amazing Book April 8 2011
By James Erickson - Published on Amazon.com
Format:Hardcover|Verified Purchase
This book brings amazing insight into the world of music and the more esoteric world of abstract music and how it all fits together. Being of a mathematical and scientific background this book really speaks my language in its explanation of music theory. However, I warn anyone interested in this book that this book uses a lot of mathematical concepts to explain music theory and link previously unrelated topics in music; so if math is a weakness of yours, this book is probably not going to be very useful to you-or at the very least much more difficult for you to plumb the depths of its knowledge.
18 of 25 people found the following review helpful
1.0 out of 5 stars Main idea not from the author Dec 9 2012
By GEB - Published on Amazon.com
Format:Hardcover
In 1739 Euler wrote a book on music, the "Tentamen novae theoriae musicae, hoping to eventually incorporate musical theory as part of mathematics." The following still seems to hold: "This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians" Source: [...]

Therefore this book has fundamentally nothing new to say. It is a bad copy of the ideas of Guerino Mazolla. Mazolla is professor at the School of Music, University of Minnesota and wrote a book in 1991 called "Geometry of music" [1] and later "Topos of music" [2]. Tymoczko is aware of this work and cites topos of music at the end, without mentioning that the main idea, music as a geometric space is not due to him. In fact he wrote the following about parts of the theory: "Like many American readers, I am fascinated by The Topos of Music; indeed, my shock upon first encountering it is comparable to my shock on first encountering the philosophy of Hegel or the music of Cecil Taylor. Here, I felt, was something new, powerful, and yet utterly beyond my comprehension: was it a great intellectual achievement, or a majestic shaggy-dog story?" [3]

In the foreword however, we can read: "My aim is to retell the history of Western music [...]". And: "It would make me happy to think that these ideas will be helpful to some young musician, brimming with excitement over the world of musical possibilities, aeger to understand how classical music, jazz, and rock all fit together - and raring to make some new contribution of muscial culture."

Publishing a book with the same title of (Geometry of music) is despicable (note, that "Töne" in German can mean sound and music in english). Perhaps the author does not know what topos means, but the original title of the first version can be directly translated as "Geometry of music, Elements of a mathematical music theory". I wish it was otherwise, but the matter is simple: this book should not have been published in this form, without proper references. With regards to the content itself. I will just quote one sentence. "Musically, a pitch can be represented by a number". Well, every number always represents something. However, numbers are part of the constructs which we call algebra. So no serious person in this field would use this expression. This is just one example from thousands. In general the theory does not provide anything, which one could not easily construct on one own. From a short overview I have identified quite a few number of ideas, which are not original.

However one wants to the view the works of Mazolla, anyone using the term "geometry of music" has to cite him, the same way, we cite Keppler and Newton for their work on physics. If I were to write a book on physics using Newtons ideas, I should cite them with proper precision, no matter what I have to say. Presenting major ideas as owns own, is what we call plagiarism. In fact the original book was considered as a breakthrough in mathematics by one of the greatest mathematicians, Grothendieck (the quote is topos of music). It's a shame that Tymoczko at least acknowledges the greatness, but does not cite Mazolla as his source. If this book would just be a good summary of the real geometric theory of music, or even a bad summary, this would be a good book to have. "Geometrie der Töne" is unfortunately currently out of print and "topos of music" indeed difficult to comprehend. However, this does not excuse this behaviour. This error should be corrected and the book either updated or the publication withdrawn. This might be a very harsh view, but the "wrongful appropriation," "close imitation," or "purloining and publication" of another author's "language, thoughts, ideas, or expressions," and the representation of them as one's own original work" (Wikpedia), is what we call plagiarism . I can only wonder what the motivation behind this behaviour was. It could have been a book which promotes the idea of the use of geometry in music. Unfortunately it misses main ingredients of the original theory, so that the substance is lost. This is very sad, because the readers could have found the original and learn a great deal. Anyone who is interested in mathematics and music should consider reading Mazolla or perhaps obtaining a copy of the first book, which is easier to understand, or wait until a summary or development for the layman is available.

[1] Guerino Mazzola: Geometrie der Töne: Elemente der mathematischen Musiktheorie Birkhäuser Verlag; Auflage: 1 (1. Februar 1990)
[2] Guerino Mazzola: The Topos of Music; incl. CD-ROM. (1368pp), Birkhäuser, Basel 2002.
[3] [...]
[4] [...]
1 of 1 people found the following review helpful
2.0 out of 5 stars A Geometry of chords and scales maybe, but not of music. May 18 2014
By Duzzi D - Published on Amazon.com
Format:Hardcover
The premise of this book is highly laudable, at least if one is of the same opinion of the author. "Tonal music" need to come back, and yes, the excesses of modernism and in particular the unforeseen consequences of strict atonality and twelve tone techniques have produced a lot of baffling music that has perfectly achieved what seemed to be the goal of at least some of their proponents: the public does not listen. (Witness the desolate emptiness of any new music concert outside of Universities or Conservatories and their captive audiences.)

The book is interesting, and besides being positively provocative it is especially stimulating because it points (by omission) to the elements that are actually missing in it and should be the focus of attention for a truly comprehensive theory of music.

I found chapter one, the introduction, an interesting one. It spells out a rational (the five features) for what constitutes "tonality" that, if far from complete, is simple and clear, and would constitute a great platform to build on.

Where I find the book less convincing is in its four claims, and in particular on the insistence on efficient voice leading, circular pitch class and the geometric space that emerges as a consequence. The main objection is that circular pitch class does not really describe how music sounds. Why? Just starting with chords: (1) contrary to the author claim, chord inversions do matter, because they do sound different. Western music from 1600 to this day treats voice leading in such a way to reach inversions or root position at certain crucial places in a composition. That is: composers treat inversion and root positions very differently, they have different functions, and their treatment and approach has consequences for the linear unfolding (counterpoint) of a piece. (2) A chord composed of any set of "pitches" (say just C, E and G) sounds very different depending on where the "pitches" are located in the 8 octaves or so available to our hearing. And that is so because the pitches are not the same: a C2, or a C3 or a C7 are very different objects. As a consequence Figure 3.5.3 is not a depiction of redundancy or inefficiency: in a piece of music each of those chord could represent and unique and vital part in the unfolding of the composition, and could not be confused with each other. Even assuming that (1) and (2) could be disregarded, because the aim is just a reductionist "harmonic analysis" of a piece, there is the problem that (3) Melody (the elephant in the room that is almost never mentioned in the book), does not work in circular pitch class. Any melody with a range of more than an octave (and there are of course millions of them in the repertoire) is completely obliterated by the little ant navigating through the Mobius strip induced by circular pitch space (Fig 3.2.1). Any melody subject to this treatment would be distorted to such an extent to be simply unrecognizable.

And finally (4) efficient voice leading seems something that frankly might happen in a lot of the inner voices of, say, a four-part counterpoint, but not in the outer ones. Take any string quartet, just in the classical period: yes maybe, occasionally, a second violin and a viola restrain their movements, but cellos and especially the violin (in its soprano-esque exuberance and being the instrument responsible for melodic exposition) do not. Not even remotely. The author seems aware of this problem (see the sub-chapter on 3+1 voice leading) but it only mentions it for the bass voice and completely misses the fact that in reality much instrumental (or vocal) music is written without any notion of, or any need for, "efficient" voice leading.

Because of these assumptions the impression is that the book constrains itself into a narrow horizon. There are a lot of interesting observations, but they mostly concern just reduced harmonies, and scales. Often such observations are mixed in with a bewildering amount of ad hoc jargon and a steady stream of some frankly unnecessary pseudo-mathematical-sounding definitions. But more fundamentally one wonders if the rather complex geometric apparatus used by the author really adds anything of substance to a type of analysis that can be achieved using traditional means with very little difficulty. It is already quite anticlimactic that when we finally reach some analysis they seem to boil down to the list of scales that Debussy, or Shostakovich, or the Beatles use. The lists explain almost nothing about the pieces, and in addition it is very questionable what they gain when they are depicted in a three dimensional lattice. A student learning about non-traditional scales could easily accomplish the same "analysis", and be utterly and completely lost in the obscure mappings used by the author. The same holds for the harmonics analyses, that are all carried out using standard chromatic harmony techniques, and then looked at as three dimensional projections: what is gained by this process is very unclear.

Far from "providing a comprehensive picture of the possibilities confronting composers" the book seems to just scratch the surface of how compositions work and are put together.

What is missing in the book, are a number of elements that make a piece of music what it is: the interplay of dissonance and consonance (which, contrary to the author somewhat restrained discussion of the topic, are real physical phenomena that determined how polyphony and tertian harmony developed in the west, and drive counterpoint much more than efficient voice leading), melody and the interplay between motives and melody, range, rhythm, texture, form and especially and first and foremost: the time oriented unfolding of those elements. With the exception of chapter 5 that uses a simplistic type of time analysis, and some even more trivial uses of "statistics", the book is just concerned with a couple of the bricks and mortar of music.

The book could be better titled "A geometry of Chords and Scales" and that title (once one accepts the simplifications forced by the use of circular pitch class), might be justified. But a piece of music is certainly not "analyzed" using the methods described in the book. Music is a very complex dynamical system. It is in movement in time and space; it exhibits memory, and recurrence, and direction and events (things happen in music, like in a novel!). A good theory, yet to be written, must describe and take those elements into account.
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