Robert W. Peck

N th roots of pitch-class inversion

In this study, we investigate the square, cubic, and other nth roots of inversion in discrete pitch-class spaces. We examine the group-theoretical structures that they inhabit, as well as various multi-dimensional regular polytopes whose symmetries model those structures. Moreover, we determine which nth roots of inversion occur in pitch-class spaces of various sizes, and their multiplicities. Because of their relevance to the majority of music in the Western canon, as well as to the transformational theories that engage this repertoire, we focus largely on inversions and their nth roots in mod-7 diatonic space and in mod-12 chromatic space. Our objective is to further the understanding of pitch-class inversion as a gesture, through an exploration of its nth roots in discrete transformational music theory.

A Hypercube-Graph Model for n-Tone Rows and Relations

We investigate the representation of n-tone rows as paths on an n-dimensional hypercube graph with vertices labeled in the power set of the aggregate. These paths run from the vertex labeled by the null set to the one labeled by the full set, passing through vertices whose labels gradually accumulate members of the aggregate. Row relations are then given as hypercube symmetries. Such a model is more sensitive to the musical process of chromatic completion than those that deal more exclusively with n-tone rows and their relations as permutations of an underlying set. Our results lead to a graph-theoretical representation of the duality inherent in the pitch-class/order-number isomorphism of serial theory.

Introduction to the special issue on tone rows and tropes

This special issue of the Journal of Mathematics and Music is devoted principally to a classification of twelve-tone rows, and secondarily to the reception and potential utility of that enterprise. The main article, “Tone Rows and Tropes,” by Harald Fripertinger and Peter Lackner (Fripertinger and Lackner 2015), presents the classification. It incorporates sophisticated counting methods based on Pólya’s Theorem. The authors apply these techniques to the set of 479,001,600 twelve-tone rows to arrive at what must be regarded as a definitive mathematical account of the classification and enumeration of twelve-tone rows and tropes (unordered pairs of complementary hexachords) under group actions. The responses by Andrew Mead (Mead 2015) and Robert Morris (Morris 2015) examine the implications of this classification from a musictheoretical and compositional perspective. In particular, they situate the results of the main article in relation to earlier musical scholarship. Whereas combinatorial applications are not new to music theory – indeed, classifying and counting musical objects have long been among music theorists’ typical endeavors – “Tone Rows and Tropes” distinguishes itself in a number of important ways. First, and perhaps foremost, the classification and enumeration methods the authors employ require a high level of mathematical involvement for a music-theoretical investigation. Such techniques are interesting and useful in their own right, but very few other musical studies have employed them (for instance, see Hook [2007], which contains an application of Pólya’s Theorem to pitch-class set theory). Second, the article deals with a comparatively large data set. The 12! tone rows comprise a sizeable collection of objects, especially when one considers the relative complexity of any single row. Few musical studies have endeavored to consider such a large corpus auf einen Blick (see also Ilomäki [2008], which defines and examines 17 similarity relations on the set of all twelve-tone rows). Nevertheless, “Tone Rows and Tropes” also presents certain challenges. Within the community of those who work in the intersection of music theory and mathematics, the symbiosis between those who consider themselves primarily music scholars and those who consider themselves primarily mathematicians is sometimes an uneasy one. The theorems and proofs of the mathematicians leave the musicians nonplussed; even those few who can follow the mathematical detail often find that it falls short in musical relevance or sensitivity, in historical or cultural reference, and in connections to the ways they have learned to think about music. The writings of the musicians, meanwhile, may be filled with revealing musical examples and analyses but frustrate the mathematicians via imprecise prose, mathematical terms and notations deployed in ways that are non-standard or outright wrong, and rambling commentary in the place of logical demonstration. This disciplinary axis is often (but not always) correlated with a trans-Atlantic one, as the preponderance of recent work in the field has been done by music theorists in North America and by mathematicians in Europe. These tensions are probably familiar to readers of this journal, and they are on display in this issue. The featured article was not commissioned but arrived as a regular submission; because of its length and its obvious importance, it was thought suitable as the centerpiece of a special issue. The editors, recognizing connections with a sizable literature on serial theory that

Mathematical music theory pedagogy and the “New Math”

In the 1960s and early 1970s, mathematics education in American primary and secondary schools underwent a short-lived, but major, reform: the “New Math.” It sought to replace rote learning with an emphasis on axiomatic (set-theoretical) conceptualization. Several aspects of this movement have implications for the pedagogy of mathematical music theory, some positive and some negative. This essay examines that relationship, and offers some consequent suggestions for the successful teaching of mathematical music-theoretical concepts and techniques.

Almost Difference Sets in Transformational Music Theory

The combinatorial theory of difference sets has prior applications in the field of mathematical music theory. The theory of almost difference sets, however, has not received similar attention from music scholars. Nevertheless, these types of structures also have significant musical applications. For instance, the well known all-interval tetrachords of pitch-class set theory are almost difference sets. To that end, we investigate the various categories of almost difference sets (cyclic, abelian, and non-abelian) in terms of their representations in Lewinian music-transformational groups.

Mathematics and Music: Reports on the American Mathematical Society Special Sessions at the 2016 Spring Southeastern Sectional Meeting and the Forthcoming 2017 Joint Mathematics Meetings

The 2016 Spring Southeastern Sectional Meeting of the American Mathematical Society was held Friday and Saturday, March 5–6, on the campus of the University of Georgia in Athens, GA. It included a special session, “Mathematics and Music,” which featured the work of 18 mathematicians, music theorists, and composers. The session organizers were Mariana Montiel, Associate Professor of Mathematics in the Department of Mathematics and Statistics at Georgia State University, and Robert Peck, Professor of Music Theory at Louisiana State University. The special session was divided into three presentation segments: Friday morning, Friday afternoon, and Saturday morning. A number of themes emerged in the session. Following is a brief description of the talks, organized according to some of these broad categories. (Full abstracts of the presentations from the special session are available via the conference website at http://www.ams.org/ meetings/sectional/2237_program_ss14.html#title.)

All-Interval Structures

All-interval structures are subsets of musical spaces that incorporate one and only one interval from every interval class within the space. This study examines the construction and properties of all-interval structures, using mathematical tools and concepts from geometrical and transformational music theories. Further, we investigate conditions under which certain all-interval structures are Z (or GISZ) related to one another. Finally, we make connections between the orbits of all-interval structures under certain interval-groups and the sets of lines and points in finite projective planes. In particular, we conjecture a correspondence that relates to the co-existence of such structures.

Generalized Voice Exchange

The notion of voice exchange in ordered pitch-class space conforms closely to that of contextual inversion in neo-Riemannian theory: the melodic dyad (a, b) in one voice inverts in another voice, and we define an axis of inversion respectively for all such pairs. We may thus apply many of the transformational concepts of neo-Riemannian theory to a study of voice exchange. We draw our musical examples from the Prelude to Richard Wagner’s Tristan und Isolde, for which a separate analytical thread exists that considers aspects of tonality in relation to the voice exchange in the resolution of the Tristan Chord.

Interval Preservation in Group- and Graph-Theoretical Music Theories: A Comparative Study

Interval preservation—wherein intervals remain unchanged among varying musical objects—is among the most basic means of manifesting coherence in musical structures. Music theorists since (1960) seminal publication of “Twelve-Tone Invariants as Compositional Determinants” have examined and generalized situations in which interval preservation obtains. In the course of this investigation, two theoretical contexts have developed: the group-theoretical, as in (1987) Generalized Interval Systems; and the graph-theoretical, as in (1991) K-net theory. Whereas the two approaches are integrally related— the latter’s being particularly indebted to the former—they have also essential differences, particularly in regard to the way in which they describe interval preservation. Nevertheless, this point has escaped significant attention in the literature. The present study completes the comparison of these two methods, and, in doing so, reveals further-reaching implications of the theory of interval preservation to recent models of voice-leading and chord spaces (Cohn 2003, Straus 2005, Tymoczko 2005, among others), specifically where the incorporated chords have differing cardinalities and/or symmetrical properties.