Modulo-7 Transformations in Extended-Tonal Music

Robert T. Kelley
The Florida State University College of Music

Welcome to Robert Kelley's dissertation webpage. My dissertation may be downloaded as a PDF (3.62 MB). To receive a CD or hard copy by mail, please . Some papers that show parts of my research include: Charting Enharmonicism on the Just-Intonation Tonnetz, In the Dark Shadow: Enharmonic Equivalence as Dramatic Subtext in Two Wolf Lieder, Reconciling Tonal Conflicts: Mod-7 Transformations in Chromatic Music, Evaluating Prolongation in Extended Tonality, and A Mathematical Model of Tonal Function. If you are interested in my work or have any comments or suggestions, feel free to email me at <>. The Music Theory Online Dissertation Listing of Modulo-Seven Transformations in Post-Functional Music

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Rob Kelly:


Many musical compositions from the end of the nineteenth century and the beginning of the twentieth century retain some elements of functional tonality but abandon others. Most analytical methods are designed to address either tonal music or atonal music, but no single method completely illuminates this body of extended-tonal music. While both tonal and post-tonal theory have been extended in various ways to address this music, the use of tonal theory for analysis of this repertoire has not been completely formalized. The main obstacle for prolongational views of extended tonality is finding sufficient conditions for establishing that certain harmonies are structural in the absence of traditional harmonic function. In this regard, acoustical measures of stability, motivic connections, and chord equivalence all may form a part in determining the structural harmonies. Prolongational analyses of music may be represented by Schenkerian notation or transformational networks based on Lewin's Generalized Musical Intervals and Transformations (1987). This study explores a number of specific graphing techniques, including the diatonic lattice (Jones 2002), the just- intonation Tonnetz, and mod-12/mod-7 prolongational networks. After using group theory to explore the relationship of diatonic scale theory and tuning theory to transformational and prolongational analysis, excerpts from Wolf, Wagner, and Ravel are analyzed using mod-7 transformations. In giving support for prolongational analyses of chromatic and neo-tonal music, this study provides a case for tonality-based approaches to post-functional harmony.

Dissertation Committee

Project History

Chapter Outline

  1. Chromatic Harmony, Diatonic Scale Theory, and Post-Tonal Prolongation
    1. Introduction
    2. Recent Theories of Chromatic Music
    3. Diatonic Theory
    4. Prolongation in Post-Tonal Music
  2. Just Intonation as Diatonic Interpretation
    1. Tuning in 5-Limit Just Intonation
    2. Diatonic Spelling based on 5-Limit Just Intonation
    3. Homomorphisms Among Scale Systems and Tuning Systems
    4. Generalizing Diatonic Structures into 5-Limit Just Intonation
  3. Spatial Graphs and Transformational Networks
    1. The Just-Intonation Tonnetz
    2. Just-Intonation and Mod-12/Mod-7 Transformational Networks
    3. Prolongational Transformational Networks
  4. Methodologies for Determining Chordal Salience in Post-Functional Music
    1. Finding Chord Roots
    2. Finding Structural Chords
    3. Impediments to Prolongation
  5. Analytical Examples
    1. Enharmonic Progressions in Wolf's "Und steht Ihr früh"
    2. Directional Tonality in Wolf's "Der Mond"
    3. Post-Functional Progressions in Wagner's Tristan und Isolde
    4. Non-Tertian Progressions in Ravel's Valses Nobles et Sentimentales
    5. Post-Functional Non-Tertian Progressions in Ravel's Gaspard de la Nuit
  6. Conclusions
    1. The Use of Diatonic Theory for Extended Tonal Music
    2. The Place of this Work within the Field of Music Theory

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