The first movement of Mozart's Piano Sonata in A Minor, K. 310, is imbued with profound pathos. Some have interpreted this movement as expressing Mozart's grief over the death of his mother, but offer little supporting evidence. Although expressive topics can explain the expression of grief in this movement in general, the work of Lakoff and Johnson in cognitive linguistics suggests a more nuanced reading. According to Lakoff and Johnson, we shape and understand reality by means of embodied image schemas, developed and extended as metaphorical projections. These are not metaphors in a literary sense, but cross-domain conceptual mappings: we use the same words to describe and understand very different experiences because we structure those experiences with the same image schemas. From the primary metaphor change is motion one obtains two closely related complex metaphors: a phrase is a motion and grieving is a motion. Thus, this movement can be heard as an expression of Kübler-Ross's first stage of grief, denial, in the way cadential closure is denied throughout this movement within textural, registral, rhythmic, and voice-leading domains. (Significantly, denial is a plausible explanation for the tone of Mozart's letter to his father on the night of his mother's death.) Just as the denial stage prevents the griever from achieving the desired state of closure or acceptance, cadential denial prevents the desired state of tonal closure.

This paper presents and discusses results of a quantitative study of the relationship between rhythmic characteristics of spoken German and French and the rhythm of musical melody in nineteenth-century German and French art song. A recent series of articles in Music Perception demonstrated a general correlation between the rhythm of spoken language and the rhythm of short incipits of instrumental music by composers of varying nationalities. This study expands dramatically upon those results, studying melodic rhythm in over six hundred French and German art songs by eighteen composers.

      Linguists study rhythmic characteristics of spoken language by measuring the amount of variability in the length of successive syllables. One such measure is known as the Normalized Pairwise Variability Index, or nPVI (Grabe and Low 2002); that measure is adapted for this study and used to study rhythmic characteristics of musical melody.

      The primary focus of this presentation is an unexpected result of the study: there is a sharply diverging trend in the nPVI of French and German songs as a function of time through the 19th century. It is unlikely that these trends are the result of changes in the spoken language over that time period; therefore, there must have been stylistic changes in French and German song that caused the nPVI to rise or decline. This presentation will focus on this trend over time exhibited by French composers, demonstrating the rhythmic changes that occurred in French song from 1840-1900 and how those changes reflect a concerted effort on the part of French composers to reflect a French musical character.

Using methodologies drawn from music theory and cognitive linguistics, this paper extends the idea of musical invariance in order to bring a broader array of musical phenomena under the umbrella of the cognitive process known as "frame-shifting." Frame-shifting occurs when semantic reanalysis recognizes contradictory information, resulting in a shift to a new frame in which the contradictory information makes sense. Critical to this shift is an invariant element that belongs to both frames, but which is interpreted differently according to the prevailing frame. Several different types of musical invariance that can facilitate frame-shifting are proposed. I then turn to several examples of invariance and frame-shifting (including Verdi, Brahms, and Tchaikovsky) in order to explore how much information is necessary in music in order for frame-shifting to occur; determine at which musical and textual levels frame-shifting occurs; and speculate on the range of aesthetic effects that frame-shifting can produce. Finally, I explore how frame-shifting can be used to model some music-theoretical assertions, thus providing another lens through which to view music theories.

Music theorists Richard Parks, Christopher Hasty, and Simon Trezise, among others, have made contributions to the study of Debussy's metric ambiguities and their immediate effects upon phrasing. In this presentation, I will not only illustrate the immediate effects that metric ambiguity has on intra- and inter-phrase relationships in this prelude, but also the manner in which the metrically supple facets of its opening phrase-group form a process that permeates and informs the formal structure of the entire work.

      Significantly, I aim to demonstrate a necessary allowance for flexibility in the perception of ongoing changes in metric orientation when studying the metric structure of Debussy's music. Specifically, I argue that brief spans of metric orientation are heard as such (that is, as "metric") rather than solely as grouping structures against a single referential meter. The primary support for this argument is that Debussy's progressive metric canvas is often so free that it greatly challenges-and in many cases even denies any strong sense of-a single, referential meter in a work. This is particularly true in the Préludes, whose relative brevity-I believe-inspires Debussy to work in a microcosmic manner whereby changing metric perceptions are compressed into a smaller overall time span and are thus more frequent than in most of his lengthier compositions. I refer to this process of changing metric orientations as "metric fluctuation," a term adopted from Wallace Berry's groundbreaking 1985 Music Theory Spectrum article "Metric and Rhythmic Articulation in Music."

An established music-theoretical vocabulary might succinctly describe the contrapuntal phrase that repeats through most of "Vaka," the opening track from the Icelandic band Sigur Rós's parenthetically titled 2002 CD, ( ). Yet an analytic focus on the minimal, looping pitch material implies we might listen elsewhere if we wish to unpack the compelling processes woven together-and ultimately unraveled-by the track's electronic manipulation of resonance. Situating "Vaka" in an alternative history of twentieth-century music, this paper sets forth a comparative analytical approach that examines the development of new performance and recording technologies alongside their metaphoric emergence in actual musical material.

      Drawing on the work of Rebecca Leydon, analysis of Debussy's "La Cathédrale engloutie" suggests that, mirroring the process in which inventors reflect on the present-day world to come up with ideas and inventions that will shape tomorrow, Debussy's compositional metaphors were reaching toward the future, suggesting not (just) cinematographic editing procedures of his own day, but audio-recording and performance technologies he would never live to see. Listening to the famous Welte piano-roll recording of the prelude, we witness a pianist/composer who de-emphasizes the tonal goals of a contrapuntal trajectory in an effort to foreground resonance in a pre-technological musical enactment of short-circuiting. When we allow the cathedral's bells to resonate through a second listen to "Vaka," the opening counterpoint's teleological motion can be heard to enact metaphorically the closing of an ultimately shorted musical circuit; in a word, re-thinking old music (theories) in light of the new.

In recent music discourse, scholars have struggled to explain the sudden harmonic shifts, remote tonal regions, and discontinuity of musical gestures in Schubert's music. In an effort to rationalize these idiosyncrasies by relating them to a unified whole, some scholars have retooled pre-existing analytical systems by extending concepts of diatony; others have sought to devise new systems altogether, or have turned to hermeneutic models. What seems to fuel this drive toward integrating disparate musical events is an aesthetic of unity. This paper asks what other options might be available to us and how pursuing an alternative to an aesthetic of unity can affect our understanding of Schubert's music.

      Using the Moment Musical, Op. 94, No. 2, as a case in point, my paper will suggest that certain pieces can be thought of in terms of irony, fostering a double-consciousness by engaging in a dialogic relationship with themselves and with formal and harmonic structures drawn from the past. The paper will (1) provide an alternative to perceiving Schubert's music as a monologic, unified consciousness, whereby idiosyncratic musical events are explained as contributing to a greater, continuous whole; (2) show how Schubert's use of tonality and form can coexist with notions of conventional diatony and form, and need not be understood either as a derivative of these customary procedures or as independent from them.

Schenker's lack of specific attention to most aspects of rhythm is commonly recognized by those familiar with Schenker's theoretical ideas. But temporality in music extends well beyond the boundaries of rhythm and meter with which we usually circumscribe musical time. Rather, the primary cause of musical motion-the impetus for most linear extension, and thus for the temporal dimension in music-is tension, which has a number of different sources.

      In this paper I will propose that Schenker's basic models explicitly describe some kinds of musical motion. By itself, this proposal is hardly original or remarkable, as the linearization of the triad that is the basis of musical extension is, by definition, driven by the passing tones that connect the triadic consonances. But there are also very far-reaching temporal indications implicit in some of Schenker's ideas. In this brief study I will focus on three of these, emphasizing the relevance for analysis and performance of using Schenker's ideas as a fulcrum for discovery of hidden tensions.

      As an example, we may consider that the elaborations-on any level-of the basic counterpoint constitute not only the prolongation of a more structural pitch but also, simultaneously, a divergence from a voice's primary path. To the extent that that path is recognized, the diversion can be conveyed, and experienced, as tensional. Similarly, we can note that to the extent norms of musical progression are embodied in Schenker's models, divergence from them reveals a tension that materially influences a work's shaping of musical time.

Schoenberg's Sechs kleine Klavierstücke, Op. 19, are a familiar part of the literature; indeed, perhaps too familiar: theoretical writing has largely either restricted itself to using decontextualized excerpts to illustrate general methodologies or presupposed and set out to demonstrate the presence of unity or hierarchy in the music. In either case, significant textural, gestural, formal, rhetorical, and historical aspects of the music have gone unheeded. Drawing on contour theory, as well as on writings of Schoenberg, I aim in this paper to defamiliarize Op. 19 by demonstrating the realization of an ideal contour at the end of No. 4, its visionary appearance in No. 1, cumulative steps taken in Nos. 2 and 3 toward its realization, and the aftermath in Nos. 5 and 6. I will argue that the spiral form of this ideal contour indicates an infinite self-similarity resulting from an instance of self-reference: the work is the realization of a vision of the realization of that same vision. The music reflects the act of its own composition ever more closely until it reaches the immediacy of a dimensionless point, at which moment the work perfectly reflects the composer. In this way, Op. 19, written at a time of crisis in Schoenberg's compositional career after a year of almost total silence, fulfills the impossible demand for artistic integrity in his Harmonielehre, completed the same month as Nos. 1-5.

Arnold Schoenberg stated in "Brahms the Progressive" that the final work of a "Great" composer should reach for "the uttermost limit of the still expressible." Schoenberg chose to write three religious choral works, "Dreimal Tausend Jahre," Op. 50a, "De Profundis," Op. 50b and "Modern Psalm," Op. 50c, as his final opus. Schoenberg's use of consonant sonorities and areas of pitch emphasis in these works provoked severe criticism from Boulez and other avant-garde composers. On the 55th anniversary of Schoenberg's death, it seems appropriate to take a fresh look at these works.

      In each of the works of Op. 50, Schoenberg takes a progressively more systematic approach to pitch emphasis that is directly related to an increasing use of symmetrical pitch structures. The first section of this paper will trace this development to its culmination in the "Modern Psalm," where an analytical model based on symmetrical pitch axes will be proposed.

      "Modern Psalm," like Schoenberg's other large religious works Die Jakobsleiter and Moses und Aron, was left incomplete. The second section of this paper will demonstrate the relationship between areas of pitch emphasis and references in the text to the use of prayer to reach toward the divine and suggest how Schoenberg's ideas about the impossibility of representing God are expressed in the structure of the music.

The relationships that exist in musical compositions that are generated by complex serial manipulations, such as Boulez's multiplication operations, have often been equated to the random associations that result from chance operations. In this paper I will demonstrate that this conclusion has been reached because an imperfect understanding of the compositional process and resultant structures has obscured the specific correlations between them and the musical surface of these works.

      Heinemann's formula for complex multiplication, though arithmetically correct, provides an unnecessarily complex and abstract explanation of pitch-class generation in these pieces. This paper provides a clarification of Boulez's multiplication technique in a way that is considerably less abstract and more intuitive (from a musical standpoint) than the current theoretical apparatus. It describes the structural properties of Boulez's multiplication tables, demonstrating that the intrinsic transformational, serial and symmetrical relationships translate into meaningful qualities of the musical surface. It demonstrates, using Boulez's sketches, that this theoretical approach reflects Boulez's compositional process. Furthermore, by presenting concrete examples which apply this theoretical tool to analyses of several different works, this paper demonstrates its significance, practicality and potential for yielding new and fascinating insights into this music.

Conventional wisdom holds that the tonal-dramatic musical language of nineteenth-century Europe-particularly Wagner's-is alive and well in the modern-day film score. Most often cited in support of this claim are the chromatic harmony and Leitmotive (associative themes) shared between the two genres. The tonal-dramatic similarities and differences between nineteenth-century dramatic music and twentieth-century film music thus provide a natural avenue into the topic of associative harmonic progression.

      Associativity itself remains under-explained in the literature on musical meaning, and this paper begins by exploring its simple definition as recurring music's tendency to recall for the listener emotions present in the drama during its earlier presentations-an effect that relies on both generic-topical and piece-specific phenomena. As a point of departure, this study begins with Wagner's notorious "Tarnhelm" music from Das Rheingold. It then proceeds to later dramatic works that include salient presentations of the opening "Tarnhelm" harmonic progression, demonstrating the progression's remarkable commonality of sinister, eerie, and eldritch associations in contexts ranging from nineteenth-century art music to scores from science-fiction, fantasy, and horror movies of the post-1975 orchestral film music renaissance.

      Methodologically, this study adopts a plurality of analytic approaches to explain the role of the "Tarnhelm" progression in various dramatic musics. These address the progression's Stufen-centric, harmonic functions; its voice-leading transformations; and its various roles in establishing formal coherence. In turn, this multi-faceted approach reveals the functional/semiotic ambiguities and multiple connections that obtain between the "Tarnhelm" progression and the textual, programmatic, visual, narrative, and dramatic contexts in which it appears.

In his 1992 essay "Some Notes on Analyzing Wagner," David Lewin examines connections between two motives in Wagner's Der Ring des Nibelungen, the Valhalla and Tarnhelm, through the lens of neo-Riemannian operations. His analysis reveals striking connections between the two musical ideas that have important dramatic implications, and provokes the idea that further analysis of this kind could penetrate deeper into the musical-dramatic fabric of Wagner's complex technique of motivic transformation. Lewin's methodology can be extended by considering not only other occurrences of the "distorted" Valhalla motive and other motives in the Ring cycle (such as the Tarnhelm and Magic Potion motives), but by broadening the analytic lens to include new neo-Riemannian relations involving non-consonant chords and considerations of the motives' Schenkerian contexts. I specifically propose a pluralistic analytical approach that takes Lewin's insights a step further and traces the process of motivic manipulation not only from a neo-Riemannian "transformation of transformations" perspective, but also from a Schenkerian reading of the motives' changing diatonic contexts. The transformational graphs used throughout the paper integrate the findings from the two different analytic approaches and contextualize appearances of the motives in "diatonic zones" and "chromatic zones." Particular emphasis is given to Siegfried and Brünnhilde's scene in Act I, Scene 3 of Götterdämmerung and its employment of rotational form.

Brahms's first String Sextet begins with an antecedent phrase whose rhythm invites multiple perspectives. The passage is mehrdeutig in that there are multiple ways to prioritize its structural constituents. Moreover, these initial conflicting priorities give rise to multiple solutions to grouping problems (both medium- and large-scale) that arise later in the piece.

      This paper explores three different re-compositions of the opening phrase, seeking to identify a "well-behaved" prototype of which it may be an expansion. Each of these attempts highlights a different feature of the phrase. The first is guided by a Schenkerian paradigm of rhythm and considers repetition largely superfluous. The second is guided by the phrase's sentence shape and considers the repetition of a head motive critical to the meaning of the phrase. The third is a dynamic, forward-oriented reading that seeks to consider the enigmatic tenth measure of the phrase as integral, not expendable.

      Two problems that arise as the piece continues are the structure of the first theme group in both the exposition and the recapitulation, and the nature of the recapitulation, whose clear thematic return is harmonically undermined. The three re-compositions of the opening phrase affect (and are retrospectively affected by) how we hear these ambiguous moments in the remainder of the movement. The richness of the piece is ultimately seen to lie not in a single "correct" answer to these points of analytic difficulty but in the openness to interpretation and dialogue among various moments of Mehrdeutigkeit that are the work's most striking features.

The research of Richard Cohn in the area of metrical consonance has placed meter on a more equal ontological footing with pitch, pitch class, duration, and many other kinds of entities that music theorists have traditionally compared with set-theoretic relations, and/or acted upon with group-theoretic transformations. One advantage of this development allows meters to be positioned in a hypothetical space, which further allows a listener to envision a path that a succession of meters creates through this space. Although such spaces and paths are largely metaphorical, they nonetheless might reveal connections among the metrical contents of a musical work that were not observable before. Cohn has put forth two such metric spaces in his research: one which is inclusive of all meters but is relatively simple in structure, and one whose structure is more complex but which admits only a certain class of meter. In an endeavor to find a compromise between these two spaces, this study introduces another metric space-the metric cube-and defines a metric transformation that acts on the contents of a metric cube. These concepts are explored through analyses of some music of Brahms: the first movement of the Third Symphony, the third movement of the Second Symphony, and a particular performance of the two concluding movements of the Second String Quartet.

Some recent developments in metric theory seem to suggest a potential for transformational theory of meter. Not only a labeling system for both grouping and displacement dissonance was provided (Harald Krebs, 1999), but also spaces of pulses were defined (Richard Cohn, 2001, and Justin London, 2002). However, it is not clear how "multiplicative" grouping can be formally combined with "additive" displacement in a single transformational framework. In this paper, I propose a unified transformational framework which represents both grouping dissonance and displacement dissonance as well as other metric relations.

      Generalizing traditional meter signatures, I define a space of semimetric layers. A semimetric layer is represented by a size of displacement, a contextual unit pulse, and a "periodicity." To clarify the relation between displacement and unit pulse in a semimetric layer, Lewin's non-commutative GIS of time spans is re-examined. Geometrically, Lewin's transposition has two components: it is a shear followed by a scaling. Likewise, a metric transformation on semimetric layers has the same two components. Metric displacements are modeled by shears behaving just like additive commutative transpositions. Then four kinds of characteristic scalings behaving just like multiplicative commutative transpositions are introduced to model other metric processes which include grouping dissonance, hypermetric dissonance, and metric augmentation/ diminution. A transformation combining a shear and a scaling, however, is a non-commutative transposition in general, leading to a non-commutative GIS.

This year marks the ninetieth year since the birth of Argentine composer, Alberto Ginastera. Several of his compositions, especially those for piano, have become staples of twentieth-century repertoire; however, analytical discussions of his music have been relatively limited. Scholars have tended to focus on more general characteristics, noting the bimodal and polychordal textures typical of his early nationalistic style. In spite of the fact that Ginastera's Piano Sonata No. 1 (1952) is in many respects representative of this style, the highly expressive and improvisatory third movement contains tightly controlled pitch structures that resonate with the style of his later Neo-expressionism period.

      The pitch components of this movement consist of motivic cells, which are composed-out (or elaborated) through progressively more abstract transformations that create relationships between formal sections and amongst different structural levels. Furthermore, the pitch-space symmetry that is a principal characteristic of the opening motive is integrated on deeper structural levels, functioning in a manner that adds support to the process of motivic composing-out. Through a detailed analysis of the third movement of this sonata, I will discuss how Ginastera uses two motives to generate the pitch material of each subsequent formal section. This paper will explore some of the formal implications of the composing-out process and discuss how both literal and more abstract motivic recurrences provide a sense of continuity amongst otherwise contrasting materials.

Despite the rising interest in transformational theory in the past two decades, few scholars have considered its relation to form (the exceptions are Lewin 1993 and Cohn 1999). Specifically, they have not studied correlations between the formal sections and the transformations that take place within them. In this paper, I discuss how changes in transformations in the first movement of George Rochberg's sixth string quartet articulate specific formal functions, in the sense of Caplin's theory of tonal form (1998). Although this movement is called "Fantasia" and lacks triadic tonality, it can be understood as being in sonata-allegro form, partly because of the nature of the transformational processes that characterize its sections. The contrast between the types of transformations (transposition versus inversion) in the first two sections is analogous to the contrast of first and second themes that is characteristic of classical sonata form. The third section functions as a development section, blending both transpositional and inversional processes of the exposition. The last two sections act as a recapitulation, in which the second-theme group is transposed. Rochberg is often criticized for imitating traditional musical structures, but this analysis demonstrates how he successfully reinvents sonata form with non-tonal transformations.

The research of Richard Cohn in the area of metrical consonance has placed meter on a more equal ontological footing with pitch, pitch class, duration, and many other kinds of entities that music theorists have traditionally compared with set-theoretic relations, and/or acted upon with group-theoretic transformations. One advantage of this development allows meters to be positioned in a hypothetical space, which further allows a listener to envision a path that a succession of meters creates through this space. Although such spaces and paths are largely metaphorical, they nonetheless might reveal connections among the metrical contents of a musical work that were not observable before. Cohn has put forth two such metric spaces in his research: one which is inclusive of all meters but is relatively simple in structure, and one whose structure is more complex but which admits only a certain class of meter. In an endeavor to find a compromise between these two spaces, this study introduces another metric space-the metric cube-and defines a metric transformation that acts on the contents of a metric cube. These concepts are explored through analyses of some music of Brahms: the first movement of the Third Symphony, the third movement of the Second Symphony, and a particular performance of the two concluding movements of the Second String Quartet.

Starting approximately with El salón México (1936), the perfect fifth and its inversion became ubiquitous features of much of Aaron Copland's music. The composer's melodic and harmonic attention to this interval class, coupled with his penchant for pitch centricity, did much to define the style most often identified with him and, eventually, with an entire "school" of composition often described as "American." On the other hand, little has been done to explore the innovative role Copland frequently grants this interval in the large-scale tonal organization of his music from this period.

      Specifically, this paper shows that interval class 5 is a crucial element in understanding not only individual melodies and harmonies, but also the large-scale tonal structure of many Copland works of the 1940s. To demonstrate the impact of perfect fifths on tonal structure, this discussion focuses on two representative works: "Nature, the gentlest mother" from Twelve Poems of Emily Dickinson (1949) and the Sonata for Violin and Piano (1943). In each composition, two potential pitch centers separated by interval class 5 are simultaneously stressed using various salience factors, creating an ambiguity that is "worked out" through the music and its organization around specific pitch centers.

Béla Bartók's music has been investigated with various descriptive methods that focus upon pitch and formal structure. Proportional cataloging techniques, such as golden mean analysis, may yield results that are aesthetically beautiful, but these data do not explain how specific organizations generate musical effects. In addition, Bartok's rhythmic ingenuity has not been studied in as much depth as other aspects of his music. Movement two of Bartók's Music for Strings, Percussion and Celesta (1936) uses evolutionary transformations of rhythm to heighten tension and explore the spaces within a metric clash in the first part of the development section (mm. 186-241). Two conflicting metric identities radically diverge and then gradually realign via a series of durational motive transformations. Local-level accent correlations operate in tandem with a large-scale plan of metric tension prolongation. Rhythmic mutations produce continuous, goal-directed variation during the journey through this polymetric grey area. Metric shifts are often resolved with an immediate metric modulation, but this passage extends the discrepancy over approximately fifty measures. Dynamic transformations look forward to the eventual goal of alignment, yet at any point in time, a single shift might stumble and regress back to the original metric area. This process is akin to evolution. A musical situation pressures the motives to adapt, yet they cannot do so instantaneously. The result is a rich interplay of rhythm and a heightening of drama as the metric dissonance unfolds.

Georges Aperghis's Les guetteurs de sons is a trio for percussion that creates an intriguing world of sensory experience by playing with a perceptual schism between the auditory and visual realms: though the performers appear to strike their drums, these drums often stubbornly refuse to produce sound. These silent or empty gestures become compositional elements in their own right and challenge our notions of expectation and perception, while also bringing a theatrical layer to the work. At the same time, by emphasizing mute physical gestures, the role of the body becomes a crucial element in mediating our understanding of the piece.

      Yet, how can analysis approach such a seemingly amorphous composition? Given the complex strata of visual images, my examination of this work begins by taking video footage of a performance as a primary text rather than focusing exclusively on the score. Further, I will adapt David Lewin's p-model as an alternative to more familiar analytical approaches to multimedia and film, which can highlight our perceptions in time while also modeling conflicts between the visual and auditory realms.

      While the focus of this paper is analyzing Les guetteurs de sons in order to highlight its unique structuring of gesture, sound, and physical presence, I ultimately hope to show how Aperghis has used musical elements to create a piece that crosses the lines separating music, theater, and performance art.

Sarah Fuller's work on harmony in the music of Guillaume de Machaut has established a useful set of terms suitable for discourse about sonority in 14th-century music. Her analytical approach-based on the premise that perfect consonances are stable and imperfect consonances unstable-provides a means to classify sonorities according to stability and instability. Fuller proposes three categories: perfect, which contain two perfect intervals and lie at the most stable end of the spectrum; doubly imperfect, which contain two imperfect intervals and are at the most unstable terminus; and, imperfect, which contain one perfect interval, and lie somewhere in between the other two.

      I propose refinements to this classification scheme that continue to be rooted in 14th-century thought, yet also respond to a 21st-century aural perspective. By looking at all sonorities sustained at least the duration of a breve in Machaut's 19 three-voice motets, I describe how each of Fuller's three categories can be further divided into subcategories. For example, I perceive a definite aural distinction between a sonority whose imperfect interval is located between the lowest and middle voices and a sonority whose imperfect interval is located between the outer voices. Within the perfect and imperfect categories, sonorities with differing intervallic constitutions, depending on context, function in decidedly different ways. My proposed nomenclature makes possible a more specific rendering of the voice leading, and allows for a richer rehearing experience and a greater understanding of sonority in Machaut's motets.

Thomas Morley has enjoyed a reputation for being the most eminent theorist of the Elizabethan era, as well as one of its most celebrated composers. However, current scholarship regards him as having been confused about modes and psalm tones. This belief has been used to promote a picture of Renaissance England as a place where the modes where not generally understood or practiced. There has been understandable difficulty in interpreting one particular line written in Morley's Elizabethan language, but once this line is interpreted correctly, his entire discussion of mode and psalm tones is insightful. Morley even gives a specific technical reason as to why psalm tones are not modes. His understanding of modal concepts supports the work of scholars who have found English Renaissance polyphony to be generally classifiable by mode. We know that Thomas Tallis knew the modes, because he masterfully demonstrated all of the traditional eight modes in order in his Psalms for Archbishop's Psalter; when his abbey was dissolved by Henry VIII, he also rescued a group of treatises including a detailed 15th-century plainchant mode treatise by John Wylde.

      Since both Tallis and Morley knew modal theory, it only stands to reason that William Byrd did too. Although he is largely progressive, his composition Deus in adiutorium demonstrates mastery of the same Phrygian polyphonic language used by Palestrina and Lassus. At the same time, the English Renaissance composers did have a slightly broader use of F[sharp] to avoid melodic and vertical dissonances than was typical of continental modal polyphony. This allowed the use of authentic cadence forms where Phrygian cadence forms would have been otherwise needed. Byrd's music is rooted in modes, with preference towards Aeolian and Ionian, but he also expanded even past the broader English practice to form early uses of modulation.

The ideas of harmonic dualists and the analysis of nineteenth-century music have been the focus of much recent transformational theory. The tools of transformational theory, however, can also be fruitfully applied to the theory and music of Jean-Philippe Rameau. Following the work of David Lewin, this paper formally defines a generalized interval system (GIS) based upon Rameau's mature theory as expressed in his Géneration Harmonique (1737) and his Démonstration du principe de l'harmonie (1750). The Rameau GIS is made possible by Rameau's insight that the fundamental bass moves via the same intervals as those of the triad, justly tuned thirds and fifths. Therefore, the space defined by the Rameau GIS is similar to that of the Tonnetz widely used by neo-Riemannian theorists.

      The Rameau GIS provides us with a powerful lens, through which we can view the harmonic progressions that Rameau discusses, especially those involving chromaticism or enharmonicism. Particularly interesting in this regard are the voice-leading cycles that constitute what Rameau calls the diatonic-enharmonic and chromatic-enharmonic genera, as they relate to harmonic progressions considered by neo-Riemannian theory. In addition to modeling Rameau's harmonic theory, the Rameau GIS also suggests many possibilities for analysis of particularly dramatic chromatic and enharmonic moments that occur in Rameau's operas, including passages from Hippolyte et Aricie, Les Indes Galantes, and Pigmalion. In this music, we see how both local harmonic progressions and longer spans of music move through the space of the Rameau GIS.

Many theorists, finding in Neo-Riemannian and transformational theory a welcome adjunct to Roman numeral and linear graphic analysis, have sought ways to incorporate them into an undergraduate curriculum. Yet the abstract, mathematical nature of Neo-Riemannian theory can make its concepts difficult to grasp, and students may find disorienting the notion of a toroidal space in which it is possible to move in any direction, and in which no tone or triad has priority over any other. This paper proposes overcoming these difficulties by introducing students to Neo-Riemannian concepts by way of a geometric model of triadic pitch space that is isomorphic to the Neo-Riemannian Tonnetz, but differs from it in ways that allow the student to grasp these concepts more easily in relation to familiar concepts of triadic tonality.

      Students construct their own model of triadic space according to the mathematical principles of tiling. They are given a blank grid upon which they place triadic tiles, one by one, beginning with C major. The addition of each new triad increases the possibilities for harmonic motion, first within diatonic space, then chromatic space, as each new triad allows for motion further away from the central key of C major. This paper will show how the early stages of construction can be used to deepen the students' understanding of tonal concepts already familiar to them, including that of modulation, while preparing them to grasp at a later stage the less familiar triadic transformations associated with late nineteenth-century chromaticism.

Proponents of harmonic dualism had been put to the pale, in the Anglo-American theory community at least, until a recent revival of interest spurred by the work of David Lewin and Daniel Harrison, and by the more-or-less tacit importation of dualist elements under neo-Riemannian cover. I intend here to recast dualism within diatonic scale theory. Although my point of embarkation is different, I land on shores explored recently by Harrison, and previously by Hugo Riemann and his many precursors: Weitzmann, Hauptmann, Rameau, and many others at least claimed as such by Riemann.

      Although contrapuntal aspects of tonality are at least as important as the harmonic aspects, and while analysis in terms of counterpoint, including larger-scale contrapuntal connections, is far more fruitful than harmonic analysis, whether of Riemannian or any other flavor, Schenkerian accounts of tonal theory make the complementary error of a flawed and underdeveloped treatment of harmony. My critique of Schenker will have to be implicit in this short presentation, as will my critique of recent function theories that do without primary triads, roots, or dualism. I will present an overview of a theory that proposes broken symmetries at all levels, built from binary oppositions with marked and unmarked terms, beginning from the binary opposition of the perfect fifth and diminished fifth. The asymmetry of pitch space is acknowledged, and in the major-minor dualism that results, minor is the marked term.