[2] My contribution is inspired in part by Patrick McCreless's
recently published paper, "Rethinking Contemporary Music Theory."
[3] When I first entered the field as a graduate student in 1979, the boundary between research and teaching at the introductory levels seemed inevitable and unbreachable. At that time, the most progressive and stimulating concepts in the field--middleground motivic parallelisms, dissonant prolongations, double-tonic complexes, Kh-networks, and hexachordal combinatoriality--were embedded in distinct repertories. Grasping these concepts required a saturation in the respective repertories, and then a long walk through preparatory concepts.
[4] I think we are at a somewhat different juncture now. A number of central concepts have emerged in the last fifteen years that can be taught at the introductory level, to students whose only prior exposure is some familiarity with basic notation. In identifying some of these areas, my primary purpose is not to advocate for a particular approach to pedagogy, but rather to establish a position from which to observe the state of music theory as a research field. I will suggest that the opportunity to transgress the boundary between introductory teaching and research reflects other ways that music theorists have been recently crossing boundaries--boundaries between repertories, between compositional parameters, and between domains of knowledge. I will suggest further that these circumstances provide an opportunity to develop a more effective broader pedagogy.
[5] I will now proceed through an imaginary syllabus for the first six units of an introductory course in the fundamentals of music in the European tradition. Course pre-requisite: ability to "read music," i.e. to passively interpret standard pitch/duration symbols in treble and bass clefs.
[6] When an oral version of this paper was presented at the Plenary Session in Phoenix, my 'pedagogical program' provoked some irritation among a subset of those present, so I reiterate, with some emphasis, that my intention is not to prescribe how music theory ought to be taught, nor even to recommend a particular pedagogical program, but rather, to call attention to the pedagogability of recent music theory--the potential for its central concepts to be explained to a population who has not made a substantial prior investment in conceptual or repertorial overhead--and to use this circumstance as a mirror for reflecting back on the current state of research in the field.
[7] UNIT 1. Rhythm and Meter. After reviewing rhythmic notation,
establishing the concept of metric hierarchy, and introducing the
strong-beat oriented durational patterns most characteristic of
art-music and its pedagogy, we turn to the topic of funky rhythms.
The concept, but not the name, comes from the work of Jeff Pressing;
he calls them "prime-generated."
[8] The concept of funky rhythms creates an opportunity to pay a visit to some students in their musical homes, and in so doing to demonstrate that productive systematic inquiries into musical organization are not the province of Western European art music alone. But there is another virtue as well: prime generativity is characteristic of pitch as well as of rhythm, and thus provides a lead-in to future topics. As Pressing shows, pentatonic and diatonic scales in chromatic space, and triads and seventh chords in diatonic space, are all prime generated, and thus isomorphic with funky rhythms.
[9] UNIT 2. Scales. Diatonic scales are over-determined, and this
presents a pedagogical advantage. They can be prime-generated by the
interval of the perfect fourth, as Schenker taught. Or they can be
taught, perhaps more practically, as a series of whole and half-steps,
where the position of the half-steps is governed by the principle of
maximal evenness.
[10] UNIT 3. The Tonic Triad as Melodic Frame. We begin this unit by
partitioning the diatonic scale into triadic and non-triadic tones.
This partition is both prime-generated and maximally even; the latter
concept will eventually serve to underwrite the complementarity of
steps and leaps, i.e., of voice-leading and harmony, as emphasized in
Eytan Agmon's work.
[11] UNIT 4. Two-Voice Counterpoint. We begin our study by
partitioning harmonic intervals into dissonance and consonance, and
further partitioning the latter category into imperfect and perfect.
We then study the syntactic model that leads dissonance to imperfect
consonance to perfect consonance. The progression toward perfection
and the cadential status of the perfect intervals are both
psychologically clarified and historically situated through the
Aristotelean notion that the imperfect strives for the perfect, and
through the adaptation of these concepts to Christian theology by
Scholasticist philosophers.
[12] UNIT 5. Triadic Voice-Leading in Root Position. Having studied
triads as the stable ingredients of a diatonic collection, it is time
to learn the twenty-four consonant triads as individual entities, and
to begin to explore inter-triadic voice-leading, restricted for the
moment to root position. We introduce the Verwandtschaftstabelle of
19th-century German theory, but adapted to equal temperament. Motion
through the table proceeds by stepwise displacement of a single voice,
moving in oblique or contrary motion with the bass. The pedagogy here
is rooted in the neo-Riemannian work of David Lewin and Brian Hyer.
The incremental moves are Lewin's Leitton, Parallel, and Relative
operations, the charting of these operations on an equal-tempered
version of Riemann's map is inspired by Brian Hyer's work, and the
focus on voice-leading parsimony has been the focus of my own recent
research.
[13] UNIT 6. Talking About Music. Having spent an intense few weeks
learning to talk about music, it's a good time to spend some time
examining the language that we've been using. In speaking of Maximal
Evenness, we appealed to a notion of objects in space. We studied
melodic succession by analogy with physical motion. For dissonance
and consonance, we invoked theology; for voice-leading parsimony, we
quietly appropriated a central concept from logic and economics. When
we spoke of melody as adhering to principles of inertia, magnetism,
and gravity, we did not mean to imply that acoustic phenomena have
weight, volume, and mass. My own understanding of these matters has
been enriched by Thomas Christensen's study of Newtonian mechanics and
Enlightenment musical concepts;
[14] The discussion of the metaphorical nature of musical language marks the end of my hypothetical musical curriculum. I turn now to a consideration of what broader trends are reflected in the circumstances that I have been outlining. Here I return to the theme of boundary crossings that I mentioned earlier. The term has acquired the aura of a buzzword lately, yet its prominence responds to something very real: the rapid migration of populations and the expedited pace of communications. Among those phenomena hurtling through the ether are artistic artifacts, including musics. On this scale, the boundaries that music theorists have crossed in the last fifteen years have been relatively modest ones, but they are nonetheless significant to my theme.
[15] First there are the boundaries between musical repertories within
the Western art music tradition. Such boundary crossings are not
without precedent: for example, the putative modernist wall of 1908
was long ago transgressed by Adele Katz and Felix Salzer in the one
direction, and in the less distant past by Benjamin Boretz and John
Clough in the other. But Lewin's Generalized Musical Intervals and
Transformations has had a crucial and probably long-lasting impact on
problematizing the boundary between the atonal and tonal.
[16] Second are the boundaries between art music and popular music,
between Western and non-Western, between written and oral musical
traditions. This is part of a broader trend away from an exclusive
focus on the aesthetic experience of the focused expert listener in
the sustained presence of the masterwork, and toward a regard for the
ordinary cognition of ordinary folks in the perhaps casual presence of
perhaps ordinary music. I don't mean to say that the one has replaced
the other, but rather that the two orientations increasingly exist
side-by-side, so that our discourse about music now travels
effortlessly and perhaps tacitly along a continuum between them. On
one end of this continuum is traditional analytic scholarship; on the
other end, work in music perception and cognition, and work by
scholars such as Jay Rahn and Kofi Agawu on popular and folk music
traditions.
[17] The third transgressed boundary is intimately related to the
second. It is the boundary between music and other human activities.
Again we can point to a substantial pre-history, including the work of
Zuckerkandl and Meyer in the 1950's, and Clifton in the 1970's. But
the 1983 book of Lerdahl and Jackendoff, with its articulate advocacy
of a cognitive orientation and its compelling demonstration of the
general cognitive basis of musical grouping judgments, marked an
important moment when musical capacities began to be viewed in the
broader framework of mental life.
[18] The transgression of these boundaries is, I think, connected to the potential for eroding the boundary between research and pedagogy. The concepts enumerated in my 'syllabus' are style- and music-specific to varying degrees, but all of them point arrows to other domains of musical and human experience. Prime-generativity is powerful because, as Pressing's evidence suggests, it crosses boundaries of musical style, and even domains of musical organization. Maximal evenness, inertia, and magnetism acquire their power because of their applicability to phenomena in the physical and social world; parsimony because it is a desideratum of logical and economic systems. Whether we take these arrows to represent an ontologically real connection is a matter of preference and ideology. From the perspective I am adopting here, what matters is the heuristic job that each arrow performs: it provides a conduit for explanation, and such conduits are central to the pedagogical process.
[19] My essay so far has adopted a traditionally narrow view of
pedagogy, as the education of undergraduate students in the
classroom. But we don't only teach our students. We teach each
other--fellow practitioners of music theory in the Anglo-American
tradition--and we teach (and in turn are taught by) scholars and
musicians who locate themselves outside that tradition. I will not
attempt an analysis of our scholarly modes of discourse, but it is
worth observing that we seem to be collectively placing a higher value
on ushering each other incrementally into our private
thought-universes. I am thinking especially of the trend toward more
targeted deployment of sparser graphic images, in particular, the epic
story told by the arrow in Figure 0.1 of Lewin's GMIT, and the
technique for graphically transmitting analytic information developed
by Lewin in the Four Analytical Essays and in Henry Klumpenhouwer's
work.
[20] The manner in which we address those outside our profession is
also changing. I have in mind here not only targeted pedagogical
works, such as those of Straus, Friedmann, and Morris, but also overt
works of outreach toward performers, such as Deborah Stein's book
(with Robert Spillman) on song, and original works of research that
make explicit attempts to maximize their audience when the topic
allows, such as the books on tonal rhythm by Joel Lester and William
Rothstein.
[21] The increasing possibility of disseminating music-theoretic
insights, without huge investment in overhead, provides a number of
opportunities for a broadened audience, and a broadened set of
dialogues which can mold and inform our own practice. Some of these
dialogues have already begun. The commerce between music theorists
and behavioral and cognitive scientists is already approaching a
healthy volume. This is the area of music theory that treats of
fundamental questions by its very nature, and the problem of
transmissability seems thereby less acute. The communication gates
between music theorists and critical and cultural musicologists are
opening a bit more creakily, but the increasing volume and
sophistication of historical approaches to music theory will likely
oil these doors as a matter of course. The challenge here will be for
those of us who deal with problems that are analytic, or ahistorically
speculative, to communicate our insights in ways that other scholars,
trained in different traditions, will want to absorb them. Martha
Hyde's admonition from ten years ago is perhaps less urgent than it
was on that similarly commemorative occasion: "If we remain exclusive,
competing among ourselves, interested only in our own issues, reading
only our own journals, in ten years or twenty we may find that we have
conceded the real competition without knowing it, for in the meantime
younger music historians are going to be learning analysis--well
enough for their purposes if not for ours."
[22] There are two sets of boundaries, however, that we have only just
begun to permeate. One of them is work on non-Western music. As
repertorial boundaries erode, and Anglo-American music scholars become
increasingly aware that music is located in knowledge and behavior as
well as in authorized texts, the line that divides our work from
research on non-Western and non-classical repertories begins to seem
increasingly artificial. Is it possible that work now wearing the
jersey "ethnomusicology," by scholars such as Paul Berliner, Simha
Arom, and Steven Feld, is no different in kind from work that sports
the colors of our own discipline?
[23] Here is a case that clearly outlines the issue. In 1978,
archaeologists working in Southern China uncovered a tomb from 433 BC,
containing sixty-five bells. Each bell preserves not only its
acoustic properties but also contains an inscription characterizing
its function in various tonal systems.
[24] The second boundary is one that divides music theory across
linguistic boundaries. There are increasing points of contact between
Anglophone theorists and those in Europe and elsewhere. Translations
of writings of Nattiez, Vieru, and others have revealed colleagues
spontaneously traveling along paths related to our own.
[25] Twenty years after the founding of its principal professional society, North American music theory is less insular than it was, but more insular than it need be. The pedagogability of many recent concepts suggests that there are emerging opportunities to communicate our best ideas in languages that can reach audiences beyond our sub-disciplinary and linguistic borders, and that we may not even need to work all that hard to meet those opportunities.
2. Jeff Pressing, "Cognitive Isomorphisms Between Pitch and Rhythm in
World Musics: West Africa, the Balkans and Western Tonality," Studies
in Music (Australia) 17 (1983): 38-61.
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3. John Clough and Jack Douthett, "Maximally Even Sets," Journal of
Music Theory 35 (1991): 93-173.
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4. Jack Douthett and Richard Krantz, "Energy Extreme and Spin
Configurations for the One-Dimensional Antiferromagnetic Ising Model
with Arbitrary-Range Interaction," Journal of Mathematical Physics
37.7 (July 1996): 3334-3353.
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5. Eytan Agmon, "Linear Transformations Between Cyclically Generated
Chords," Musikometrika (1991): 15-40.
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6. Steve Larson, "Scale-Degree Function: A Theory of Expressive
Meaning and its Application to Aural-Skills Pedagogy," Journal of
Music Theory Pedagogy 7 (1993): 69-84.
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7. David E. Cohen, "Aristotelian Physics and the Early Concept of
Harmonic Progression," unpublished paper presented at the joint
meeting of the American Musicological Society and the Society for
Music Theory, October 30, 1997.
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8. David Lewin, Generalized Musical Intervals and Transformations
(New Haven: Yale University Press, 1987), pp. 175-180; idem., "Some
Notes on Analyzing Wagner: The Ring and Parsifal," 19th-Century
Music 16 (1992): 49-58; Brian Hyer, "Reimag(in)ing Riemann," Journal
of Music Theory 39 (1995): 101-138; Richard Cohn, "Neo-Riemannian
Operations, Parsimonious Trichords, and their Tonnetz
Representations," Journal of Music Theory 41 (1997): 1-66.
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9. Thomas Christensen, Rameau and Musical Thought in the
Enlightenment (Cambridge: Cambridge University Press, 1993).
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10. Janna Saslaw, "Forces, Containers, and Paths: The Role of
Body-Derived Image Schemas in the Conceptualization of Music,"
Journal of Music Theory 40 (1996): 217-243; Lawrence Zbikowski,
"Conceptual Models and Cross-Domain Mapping: New Perspectives on
Theories of Music and Hierarchy," Journal of Music Theory 41, no. 2
(1997; in press).
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11. Marion Guck, "Two Types of Metaphoric Transference," in
Metaphor--A Musical Dimension, ed. Jamie C. Kassler (Sydney,
Australia: The Currency Press), 1-12; Judy Lochhead, "The Metaphor of
Musical Motion: Is There an Alternative?" Theory & Practice 14/15
(1989/1990), 83-104; Nicholas Cook, "Music Theory and 'Good
Comparison': A Viennese Perspective," Journal of Music Theory 33
(1989): 117-141. Zbikowski's recent "Metaphor and Music Theory:
Reflections from Cognitive Science" (Music Theory Online 4.1, 1998)
and the electronic dialog that it stimulated on mto-talk are but the
most recent manifestations.
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12. Adele Katz, Challenge to Musical Tradition (New York: Alfred A.
Knopf, 1945); Felix Salzer, Structural Hearing (New York: Charles
Boni, 1952); Benjamin Boretz, "Meta-Variations: Studies in the
Foundations of Musical Thought," Ph.D. diss, Princeton University,
1970; John Clough, "Diatonic Interval Sets and Transformational
Structures," Perspectives of New Music 18 (1979-80): 461-482.
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13. Jay Rahn, A Theory for All Music (Toronto: University of Toronto
Press, 1982); Kofi Agawu, African Rhythm: a Northern Ewe
Perspective (Cambridge: Cambridge University Press, 1995).
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14. Fred Lerdahl and Ray Jackendoff, A Generative Theory of Tonal
Music (Cambridge: MIT Press, 1983).
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15. Lewin, Generalized Musical Intervals, op. cit.; idem, Musical
Form and Transformation: Four Analytical Essays (New Haven: Yale
University Press, 1994); Henry Klumpenhouwer, "An Instance of
Parapraxis in the Gavotte of Schoenberg's Opus 25," Journal of Music
Theory 38 (1994): 217-248.
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16. Deborah Stein and Robert Spillman, Poetry into Song: Performance
and Analysis of Lieder (New York: Oxford University Press, 1996);
Joel Lester, The Rhythms of Tonal Music (Carbondale: Southern
Illinois University Press, 1986); William Rothstein, Phrase Rhythm in
Tonal Music (New York: Schirmer, 1990).
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17. Martha M. Hyde, "Twentieth-Century Analysis during the Past
Decade: Achievements and New Directions," Music Theory Spectrum 11
(1989): 38-39.
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18. Paul Berliner, Thinking in Jazz (Chicago: University of Chicago
Press, 1994); Simha Arom, African Polyphony and Polyrhythm: Musical
Structure and Methodology, trans. Martin Thom, Barbara Tuckett, and
Raymond Boyd (Cambridge: Cambridge University Press; Paris: Editions
de la Maison des Sciences de l'Homme, 1991); Steven Feld, Sound and
Sentiment: Birds, Weeping, Poetics and Song in Kaluli Expression
(Philadelphia: University of Pennsylvania Press, 1982).
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19. The initial English-language report of this discovery is Lee
Yuan-Yuan, "An Amazing Discovery in Chinese Music," Chinese Music
2.2 (1979): 16-17. A more comprehensive report by the same author is
"The Music of the Zenghou Zhong," Chinese Music 3.1 (1980): 3-15.
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20. See Lothar von Falkenhausen, Suspended Music: Chime-Bells in the
Culture of Bronze-Age China (Berkeley and Los Angeles: University of
California Press, 1993) and Cheng-Yih Chen, ed., Two-Tone Set-Bells
of Marquis Yi (Singapore: World Scientific, 1994).
Falkenhausen's book has been reviewed by a music theorist, Lewis
Rowell, although not in a music journal. (See Journal of Asian
History 29.1 (1995): 91).
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21. Jean-Jacques Nattiez, Music and Discourse, trans. Carolyn Abbate
(Princeton: Princeton University Press, 1990); Anatol Vieru, The Book
of Modes (Bucharest: Editura Musicala, 1993).
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22. Loris Azzaroni, Canone Infinito: Lineamenti di teoria della
musica (Bologna: Cooperativa Libreria Universitaria Editrice Bologna,
1997).
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