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A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice Hardcover – Mar 4 2011
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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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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.
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.
1. I've never encountered a more comprehensive explanation of tonality. In particular, Tymoczko's "Five Components" are extremely enlightening in showing that tonality's incredible versatility is in fact derived from the confluence of multiple constraining musical forces. He sums it up pretty well with the clever analogy of God and the suitcases on page 64.
2. The author makes a compelling case that there is a continuum tying the tonality of the Western classical tradition to the tonality in modern genres such as rock and jazz. I have college degrees in music composition but I don't recall a single professor offering an analytical tool that could be applied to modern pop music as readily as to Chopin or Debussy. For that matter, I'm not sure they would have taken the endeavor seriously. I tip my hat to Tymoczko for bringing John Lennon and Roger Daltrey into the same compositional arena as Stravinsky.
3. As a composer, viewing the musical relationships in three-dimensional cubic lattices brought a fascinating new perspective. Perhaps it's ironic that musicians speak so commonly in terms of musical space ("intervallic distances", "key areas", etc), but it always remains two-dimensional in standard notation. The geometric mappings lend an almost tangible quality to the musical relationships, and I found it very revealing to visually gauge and compare the proximities of various notes and chords to one another in 3-D space.
4. Personally I love the data graphs. For example, the graph showing the prevalence of various chord root motions in the music of Schubert vs. Chopin (pg. 99), or the speed of pitch-class circulation across composers spanning different centuries (pg. 160). Again from my perspective as a composer and sometime theorist, I think a snapshot of compiled "musical metrics" can be worth a thousand words. You rarely see these in theory textbooks, but I find them extremely informative for defining some aspect of the compositional decision-making. I think these meticulous data collections are likewise a tribute to the author's thoroughness in his research methods.
5. Lastly, I enjoyed the author's candid tone. Theoretical writings can be so clinical, but Tymoczko's personal style made this work a real pleasure to read. I got a kick out of the clever quips, the jokes, the personal anecdotes, and other moments of levity so often missing in works of dense subject matter.
Overall I highly recommend Tymoczko's A Geometry of Music.
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.