A Tonnetz Model for pentachords

This article deals with the construction of surfaces that are suitable for representing pentachords or 5-pitch segments that are in the same $T/I$ class. It is a generalization of the well known \"Ottingen-Riemann torus for triads of neo-Riemannian theories. Two pentachords are near if they differ by a particular set of contextual inversions and the whole contextual group of inversions produces a Tiling (Tessellation) by pentagons on the surfaces. A description of the surfaces as coverings of a particular Tiling is given in the twelve-tone enharmonic scale case.Comment: 27 pages, 12 figure

A Model of the Tonal-Chromatic System and Its Application to Selected Works of Gustav Mahler

During the latter half of the twentieth century there was a marked shift in the way that many scholars approached analysis of late-nineteenth-century tonality. This shift in approach was motivated by the behavior and interaction of harmonic and melodic entities encountered in the music of the great composers of the nineteenth century, such as Schubert, Schumann, Brahms, Mahler, Wagner, and Strauss. The question arose: did the diatonic system remain at the heart of tonality during the late nineteenth century, or—as some propose—did a new chromatic-based tonality emerge? The acceptance of this new chromatic-based tonality is at the heart of this project. In order to address the acceptance of this premise, this dissertation brings together two disparate strands of research and posits a system termed the tonal-chromatic scale. First, the writings of Mitchell, Marra, and Proctor help define the need for and structure of a chromatic scale that implies tonal function; second, the mechanics of transformational theory, stemming from the work of Lewin underpins the methodology of the system. The recent work of Rings is an influential aspect of the methodology. Throughout past century, different analysts have used a myriad of methodologies to explore the complex musical language of Gustav Mahler. Mahler’s work represents an important turning point in the evolution of tonal music. Specifically, his work contributed to the onset of and evolution toward the complete breakdown of tonality. The chromatic tendencies in Mahler’s music make it an especially relevant candidate for analysis using this type of system. Even though his music exhibits an inordinate amount of chromaticism, the pillars of functional tonality are still operative. Three selected works of Gustav Mahler serve as the examples of the analytical usefulness of the proposed system: 1) the first movement of the Piano Quartet in A minor, which was composed as a student at the Vienna conservatory; 2) the fourth movement of Symphony no. 5, Adagietto; and 3) Kindertotenlieder, no. 2, “Nun seh’ ich wohl, warum so dunkle Flammen.

Andrew Mead. An Introduction to the Music of Milton Babbitt

This review addresses the multifacted aspect of Babbitt's work. Eckardt addresses Babbitt's composition methods, and critiques Mead's analyses

A permutational triadic approach to jazz harmony and the chord/scale relationship

This study provides an original triadic theory that combines existing jazz theory, in particular the chord/scale relationship, and mathematical permutation group theory to analyze repertoire, act as a pedagogical tool, and provide a system to create new music. Permutations are defined as group actions on sets, and the sets used here are the constituent consonant triads derived from certain scales. Group structures provide a model by which to understand the relationships held between the triadic set elements as defined by the generating functions. The findings are both descriptive and prescriptive, as triadic permutations offer new insights into existing repertoire. Further, the results serve as an organizational tool for the improviser and composer/arranger. In addition to the ability to describe individual triadic musical events as group actions, we also consider relationships held among the musical events by considering subgroups, conjugacy classes, direct products and semidirect products. As an interdisciplinary study, it is hoped that this work helps to increase the discourse between those in the music subdisciplines of mathematical music theory and jazz studies