Introduction: theories of theory

What is theory?

What is music theory, and why is it important? In my experience, these core philosophical questions are seldom addressed by teachers of music theory, who, after all, must attend to a daunting body of content and skills. Admittedly, the latter question—the "why" of music theory—might occasionally inspire meaningful, albeit limited and too often private, pedagogical reflection. But oddly, the more foundational first question—the "what" of music theory, and consequently the "how"—is scarcely even broached in music theory textbooks and classrooms.

This disinclination to consider the foundations of music theory has allowed naive views to persist among students and teachers alike. For many students (and even teachers), "theory" connotes only the most uninspiring and technical necessities of musical study, such as notation, scales, chords, and other fundamentals. On the other hand, for many professional music theorists, "theory" comprises only the most advanced, rigorous, and sophisticated topics, such as Schenkerian, neo-Riemannian, or pitch-class theories. They are both wrong.

I offer this essay as a corrective to such widespread and small-minded conceptions. I will describe a subtle pedagogical reformulation of elementary "music theory" that celebrates its true theoretical essence and methodological complexities. I will resituate music theory as a bona fide epistemological enterprise akin to other sorts of theory, thereby buttressing curricular efforts aimed at the integration of knowledge across the disciplines. Importantly, this approach easily supplements the content of a traditional "theory" curriculum without compromising the teaching of that content itself: indeed, I will describe opportunities to engage with numerous venerable topics in tonal music while centering the theoretical underpinnings of each. Such an enterprise not only fosters a more realistic, reflective, and accurate understanding of music theory as a discipline but also enriches students' appreciation of music theory and its resonance with other fields. The study of music theory—even from the very first rudiments—is thus transformed from a stern rite of passage mired in rules and technicalities, into an expansive intellectual endeavor, adding yet another educational component to music theory's manifold musical benefits (which I will also briefly enumerate at the end of this essay).

An illustrative tale: the case of the hidden theory

To motivate this recontextualization of theory, I begin by offering my students the following tale—part allegory, part shaggy-dog story.

Sally Sophomore returns to campus after a weekend away and finds that the laptop she left in her dorm room won't boot up. She is puzzled, as the computer worked just fine before she left campus. After a moment's thought, she develops a theory: perhaps her roommate had carelessly run down the battery. But upon further investigation, Sally finds that the computer is plugged in and appears to be fully charged. A moment later another theory occurs to her: maybe a virus is to blame. But her attempts to boot in "safe mode" are similarly unsuccessful, so she rules out that idea as well. Finally, Sally becomes aware of a large sticky spill across the keyboard and the unmistakable smell of Cherry Garcia ice cream. Just as she begins to make sense of this, her roommate rushes in with a package under her arms:

"Sally, I'm so sorry! I can explain: we had a party while you were gone, and things got a little out of hand. Before we knew it, a pint of ice cream had melted all over your laptop and wrecked it. I should have kept a better eye on things. I'm really sorry." Presenting the package, she adds, "But don't worry: I just bought you a new computer—the next higher model, actually—and I had all of your files transferred. It's all OK now. I hope you can forgive me."

Sally is relieved, and touched—her roommate didn't have to go to all that trouble and expense to rectify such an innocent mistake. "It's just like her," Sally muses, affectionately: "She's a middle child!" And the two happily embrace, before settling down to work on their music theory homework.

The moral: theory versus hypothesis

Where is "theory" in this tale? Did you spot it? I like to lead students in an in-class discussion, exploring the difference between the casual, popular use of the word "theory" and the strict, more formal use of that word. Students will easily identify two examples of the former: Sally's "theories" that the computer had a dead battery or was stricken by a virus. Such ideas are more properly called "hypotheses"—guesses about a particular state of affairs. A proper theory, on the other hand, may be defined as a conceptual framework that helps make sense of some broad set of phenomena. A hypothesis will prove to be either true or false; a theory, on the other hand, is a way of seeing, for which truth and falsehood are largely beside the point.

Students can be pressed to discern the proper "theory" in the tale above, but it is one that had nothing to do with solving the mystery: the concept of "middle child." This "birth order" theory of personality emerged in the early twentieth century and was popularized more recently by Kevin Leman, whose The Birth Order Book (2009) contains a telling subtitle: "Why you are the way you are." Sally might have said of her roommate, "It's just like her: she's a Libra." That too is a theory of personality, even if its theoretical primitives are very different: instead of "eldest child," "middle child," and "youngest child," astrological theory postulates personality categories based on the month of one's birth.

From here, students can be encouraged to brainstorm and/or research other theories. And in each case, the theory can be shown to have certain basic elements: at the very least, a scope of study and a set of theoretical concepts and categories. Economic theory, for instance, studies the behavior of agents in a market economy, and to that end it invokes such concepts as supply, demand, choice, utility, etc. Likewise for countless other theories, such as various psychological theories, political theory, feminist theory, game theory, aesthetic theory, and of course a multitude of scientific theories—including, notably, obsolete scientific theories like Aristotelian physics or medieval alchemy, each of which were simply the best means of conceptualizing physical phenomena until they were replaced by ones with more explanatory power. Non-majors, double-majors, students in liberal-arts institutions, and broad-minded musicians of all stripes, will happily furnish examples of theories from other disciplines and fields of inquiry. (And some students may be inspired to delve deeper into meta-theory, via such seminal works as Popper 1935/2002, Kuhn 1962/2012, or Thagard 1992.) Music students who breezily speak of "theory class" or "theory homework" rarely consider that music theory, too, is such an example.

* * *

A well constructed theory is a powerful thing; where there is understanding, a theory is surely at play. As Leonard Meyer wrote, "Like air, theories may be unsubstantial; but, as with air, we can't live and act without them" (Meyer, 1998, p. 18 n. 45). In the case of music theory, it might seem that intervals, scales, chords, meter, form, and other basic elements are uncontroversial, even self-evident, "facts" to be memorized. But students should learn that even the most seemingly obvious music-theoretical constructs are laden with abstraction and artifice. True theory (unlike the hypotheses of a clever fictional detective) needn't be surprising or esoteric. And by the same token, students should recognize that some portions of their "theory" studies fall strictly outside the realm of true theory: for instance, reading clefs and deciphering transposing instruments, however valuable and challenging they may be, are matters of mere notation, not theory.

What follows are some specific ways of leveraging these initial explorations, to help students put the theory back in "theory."

Theorizing in the theory classroom

I will now focus on four meta-theoretical issues that can be seamlessly introduced early in the music theory curriculum: explanatory power, listening as theorizing, theory-building, and symbolic representation. I will discuss these with an emphasis on their relevance to familiar elementary musical topics, each one furnished with musical illustrations. The issues have been chosen both for their general epistemological applicability and for their practicality as touchstones throughout a student's musical studies.

Explanatory power: measuring intervals

Since theory is, as I wrote earlier, "a way of seeing, for which truth and falsehood are largely beside the point," a theory should be judged according to its explanatory power—its efficacy in coherently describing its subject and facilitating fresh insight. The classic tradeoff between theoretical simplicity and explanatory power is well illustrated by a heavily theorized (it would seem, even excessively theorized) elementary musical concept: interval. The concept of interval also offers a perfect opportunity for a "devil's advocate" approach that helps our students dig deeper as they begin to imagine themselves as theorists.

The theory of interval in tonal music postulates categories of distance (second, third, fourth, etc.) that are modified by categories of quality (major, minor, perfect, augmented, diminished). As soon as I introduce this system, I (disingenuously) encourage my students to bristle against such an apparent theoretical contortion and to long for the abundantly straightforward, one-dimensional metric of semitone distance. I lead them to confront the question, Why is semitone distance alone insufficient to fully describe musical space? To answer that question, I invoke the curious reality of musical context, in the form of two simple melodies (Figure 1): the opening of Chopin's Nocturne in E♭, op. 9 no. 2 (Figure 1a), and a subsidiary theme from Haydn's Symphony no. 104 Finale (Figure 1b). I isolate the interval B♭–G (having established a suitable E♭-major context, per the first excerpt) and then its enharmonic equivalent A♯–G (having established an alternate B-minor context, per the second excerpt), easily illustrating at the piano the very different sounds of these two intervals. Students discover that the tempting but theoretically naive conceptualization of semitone distance—9 semitones in either case—elides what is most musically essential about the two situations. By contrast, our theoretically sophisticated concept of interval precisely captures the crucial distinction between a major sixth and a diminished seventh—the one harmonious and unexceptional, the other dissonant and striking.

I then invite the students to sing the two intervals in question, in each case vocally tracing a stepwise path from note to note (Figure 2). Given sufficient tonal context, students will naturally sing B♭–C–D–E♭–F–G in the first case (six notes, hence a "sixth," Figure 2a) and A♯–B–C♯–D–E–F♯–G in the other case (seven notes, hence a "seventh," Figure 2b). The difference between the two intervals, which can otherwise strike some students as senselessly pedantic or doctrinaire, becomes "real," even embodied. A similar aural trick involving ambiguous tritones is perhaps even more compelling—see Figure 3. In contrast to the dispassionate yardstick of semitone distance, the theoretical contrivance of "interval" proves to be robust and human, a triumph of "sense-making." Explanatory power must always prevail.

Figure 1. Two excerpts illustrating the enharmonically equivalent interval (B♭–G versus A♯–G).

(a) Chopin, Nocturne in E♭, op. 9 no. 2

The opening measure (with anacrusis) of Chopin, Nocture in E-flat, op.9, no.2

(b) Haydn, Symphony no. 104, Finale, m. 84

Measure 84 of Haydn, Symphony no. 104, Finale.

Figure 2. A directed "sung analysis" of the intervals in Figure 1.

(a) Major 6th, after Figure 1a.

A directed 'sung analysis' of the intervals in Figure 1. First a Major 6th.

(b) Diminished 7th, after Figure 1b.

A directed 'sung analysis' of the intervals in Figure 1. Now a diminished 7th.

Figure 3. An analogous "sung analysis" of a chameleonic tritone (F–B versus F–C♭).

(a) Augmented 4th.

A directed 'sung analysis' of the intervals in Figure 1. Augmented 4th.

(b) Diminished 5th.

A directed 'sung analysis' of the intervals in Figure 1. Diminished 5th.

Listening as theorizing: hearing rhythm and meter

In the prior illustration, and ideally throughout the music theory curriculum, theory is tested against the listener's real-time experience of music. Such aural verification underscores the crucial truth that successful theorizing mirrors human cognition itself. And indeed, some theorizing even happens automatically: to operate in a culture is to subscribe to existing theories, most of which are acquired without effort or even awareness. It could be said that music theory's raison d'être is to make plain the implicit conceptualizations that any competent listener brings to the act of listening. The enharmonic interval demonstration makes that point vividly (and for many students, astonishingly): not only is the difference between a major sixth and a diminished seventh real, but it's a difference that the ears knew even before the mind was taught. "Your ears are smarter than you thought they were," I tell my students at such moments. Rhythm and meter provide further exquisite demonstrations of the psychological inescapability of such structured (i.e., theory-driven) hearing—listening as theorizing.

Even the simplest musical stimulus—a lone hand-clap—inevitably acquires meaning through the structures assumed by a listener. The difference between Figures 4a and 4b arises not from the stimulus itself (the "music") but from the existence of conceptual categories—an implicit theory of meter. In this case, the listener infers "beat" versus "off-beat" through a real-time application of those categories—which is to say, through implicit musical analysis. As Goethe insisted, "With every intent glance at the world, we theorize." (Goethe 1810, Vorwort; thus quoted, prominently, in Schenker [1935] 2001, p. 3)

Students must also confront the limits of such theorizing, as when they come to terms with musical situations that are rhythmically challenging, under-determined, and/or unfamiliar. I like to ask students to tap their toes to the opening of Leila Pinheiro's "Chega de Saudate"; it is a task that many students find difficult or impossible, but repeated and directed listening can help them to orient their rhythmic understanding to the stylistically idiosyncratic metrical framework. Even more challenging and ear-opening is the rhapsodic 7/8 of Karolina Goceva's "Mojot svet" (Macedonia's 2007 Eurovision song entry): students who have a hard time finding the down-beat will marvel at an informal live performance during which an audience of (possibly inebriated) amateurs spontaneously entrain to the non-isochronous meter through participatory clapping.

Figure 4. The same rhythmic stimulus (a hand-clap) in two contexts.

(a)

Steady quarter note beats in first measure and clap on down beat.

(b)

Steady quarter note beats in first measure and clap on the 'and' of the first beat of measure 2.

Such seemingly rudimentary concepts as beat, off-beat, and down-beat, often dispensed with unceremoniously in the first weeks of a theory curriculum, should instead be honored as the cognitive miracles that they are—profoundly meaningful and profoundly constructed.

Theory-building: confronting anomalies in harmonic analysis

As a tool for understanding, theory must be responsive to the full range of specimens under its purview. For that reason, successful theories are not ordained but rather developed: theory-building is a dialectic process of testing and refining a theoretical system in light of a fulsome set of data. The earlier enharmonic demonstration hinted at this process (as did the beat-finding exercises, in a different sort of way); here I will more explicitly invite my students to partake in theory-building, in the course of a pedagogical turn from chordal identification to full-fledged harmonic analysis.

Tonal harmonic theory postulates a very limited set of harmonic entities—a handful of triads and seventh chords—that are easily learned. (Compare the systematic completeness of pc-set theory; the difference between these two theories reflects the differing demands of the respective repertories.) In an early exercise in chord identification, I present my students with a homophonic choral texture from Mozart's Requiem, the Hostias, which provides abundant examples of various chord roots, qualities, and inversions. My instructions are simple: "On every beat, indicate the chord using 'fakebook' notation. When the chord fails to conform to our inventory of chord-types, simply place an X."

There are indeed plenty of pesky X's, which would seem to besmirch what are some of the most sonically appealing moments in the passage. I used to avoid such inconvenient distractions by micro-managing my choice of repertoire—or else by summarily dismissing them ("Never mind … we'll get to that later"). But in the context of a "theory-aware" pedagogy, I have come to embrace these moments as important teaching tools. I find it stimulating (and fun) to feign exasperation: "Our theory of chords isn't all that good, is it? It has nothing useful to say about many simultaneities in this apparently straightforward piece of music!" A closer look—and listen—will suggest satisfying ways of accounting for those anomalous simultaneities, of course, and thus will emerge the concept of the non-harmonic tone. As my students and I work our way toward a richer understanding of harmony and counterpoint—figure and ground, structure and elaboration, tension and resolution—theory-building comes to the fore.

Symbolic representation: data compression and theoretical priorities in chord shorthand

Finally, it behooves us to consider another common metatheoretical issue: the use of symbolic representation and notation. Theory necessarily simplifies our world, and that simplification often goes hand in hand with exigencies of notation. Chord shorthand is a case in point, and a telling one.

Students find it provocative to learn that an 18th-century keyboardist or fretboardist would have faced a decoding challenge analogous to that faced by a modern-day reader of a jazz or pop "fakebook." The specifics of those two notational traditions, however, reveal how particular theoretical priorities and affordances shape symbolic systems. Students can compare the "native" chordal shorthand schemes for an early-18th-century solo sonata and a mid-20th-century popular ballad, discovering the conceptual traces embodied in each (Figure 5). Baroque figured bass emphasizes intervals and voice-leading within a particular key signature (Figure 5a), whereas modern fakebook notation, agnostic with respect to key, emphasizes instead chord structures in isolation (Figure 5b). These systems consequently elicit very different cognitive work from the performer. (Indeed, the most adept musician of three hundred years ago would have scarcely had a concept of chord root, the 'bread and butter' of even the most casual dorm-room guitarist today.)

An anachronistic swapping of those notational systems helps to recover otherwise hidden elements of the harmonic picture: a hypothetical fakebook to the sonata, for instance (Figure 6a), immediately reveals chord roots and qualities undisclosed by the figured bass, while a figured bass to the ballad (Figure 6b) immediately foregrounds the salient out-of-key chord in m. 4, inconspicuous in the fakebook notation.

Figure 5. Two notational traditions of chordal shorthand.

(a) Figured bass for Handel Sonata op. 1 no. 7, iii, mm. 1-5.

Figured bass for Handel Sonata op. 1 no. 7, iii, mm. 1-5. More description above.

(b) Fakebook chords for "What a Wonderful World" (Thiele/Weiss), mm. 1-4.

Fakebook chords for 'What a Wonderful World': F Am B♭ Am Gm F A7 Dm

Figure 6. Hypothetical anachronistic chordal shorthand for Figures 5a and 5b.

(a) Fakebook chords for Handel Sonata op. 1 no. 7, iii, mm. 1-5.

Fakebook chords for Handel Sonata op. 1 no. 7, iii, mm. 1-5: Am Am/C G♯°/B Am Em/G Dm/F E E7/D Am/C B𝆩7/D Am/E E7 Am

(b) Figured bass for "What a Wonderful World" (Thiele/Weiss), mm. 1-4.

Figured bass for 'What a Wonderful World.' More description above.

Those two symbolic systems correspond to "prescriptive" realms of harmonic theory. By contrast, Roman numerals generally represent tonal harmonic structures with a more "descriptive" purpose in mind. Here too, much is at stake as we develop our notational details and decisions. The explanatory power of the nomenclature "V/V," for instance, points to very different structural aspects of that chord than does the more tempting and straightforward "II". Similarly, nomenclature looms large with respect to that familiar pedagogical bugbear, the cadential six-four chord: whatever may be a teacher's stance on the question (of the chord's function as tonic versus dominant), s/he would be remiss not to draw attention to the ways that one's theoretical commitments shape (and are shaped by) choices of analytical notation (" I46 – V " versus " V[4–36–5] "). And in reflecting on their own intuitions about the cadential six-four, students will see that this descriptive nomenclature (no less than prescriptive nomenclature) ultimately represents a compromise in an attempt to capture the fullness of musical meaning.

Conclusion: learning goals in the theory classroom

I began this essay with the question, "What is theory, and why is it important?" Having explored some approaches to the first half of that question, I will conclude with a brief discussion of the second half, affirming the many and varied educational benefits of music theory. The study of music theory, needless to say, helps students to better understand the mechanics of music and the construction of musical works. It enables students to cogently talk about and write about music while exposing them to a large body of repertoire. By fostering intimacy with the details of musical construction, it leads to a deeper appreciation of the artistry of composers and performers—an insight into great minds. Music theory also offers many frankly practical benefits to musicians, in the form of musicianship: it facilitates the learning and memorization of new pieces; it is indispensable to the conductor or, indeed, to any ensemble musician; it informs composition and can be applied to the art of improvisation; and it shapes a performer's interpretation of a piece. More generally, music theory stands to foster a broader disposition of attentiveness: in a world marked by passivity and saturated by distraction, music's great gift is that it invites us to engage with pure sound, and theory's great gift is that it helps us to engage.

Note, however, that this prodigious list of educational payoffs relates to the categories of analysis and musicianship. In the course of those essential and deeply rewarding educational experiences, I find it valuable to also remind students of the intellectual marvel that theory is unto itself, and to remind students that they themselves are theorists, both in class and in life. Reclaiming the theory in what we do as teachers and students will only add a unique layer of richness to students' musical formation.

Acknowledgements

I would like to thank Sarah Day-O'Connell, Gretchen Foley, and Jennifer Shafer for their careful reading and helpful suggestions.

Bibliography

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