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.