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

Most listeners prefer tonal music to atonal music, but what exactly is the difference between them? In this groundbreaking work, author Dmitri Tymoczko identifies five basic musical features that jointly contribute to the sense of tonality, and shows how these features recur throughout the history of Western music. Tymoczko creates for the reader a new framework for thinking about music, one that emphasizes the commonalities among styles from Medieval polyphony to contemporary jazz. A Geometry of Musicprovides an accessible introduction to Tymoczko's revolutionary geometrical approach to music theory. The book shows how to construct simple diagrams representing the relationships among familiar chords and scales. This gives readers the tools to translate between the musical and visual realms, revealing surprising degrees of structure in otherwise hard-to-understand pieces. Tymoczko uses these theoretical ideas to retell the history of Western music from the eleventh century to the present day. Arguing that traditional histories focus too narrowly on the "common practice" period from 1680-1850, he proposes instead that Western music comprises an extended common practice stretching from the late middle ages to the present. Using analysis to make his argument clear, he discusses a host of familiar pieces by Bach, Mozart, Chopin, Debussy, Stravinsky, Shostakovich, Miles Davis, Bill Evans, and others. A Geometry of Musicis accessible to a range of readers, from undergraduate music majors to scientists and mathematicians with an interest in music. Defining its terms along the way, it presupposes no special mathematical background and only a basic familiarity with Western music theory. The book also contains exercises designed to reinforce and extend readers' understanding, along with a series of appendices that explore the technical details of this exciting new theory.

Dmitri Tymoczko is a composer and music theorist who teaches at Princeton University. His 2006 article "The Geometry of Musical Chords" was the first music theory article published in the 127-year history of Science magazine, and was widely covered in the popular press. His music has been performed by ensembles throughout the country, and he has received a Rhodes scholarship, a Guggenheim fellowship, and numerous other awards.

PREFACE

PART I. Theory

CHAPTER 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.

CHAPTER 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

CHAPTER 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.

CHAPTER 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.

CHAPTER 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

CHAPTER 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.

CHAPTER 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.

CHAPTER 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

CHAPTER 9. Scales in Twentieth-Century Music
9.1 Three scalar techniques.
9.2 Chord-first composition.
A. Grieg's "Drömmesyn," (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. Janácek'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.

CHAPTER 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