Exploring Musical Spaces: A Synthesis of Mathematical Approaches

the title of this book might well feel misleading to buyers who do not peruse the text on the back of the jacket. Certainly, many musical spaces of note are studied here: chromatic circles and the circle of fifths, the pitch line, hexagonal cycles, tonnetze, OPTIC orbifolds, scale spaces—all of these are neatly described and very clearly explained in this book. However, this project faces a serious challenge, one that has been addressed fairly often in recent years, but so far with limited success: the challenge of bridging the gap between those music theorists without significant mathematical training and current mathematical music theory. To quote the author in the preface, “The price of admission to the field. . . can seem exorbitant” (ix). When writing about or teaching mathematical music theory, it is quite difficult to avoid taking for granted some notion or other that the author deems everyone should know about. Take, for example, functions, their composition, and the mysteries of the notation f∘g, the notion of algebraic group and group actions, computation modulo 12, or even the fundamental differences between sets and other types of collections of objects, to mention only a few examples among the myriad carefully treated by Julian Hook in this book. As he states in the preface, he hopes that music theorists at large could know the substance of at least the first five chapters. The author manages quite well in making these concepts approachable, though he presupposes that the reader is already (very) well trained in (nonmathematical) music theory. Casual allusions to music theory ideas include tone row transformations, bits of Schenkerian analysis, German augmented sixth chords (as distinct from dominant seventh chords), just intonation, pitch class sets and their normal forms, and many other advanced theoretical concepts that many graduate students may not fully grasp. These are indeed probably necessary prerequisites, since the topics explored in this book are far from trivial, many of them building on current research in the field and out of reach for complete novices. These prerequisites enable the author to develop interesting and significant examples to a greater extent than one usually finds in pure math textbooks that explore the same concepts at the same level. A common pitfall in a would-be comprehensive textbook is a researcher's tendency to highlight their subject-matter specialties. By and large, Hook avoids this trap, although it could be argued that he stopped short of developments by other researchers of some of his own work (e. g. , regarding signed diatonic sequences or Uniform Triadic Transformations). Of course, a synthetic project like this necessitates that the scope be defined and boundaries drawn; indeed, Hook expresses his regrets for not mentioning currently rapidly developing fields (such as Discrete Fourier Analysis of musical structures). I concur with his decision to leave this out, as this subject would have required several more preparatory chapters. While readers may lament the absence of recent developments in category theories (which can simplify a lot the exposition of K-networks, generalize GIS, and provide a more general setting than semigroups as they appear in the book), the book is already quite long and already brings less mathematically inclined readers an impressive distance out of their comfort zones and into the mathematical weeds. The pedagogical methods Hook uses in this book are well suited to his intended readers. He tends to begin with examples, informally presenting ideas that he will rigorously define later on, and subsequently generalize. This nonmathematician-friendly approach does not prevent mathematicians from reading the book—and indeed, from taking pleasure in doing so—though the author goes as far as adopting “natural” conventions at odds with mainstream mathematical usage. For example, he writes the composition of functions f∘g as gf (or g ∙ f) so that functions are applied in the order we read them (meaning that gf (x) = f (g (x) ) ), a usage quite uncommon in math, though it may look more natural in neo-Riemannian theory when combining LPR operations. (Indeed, what do you mean by applying PR to the C major triad? 1 Is PR first P, making C minor, and then R to reach E♭ major? Or is PR first R to A minor, and then P to A major? ) Relatedly, Hook avoids falling into confusion with words like “transposition” or “metric, ” which have distinct meanings in math and in music, by avoiding their use in the former field (using, for example, distance function instead of metric). Unavoidably, some proofs of mathematical propositions are missing or incomplete and some notions left aside (topology is mostly absent, notably, apart from some metric properties and a lavish use of Euler's notion of genus of a surface), 2 but the author does his best to defer these matters to exercises, or to the abundant list of references, be they to mathematical textbooks or to current music theory research. Indeed, these suggested readings are sometimes ambitious: For instance, the introduction to pitch classes and modular arithmetic may not necessitate taking the plunge into Hardy and Wright's famous Theory of Numbers! In this reviewer's opinion, leaving technical and barren proofs to the side in most cases is a commendable move for a book aiming at nonmathematicians, who can satisfy their eventual curiosity by an internet search, or much more thoroughly through the references provided. Readers confident in their knowledge of mathematical music theory might feel that this book is not for them, able as they may be to peruse the existing technical literature, which is more directly presented and sometimes more comprehensive. Yet this book contains an incredible wealth of exercises and examples that are interesting, often original, and varied in topics, style, and level, as well as copious footnotes, which together account for half the book's size. Indeed, some music theory teachers may wish to buy the book for the examples and exercises alone (some lazy students will deplore the absence of solutions to the exercises, though footnotes provide adequate indications for the most technical ones). Less confident readers will of course find a clear and fresh presentation of many essential notions in contemporary research. As advertised in the preface, the five chapters composing the first part of the book introduce essential mathematical objects that play vital roles in virtually any mathematical developments in music theory. As mentioned above, the author always starts with musical examples. For instance, sets and functions and modular arithmetic appear only in chapter 2 after several foundational topics are introduced in chapter 1, such as pitch and pitch-class spaces and a few tonnetze. This initial chapter begins slowly and gently but should nonetheless be read with care and thoroughness, as many of the examples tackled there will be developed in exacting detail later on. In any case, some of Hook's perspectives are original, or at least distinct from common trends. For instance, the author focuses on diatonic and generic pitch spaces at least as much as familiar pitch and pitch-class spaces (generic pitches being the seven note names A to G waiting for their accidentals). And the usual tonnetz is only one case of discrete space generated by two different directions, on a par with, say, signed pitches (a cross product of generic pitches and the whole sequence of accidentals). Chapter 3 is devoted to graphs, with a very thorough study of a “hexatonic triadic graph” seen from many angles and representations (it is the cubic graph of all minor, major, and augmented triads included in a hexatonic collection, not to be confused with the usual hexatonic cycle). The author introduces subgraphs, isomorphisms, dual graphs, Euler genus formula (used several times for proofs of nonplanarity, which may be trying at this point), directed graphs (as exemplified by lattices for partial orders, notably ski-hill graphs), and all of the concepts necessary for the study of tonnetze (and transformational graphs more generally). Some examples of the latter are given informally, although Klumpenhouver networks will only be formally tackled in chapter 9. Here the choice of a Webern example with a cubic graph of transformations between tone rows isomorphic to the first one studied in the chapter is a convincing illustration of isomorphic graphs of transformations through very different musical contexts. The next chapter extends similar ideas to more complicated musical spaces whose elements are chords or tonalities or tone rows. These are approached by abstracting from intuitive knowledge to algebraic objects, focusing on their relationships—it reminds one of Hilbert's famous statement that “points, lines, and planes could be replaced by tables, chairs, and beer tankards, ” only we are gently led to replace pitches (or pitch-classes) with chords or tone rows. There is already enough material to discuss some cycles in the Riemannian tonnetz (or rather its dual) and diverse graph distances between abstract objects, with some detailed examples of neo-Riemannian analyses. Significant emphasis is given to generic space (s) (like A B C D E F G), which will of course be useful when discussing advanced concepts like signature transformations and the spelling of pc-collections in the final chapters. The chapter ends with a sparkling variety of musical examples, from Webern to Schubert, which convincingly demonstrate the flexibility of thinking with spaces (mostly graphs so far). Part 1 of the book is crowned with the long-expected chapter on groups. “At last!” some may say with a sigh. At this point, the seminal cases of T and T/I have been introduced, but the main focus remains on commutative groups. From the already studied actions and orbits of these fairly simple groups on pitches or triads, it is a small step to defining (not yet too rigorously) general group actions—and especially interval groups—focusing around Lewin's initial presentation of Generalized Interval Systems. This is explained in very thorough detail, in contrast to the concise, purely mathematical definition (as a simply transitive group action) that is provided in many contemporary papers. Indeed, one avowed intention of the author is to provide an accessible way (for the mathematically uninitiated, and maybe some others) into Lewin's magnum opus—an admirable though not easy task. The notion of transformation (always asking oneself whether some notion of interval lurks behind) with transformation groups and transformation spaces will be the thread of the next part. Recall that the first part is self-contained; it sums up what the author deemed necessary for any musician seriously considering the study of mathematics in music theory. The second part begins with a study of transformations and musical invariants: mostly intervals, with an increasingly broader definition. At last, groups themselves, quotients, products, 3 and isomorphisms are properly and formally defined. True to form, the author uses many graphic illustrations, such as Cayley graphs. This material is followed by a chapter employing all this machinery in the first general setting of musical spaces, namely interval spaces. Some American readers may be surprised by the cursory treatment of interval contents and Babbitt's theorem (though the proof provided is a well-chosen one, short and understandable). Hook had noted in the preface that he is not following historical trends or traditions but rather trying to present a coherent approach of the spaces that he deems useful in the field today. This leads to setting aside several notions that others might have deemed essential. However, pointers to developments on these voluntary omissions (homomorphisms, Z-relation, criterion for hereditary quotient structures or metrics, Messiaen's Modes à transposition limitée, Fortean Set Theory, etc. ) can often be found, by careful readers, in the footnotes—and of course the bibliography. The core chapter of the whole book is probably the meticulously written chapter 7, titled “Intervals. ” It is obvious here how much the author wants to clarify Lewin's theory and the relationship between generalized intervals and transformations. Shorter definitions could and have been used for products or quotients of interval spaces, but they depend on deep mathematical knowledge, while this chapter remains accessible to nonmathematicians (if they have read carefully the previous chapters and worked out enough exercises!). The book really takes the plunge into musical spaces with chapter 8, where PLR transformations are more rigorously defined and generalized to uniform triadic transformations (UTTs) (Scritt/Wechsel). Paths in tonnetze are further generalized in the next chapter about graphs and networks (actually mostly digraphs), among which Klumpenhouver networks and their transformations are briefly introduced, alongside tone rows and their specific UTTs. The material is dense here, but the abundance of musical examples helps to alleviate the effort required of the reader. Part 3 is made of three chapters only, and again the author makes the choice of starting with case studies and leaving the formal definitions for later. This time it means that the comparatively easy chapter 12 on distances (meaning, really, metrics, in the mathematical sense) comes after the battle against OPTIC spaces that covers chapters 10 and 11. The exposition is again very clear and example based, and the reader is warned about the difficulties of figuring out these spaces when the number of dimensions begins to grow. Better still, these difficulties are in some ways explained, with many detailed examples of paths in varying spaces where every bounce on a frontier or reflection on a singularity is highlighted. There is probably no good way to simply explain the OPTIC orbifolds, but this is a solid effort, though it perhaps obfuscates the general concepts (quotient metric spaces) through the multiplicity of cases. It is here that the abundant and painstakingly detailed figures are most useful, though they prove helpful everywhere they appear. 4 To better explain the idiosyncrasies of those spaces, Hook develops explicit rules for deriving a unique “normal form”—a generic element of a fundamental domain, to use topologist vocabulary—starting from any collection of pitches (repetitions allowed) and reducing modulo each of the thirty-two OPTIC equivalences taken in any combination. This is valuable, original work, and probably the best way to understand the geometry and topology of these convoluted spaces/orbifolds, though it requires a pen, paper, time, and a modicum of aspirin. It appears surprising at first that scales are addressed last of all in part 4, although much about them could have been said much earlier in the book. This approach enables the author to lean on the solidly constructed notions of transformation groups and voice-leading spaces to enrich the presentation of his diatonic spaces and signature transformations, at the price perhaps of deeper discussions of maximal evenness or well-formedness. Hook is justified in feeling some pride over his original proof of “Cardinality = Variety, ” not because it is shorter than the seminal one, but because it is a good illustration of the power of transformational theory as outlined throughout the book. 5 The last chapter ventures into other cardinalities than 7 or 12 with several interesting arithmetical considerations synthetizing the large research field about generalizations of the diatonic scale. All in all, these last three chapters are a welcome relief after the much more difficult parts 2 and 3. This book may not be the quickest approach to the first part of its title, “Musical Spaces, ” as some articles, presupposing the required mathematical knowledge in the reader or assuming a lack of mathematical rigor, offer more direct access. Nonetheless, even as a reference opus on general tonnetze or OPTIC spaces or scales/signatures systems, this book offers some novel connections that may well supersede the classical ones in time. But this project's strongest achievement is the fulfillment of the goal outlined in the preface: initiating novices into the beauties and efficacy of mathematical machinery in musical spaces. Keep in mind that this book is intended for serious readers; a solid knowledge of nonmathematical music theory is an absolute prerequisite, and so is a willingness to work with paper and pencil. That said, taking into account the quality and diversity of examples and exercises, the pedagogical value of this book is considerable, especially (as intended) the first five chapters—I would go further and recommend adding the sixth chapter to any regular curriculum in mathematical music. This book will likely find a good place on many university shelves and in the personal libraries of curious music theorists.