Understanding Post Tonal Music Miguel Roig Francoli

What This Book Actually Covers

Post-tonal music theory is one of those subjects that sounds intimidating but mostly just requires patience and a willingness to accept that not everything has a functional harmony home. Miguel Roig-Francoli's textbook is a standard entry point for this material. It covers set theory, serialism, twelve-tone technique, and spectral approaches to composition and analysis. The book is widely used in undergraduate music theory programs across the United States, and if you're taking a graduate course or preparing for a comprehensive exam, you will likely encounter its frameworks directly. The book is structured in a way that assumes you already know tonal harmony. Chapter 1 moves quickly through pitch-class sets and normal form, which means if you are coming from a strictly classical theory background, you should be prepared to spend extra time on the mathematics side before the musical applications start making sense. The later chapters on serialism and twelve-tone analysis are where most students hit real obstacles, and that is worth knowing before you commit to it. I remember working through one specific problem with a student who was analyzing Webern's Op. 27, first movement. The score shows a prime form labeled I, and at first glance the row seems straightforward. But when you derive the inversion, the transpose levels, and then check the combinatorial properties, things get messy very quickly. The issue was that the row form used in the cello line does not align with the expected intervallic content from the basic form because Webern applies a retrograde inversion simultaneously. The workaround was to label the row forms using both the prime/inversion designation and the literal pitch-class content side by side, rather than trusting the analytic shorthand alone. That habit of writing out the actual pitches next to the formal labels saves you from several common misreadings.

How the Set Theory Section Actually Works in Practice

The pitch-class set theory portion uses Forte numbers and interval vector calculations. The process is mechanical, which is partly why people find it tedious. You take a chord or melodic segment, reduce it to pitch classes, order it in normal form, find the prime form, and then match it to Forte's catalog. The interval vector is a six-number sequence representing how many of each interval class appears between the pitch classes in the set. A practical detail that textbooks sometimes gloss over: transposition and inversion are operations, not properties. When Roig-Francoli presents them, he treats them as tools for reducing and comparing sets. A counter-intuitive thing beginners often miss is that two segments from different pieces can share the same prime form but function completely differently because context matters more than the set itself. The book does acknowledge this, but it does not always hammer home that the analytic label is descriptive, not prescriptive. A set class does not determine how a passage sounds in isolation. When you are actually doing analysis, the most useful shortcut is learning to recognize common set classes by ear and by shape rather than recalculating normal form every time. Sets like [0,1,4,6] and [0,3,4,7] come up constantly in Stravinsky and Bartok. If you can identify them at sight, you save perhaps ten to fifteen minutes per passage in an exam setting. That is a real difference when you are working under time pressure.

Twelve-Tone and Serialism Coverage

The serialism chapters cover basic row construction, operations on the row, matrix building, and analysis of tone rows in works by Schoenberg, Webern, and Berg. The matrix is the core tool here. You build it by placing the prime form in the top row and then deriving each inversion by transposing it according to its index number. The result is a grid where every row and column represents a specific row form. Here is where students typically run into trouble. The book presents the matrix as a neat, logical system, but real compositions rarely use it cleanly. Composers truncate rows, combine row forms, nest rows inside rows, and occasionally violate their own stated rules for expressive reasons. I once spent an afternoon trying to force a Schoenberg piano piece into the matrix because a single measure did not seem to correspond to any row form. The resolution was simpler than expected: the passage was using a segment of the row plus a non-row accompaniment figure that the matrix could not account for. The matrix is a reference tool, not a complete analytical map for every surface event. Roig-Francoli's treatment of combinatoriality is one of the stronger sections in the book. The concept is not difficult but requires careful attention to which hexachords pair with which. The practical tip here is to write out the hexachordal partitions explicitly before checking for combinatorial properties. Skipping that step leads to errors in nearly every second attempt.

Limitations and Where the Book Falls Short

The book has real limitations that matter if you plan to use it for serious analysis. It covers only a narrow range of twentieth and twenty-first century repertoire. Spectral music, minimalism, and many important post-tonal traditions from outside the European art music canon are not addressed. If your work involves Ligeti, Messiaen's limited transposition scales, or non-Western post-tonal practices, this text will not guide you there. Another honest note: the exercises assume access to a teacher or a study group. Several of the later problems in set theory and twelve-tone analysis are the kind where you can produce a mechanically correct answer that is still analytically meaningless. Without someone to review your work, you can easily build bad habits around labeling. A second set of ears for the first month of study usually prevents that. For students who want more depth on serialism, I would recommend pairing this book with Allen Forte's Structural Functions of the Chromatic Language or Lewis Lockwood's work on Schoenberg. Those texts complement Roig-Francoli without duplicating the same examples. If your goal is computational music analysis or algorithmic row generation, you will also need supplementary resources since this book does not cover software implementations at all.

What the Book Gets Right

The clear notation and consistent terminology are genuinely useful. Roig-Francoli defines terms before using them in ways that require prior knowledge, which reduces the back-and-forth confusion that happens in other theory textbooks. The appendix with Forte number tables and interval vector summaries is functional and accurate. The musical examples are well chosen for introductory-level exposure, even if they lean heavily on familiar composers. The section on pitch-class set equivalence is perhaps the most technically precise part of the book. It handles transpositional and inversional equivalence without hand-waving, and it presents the reduction process in steps that are easy to follow if you do each step manually before relying on calculators or software tools.

How to Use This Material Effectively

The most practical approach is to work through the set theory chapters first, make sure your normal form and prime form calculations are automatic, and only then move into the twelve-tone sections. Doing them in reverse order creates unnecessary friction because set theory underpins most of the serial analysis methods presented later. Plan for roughly eight to ten weeks of study if you are covering the book in a single semester, with additional time allocated for the analysis projects that follow each major section. Buy or access the book through your university library or standard academic retailers. There is no official free PDF, and downloading unauthorized copies is both unreliable and ethically unnecessary given how widely the text is available through institutional channels. The solution manual exists separately and is intended for instructor use, so expect to work through problems without immediate answer checks unless your course provides it. If you are approaching this material on your own, keep a running log of the set classes you encounter and the contexts in which they appear. That personal reference becomes more useful over time than any static chart in the book, and it is the kind of thing that only develops through repeated direct engagement with scores rather than abstract exercise completion.