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Book Review

Generalized Musical Intervals and Transformations

Generalized Musical Intervals and Transformations by Lewin
David Lewin
Oxford University Press
ISBN-10: 0199759944
ISBN-13: 978-0199759941
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David Lewin (1933 - 2003), was an American musicologist, music theorist, critic and composer. Widely acknowledged as one of our most influential and profound thinkers on (contemporary) music, he concentrated on the application of mathematics to music, specifically on the relationship of transformational and group theory with each other and with music. Born in New York, Lewin graduated from Harvard in 1954 before studying (music) with Sessions and Babbitt. His academic, publishing and compositional achievements are striking; they reflect radical, sustainable and a comprehensive approaches to many (often quite disparate) aspects of music theory – from text and literary theory to mathematical, for instance.

Perhaps his most important contribution to the latter area is the treatise "Generalized Musical Intervals and Transformations", known affectionately as "GMIT". It was first published in 1987 (by Yale University Press) and reprinted by Oxford University Press 20 years later, sufficiently enduring was its appeal and important its contribution to the role of group theory to understanding both the basic concepts of intervals and transposition; and by extension to explaining musical entities other than pitch.

Now there is a new edition of the book (in paperback, again from OUP, 2011) which ought to bring Lewin's inspiring ideas to a new generation of music lovers, theorists, students, composers and performers. For all the book's ostensibly forbidding subject matter and density, it's a key work and one that deserves a place on every music-lover's shelf – especially at the very reasonable price for which it's now available again, and even more cheaply electronically.

"Generalized Musical Intervals and Transformations" advances (then convincingly and successfully supports) theories based on mathematics which can be seen to apply not only both to tonal as well as atonal music. But also to rhythm, meter and even timbre. Add these aspects of music amenable of such analysis – as well as pitch, of course and the consequences are quite far-reaching. At the heart of the theory is the compelling metaphor of "musical space", which is a concept able to explain much more than mere passing in time from one pitch or pitch class to another. Transformation rather than cadence.

One of the aspects of Lewin's work that is so striking, and which makes "GMIT" such a useful book – even to non-specialists – is that he has the gift of making complex ideas accessible. Perhaps such a gift derives from his attachment to literature. His style has been described as "poetical". To read, for example, a typical description of how pieces by Ligeti or Carter work in terms of the most complex mathematical development [page 67], is to follow an open, transparent narrative describing the music from the music's perspective… instruments, sounds, the familiar. Rather than how it can be made to fit into graphs and diagrams.

There are plenty of those too in the book! In fact, "GMIT" looks at first sight like an advanced math textbook. We're saturated immediately by arithmetical concepts. And it would be disingenuous to suggest that someone with little or no mathematical understanding can get the most from the book. At the same time, Lewin is as "gentle" as he is precise: disambiguation is at the core of all his explanations: "A mathematician would begin by saying, 'Let S be a set'. Unfortunately, music theory today has expropriated the word 'set' to denote special music-theoretical things in a few special contexts. So I shall avoid the word here. Instead I shall speak of a 'family' or 'collection' of objects or members. When I do so, I mean just what mathematicians mean by a 'set'." [page 1] Similarly, Lewin's Introduction suggests a passage through the book's chapters that includes the exhortation to move on from the purely mathematical subject matter of Chapter 1 (and return to it) if it proves "fatigu[ing"] [xxix]. Could we be in more sympathetic hands than that?

Again, some experience of the ground covered by books such as the classic "The Structure of Atonal Music" by Allen Forte (ISBN-10: 0300021208 ISBN-13: 978-0300021202), "Introduction to Post-Tonal Theory" by Joseph N. Straus (ISBN-10: 0131898906 ISBN-13: 978-0131898905) and "Basic Atonal Theory" by John Rahn (ISBN-10: 0582281172 ISBN-13: 978-0582281172) would be useful. Though as complement as much as preliminary. Similarly Lewin's own "Musical Form and Transformation: Four Analytic Essays" (ISBN-10: 0199759952 ISBN-13: 978-0199759958) forms a useful extension of the ground covered in "Generalized Musical Intervals and Transformations".

Read "GMIT" carefully and you will come to understand how many composers now think. How musical sounds actually work on our perception. How music can be described in terms much broader and deeper than those of static relationships between the elements of intervals. Most crucially, transformational theory (a sub-discipline effectively invented, and successfully developed, by Lewin) demonstrates that musical elements, because they co-exist in a "space", should be thought of as entities constantly transforming, the one into the other. Motion is key. While conventional set theorists emphasize the existence of independent musical objects – albeit with their own roots and potential for resolution and shifts in pitch etc – for Lewin chords, notes, timbres and so on exist chiefly in complete and integrated relationships with one another. A G isn't followed by a B. It is transformed into it. The difference is not trivial. And "GMIT" uses a myriad of examples to make, support and illustrate the process… "The transformational attitude does not ask for some observed measure of extension between reified 'points'; rather it asks: 'If I am at s and wish to get to t, what characteristic gesture… should I perform in order to arrive there?'" [159].

While "Generalized Musical Intervals and Transformations" is not perhaps the most descriptive of titles, and maybe an off-putting one to some, it's a book that has rightly earned its place as one of the most important and highly esteemed of its kind in the last half century or so. As Edward Gollin explains in his forward to the new edition, Lewin knew of errors found and corrections needed in the earlier edition. If he had had his way, we would have had a second edition. As it is, this OUP volume makes what truly is a revolutionary theory on the way music works (and not just contemporary music… Bach and Wagner people the opening pages in no small way) available to us now. Put in the work to understand the admittedly often dense (but always lucidly set out) material which Lewin handles so deftly, and the rewards are huge. The technicalities of the book's production, indexing, footnoting, clarity of reproduction of the examples (musical and mathematical), proofing and so on are all just as one would expect from OUP. Unreservedly recommended.

Copyright © 2012 by Mark Sealey.