A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice

Contents

• PREFACE
• PART I. Theory
• 1 Five Components of Tonality
• 1.1 The five features.
• 1.2. Perception and the five features.
• 1.3 Four Claims.
• A. Harmony and counterpoint constrain each other.
• B. Scale, macroharmony, and centricity are independent.
• C. Modulation involves voice leading.
• D. Music can be understood geometrically.
• 1.4 Music, magic, and language.
• 1.5 Outline of the book, and a suggestion for impatient readers.
• 2. Harmony and Voice Leading
• 2.1 Linear pitch space.
• 2.2 Circular pitch-class space.
• 2.3 Transposition and inversion as distance-preserving functions.
• 2.4 Musical objects.
• 2.5 Voice leadings and chord progressions.
• 2.6 Comparing voice leadings.
• 2.7 Voice-leading size.
• 2.8 Near identity.
• 2.9 Harmony and counterpoint revisited.
• 2.10 Acoustic consonance and near-evenness
• 3. The Geometry of Chords
• 3.1 Ordered pitch space.
• 3.2 The Parable of the Ant.
• 3.3 Two-note chord space.
• 3.4 Chord progressions and voice leadings in two-note chord space.
• 3.5 Geometry in analysis.
• 3.6 Harmonic consistency and efficient voice leading.
• 3.7 Pure parallel and pure contrary motion.
• 3.8 Three-dimensional chord space.
• 3.9 Higher-dimensional chord spaces.
• 3.10 Voice leading lattices.
• 3.11 Triads are from Mars, seventh chords are from Venus.
• 3.12 Two musical geometries.
• 3.13 Study guide.
• 4. Scales
• 4.1 A scale is a ruler.
• 4.2 Scale degrees, scalar transposition, scalar inversion.
• 4.3 Evenness and scalar transposition.
• 4.4 Constructing common scales.
• 4.5 Modulation and voice leading.
• 4.6 Voice leading between common scales .
• 4.7 Two examples.
• 4.8 Scalar and interscalar transposition.
• 4.9 Interscalar transposition and voice leading.
• 4.10 Combining interscalar and chromatic transpositions.
• 5. Macroharmony and Centricity
• 5.1 Macroharmony.
• 5.2 Small-gap macroharmony.
• 5.3 Pitch-class circulation.
• 5.4 Modulating the rate of pitch-class circulation.
• 5.5 Macroharmonic consistency.
• 5.6 Centricity.
• 5.7 Where does centricity come from?
• 5.8 Beyond tonal and atonal.
• PART II. History and Analysis
• 6. The Extended Common Practice
• 6.1 Disclaimers.
• 6.2 Two-voice medieval counterpoint.
• 6.3 Triads and the Renaissance.
• 6.4 Functional harmony.
• 6.5 Schumann's Chopin.
• 6.6 Chromaticism.
• 6.7 Twentieth-century scalar music.
• 6.8 The extended common practice.
• 7. Functional Harmony
• 7.1 The thirds-based grammar of elementary tonal harmony.
• 7.2 Voice leading in functional harmony.
• 7.3 Sequences.
• 7.4 Modulation and key distance.
• 7.5 The two lattices.
• 7.6 A challenge from Schenker.
• 8. Chromaticism
• 8.1 Decorative chromaticism.
• 8.2 Generalized augmented sixths.
• 8.3 Brahms and Schoenberg.
• 8.4 Schubert and the major-third system.
• 8.5 Chopin's tesseract.
• 8.6 The Tristan Prelude.
• 8.7 Alternative approaches.
• 8.8 Conclusion
• 9. Scales in Twentieth-Century Music
• 9.1 Three scalar techniques.
• 9.2 Chord-first composition.
• A. Grieg's Drommesyn, (Vision), Op. 62 no. 5 (1895).
• B. Debussy's Fetes (1899).
• C. Michael Nyman's The Mood That Passes Through You (1993).
• 9.3 Scale-first composition.
• A. Debussy's Des pas sur la neige (1910).
• B. Janacek's On an Overgrown Path, Series II, no. 1 (1908).
• C. Shostakovich's Fs minor Prelude and Fugue, Op. 87 (1950).
• D. Reich's New York Counterpoint (1985).
• E. Reich's The Desert Music, movement 1 (1984).
• F. The Who's Can't Explain (1965) and Bob Seger's Turn the Page (1973).
• 9.4 The Subset Technique.
• A. Grieg's Klokkeklang, (Bell Ringing), Op. 54 no. 6 (1891).
• B. Petit Airs, from Stravinsky's Histoire du Soldat (1918).
• C. Reich's City Life (1995).
• D. Stravinsky's Dance of the Adolescents (1913).
• E. The Miles Davis Group's Freedom Jazz Dance (1966).
• 9.5 Conclusion.
• 10. Jazz.
• 10.1 Basic jazz voicings.
• 10.2 From thirds to fourths.
• 10.3 Tritone substitution.
• 10.4 Altered chords and scales.
• 10.5 Bass and upper-voice tritone substitutions.
• 10.6 Polytonality, sidestepping, and playing out.
• 10.7 Bill Evans's Oleo.
• 10.8 Jazz as modernist synthesis.
• CONCLUSION
• APPENDIX A. Measuring voice-leading size
• APPENDIX B. Chord geometry: a technical look.
• APPENDIX C. Discrete voice leading lattices.
• APPENDIX D. The interscalar interval matrix.
• APPENDIX E. Scale, macroharmony, and Lerdahl's basic space.
• APPENDIX F. Some study questions, problems, and activities.
• BIBLIOGRAPHY
• INDEX