An old puzzle asks, "What is the smallest party one can have, such that necessarily three of those present are either mutual acquaintances, or complete strangers?" The puzzle is an instance of a class of problems for which a theorem of Frank Ramsey (1904-1930) guarantees a solution. It is possible to reformulate this and related puzzles to exhibit a class of pitch-class sets that share a graph-theoretical property, a property which is based upon the character of the set of trichordal subsets of a given set. A refinement on this property isolates five set classes: an altered diatonic set (7-29), the usual diatonic (7-35), the octatonic (8-28), the diatonic plus one fifth (8-23), and Messiaen's fourth mode of limited transposition (8-9). A further refinement on the property uniquely isolates the diatonic and octatonic set classes.
This paper attempts to understand what is being captured by this property and its refinements, by examining what these seemingly disparate set classes have in common. The three eight-note minimal pure all-small set classes mentioned above have seven-note subsets that in turn hold hexachords invariant under some Tn or In (the Q-relation). This property of the two types of seven-note subsets of set class 8-9 leads to some insights into the structure of the fourth movement of Webern's op. 5.

Throughout much of modern history--certainly since Tinctoris in the fifteenth century--two central concerns of music theorists have been "vertical" pitch structures (chords or simultaneities) and the "horizontal" connections (voice leading) among them. In the study of twentieth-century music the former has received extensive coverage, while the latter remains substantially less explored. Expanding on recent work by Lewin, Klumpenhouwer, Forte, and Straus, I use networks, traditional set-theoretical operations, and an original transformation called split transposition, to generate mappings which I interpret as voice-leading lines. I present this composite theoretical model in the context of two analyses of songs by Charles Ives: "Serenity" and "'The Cage." Multiple mappings in these analyses coexist in "levels" called "Adjacencies" and "Recursive structure." Despite a sense of varying distance from the musical surface, these are not hierarchical levels, but instead compare and contrast alternative interpretations of the voice leading in each song.

Webern's works for string quartet are among the most studied of the atonal literature. These studies have been mostly concerned with pitch structures. Some of Webern's movements, such as the fifth Bagatelle of Opus 9, have yielded their harmonic secrets willingly; others, such as the second of the Opus 5 Movements have proven more resistant. In this paper, these two pieces will be examined using two recent analytical tools, Robert Morris's contour reduction algorithm and minimal interval content descriptions. The two works have been chosen because their homophonic textures allow the application of the contour reduction algorithm to the entire melodic line.

In Op. 9, No. 5, we find that the reduced contour of the entire melody of the piece is replicated at smaller levels by the contours of individual phrases and combinations of phrases. The intervallic relations between structurally deeper notes (as determined by the contour reduction algorithm) are themselves replicated in more surface-level relations. In Op. 5, No. 2, the work's three sections present a contour palindrome, and the final section itself reduces to the same contour as the entire melody. Here, again, intervallic relations at deep levels of structure provide the basis for a description of the surface of the melody.

When examining a composition containing many discrete melodies or contour segments, one can assert various types of relations between the melodies or segments. For example, a prominent melodic shape in one section of a piece could be said to be the inverse, the retrograde, or a registrar expansion of a contour segment from another section. But in a musical setting where all contour segments are brief and unidirectional, only a limited number of associations based on salient points of contour can be drawn. Because of this dilemma, an examination from a different perspective is needed. I propose an intermediate level of structure where the "pitched event" of a contour segment is not the single note of a melodic construct, nor is it a section of an entire piece; its scope lies between the two. I call this type of structure textural contour to differentiate it from melodic contour. This paper details this type of analytic approach in a specific context, the first of Webern's Op. 11 pieces for piano and cello. In this piece, most melodies are brief; they are significant not because of their pitch- or set-class membership, but because of their projection of long-range contour relationships. When discrete simultaneities and the motions that connect them are examined, though, the textural contour they delineate reveals an intricate textural framework that controls much of the spatial orientation of musical elements.

The serialization of rhythm is one of the most significant musical developments of the post-war period. Integral serialism represents an attempt to organize rhythm in a systematic fashion comparable to the pre-war serialization of pitch by Schoenberg, Webern and Berg. By partitioning musical time into a recurrent module of twelve equidistant timepoints paralleling the semitonal division of the octave, Milton Babbitt, the vanguard of this movement, was able to structure rhythm and pitch isomorphically using a single series.

This paper utilizes concepts from set theory to create a new model for analyzing serial rhythm. The model's application is illustrated using Babbitt's String Quartets Nos. 3 and 4. The minimum number of modules required for the unfolding or aggregation of a timepoint series represents its minimum aggregate partition or MAP, comparable to the "normal form" of a pitch class set. Comparing this abstraction with the actual number of modules over which the timepoint aggregate unfolds in a specific musical context yields a comparative index of its aggregational efficiency. Variations in minimum partitions and aggregational efficiency model our aural apprehension of integral serial music. Analytically graphing or "mapping" timepoint aggregates in Babbitt's quartets reveals significant perceptual variations in modular structure, expansion and efficiency that corroborate our musical intuitions. MAP theory provides a valuable supplement to the analyst's toolbox in understanding one of the most comprehensive attempts to organize rhythm in the history of music.

Most of the Mississippi Delta Blues recorded in the late 1920s and 1930s is based on the well-known 12-bar blues archetype, with each 4- measure phrase clearly articulated by harmonic, textural, and melodic change. In the hands of the first generation of Delta Bluesmen, however, this paradigm was not a strict pattern, but a more fluid formal plan. This presentation will examine the formal and metric structure of Robert Johnson's "Ramblin' on My Mind." "Ramblin'" (for solo guitar and vocalist, both parts performed by Johnson in one "live" take) displays normative 12-bar structure in some verses, expansions of individual measures in other verses, interpolations of extra foreground measures in one verse, and in another verse a break in the metric flow entirely. Thus it exhibits a wide range of metric and hypermetric phenomena. Indeed, irregularities which might destroy a sense of hypermeter in other contexts can, in the case of "Ramblin'" be linked to the 12-bar norm, and hence preserve a sense of metrical order. In examining Johnson's composition another example (by Charlie Patton, one of Johnson's contemporaries) will also be considered.

Metric and hypermetric structure in a well-known (and much-analyzed) Mozart composition are examined. A bottom-up, relatively flat metric structure is proposed to explain how the listener experiences this passage. This flat model stands in contrast to commonly suggested "hypermetric" explanations. The advantage of a bottom-up, foreground rhythm-generated model of meter is that it helps explain conflicting hypermetric structures that arise in and around measure 22. Rather than a point of accentual strength, the paper argues that measure 22 is a point of accentual ambiguity and weakness; as such, it plays a key role in the formal structure of the exposition at that point. The paper discusses theories of metric induction from the psychological literature and music-theoretical models of hypermeter in this passage and proposes an approach to rhythmic and metric analysis that reflects the experience of both knowledgeable and naive listeners.

Motion in music has been addressed in many theoretical writings; however, the sources of musical momentum have yet to be thoroughly examined. This paper draws upon the work of Harald Krebs and expands the modern definition of "metric dissonance" to include irregular phrase length as a metric conflict. Two new classifications of metric dissonance are introduced: one that results from vertical nonalignment (such as triplets and syncopation) and one that results from horizontal irregularity (phrase length alteration). Following a clarification of these terms, the paper addresses how the context of rhythmic conflicts within a passage creates a compelling force that propels motion forward and, consequently, increases the strength of the resolution to consonance and regularity. An analysis of Brahms's "Intermezzo" in Ab major, op. 76, no. 3, with its five-bar phrasing and multi-leveled rhythmic conflicts, is the focus of the final section of the paper. The analysis clarifies the relationships of the rhythmic conflicts and extracts a regular prototype by reducing and normalizing the rhythmic irregularities.

The Beatles' recorded output (1962-70), augmented by recordings of compositional drafts and outtakes, contains thousands of sources for this study of their many career-spanning techniques for the manipulation of phrase rhythm. The presentation follows methods pioneered by others but suggests a few newly classified techniques that may apply to other tonal repertoires. A large number of examples are categorized in a taxonomy of phrase rhythm techniques, but a few ("Lucy in the Sky With Diamonds," "A Day in the Life," "Because," "The End," "I Should Have Known Better" and "Not a Second Time") will be chosen for closer study.
While the Beatles' phrase rhythms are foursquare often enough to permit the establishment of regular norms against which abnormalities can be measured, a large number of their songs feature contrasting unit lengths, expanded prototypes, reinterpretations of accentual function at the phrase level, tonicization-related stretching and elision, adjustments required by changes in harmonic rhythm, metric modulation or thoroughly asymmetrical patterns. In many cases, the rhythmic technique is found to be closely tied to the phrase's central formal function within the song as a whole. While all three of the composing Beatles experimented with free phrase rhythms, Lennon was adventurous most often--though Harrison most consistently--in this regard. 

The theme of ambiguity is addressed by Leonard Bernstein in The Unanswered Question. One particularly lengthy and interesting musical case study is taken from Berlioz' Roméo et Juliette (the masked ball of the Capulets). The purpose of this study is to examine Bernstein's analysis of this passage in light of his own attempt to grapple with the themes of ambiguity in musical and dramatic terms in his own composition, the Dance at the Gym from West Side Story. In analyzing Berlioz' work, Bernstein finds that ambiguity is created in both a musical and a programmatic sense, in ways appropriate to Berlioz' musical language. Turning to West Side Story, striking parallels emerge between the two compositions as Bernstein marshals an array of contemporary techniques for the creation of ambiguity in the service of the drama. These include metric ambiguities, tonal patterns manipulated to alter their typical ordering and content, and chords or motives based on sets which display different kinds of invariance under transposition and inversion.

One gratifying area of popular music analysis is the discovery of works which exhibit sophistication in formal and thematic procedures. "Suite Sister Mary," the climax of Queensryche's Operation: mindcrime, is such an example. "Suite Sister Mary" is not only important because it exhibits aspects of sonata form, but that the form supports the unfolding drama. Such formal architecture is unusual in popular song. A double exposition of the primary theme reflects emotional intensification in the protagonist. This is achieved through reorchestration, which magnifies the dissonances in the primary theme. The musical and dramatic climax of the opera appears at the most motivically saturated and rhythmically unstable moment of the development section. Once the story line of the rock opera is established, figures and musical examples are used to illustrate the thematic unity and form of "Suite Sister Mary." Queensryche's "Suite Sister Mary" proves that popular music does not have to be devoid of formal and thematic sophistication.

Popular music from the 1920's to the 1950's, the period characterized by the "American popular ballad," carried as its hallmark the use of descriptive texts that portrayed singular emotions, most commonly love. But the texts of present-day alternative songs push these simple emotions "underground"; the harsh society of alternative rock resists vulnerability by referring only obliquely to these softer feelings. Because these oblique references are made through the use of irony, sarcasm and other forms of ambiguity, they intentionally confuse the audience's understanding of the singer's message.

The decoding of alternative texts presents a problem for analysts since the texts can inform analysis only to the extent that we understand their ambiguous intent. Traditional methods of text-based analysis, even those that address ambiguity, often prove ineffective in analyzing alternative songs since they fail to recognize the crucial influence of performance practice. In alternative songs the text alone rarely reveals the irony portrayed in performance. And often, in these settings, the dialogue between band members serves to orchestrate the text.

In this paper I argue that the texts of two songs by the alternative band Offspring, Come Out and Play and Self-Esteem, use backup singers to express irony and to set up dramatic action between themselves and the lead singer. From an historical perspective, the backup singers function much as a chorus does in Greek and Shakespearean drama. An examination of the text alone fails to reveal the conflict this drama portrays; it is only apparent when we consider the function of the chorus and its relationship to that text. My analysis of these two songs suggests that harmonic and textual analyses in alternative songs must consider the use of the chorus in performance practice, especially when irony so often works to subvert textual meaning.

This paper proceeds from the hypothesis that, in an opera, all musical elements, both abstract and audible, are linked to the drama. Building upon the work of Babbitt, Lewin, Cherlin, and others, the paper explores the relationship between the 12-tone pitch materials of the opera and the characters of Moses and Aron, focusing on Act 1, scene 2. A third element, that of aurally prominent features of the music (including texture, dynamics, register, tempo, meter, and timbre) is added, and the three are examined for their individual contributions to the antithetical relationship between Moses and Aron, which is called the "dialogue of contrast."

The results of the analysis, which combines traditional row identification, Lewin's notion of areas, Cherlinian partitions, and observations on the aurally salient features of the scene, reveal a tightly interwoven structure, where all elements work together to highlight the opposition of the two characters. Structural evidence is also presented for a catalytic view of the scene, in which Aron's question to Moses concerning the worshipping capabilities of the Volk ("People chosen by the only one, can you worship what you dare not even conceive?") serves simultaneously as a climax and a turning point. The paper closes with some general remarks on the usefulness of the analytical approach taken, both for Moses und Aron in particular, and twentieth-century opera in general.

Morton Subotnick's A Key to Songs (1985) is the first piece in a trilogy of works each based on a separate collage "novel" by the Surrealist artist Max Ernst; A Key to Songs is based on the 1934 novel Une semaine de bonté, ou les sept éléments capitaux (A Week of Kindness, of the Seven Deadly Elements). In each piece of the trilogy, Subotnick attempts to represent musically the bizarre worlds of Ernst's collage novels. In A Key to Songs, the methods of representation include imitating Ernst's collage technique and drawing upon some musical corollaries of the primary influences on Ernst's collages. Subotnick collages direct quotations and the stylistic variations from 19th century German Romantic lieder together in rapidly alternating sections, creating a formal collage of themes which is camouflaged by the surface smoothness of the music; in addition, Subotnick collages the acoustic and electronic instruments together, creating hybrid sounds. The result is an overall collage of 19th century Romanticism with Subotnick's contemporary computer music, and a collage technique analogous to that of Ernst's and including influences ranging from Freudian psychology to literary theory.