Inversion as Rotation in Ordered Pitch-Class Sets

The vast majority of pitch-class sets are 'rotationally symmetric' about their normal form, meaning that at set in normal form will remain in normal form when inverted (though read backwards). In the process of finding a set's prime form it is necessary to compare the normal form of the set with the normal form of its inversion to find the 'best normal form.' For rotationally-symmetric sets, one can simply retrograde the inverted set to find its normal form. There exactly 50 non-trivial cases, however, in which the retrograde of the inverted set is not the normal form of the inverted set. In some cases these sets are still rotationally symmetric but about a rotation other than the normal form, but for sets of even cardinality (4, 6, 8), it is possible for a set to have no axis of rotational symmetry (like the major-major seventh chord, 4-20). This paper attempts to classify the behavior of these sets and to explain why some sets exhibit this property while others do not.

The Study of Music at the University of Oxford in the Twelfth and Thirteenth Centuries

Music, as one of the quadrivial disciplines of the seven liberal arts, had played a vital role in early-medieval education by preparing the mind of the student for theological study. With the re-discovery of the writings of Greek writers like Aristotle in the twelfth century, however, came a educational renaissance of sorts that appears to have pushed the quadrivial arts toward the background of the arts curriculum, at least at the University of Paris. Indeed, by 1255, it appears that lectures on musical texts are no longer required of in the liberal arts course at Paris. Because of the similarities between Paris and the University of Oxford, most scholars to date have tended to assume that this situation was true of Oxford as well. By 1431, however, music appears to have re-entered the arts curriculum at Oxford—something that did not happen at Paris. This, along with a thriving mathematical culture at Oxford during its early days, suggests that music continued to be studied at Oxford, perhaps unofficially, throughout the twelfth and thirteenth centuries even when it was not required in the official university statutes.

Non-Syntactical Closural Features in Atonal Music by Schoenberg, Webern, and Berg

This paper examines the role that secondary parameters like duration, pauses, dynamics, texture, pitch height, and consonance play in the articulation of closure in a corpus of 33 works by Schoenberg, Webern, and Berg. For each of these pieces a “closural moment” was defined, a priori, as any sonority: 1) immediately before the final double bar at the end of a movement or whole work, 2) immediately before or after an internal double bar, 3) immediately before or during a fermata, or 4) immediately before a rest in all parts of a half duration or longer. At each of these deemed closural moments, data was collected about duration, the presence or absence of rests (except when the closural moment was identified by the presence of a rest), dynamics, textural density, pitch height, and harmonic consonance. Upon analysis, this data indicates that there is a statistically-significant correlation between moments of closure and sonorities that exhibit comparatively longer durations, quieter dynamics, thinner textures, and that are followed by a rest in all parts. That these same characteristics are also found at moments of closure in tonal music perhaps indicates that composers of non-tonal music were unable to find a non-tonal analogue to the tonal cadence and were thus forced to rely upon those cadential characteristics not explicitly associated with a tonal center. Furthermore, this also speaks to the universality with which secondary parameters like longer durations, quieter dynamics, thinner textures, and pauses are associated with closure in the Western mind.

Exploring Cohn's Sum-Class System Through Charles Villiers Stanford's La belle Dame sans merci

A survey of transformational literature dealing with triadic music—especially the work of Richard Cohn might give one the impression that transformational theory is only relevant in triadic contexts when major and minor triads alternate systematically. Such progressions invite a transformational approach because they produce highly organized voice leading that cannot be explained by tonal syntax. While these progressions do occur frequently in late-romantic music, it should be obvious that they do not account for all non-tonal triadic progressions. Nor can it be said that progressions of alternating major and minor triads provide the only means for systematically traversing voice-leading space. In this paper, I suggest that Cohn’s (1998) work with “sum classes” offers the beginnings of a system for understanding virtually any non-tonal triadic progression in terms of its voice leading and demonstrates that different background voice-leading structures can be expressed in various, seemingly different, surface realizations. In particular, I show that progressions of same-mode triads can also be used to produce consistent, small-interval voice leading. Charles Villiers Stanford’s dramatic ballade La belle Dame sans merci provides two prominent examples of triadic progressions in which same-mode triads produce systematic voice leading that suggests a transformational approach. Here, I explore the possibilities and implications of the sum-class system in the context of La belle Dame and propose small modifications to that system that make its generalizations more universally applicable. In particular, I note that, while neo-Riemannian transformations are a valuable tool for comparing triads of different mode, I argue that they do not provide an intuitive notion of the “distance” between triads of the same mode. Perhaps the greatest strength of this modified system lies in its ability to describe those non-tonal triadic succession that appear to be highly ordered but do not make sense in a hexatonic, octatonic, or other previously defined transformational space. Progressions of same-mode triads—as explored here—are one possible context, but such a concept might be applied to a variety of tonal and post-tonal settings.

Form-Generating Elements in Anton Webern's String Quartet (1905)

This paper takes as its starting point Sebastian Wedler’s suggestion in his 2015 article, “Thus Spoke the Early Modernist: Zarathustra and Rotational Form in Webern’s String Quartet (1905)” that Anton Webern’s 1905 string quartet might be profitably examined through the lens of sonata form—particularly the theories of Hepokoski and Darcy. Even a cursory examination of the work, however, reveals that such an interpretation is fraught with perils. Indeed, anyone attempting to understand the quartet from this perspective is forced to wrestle with significant formal deformations that strike at the very heart of a sonata’s fundamental structure and rhetorical power—the conflict between the tonal centers of the primary and secondary thematic areas. Where Welder is content to merely label this work as some sort of vague “dialogue with the sonata paradigm” and continue on to a discussion of the quartet’s place within the philosophical Zeitgeist of the early-twentieth century, the present paper seeks, instead, to investigate the implications of such deformations on Webern’s later music, the music of Schoenberg, and the development of atonal composition as a whole. Specifically, this paper finds that the quartet relies heavily upon inversional symmetry and the composing out of a fundamental motive—both highly important techniques in later atonal music—for the generation of form.

Serialism in Three Chamber Works of Reginald Smith Brindle

This paper examines the usage of serial techniques in three of Reginald Smith Brindle’s chamber pieces for guitar—Tierra Seca from Four Poems of Garcia Lorca, the first movement of Ten-String Music for cello and guitar, and Lachrimae from Five Sketches for violin and guitar—in light of Brindle’s own writings about serialism in his 1966 book Serial Composition. The attitude toward serialism that Brindle articulates in this book is a very skeptical one, and he mainly advocates for the use of serial techniques for their practical compositional and pedagogical value rather than the ideological merit often attached to them by the composers of the Second Viennese School. Specifically, Brindle finds that the absorption of serial techniques is a very effective way to teach the budding composer to write in an atonal musical language and that a tone row can provide the more experienced composer with musical material with which to work during the early stages of the compositional process. But Brindle also identifies several “defects” in music produced by the strict adherence to serial techniques and thus advocates for a more liberal approach to the ordering of pitch classes in a tone row. This practice is particularly evident in Ten-String Music where the presence of tone rows are quite evident but the pitch classes within them frequently appear out of order. From the pieces examined in this presentation, it seems that whether Brindle follows serial practices strictly—as in Tierra Seca where he utilizes a tone row and derives all pitch material from that row or one of its transformations—or liberally—as in Ten-String Music where only portions of the work are constructed from a tone row and those that are often contain modifications to the row ordering—, his reasons for doing so arise from the needs of the individual piece, rather than from some belief in the superiority of such methods. The freedom with which Brindle treats the series suggests that serial practices may have been at play during the compositional process of many of Brindle’s works, even those that do not seem to exhibit serial characteristics on the surface. While the row from which Brindle worked may not always be present on a note-to-note level in such cases, the analysis of Lachrimae suggests that many of the row’s larger pitch structures may still be present in the music. Above all, the findings of this presentation suggest that serial techniques need not always be used in the strict and uncompromising way that they are normally presented and that we as theorists might look for traces of serialism in a variety of musical contexts.

Sonata Form in the First Movement of Edvard Grieg's String Quartet in G Minor, Op. 27

This paper examines the form of the first movement of Edvard Grieg's G-minor string quartet in terms of Hepokoski and Darcy's three-part, "tri-modular block" expositional structure. Approaching the movement from this paradigm helps to reconcile the profusion of seemingly unrelated themes into a single unified sonata whole. Such an interpretation also allows for suggestive readings of the quartet's narrative in light of many interesting connections to Grieg's biography and oeuvre.

The Visualization of Tonality Through Color in the Animations of Stephen Malinowski

Stephen Malinowksi's musical animations of scrolling color and shapes have delighted viewers for over a decade. Yet these animations go far deeper than many people likely realize. Through these animations, Malinowski hopes to help people to hear and understand the music in a new and more meaningful way. In this paper, I examine in detail Malinowski's use of color to depict tonal structures in the music he animates. I argue that these colored animations, rather than being a mere gimick, can actually help us as listeners to hear important tonal structures in the music that would otherwise go unnoticed without the visuals. In this way, Malinowski helps us to actually hear and experience in real time tonal structures that could otherwise only be brought out through careful analysis.

Ex Nihilo: An Inquiry into the Nature of Musical Composition

As of late there has been much interest in the creative process of musical composition. Most studies to date have taken the form of quantitative and/or psychological analyses performed upon composers by researchers who are not themselves composers. In contrast, this project takes a more philosophical and historical approach in that it takes time to reflect upon the context and manner in which music has been composed throughout history. Additionally, this study offers a more personal perspective as it draws information from the author’s own compositional process–as recorded in a composition journal–as well as the thoughts and reflections of composers throughout history. From this information, the essential elements of the compositional process–that is, the elements present in the compositional processes of all composers throughout time–are delineated and several conclusions are drawn about the nature of the compositional process and creativity in general. From these conclusions, several practical suggestions may be made, which might enhance compositional creativity. Of the conclusions reached in this study, perhaps the most important is the pervasiveness of boundaries–limitations intentionally or unintentionally placed upon the compositional process–within the compositional process. Indeed, it has been concluded that without boundaries of some form, composition would be virtually impossible. Beyond this, however, boundaries help to focus a composer’s creative efforts and, in fact, make the composer more creative than they would have been otherwise. This would seem to suggest that, in sharp contrast to popular opinion, creativity lies not in freedom but rather in the manner in which freedom is limited. This, in turn, might suggest a new paradigm through which musical composition–and possibly all creative practices–might be understood and taught.


Please feel free to contact me at the email address below with any bugs you find on this site, questions you might have, or with inquiries about private lessons in music theory, composition, or guitar.