Listening with Two Ears: Conflicting Perceptions of Space in Tonal Music Justin Hoffman Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2011 © 2011 Justin Hoffman All Rights Reserved ABSTRACT Listening with Two Ears: Conflicting Perceptions of Space in Tonal Music Justin Hoffman The Tonnetz is a spatial model of tonal pitch, constructed by placing fifths along the horizontal axis of a coordinate plane and thirds along the vertical axis. This dissertation examines the ways in which different conceptions of interval, including just- intonation ratios, diatonic scalar intervals, and pitch-class intervals, result in different Tonnetz geometries, representing different, and sometimes conflicting, modes of musical perception, and argues for treating conflicts between these often unexamined conceptions of interval as an explicit part of musical analysis. Chapter One considers relationships between a number of Tonnetz spaces, as well as the groups of intervals they model, using harmonic function theory. Chapter Two examines ways in which pitches may project multiple functions in Tonnetz spaces and uses these spaces to model some aspects of the harmonic theory of Jean-Philippe Rameau. Chapter Three considers the ways in which neo-Riemannian transformations, as ways of relating triads and seventh chords to one another, might be associated with changes of harmonic function in different Tonnetz spaces, and culminates in an analysis of Chopin’s E Minor Prelude. Chapter Four explores primary triads with chromatically altered roots and fifths and, in this context, analyzes an unusual modulation from A-flat major to E major in Hugo Wolf’s song “An den Schlaf.” Finally, Chapter Five considers harmonic function in nontriadic music, examining the beginning of the final movement of Bartók’s Fourth String Quartet. Contents List of Musical Examples ................................................................................................... ii List of Figures.................................................................................................................... iv Acknowledgements............................................................................................................ ix Introduction Listening with Two Ears.......................................................................... 1 Chapter One Tonal Spaces and Tonal Function.............................................................8 Chapter Two Plural Function in Letter-Class and Pitch-Class Spaces.........................52 Chapter Three Neo-Riemannian Transformations and Harmonic Function.................102 Chapter Four Functional Intensification of Bases and Associates..............................160 Chapter Five Inversion and Dualist Harmonic Function in the Last Movement of Bartók’s Fourth String Quartet .............................................................208 Works Cited .....................................................................................................................239 i Musical Examples Example 1-1 Schubert, String Quartet in G Major, Op. 161, I, mm. 15-24....................20 Example 1-2 Wagner, chromatic grail motif, from Parsifal (with three hearings).........28 Example 1-3 Wagner, diatonic grail motif, from Parsifal ..............................................34 Example 1-4 Wagner, chromatic grail motif, from Parsifal (two hearings)...................43 Example 1-5 Wagner, chromatic grail motif, from Parsifal (two hearings)...................44 Example 1-6 Wagner, chromatic grail motif, from Parsifal (two hearings)...................45 Example 1-7 Schubert, Sanctus, from Mass in E-flat Major, mm. 1-8 ...........................47 Example 2-1 Beethoven, Trio in D Major, Op. 70, no. 1 (Ghost), I, mm. 48-73............59 Example 2-2 Beethoven, Trio in D Major, Op. 70, no. 1 (Ghost), mm. 232-253 ...........62 Example 2-3 Beethoven, Trio in D Major, Op. 70, no. 1 (Ghost), mm. 8-21 .................64 Example 2-4 Beethoven, Trio in D Major, Op. 70, no. 1 (Ghost), mm. 1-15 .................67 Example 2-5 Rameau, Les Indes galantes, earthquake scene, mm. 21-36......................86 Example 2-6 Rameau, Hippolyte et Aricie, Act II, Second Trio of the Fates .................95 Example 2-7 Rameau, Les Indes galantes, earthquake scene, mm. 56-68......................99 Example 3-1 Schubert, Sanctus, from Mass in E-Flat Major, mm. 1-8 ........................108 Example 3-2 Liszt, “Un Sospiro,” from Trois Caprices Poetiques, mm. 66-70...........110 Example 3-3 Beethoven Piano Sonata in A-flat Major, Op. 26, III, mm. 1-29.............113 Example 3-4 Schumann, Novelette, Op. 21, no. 2, mm. 83-91 .....................................119 Example 3-5 Sanctus, from Schubert Mass in A-flat Major, mm. 1-20........................122 Example 3-6 Brahms, Concerto for Violin and Cello, I, mm. 270-278 ........................140 Example 3-7 Chopin, Prelude in E Minor, Op. 28, no. 4 ..............................................147 Example 4-1 Liszt, Organ Mass, Kyrie.........................................................................170 ii Example 4-2 Beethoven, Symphony No. 8 in F Major, IV, mm. 372-292 ...................175 Example 4-3 Chopin, Mazurka in F Minor, op. 68, no. 4 .............................................179 Example 4-4 Wolf, “An den Schlaf”.............................................................................187 iii Figures Figure 0-1 The just-intonation Tonnetz........................................................................... 2 Figure 0-2 The pitch-class Tonnetz ................................................................................. 2 Figure 1-1 Functional roles of each of the seven scale degrees .....................................11 Figure 1-2 Paradigmatic functional discharges ..............................................................12 Figure 1-3 Paradigmatic functional intensifications.......................................................13 Figure 1-4 Discharge and intensification in tonal space.................................................14 Figure 1-5 A segment of the just-intonation Tonnetz.....................................................16 Figure 1-6 Lewin’s just-intonation analysis of Schubert’s quartet.................................20 Figure 1-7 Chromatic and diatonic semitones................................................................22 Figure 1-8 Harmonic progressions illustrating diatonic and chromatic semitones ........22 Figure 1-9 A segment of note-class space ......................................................................23 Figure 1-10 The pitch-class Tonnetz ................................................................................26 Figure 1-11 Chromatic and diatonic semitones in the Tonnetz........................................27 Figure 1-12 The letter-class Tonnetz................................................................................31 Figure 1-13 The scale-degree Tonnetz .............................................................................32 Figure 1-14 Voice leadings in the letter-class Tonnetz ....................................................33 Figure 1-15 Harmonic functions in letter-class and pitch-class space .............................36 Figure 1-16 Mean distances between points in just-intonation and note-class space.......40 Figure 2-1 Dominant and subdominant seventh chords in scale-degree space ..............55 Figure 2-2 Secondary functional identities of V7 and II7 ...............................................56 Figure 2-3 Tonnetz inversions of seventh chords...........................................................57 Figure 2-4 Inversion of an E dominant seventh chord ...................................................63 iv Figure 2-5 Letter-class space centered on D and A triads..............................................65 Figure 2-6 Discharge patterns for dominant and subdominant harmonies.....................68 Figure 2-7 A dominant seventh chord transformed to a French sixth chord..................69 Figure 2-8 The augmented-sixth chord in the Tonnetz ..................................................69 Figure 2-9 Subdominant-to-dominant voice leading......................................................70 Figure 2-10 Revised discharge patterns............................................................................70 Figure 2-11 Rameau’s dominante and sous-dominante harmonies..................................72 Figure 2-12 The dominante and sous-dominante harmonies in Tonnetz space................73 Figure 2-13 Two different just-intonation contexts for a series of pitch classes..............76 Figure 2-14 Riemann’s illustration of connections between harmonies ..........................77 Figure 2-15 Tonnetz pathways associated with each of the six interval classes ..............80 Figure 2-16 Four harmonic contexts for interval class 1..................................................80 Figure 2-17 Harmonic progression with chromatic thirds................................................83 Figure 2-18 Two different just-intonation contexts for a series of pitch classes..............84 Figure 2-19 A Tonnetz representation of mm. 21-36 of the earthquake scene ................88 Figure 2-20 Tonnetz inversion of an A diminished seventh chord ..................................89 Figure 2-21 Rameau’s diatonique-enharmonique progression ........................................91