Society for Music Theory

New Directions in the Theory and Analysis of Musical ContourAuthor(s): Robert D. MorrisSource: Music Theory Spectrum, Vol. 15, No. 2 (Autumn, 1993), pp. 205-228Published by: on behalf of the Society for Music TheoryStable URL: http://www.jstor.org/stable/745814 .Accessed: 03/04/2014 20:05

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New Directions in the Theory and Analysis of Musical Contour

ROBERT D. MORRIS

New Directions in the Theory and Analysis of Musical Contour

ROBERT D. MORRIS

New Directions in the Theory and Analysis of Musical Contour

ROBERT D. MORRIS

New Directions in the Theory and Analysis of Musical Contour

ROBERT D. MORRIS

New Directions in the Theory and Analysis of Musical Contour

ROBERT D. MORRIS

New Directions in the Theory and Analysis of Musical Contour

ROBERT D. MORRIS

New Directions in the Theory and Analysis of Musical Contour

ROBERT D. MORRIS

New Directions in the Theory and Analysis of Musical Contour

ROBERT D. MORRIS

New Directions in the Theory and Analysis of Musical Contour

ROBERT D. MORRIS

New Directions in the Theory and Analysis of Musical Contour

ROBERT D. MORRIS

New Directions in the Theory and Analysis of Musical Contour

ROBERT D. MORRIS

Musical contour is one of the most general aspects of pitch perception, prior to the concept of pitch or pitch class, for it is grounded only in a listener's ability to hear pitches as relatively higher, equal, or lower, without discerning the ex- act differences between and among them. Yet, until rather recently, music theorists have paid little attention to contour relations. This is somewhat understandable since contour, especially melodic contour, remained largely an aspect of musical diction until the beginning of the twentieth century. But in much twentieth-century music, especially in the works of Edgard Varese, Iannis Xenakis, and Gy6rgy Ligeti, con- tour has been generalized beyond melody and may play an important structural role in a specific composition or reper- toire.

Apart from work by a few ethnomusicologists,' and a text- book by Robert Cogan and Pozzi Escot,2 the foundations for a theory of contour were advanced in the later 1980s by

1See Mieczyslaw Kolinski, "The Structure of Melodic Movement: A New Method of Analysis," Studies in Ethnomusicology 2 (1965): 96-120; Charles R. Adams, "Melodic Contour Typology," Ethnomusicology 20 (1976): 179- 215; and Charles Seeger, "On the Moods of a Music-Logic," Journal of the American Musicology Society 8 (1960): 224-61. For a review of these articles and other earlier discussions of melodic contour see Elizabeth West Marvin, "A Generalization of Contour Theory to Diverse Musical Spaces: Analytical Applications to the Music of Dallapiccola and Stockhausen" in Musical Plu- ralism: Aspects of Aesthetics and Structure Since 1945 (forthcoming).

2See Cogan and Escot, Sonic Design: The Nature of Sound and Music (Englewood Cliffs, NJ: Prentice-Hall, 1976).

Musical contour is one of the most general aspects of pitch perception, prior to the concept of pitch or pitch class, for it is grounded only in a listener's ability to hear pitches as relatively higher, equal, or lower, without discerning the ex- act differences between and among them. Yet, until rather recently, music theorists have paid little attention to contour relations. This is somewhat understandable since contour, especially melodic contour, remained largely an aspect of musical diction until the beginning of the twentieth century. But in much twentieth-century music, especially in the works of Edgard Varese, Iannis Xenakis, and Gy6rgy Ligeti, con- tour has been generalized beyond melody and may play an important structural role in a specific composition or reper- toire.

Apart from work by a few ethnomusicologists,' and a text- book by Robert Cogan and Pozzi Escot,2 the foundations for a theory of contour were advanced in the later 1980s by

1See Mieczyslaw Kolinski, "The Structure of Melodic Movement: A New Method of Analysis," Studies in Ethnomusicology 2 (1965): 96-120; Charles R. Adams, "Melodic Contour Typology," Ethnomusicology 20 (1976): 179- 215; and Charles Seeger, "On the Moods of a Music-Logic," Journal of the American Musicology Society 8 (1960): 224-61. For a review of these articles and other earlier discussions of melodic contour see Elizabeth West Marvin, "A Generalization of Contour Theory to Diverse Musical Spaces: Analytical Applications to the Music of Dallapiccola and Stockhausen" in Musical Plu- ralism: Aspects of Aesthetics and Structure Since 1945 (forthcoming).

2See Cogan and Escot, Sonic Design: The Nature of Sound and Music (Englewood Cliffs, NJ: Prentice-Hall, 1976).

Musical contour is one of the most general aspects of pitch perception, prior to the concept of pitch or pitch class, for it is grounded only in a listener's ability to hear pitches as relatively higher, equal, or lower, without discerning the ex- act differences between and among them. Yet, until rather recently, music theorists have paid little attention to contour relations. This is somewhat understandable since contour, especially melodic contour, remained largely an aspect of musical diction until the beginning of the twentieth century. But in much twentieth-century music, especially in the works of Edgard Varese, Iannis Xenakis, and Gy6rgy Ligeti, con- tour has been generalized beyond melody and may play an important structural role in a specific composition or reper- toire.

Apart from work by a few ethnomusicologists,' and a text- book by Robert Cogan and Pozzi Escot,2 the foundations for a theory of contour were advanced in the later 1980s by

1See Mieczyslaw Kolinski, "The Structure of Melodic Movement: A New Method of Analysis," Studies in Ethnomusicology 2 (1965): 96-120; Charles R. Adams, "Melodic Contour Typology," Ethnomusicology 20 (1976): 179- 215; and Charles Seeger, "On the Moods of a Music-Logic," Journal of the American Musicology Society 8 (1960): 224-61. For a review of these articles and other earlier discussions of melodic contour see Elizabeth West Marvin, "A Generalization of Contour Theory to Diverse Musical Spaces: Analytical Applications to the Music of Dallapiccola and Stockhausen" in Musical Plu- ralism: Aspects of Aesthetics and Structure Since 1945 (forthcoming).

2See Cogan and Escot, Sonic Design: The Nature of Sound and Music (Englewood Cliffs, NJ: Prentice-Hall, 1976).

Musical contour is one of the most general aspects of pitch perception, prior to the concept of pitch or pitch class, for it is grounded only in a listener's ability to hear pitches as relatively higher, equal, or lower, without discerning the ex- act differences between and among them. Yet, until rather recently, music theorists have paid little attention to contour relations. This is somewhat understandable since contour, especially melodic contour, remained largely an aspect of musical diction until the beginning of the twentieth century. But in much twentieth-century music, especially in the works of Edgard Varese, Iannis Xenakis, and Gy6rgy Ligeti, con- tour has been generalized beyond melody and may play an important structural role in a specific composition or reper- toire.

Apart from work by a few ethnomusicologists,' and a text- book by Robert Cogan and Pozzi Escot,2 the foundations for a theory of contour were advanced in the later 1980s by

1See Mieczyslaw Kolinski, "The Structure of Melodic Movement: A New Method of Analysis," Studies in Ethnomusicology 2 (1965): 96-120; Charles R. Adams, "Melodic Contour Typology," Ethnomusicology 20 (1976): 179- 215; and Charles Seeger, "On the Moods of a Music-Logic," Journal of the American Musicology Society 8 (1960): 224-61. For a review of these articles and other earlier discussions of melodic contour see Elizabeth West Marvin, "A Generalization of Contour Theory to Diverse Musical Spaces: Analytical Applications to the Music of Dallapiccola and Stockhausen" in Musical Plu- ralism: Aspects of Aesthetics and Structure Since 1945 (forthcoming).

2See Cogan and Escot, Sonic Design: The Nature of Sound and Music (Englewood Cliffs, NJ: Prentice-Hall, 1976).

Musical contour is one of the most general aspects of pitch perception, prior to the concept of pitch or pitch class, for it is grounded only in a listener's ability to hear pitches as relatively higher, equal, or lower, without discerning the ex- act differences between and among them. Yet, until rather recently, music theorists have paid little attention to contour relations. This is somewhat understandable since contour, especially melodic contour, remained largely an aspect of musical diction until the beginning of the twentieth century. But in much twentieth-century music, especially in the works of Edgard Varese, Iannis Xenakis, and Gy6rgy Ligeti, con- tour has been generalized beyond melody and may play an important structural role in a specific composition or reper- toire.

Apart from work by a few ethnomusicologists,' and a text- book by Robert Cogan and Pozzi Escot,2 the foundations for a theory of contour were advanced in the later 1980s by

1See Mieczyslaw Kolinski, "The Structure of Melodic Movement: A New Method of Analysis," Studies in Ethnomusicology 2 (1965): 96-120; Charles R. Adams, "Melodic Contour Typology," Ethnomusicology 20 (1976): 179- 215; and Charles Seeger, "On the Moods of a Music-Logic," Journal of the American Musicology Society 8 (1960): 224-61. For a review of these articles and other earlier discussions of melodic contour see Elizabeth West Marvin, "A Generalization of Contour Theory to Diverse Musical Spaces: Analytical Applications to the Music of Dallapiccola and Stockhausen" in Musical Plu- ralism: Aspects of Aesthetics and Structure Since 1945 (forthcoming).

2See Cogan and Escot, Sonic Design: The Nature of Sound and Music (Englewood Cliffs, NJ: Prentice-Hall, 1976).

Musical contour is one of the most general aspects of pitch perception, prior to the concept of pitch or pitch class, for it is grounded only in a listener's ability to hear pitches as relatively higher, equal, or lower, without discerning the ex- act differences between and among them. Yet, until rather recently, music theorists have paid little attention to contour relations. This is somewhat understandable since contour, especially melodic contour, remained largely an aspect of musical diction until the beginning of the twentieth century. But in much twentieth-century music, especially in the works of Edgard Varese, Iannis Xenakis, and Gy6rgy Ligeti, con- tour has been generalized beyond melody and may play an important structural role in a specific composition or reper- toire.

Apart from work by a few ethnomusicologists,' and a text- book by Robert Cogan and Pozzi Escot,2 the foundations for a theory of contour were advanced in the later 1980s by

1See Mieczyslaw Kolinski, "The Structure of Melodic Movement: A New Method of Analysis," Studies in Ethnomusicology 2 (1965): 96-120; Charles R. Adams, "Melodic Contour Typology," Ethnomusicology 20 (1976): 179- 215; and Charles Seeger, "On the Moods of a Music-Logic," Journal of the American Musicology Society 8 (1960): 224-61. For a review of these articles and other earlier discussions of melodic contour see Elizabeth West Marvin, "A Generalization of Contour Theory to Diverse Musical Spaces: Analytical Applications to the Music of Dallapiccola and Stockhausen" in Musical Plu- ralism: Aspects of Aesthetics and Structure Since 1945 (forthcoming).

2See Cogan and Escot, Sonic Design: The Nature of Sound and Music (Englewood Cliffs, NJ: Prentice-Hall, 1976).

Musical contour is one of the most general aspects of pitch perception, prior to the concept of pitch or pitch class, for it is grounded only in a listener's ability to hear pitches as relatively higher, equal, or lower, without discerning the ex- act differences between and among them. Yet, until rather recently, music theorists have paid little attention to contour relations. This is somewhat understandable since contour, especially melodic contour, remained largely an aspect of musical diction until the beginning of the twentieth century. But in much twentieth-century music, especially in the works of Edgard Varese, Iannis Xenakis, and Gy6rgy Ligeti, con- tour has been generalized beyond melody and may play an important structural role in a specific composition or reper- toire.

Apart from work by a few ethnomusicologists,' and a text- book by Robert Cogan and Pozzi Escot,2 the foundations for a theory of contour were advanced in the later 1980s by

1See Mieczyslaw Kolinski, "The Structure of Melodic Movement: A New Method of Analysis," Studies in Ethnomusicology 2 (1965): 96-120; Charles R. Adams, "Melodic Contour Typology," Ethnomusicology 20 (1976): 179- 215; and Charles Seeger, "On the Moods of a Music-Logic," Journal of the American Musicology Society 8 (1960): 224-61. For a review of these articles and other earlier discussions of melodic contour see Elizabeth West Marvin, "A Generalization of Contour Theory to Diverse Musical Spaces: Analytical Applications to the Music of Dallapiccola and Stockhausen" in Musical Plu- ralism: Aspects of Aesthetics and Structure Since 1945 (forthcoming).

2See Cogan and Escot, Sonic Design: The Nature of Sound and Music (Englewood Cliffs, NJ: Prentice-Hall, 1976).

Musical contour is one of the most general aspects of pitch perception, prior to the concept of pitch or pitch class, for it is grounded only in a listener's ability to hear pitches as relatively higher, equal, or lower, without discerning the ex- act differences between and among them. Yet, until rather recently, music theorists have paid little attention to contour relations. This is somewhat understandable since contour, especially melodic contour, remained largely an aspect of musical diction until the beginning of the twentieth century. But in much twentieth-century music, especially in the works of Edgard Varese, Iannis Xenakis, and Gy6rgy Ligeti, con- tour has been generalized beyond melody and may play an important structural role in a specific composition or reper- toire.

Apart from work by a few ethnomusicologists,' and a text- book by Robert Cogan and Pozzi Escot,2 the foundations for a theory of contour were advanced in the later 1980s by

1See Mieczyslaw Kolinski, "The Structure of Melodic Movement: A New Method of Analysis," Studies in Ethnomusicology 2 (1965): 96-120; Charles R. Adams, "Melodic Contour Typology," Ethnomusicology 20 (1976): 179- 215; and Charles Seeger, "On the Moods of a Music-Logic," Journal of the American Musicology Society 8 (1960): 224-61. For a review of these articles and other earlier discussions of melodic contour see Elizabeth West Marvin, "A Generalization of Contour Theory to Diverse Musical Spaces: Analytical Applications to the Music of Dallapiccola and Stockhausen" in Musical Plu- ralism: Aspects of Aesthetics and Structure Since 1945 (forthcoming).

2See Cogan and Escot, Sonic Design: The Nature of Sound and Music (Englewood Cliffs, NJ: Prentice-Hall, 1976).

Musical contour is one of the most general aspects of pitch perception, prior to the concept of pitch or pitch class, for it is grounded only in a listener's ability to hear pitches as relatively higher, equal, or lower, without discerning the ex- act differences between and among them. Yet, until rather recently, music theorists have paid little attention to contour relations. This is somewhat understandable since contour, especially melodic contour, remained largely an aspect of musical diction until the beginning of the twentieth century. But in much twentieth-century music, especially in the works of Edgard Varese, Iannis Xenakis, and Gy6rgy Ligeti, con- tour has been generalized beyond melody and may play an important structural role in a specific composition or reper- toire.

Apart from work by a few ethnomusicologists,' and a text- book by Robert Cogan and Pozzi Escot,2 the foundations for a theory of contour were advanced in the later 1980s by

1See Mieczyslaw Kolinski, "The Structure of Melodic Movement: A New Method of Analysis," Studies in Ethnomusicology 2 (1965): 96-120; Charles R. Adams, "Melodic Contour Typology," Ethnomusicology 20 (1976): 179- 215; and Charles Seeger, "On the Moods of a Music-Logic," Journal of the American Musicology Society 8 (1960): 224-61. For a review of these articles and other earlier discussions of melodic contour see Elizabeth West Marvin, "A Generalization of Contour Theory to Diverse Musical Spaces: Analytical Applications to the Music of Dallapiccola and Stockhausen" in Musical Plu- ralism: Aspects of Aesthetics and Structure Since 1945 (forthcoming).

2See Cogan and Escot, Sonic Design: The Nature of Sound and Music (Englewood Cliffs, NJ: Prentice-Hall, 1976).

Musical contour is one of the most general aspects of pitch perception, prior to the concept of pitch or pitch class, for it is grounded only in a listener's ability to hear pitches as relatively higher, equal, or lower, without discerning the ex- act differences between and among them. Yet, until rather recently, music theorists have paid little attention to contour relations. This is somewhat understandable since contour, especially melodic contour, remained largely an aspect of musical diction until the beginning of the twentieth century. But in much twentieth-century music, especially in the works of Edgard Varese, Iannis Xenakis, and Gy6rgy Ligeti, con- tour has been generalized beyond melody and may play an important structural role in a specific composition or reper- toire.

Apart from work by a few ethnomusicologists,' and a text- book by Robert Cogan and Pozzi Escot,2 the foundations for a theory of contour were advanced in the later 1980s by

1See Mieczyslaw Kolinski, "The Structure of Melodic Movement: A New Method of Analysis," Studies in Ethnomusicology 2 (1965): 96-120; Charles R. Adams, "Melodic Contour Typology," Ethnomusicology 20 (1976): 179- 215; and Charles Seeger, "On the Moods of a Music-Logic," Journal of the American Musicology Society 8 (1960): 224-61. For a review of these articles and other earlier discussions of melodic contour see Elizabeth West Marvin, "A Generalization of Contour Theory to Diverse Musical Spaces: Analytical Applications to the Music of Dallapiccola and Stockhausen" in Musical Plu- ralism: Aspects of Aesthetics and Structure Since 1945 (forthcoming).

2See Cogan and Escot, Sonic Design: The Nature of Sound and Music (Englewood Cliffs, NJ: Prentice-Hall, 1976).

Musical contour is one of the most general aspects of pitch perception, prior to the concept of pitch or pitch class, for it is grounded only in a listener's ability to hear pitches as relatively higher, equal, or lower, without discerning the ex- act differences between and among them. Yet, until rather recently, music theorists have paid little attention to contour relations. This is somewhat understandable since contour, especially melodic contour, remained largely an aspect of musical diction until the beginning of the twentieth century. But in much twentieth-century music, especially in the works of Edgard Varese, Iannis Xenakis, and Gy6rgy Ligeti, con- tour has been generalized beyond melody and may play an important structural role in a specific composition or reper- toire.

Apart from work by a few ethnomusicologists,' and a text- book by Robert Cogan and Pozzi Escot,2 the foundations for a theory of contour were advanced in the later 1980s by

1See Mieczyslaw Kolinski, "The Structure of Melodic Movement: A New Method of Analysis," Studies in Ethnomusicology 2 (1965): 96-120; Charles R. Adams, "Melodic Contour Typology," Ethnomusicology 20 (1976): 179- 215; and Charles Seeger, "On the Moods of a Music-Logic," Journal of the American Musicology Society 8 (1960): 224-61. For a review of these articles and other earlier discussions of melodic contour see Elizabeth West Marvin, "A Generalization of Contour Theory to Diverse Musical Spaces: Analytical Applications to the Music of Dallapiccola and Stockhausen" in Musical Plu- ralism: Aspects of Aesthetics and Structure Since 1945 (forthcoming).

2See Cogan and Escot, Sonic Design: The Nature of Sound and Music (Englewood Cliffs, NJ: Prentice-Hall, 1976).

Michael Friedmann, Larry Polansky, and myself.3 My work, which defines "contour-spaces" with its various entities and relations, has been extended by Elizabeth Marvin and Paul Laprade.4 While my aim was in part to provide a secure foundation for the articulation of compositional designs of pitch classes, Friedmann and Marvin have been more con- cerned with developing a methodology for the study of con- tour associations in post-tonal music.5 In this way, their work mirrors the extents and limits of atonal theory of pitch-class (pc) sets. In addition, David Lidov and Jim Gabura have considered certain formal aspects of contour in their con- struction of an algorithm for writing melodies.6

3See Friedmann, "A Methodology for the Discussion of Contour: Its Application to Schoenberg's Music," Journal of Music Theory 29 (1985): 223-48; Morris, Composition with Pitch-Classes: A Theory of Compositional Design (New Haven and London: Yale University Press, 1987); and Polansky, "Morphological Metrics: An Introduction to a Theory of Formal Distances" in Proceedings of the International Computer Music Conference (San Fran- cisco: Computer Music Association, 1987).

4See Marvin and Laprade, "Relating Musical Contours: Extensions of a Theory for Contour," Journal of Music Theory 31 (1987): 225-67.

SSee Friedmann, "Methodology"; Friedmann, "A Response: My Con- tour, Their Contour," Journal of Music Theory 31 (1987): 268-74; Elizabeth West Marvin, "The Perception of Rhythm in Non-Tonal Music: Rhythmic Contours in the Music of Edgard Varese," Music Theory Spectrum 13 (1991): 61-78; and Marvin, "A Generalization of Contour Theory."

6See David Lidov and Jim Gabura, "A Melody Writing Algorithm Using a Formal Language Model," Computers in the Humanities 3-4 (1973): 138-48.

Michael Friedmann, Larry Polansky, and myself.3 My work, which defines "contour-spaces" with its various entities and relations, has been extended by Elizabeth Marvin and Paul Laprade.4 While my aim was in part to provide a secure foundation for the articulation of compositional designs of pitch classes, Friedmann and Marvin have been more con- cerned with developing a methodology for the study of con- tour associations in post-tonal music.5 In this way, their work mirrors the extents and limits of atonal theory of pitch-class (pc) sets. In addition, David Lidov and Jim Gabura have considered certain formal aspects of contour in their con- struction of an algorithm for writing melodies.6

3See Friedmann, "A Methodology for the Discussion of Contour: Its Application to Schoenberg's Music," Journal of Music Theory 29 (1985): 223-48; Morris, Composition with Pitch-Classes: A Theory of Compositional Design (New Haven and London: Yale University Press, 1987); and Polansky, "Morphological Metrics: An Introduction to a Theory of Formal Distances" in Proceedings of the International Computer Music Conference (San Fran- cisco: Computer Music Association, 1987).

4See Marvin and Laprade, "Relating Musical Contours: Extensions of a Theory for Contour," Journal of Music Theory 31 (1987): 225-67.

SSee Friedmann, "Methodology"; Friedmann, "A Response: My Con- tour, Their Contour," Journal of Music Theory 31 (1987): 268-74; Elizabeth West Marvin, "The Perception of Rhythm in Non-Tonal Music: Rhythmic Contours in the Music of Edgard Varese," Music Theory Spectrum 13 (1991): 61-78; and Marvin, "A Generalization of Contour Theory."

6See David Lidov and Jim Gabura, "A Melody Writing Algorithm Using a Formal Language Model," Computers in the Humanities 3-4 (1973): 138-48.

Michael Friedmann, Larry Polansky, and myself.3 My work, which defines "contour-spaces" with its various entities and relations, has been extended by Elizabeth Marvin and Paul Laprade.4 While my aim was in part to provide a secure foundation for the articulation of compositional designs of pitch classes, Friedmann and Marvin have been more con- cerned with developing a methodology for the study of con- tour associations in post-tonal music.5 In this way, their work mirrors the extents and limits of atonal theory of pitch-class (pc) sets. In addition, David Lidov and Jim Gabura have considered certain formal aspects of contour in their con- struction of an algorithm for writing melodies.6

3See Friedmann, "A Methodology for the Discussion of Contour: Its Application to Schoenberg's Music," Journal of Music Theory 29 (1985): 223-48; Morris, Composition with Pitch-Classes: A Theory of Compositional Design (New Haven and London: Yale University Press, 1987); and Polansky, "Morphological Metrics: An Introduction to a Theory of Formal Distances" in Proceedings of the International Computer Music Conference (San Fran- cisco: Computer Music Association, 1987).

4See Marvin and Laprade, "Relating Musical Contours: Extensions of a Theory for Contour," Journal of Music Theory 31 (1987): 225-67.

SSee Friedmann, "Methodology"; Friedmann, "A Response: My Con- tour, Their Contour," Journal of Music Theory 31 (1987): 268-74; Elizabeth West Marvin, "The Perception of Rhythm in Non-Tonal Music: Rhythmic Contours in the Music of Edgard Varese," Music Theory Spectrum 13 (1991): 61-78; and Marvin, "A Generalization of Contour Theory."

6See David Lidov and Jim Gabura, "A Melody Writing Algorithm Using a Formal Language Model," Computers in the Humanities 3-4 (1973): 138-48.

Michael Friedmann, Larry Polansky, and myself.3 My work, which defines "contour-spaces" with its various entities and relations, has been extended by Elizabeth Marvin and Paul Laprade.4 While my aim was in part to provide a secure foundation for the articulation of compositional designs of pitch classes, Friedmann and Marvin have been more con- cerned with developing a methodology for the study of con- tour associations in post-tonal music.5 In this way, their work mirrors the extents and limits of atonal theory of pitch-class (pc) sets. In addition, David Lidov and Jim Gabura have considered certain formal aspects of contour in their con- struction of an algorithm for writing melodies.6

3See Friedmann, "A Methodology for the Discussion of Contour: Its Application to Schoenberg's Music," Journal of Music Theory 29 (1985): 223-48; Morris, Composition with Pitch-Classes: A Theory of Compositional Design (New Haven and London: Yale University Press, 1987); and Polansky, "Morphological Metrics: An Introduction to a Theory of Formal Distances" in Proceedings of the International Computer Music Conference (San Fran- cisco: Computer Music Association, 1987).

4See Marvin and Laprade, "Relating Musical Contours: Extensions of a Theory for Contour," Journal of Music Theory 31 (1987): 225-67.

SSee Friedmann, "Methodology"; Friedmann, "A Response: My Con- tour, Their Contour," Journal of Music Theory 31 (1987): 268-74; Elizabeth West Marvin, "The Perception of Rhythm in Non-Tonal Music: Rhythmic Contours in the Music of Edgard Varese," Music Theory Spectrum 13 (1991): 61-78; and Marvin, "A Generalization of Contour Theory."

6See David Lidov and Jim Gabura, "A Melody Writing Algorithm Using a Formal Language Model," Computers in the Humanities 3-4 (1973): 138-48.

Michael Friedmann, Larry Polansky, and myself.3 My work, which defines "contour-spaces" with its various entities and relations, has been extended by Elizabeth Marvin and Paul Laprade.4 While my aim was in part to provide a secure foundation for the articulation of compositional designs of pitch classes, Friedmann and Marvin have been more con- cerned with developing a methodology for the study of con- tour associations in post-tonal music.5 In this way, their work mirrors the extents and limits of atonal theory of pitch-class (pc) sets. In addition, David Lidov and Jim Gabura have considered certain formal aspects of contour in their con- struction of an algorithm for writing melodies.6

3See Friedmann, "A Methodology for the Discussion of Contour: Its Application to Schoenberg's Music," Journal of Music Theory 29 (1985): 223-48; Morris, Composition with Pitch-Classes: A Theory of Compositional Design (New Haven and London: Yale University Press, 1987); and Polansky, "Morphological Metrics: An Introduction to a Theory of Formal Distances" in Proceedings of the International Computer Music Conference (San Fran- cisco: Computer Music Association, 1987).

4See Marvin and Laprade, "Relating Musical Contours: Extensions of a Theory for Contour," Journal of Music Theory 31 (1987): 225-67.

SSee Friedmann, "Methodology"; Friedmann, "A Response: My Con- tour, Their Contour," Journal of Music Theory 31 (1987): 268-74; Elizabeth West Marvin, "The Perception of Rhythm in Non-Tonal Music: Rhythmic Contours in the Music of Edgard Varese," Music Theory Spectrum 13 (1991): 61-78; and Marvin, "A Generalization of Contour Theory."

6See David Lidov and Jim Gabura, "A Melody Writing Algorithm Using a Formal Language Model," Computers in the Humanities 3-4 (1973): 138-48.

Michael Friedmann, Larry Polansky, and myself.3 My work, which defines "contour-spaces" with its various entities and relations, has been extended by Elizabeth Marvin and Paul Laprade.4 While my aim was in part to provide a secure foundation for the articulation of compositional designs of pitch classes, Friedmann and Marvin have been more con- cerned with developing a methodology for the study of con- tour associations in post-tonal music.5 In this way, their work mirrors the extents and limits of atonal theory of pitch-class (pc) sets. In addition, David Lidov and Jim Gabura have considered certain formal aspects of contour in their con- struction of an algorithm for writing melodies.6

3See Friedmann, "A Methodology for the Discussion of Contour: Its Application to Schoenberg's Music," Journal of Music Theory 29 (1985): 223-48; Morris, Composition with Pitch-Classes: A Theory of Compositional Design (New Haven and London: Yale University Press, 1987); and Polansky, "Morphological Metrics: An Introduction to a Theory of Formal Distances" in Proceedings of the International Computer Music Conference (San Fran- cisco: Computer Music Association, 1987).

4See Marvin and Laprade, "Relating Musical Contours: Extensions of a Theory for Contour," Journal of Music Theory 31 (1987): 225-67.

SSee Friedmann, "Methodology"; Friedmann, "A Response: My Con- tour, Their Contour," Journal of Music Theory 31 (1987): 268-74; Elizabeth West Marvin, "The Perception of Rhythm in Non-Tonal Music: Rhythmic Contours in the Music of Edgard Varese," Music Theory Spectrum 13 (1991): 61-78; and Marvin, "A Generalization of Contour Theory."

6See David Lidov and Jim Gabura, "A Melody Writing Algorithm Using a Formal Language Model," Computers in the Humanities 3-4 (1973): 138-48.

Michael Friedmann, Larry Polansky, and myself.3 My work, which defines "contour-spaces" with its various entities and relations, has been extended by Elizabeth Marvin and Paul Laprade.4 While my aim was in part to provide a secure foundation for the articulation of compositional designs of pitch classes, Friedmann and Marvin have been more con- cerned with developing a methodology for the study of con- tour associations in post-tonal music.5 In this way, their work mirrors the extents and limits of atonal theory of pitch-class (pc) sets. In addition, David Lidov and Jim Gabura have considered certain formal aspects of contour in their con- struction of an algorithm for writing melodies.6

3See Friedmann, "A Methodology for the Discussion of Contour: Its Application to Schoenberg's Music," Journal of Music Theory 29 (1985): 223-48; Morris, Composition with Pitch-Classes: A Theory of Compositional Design (New Haven and London: Yale University Press, 1987); and Polansky, "Morphological Metrics: An Introduction to a Theory of Formal Distances" in Proceedings of the International Computer Music Conference (San Fran- cisco: Computer Music Association, 1987).

4See Marvin and Laprade, "Relating Musical Contours: Extensions of a Theory for Contour," Journal of Music Theory 31 (1987): 225-67.

SSee Friedmann, "Methodology"; Friedmann, "A Response: My Con- tour, Their Contour," Journal of Music Theory 31 (1987): 268-74; Elizabeth West Marvin, "The Perception of Rhythm in Non-Tonal Music: Rhythmic Contours in the Music of Edgard Varese," Music Theory Spectrum 13 (1991): 61-78; and Marvin, "A Generalization of Contour Theory."

6See David Lidov and Jim Gabura, "A Melody Writing Algorithm Using a Formal Language Model," Computers in the Humanities 3-4 (1973): 138-48.

Michael Friedmann, Larry Polansky, and myself.3 My work, which defines "contour-spaces" with its various entities and relations, has been extended by Elizabeth Marvin and Paul Laprade.4 While my aim was in part to provide a secure foundation for the articulation of compositional designs of pitch classes, Friedmann and Marvin have been more con- cerned with developing a methodology for the study of con- tour associations in post-tonal music.5 In this way, their work mirrors the extents and limits of atonal theory of pitch-class (pc) sets. In addition, David Lidov and Jim Gabura have considered certain formal aspects of contour in their con- struction of an algorithm for writing melodies.6

3See Friedmann, "A Methodology for the Discussion of Contour: Its Application to Schoenberg's Music," Journal of Music Theory 29 (1985): 223-48; Morris, Composition with Pitch-Classes: A Theory of Compositional Design (New Haven and London: Yale University Press, 1987); and Polansky, "Morphological Metrics: An Introduction to a Theory of Formal Distances" in Proceedings of the International Computer Music Conference (San Fran- cisco: Computer Music Association, 1987).

4See Marvin and Laprade, "Relating Musical Contours: Extensions of a Theory for Contour," Journal of Music Theory 31 (1987): 225-67.

SSee Friedmann, "Methodology"; Friedmann, "A Response: My Con- tour, Their Contour," Journal of Music Theory 31 (1987): 268-74; Elizabeth West Marvin, "The Perception of Rhythm in Non-Tonal Music: Rhythmic Contours in the Music of Edgard Varese," Music Theory Spectrum 13 (1991): 61-78; and Marvin, "A Generalization of Contour Theory."

6See David Lidov and Jim Gabura, "A Melody Writing Algorithm Using a Formal Language Model," Computers in the Humanities 3-4 (1973): 138-48.

Michael Friedmann, Larry Polansky, and myself.3 My work, which defines "contour-spaces" with its various entities and relations, has been extended by Elizabeth Marvin and Paul Laprade.4 While my aim was in part to provide a secure foundation for the articulation of compositional designs of pitch classes, Friedmann and Marvin have been more con- cerned with developing a methodology for the study of con- tour associations in post-tonal music.5 In this way, their work mirrors the extents and limits of atonal theory of pitch-class (pc) sets. In addition, David Lidov and Jim Gabura have considered certain formal aspects of contour in their con- struction of an algorithm for writing melodies.6

3See Friedmann, "A Methodology for the Discussion of Contour: Its Application to Schoenberg's Music," Journal of Music Theory 29 (1985): 223-48; Morris, Composition with Pitch-Classes: A Theory of Compositional Design (New Haven and London: Yale University Press, 1987); and Polansky, "Morphological Metrics: An Introduction to a Theory of Formal Distances" in Proceedings of the International Computer Music Conference (San Fran- cisco: Computer Music Association, 1987).

4See Marvin and Laprade, "Relating Musical Contours: Extensions of a Theory for Contour," Journal of Music Theory 31 (1987): 225-67.

SSee Friedmann, "Methodology"; Friedmann, "A Response: My Con- tour, Their Contour," Journal of Music Theory 31 (1987): 268-74; Elizabeth West Marvin, "The Perception of Rhythm in Non-Tonal Music: Rhythmic Contours in the Music of Edgard Varese," Music Theory Spectrum 13 (1991): 61-78; and Marvin, "A Generalization of Contour Theory."

6See David Lidov and Jim Gabura, "A Melody Writing Algorithm Using a Formal Language Model," Computers in the Humanities 3-4 (1973): 138-48.

Michael Friedmann, Larry Polansky, and myself.3 My work, which defines "contour-spaces" with its various entities and relations, has been extended by Elizabeth Marvin and Paul Laprade.4 While my aim was in part to provide a secure foundation for the articulation of compositional designs of pitch classes, Friedmann and Marvin have been more con- cerned with developing a methodology for the study of con- tour associations in post-tonal music.5 In this way, their work mirrors the extents and limits of atonal theory of pitch-class (pc) sets. In addition, David Lidov and Jim Gabura have considered certain formal aspects of contour in their con- struction of an algorithm for writing melodies.6

3See Friedmann, "A Methodology for the Discussion of Contour: Its Application to Schoenberg's Music," Journal of Music Theory 29 (1985): 223-48; Morris, Composition with Pitch-Classes: A Theory of Compositional Design (New Haven and London: Yale University Press, 1987); and Polansky, "Morphological Metrics: An Introduction to a Theory of Formal Distances" in Proceedings of the International Computer Music Conference (San Fran- cisco: Computer Music Association, 1987).

4See Marvin and Laprade, "Relating Musical Contours: Extensions of a Theory for Contour," Journal of Music Theory 31 (1987): 225-67.

SSee Friedmann, "Methodology"; Friedmann, "A Response: My Con- tour, Their Contour," Journal of Music Theory 31 (1987): 268-74; Elizabeth West Marvin, "The Perception of Rhythm in Non-Tonal Music: Rhythmic Contours in the Music of Edgard Varese," Music Theory Spectrum 13 (1991): 61-78; and Marvin, "A Generalization of Contour Theory."

6See David Lidov and Jim Gabura, "A Melody Writing Algorithm Using a Formal Language Model," Computers in the Humanities 3-4 (1973): 138-48.

Michael Friedmann, Larry Polansky, and myself.3 My work, which defines "contour-spaces" with its various entities and relations, has been extended by Elizabeth Marvin and Paul Laprade.4 While my aim was in part to provide a secure foundation for the articulation of compositional designs of pitch classes, Friedmann and Marvin have been more con- cerned with developing a methodology for the study of con- tour associations in post-tonal music.5 In this way, their work mirrors the extents and limits of atonal theory of pitch-class (pc) sets. In addition, David Lidov and Jim Gabura have considered certain formal aspects of contour in their con- struction of an algorithm for writing melodies.6

3See Friedmann, "A Methodology for the Discussion of Contour: Its Application to Schoenberg's Music," Journal of Music Theory 29 (1985): 223-48; Morris, Composition with Pitch-Classes: A Theory of Compositional Design (New Haven and London: Yale University Press, 1987); and Polansky, "Morphological Metrics: An Introduction to a Theory of Formal Distances" in Proceedings of the International Computer Music Conference (San Fran- cisco: Computer Music Association, 1987).

4See Marvin and Laprade, "Relating Musical Contours: Extensions of a Theory for Contour," Journal of Music Theory 31 (1987): 225-67.

SSee Friedmann, "Methodology"; Friedmann, "A Response: My Con- tour, Their Contour," Journal of Music Theory 31 (1987): 268-74; Elizabeth West Marvin, "The Perception of Rhythm in Non-Tonal Music: Rhythmic Contours in the Music of Edgard Varese," Music Theory Spectrum 13 (1991): 61-78; and Marvin, "A Generalization of Contour Theory."

6See David Lidov and Jim Gabura, "A Melody Writing Algorithm Using a Formal Language Model," Computers in the Humanities 3-4 (1973): 138-48.

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This study begins with a review of the state of contour theory and its application to analysis. A brief study of aspects of Arnold Schoenberg's Piano Piece Op. 19 No. 4 will show how contour theory and standard versions of pc-set and trans- formational theories can interact.

The second section of the study introduces a new analytic tool, the contour reduction algorithm. Relations among the contour reductions of the six phrases of the Schoenberg piece help unify the composition in a satisfying and remarkable way: pitch classes and their sets brought out by contour hi- erarchies are related to each other as well as to other adjacent pc sets by abstract intersection and complement relations.

The issues raised in the analysis warrant a comparison of the contour reduction algorithm with the familiar family of reductive models for tonal music. In addition, the properties of an algorithm developed by James Tenney and Larry Po- lansky (which parses sequences of musical "events" into par- titioned and hierarchically embedded streams) are cited as both an adjunct and foil to contour analysis.7

The third part of the study is an elaboration of the most general definition of contour: the association of two sets. Contours that include replication and simultaneities are de- fined by various categories of mathematical relation theory. The result is a complete taxonomy of all contour types.

The present image of contour theory involves the contour, an ordered set of n distinct (contour-)pitches, with or without repetitions, numbered (not necessarily adjacently) in ascent from x to y (x< y). Normalized contours are numbered from 0 to n - 1. Contours can be written as strings of integers or as graphs (see Ex. 1). Contour intervals, denoted by the signs - (descending), 0 (equal), or + (ascending), can be con- catenated to produce an INT, or what Friedmann calls a

This study begins with a review of the state of contour theory and its application to analysis. A brief study of aspects of Arnold Schoenberg's Piano Piece Op. 19 No. 4 will show how contour theory and standard versions of pc-set and trans- formational theories can interact.

The second section of the study introduces a new analytic tool, the contour reduction algorithm. Relations among the contour reductions of the six phrases of the Schoenberg piece help unify the composition in a satisfying and remarkable way: pitch classes and their sets brought out by contour hi- erarchies are related to each other as well as to other adjacent pc sets by abstract intersection and complement relations.

The issues raised in the analysis warrant a comparison of the contour reduction algorithm with the familiar family of reductive models for tonal music. In addition, the properties of an algorithm developed by James Tenney and Larry Po- lansky (which parses sequences of musical "events" into par- titioned and hierarchically embedded streams) are cited as both an adjunct and foil to contour analysis.7

The third part of the study is an elaboration of the most general definition of contour: the association of two sets. Contours that include replication and simultaneities are de- fined by various categories of mathematical relation theory. The result is a complete taxonomy of all contour types.

The present image of contour theory involves the contour, an ordered set of n distinct (contour-)pitches, with or without repetitions, numbered (not necessarily adjacently) in ascent from x to y (x< y). Normalized contours are numbered from 0 to n - 1. Contours can be written as strings of integers or as graphs (see Ex. 1). Contour intervals, denoted by the signs - (descending), 0 (equal), or + (ascending), can be con- catenated to produce an INT, or what Friedmann calls a

This study begins with a review of the state of contour theory and its application to analysis. A brief study of aspects of Arnold Schoenberg's Piano Piece Op. 19 No. 4 will show how contour theory and standard versions of pc-set and trans- formational theories can interact.

The second section of the study introduces a new analytic tool, the contour reduction algorithm. Relations among the contour reductions of the six phrases of the Schoenberg piece help unify the composition in a satisfying and remarkable way: pitch classes and their sets brought out by contour hi- erarchies are related to each other as well as to other adjacent pc sets by abstract intersection and complement relations.

The issues raised in the analysis warrant a comparison of the contour reduction algorithm with the familiar family of reductive models for tonal music. In addition, the properties of an algorithm developed by James Tenney and Larry Po- lansky (which parses sequences of musical "events" into par- titioned and hierarchically embedded streams) are cited as both an adjunct and foil to contour analysis.7

The third part of the study is an elaboration of the most general definition of contour: the association of two sets. Contours that include replication and simultaneities are de- fined by various categories of mathematical relation theory. The result is a complete taxonomy of all contour types.

The present image of contour theory involves the contour, an ordered set of n distinct (contour-)pitches, with or without repetitions, numbered (not necessarily adjacently) in ascent from x to y (x< y). Normalized contours are numbered from 0 to n - 1. Contours can be written as strings of integers or as graphs (see Ex. 1). Contour intervals, denoted by the signs - (descending), 0 (equal), or + (ascending), can be con- catenated to produce an INT, or what Friedmann calls a

This study begins with a review of the state of contour theory and its application to analysis. A brief study of aspects of Arnold Schoenberg's Piano Piece Op. 19 No. 4 will show how contour theory and standard versions of pc-set and trans- formational theories can interact.

The second section of the study introduces a new analytic tool, the contour reduction algorithm. Relations among the contour reductions of the six phrases of the Schoenberg piece help unify the composition in a satisfying and remarkable way: pitch classes and their sets brought out by contour hi- erarchies are related to each other as well as to other adjacent pc sets by abstract intersection and complement relations.

The issues raised in the analysis warrant a comparison of the contour reduction algorithm with the familiar family of reductive models for tonal music. In addition, the properties of an algorithm developed by James Tenney and Larry Po- lansky (which parses sequences of musical "events" into par- titioned and hierarchically embedded streams) are cited as both an adjunct and foil to contour analysis.7

The third part of the study is an elaboration of the most general definition of contour: the association of two sets. Contours that include replication and simultaneities are de- fined by various categories of mathematical relation theory. The result is a complete taxonomy of all contour types.

The present image of contour theory involves the contour, an ordered set of n distinct (contour-)pitches, with or without repetitions, numbered (not necessarily adjacently) in ascent from x to y (x< y). Normalized contours are numbered from 0 to n - 1. Contours can be written as strings of integers or as graphs (see Ex. 1). Contour intervals, denoted by the signs - (descending), 0 (equal), or + (ascending), can be con- catenated to produce an INT, or what Friedmann calls a

This study begins with a review of the state of contour theory and its application to analysis. A brief study of aspects of Arnold Schoenberg's Piano Piece Op. 19 No. 4 will show how contour theory and standard versions of pc-set and trans- formational theories can interact.

The second section of the study introduces a new analytic tool, the contour reduction algorithm. Relations among the contour reductions of the six phrases of the Schoenberg piece help unify the composition in a satisfying and remarkable way: pitch classes and their sets brought out by contour hi- erarchies are related to each other as well as to other adjacent pc sets by abstract intersection and complement relations.

The issues raised in the analysis warrant a comparison of the contour reduction algorithm with the familiar family of reductive models for tonal music. In addition, the properties of an algorithm developed by James Tenney and Larry Po- lansky (which parses sequences of musical "events" into par- titioned and hierarchically embedded streams) are cited as both an adjunct and foil to contour analysis.7

The third part of the study is an elaboration of the most general definition of contour: the association of two sets. Contours that include replication and simultaneities are de- fined by various categories of mathematical relation theory. The result is a complete taxonomy of all contour types.

The present image of contour theory involves the contour, an ordered set of n distinct (contour-)pitches, with or without repetitions, numbered (not necessarily adjacently) in ascent from x to y (x< y). Normalized contours are numbered from 0 to n - 1. Contours can be written as strings of integers or as graphs (see Ex. 1). Contour intervals, denoted by the signs - (descending), 0 (equal), or + (ascending), can be con- catenated to produce an INT, or what Friedmann calls a

This study begins with a review of the state of contour theory and its application to analysis. A brief study of aspects of Arnold Schoenberg's Piano Piece Op. 19 No. 4 will show how contour theory and standard versions of pc-set and trans- formational theories can interact.

The second section of the study introduces a new analytic tool, the contour reduction algorithm. Relations among the contour reductions of the six phrases of the Schoenberg piece help unify the composition in a satisfying and remarkable way: pitch classes and their sets brought out by contour hi- erarchies are related to each other as well as to other adjacent pc sets by abstract intersection and complement relations.

The issues raised in the analysis warrant a comparison of the contour reduction algorithm with the familiar family of reductive models for tonal music. In addition, the properties of an algorithm developed by James Tenney and Larry Po- lansky (which parses sequences of musical "events" into par- titioned and hierarchically embedded streams) are cited as both an adjunct and foil to contour analysis.7

The third part of the study is an elaboration of the most general definition of contour: the association of two sets. Contours that include replication and simultaneities are de- fined by various categories of mathematical relation theory. The result is a complete taxonomy of all contour types.

The present image of contour theory involves the contour, an ordered set of n distinct (contour-)pitches, with or without repetitions, numbered (not necessarily adjacently) in ascent from x to y (x< y). Normalized contours are numbered from 0 to n - 1. Contours can be written as strings of integers or as graphs (see Ex. 1). Contour intervals, denoted by the signs - (descending), 0 (equal), or + (ascending), can be con- catenated to produce an INT, or what Friedmann calls a

This study begins with a review of the state of contour theory and its application to analysis. A brief study of aspects of Arnold Schoenberg's Piano Piece Op. 19 No. 4 will show how contour theory and standard versions of pc-set and trans- formational theories can interact.

The second section of the study introduces a new analytic tool, the contour reduction algorithm. Relations among the contour reductions of the six phrases of the Schoenberg piece help unify the composition in a satisfying and remarkable way: pitch classes and their sets brought out by contour hi- erarchies are related to each other as well as to other adjacent pc sets by abstract intersection and complement relations.

The issues raised in the analysis warrant a comparison of the contour reduction algorithm with the familiar family of reductive models for tonal music. In addition, the properties of an algorithm developed by James Tenney and Larry Po- lansky (which parses sequences of musical "events" into par- titioned and hierarchically embedded streams) are cited as both an adjunct and foil to contour analysis.7

The third part of the study is an elaboration of the most general definition of contour: the association of two sets. Contours that include replication and simultaneities are de- fined by various categories of mathematical relation theory. The result is a complete taxonomy of all contour types.

The present image of contour theory involves the contour, an ordered set of n distinct (contour-)pitches, with or without repetitions, numbered (not necessarily adjacently) in ascent from x to y (x< y). Normalized contours are numbered from 0 to n - 1. Contours can be written as strings of integers or as graphs (see Ex. 1). Contour intervals, denoted by the signs - (descending), 0 (equal), or + (ascending), can be con- catenated to produce an INT, or what Friedmann calls a

This study begins with a review of the state of contour theory and its application to analysis. A brief study of aspects of Arnold Schoenberg's Piano Piece Op. 19 No. 4 will show how contour theory and standard versions of pc-set and trans- formational theories can interact.

The second section of the study introduces a new analytic tool, the contour reduction algorithm. Relations among the contour reductions of the six phrases of the Schoenberg piece help unify the composition in a satisfying and remarkable way: pitch classes and their sets brought out by contour hi- erarchies are related to each other as well as to other adjacent pc sets by abstract intersection and complement relations.

The issues raised in the analysis warrant a comparison of the contour reduction algorithm with the familiar family of reductive models for tonal music. In addition, the properties of an algorithm developed by James Tenney and Larry Po- lansky (which parses sequences of musical "events" into par- titioned and hierarchically embedded streams) are cited as both an adjunct and foil to contour analysis.7

The third part of the study is an elaboration of the most general definition of contour: the association of two sets. Contours that include replication and simultaneities are de- fined by various categories of mathematical relation theory. The result is a complete taxonomy of all contour types.

The present image of contour theory involves the contour, an ordered set of n distinct (contour-)pitches, with or without repetitions, numbered (not necessarily adjacently) in ascent from x to y (x< y). Normalized contours are numbered from 0 to n - 1. Contours can be written as strings of integers or as graphs (see Ex. 1). Contour intervals, denoted by the signs - (descending), 0 (equal), or + (ascending), can be con- catenated to produce an INT, or what Friedmann calls a

This study begins with a review of the state of contour theory and its application to analysis. A brief study of aspects of Arnold Schoenberg's Piano Piece Op. 19 No. 4 will show how contour theory and standard versions of pc-set and trans- formational theories can interact.

The second section of the study introduces a new analytic tool, the contour reduction algorithm. Relations among the contour reductions of the six phrases of the Schoenberg piece help unify the composition in a satisfying and remarkable way: pitch classes and their sets brought out by contour hi- erarchies are related to each other as well as to other adjacent pc sets by abstract intersection and complement relations.

The issues raised in the analysis warrant a comparison of the contour reduction algorithm with the familiar family of reductive models for tonal music. In addition, the properties of an algorithm developed by James Tenney and Larry Po- lansky (which parses sequences of musical "events" into par- titioned and hierarchically embedded streams) are cited as both an adjunct and foil to contour analysis.7

The third part of the study is an elaboration of the most general definition of contour: the association of two sets. Contours that include replication and simultaneities are de- fined by various categories of mathematical relation theory. The result is a complete taxonomy of all contour types.

The present image of contour theory involves the contour, an ordered set of n distinct (contour-)pitches, with or without repetitions, numbered (not necessarily adjacently) in ascent from x to y (x< y). Normalized contours are numbered from 0 to n - 1. Contours can be written as strings of integers or as graphs (see Ex. 1). Contour intervals, denoted by the signs - (descending), 0 (equal), or + (ascending), can be con- catenated to produce an INT, or what Friedmann calls a

This study begins with a review of the state of contour theory and its application to analysis. A brief study of aspects of Arnold Schoenberg's Piano Piece Op. 19 No. 4 will show how contour theory and standard versions of pc-set and trans- formational theories can interact.

The second section of the study introduces a new analytic tool, the contour reduction algorithm. Relations among the contour reductions of the six phrases of the Schoenberg piece help unify the composition in a satisfying and remarkable way: pitch classes and their sets brought out by contour hi- erarchies are related to each other as well as to other adjacent pc sets by abstract intersection and complement relations.

The issues raised in the analysis warrant a comparison of the contour reduction algorithm with the familiar family of reductive models for tonal music. In addition, the properties of an algorithm developed by James Tenney and Larry Po- lansky (which parses sequences of musical "events" into par- titioned and hierarchically embedded streams) are cited as both an adjunct and foil to contour analysis.7

The third part of the study is an elaboration of the most general definition of contour: the association of two sets. Contours that include replication and simultaneities are de- fined by various categories of mathematical relation theory. The result is a complete taxonomy of all contour types.

The present image of contour theory involves the contour, an ordered set of n distinct (contour-)pitches, with or without repetitions, numbered (not necessarily adjacently) in ascent from x to y (x< y). Normalized contours are numbered from 0 to n - 1. Contours can be written as strings of integers or as graphs (see Ex. 1). Contour intervals, denoted by the signs - (descending), 0 (equal), or + (ascending), can be con- catenated to produce an INT, or what Friedmann calls a

This study begins with a review of the state of contour theory and its application to analysis. A brief study of aspects of Arnold Schoenberg's Piano Piece Op. 19 No. 4 will show how contour theory and standard versions of pc-set and trans- formational theories can interact.

The second section of the study introduces a new analytic tool, the contour reduction algorithm. Relations among the contour reductions of the six phrases of the Schoenberg piece help unify the composition in a satisfying and remarkable way: pitch classes and their sets brought out by contour hi- erarchies are related to each other as well as to other adjacent pc sets by abstract intersection and complement relations.

The issues raised in the analysis warrant a comparison of the contour reduction algorithm with the familiar family of reductive models for tonal music. In addition, the properties of an algorithm developed by James Tenney and Larry Po- lansky (which parses sequences of musical "events" into par- titioned and hierarchically embedded streams) are cited as both an adjunct and foil to contour analysis.7

The third part of the study is an elaboration of the most general definition of contour: the association of two sets. Contours that include replication and simultaneities are de- fined by various categories of mathematical relation theory. The result is a complete taxonomy of all contour types.

The present image of contour theory involves the contour, an ordered set of n distinct (contour-)pitches, with or without repetitions, numbered (not necessarily adjacently) in ascent from x to y (x< y). Normalized contours are numbered from 0 to n - 1. Contours can be written as strings of integers or as graphs (see Ex. 1). Contour intervals, denoted by the signs - (descending), 0 (equal), or + (ascending), can be con- catenated to produce an INT, or what Friedmann calls a

7See Tenney and Polansky, "Temporal Gestalt Perception in Music," Journal of Music Theory 24 (1980): 205-41.

7See Tenney and Polansky, "Temporal Gestalt Perception in Music," Journal of Music Theory 24 (1980): 205-41.

7See Tenney and Polansky, "Temporal Gestalt Perception in Music," Journal of Music Theory 24 (1980): 205-41.

7See Tenney and Polansky, "Temporal Gestalt Perception in Music," Journal of Music Theory 24 (1980): 205-41.

7See Tenney and Polansky, "Temporal Gestalt Perception in Music," Journal of Music Theory 24 (1980): 205-41.

7See Tenney and Polansky, "Temporal Gestalt Perception in Music," Journal of Music Theory 24 (1980): 205-41.

7See Tenney and Polansky, "Temporal Gestalt Perception in Music," Journal of Music Theory 24 (1980): 205-41.

7See Tenney and Polansky, "Temporal Gestalt Perception in Music," Journal of Music Theory 24 (1980): 205-41.

7See Tenney and Polansky, "Temporal Gestalt Perception in Music," Journal of Music Theory 24 (1980): 205-41.

7See Tenney and Polansky, "Temporal Gestalt Perception in Music," Journal of Music Theory 24 (1980): 205-41.

7See Tenney and Polansky, "Temporal Gestalt Perception in Music," Journal of Music Theory 24 (1980): 205-41.

CAS.8 An array of contour intervals, called a COM-matrix, has the function of the interval-vector in pc-set theory. The contour interval from the ath member to the bth member of a contour is located in the intersection of the COM-matrix's ath row and bth column.

Contours need not be of pitches in time. As Example 2 shows, the contour can be interpreted as pitches, dynamics, or chord densities in time. As we will see later, contours need not even be temporal.

Some of the properties of a COM-matrix are illustrated in Example 3. The matrix's main diagonal is the series of zeros descending from its upper left hand corner. The CAS or INT, is always given on the diagonal to the right of the main di- agonal. These features and others, like the plus/minus sym- metry around the main diagonal, show that the COM-matrix is exactly analogous to a row table or T-matrix in atonal theory.

Given the wealth of all possible contours, it is useful to define equivalence relations in order to group contours into types or equivalence classes. The most important kinds of equivalence are two. The first is founded on similitude. This kind of similarity is based on identical COM-matrices, since two contours that have the same overall shape share the same COM-matrix. A set of equivalent contours is called a cseg.9 See Example 4.

The other criterion for equivalence involves transforma- tional relations. Two different contours or csegs are trans- formationally equivalent if they are related by identity, ret- rograde, inversion, and/or retrograde-inversion, to within the

8The INT1 of a contour corresponds to the INTi of pc-set theory; INT, of the ordered pc set X lists the successive ordered pc intervals between the pcs of X. See Morris, Composition. Friedmann's equivalent term CAS stands for Contour Adjacency Series. See Friedmann, "Methodology."

9All equivalent contours reduce to the same contour when normalized. This normalized contour can be used to name the cseg. In Example 4. the three equivalent contours reduce to .

CAS.8 An array of contour intervals, called a COM-matrix, has the function of the interval-vector in pc-set theory. The contour interval from the ath member to the bth member of a contour is located in the intersection of the COM-matrix's ath row and bth column.

Contours need not be of pitches in time. As Example 2 shows, the contour can be interpreted as pitches, dynamics, or chord densities in time. As we will see later, contours need not even be temporal.

Some of the properties of a COM-matrix are illustrated in Example 3. The matrix's main diagonal is the series of zeros descending from its upper left hand corner. The CAS or INT, is always given on the diagonal to the right of the main di- agonal. These features and others, like the plus/minus sym- metry around the main diagonal, show that the COM-matrix is exactly analogous to a row table or T-matrix in atonal theory.

Given the wealth of all possible contours, it is useful to define equivalence relations in order to group contours into types or equivalence classes. The most important kinds of equivalence are two. The first is founded on similitude. This kind of similarity is based on identical COM-matrices, since two contours that have the same overall shape share the same COM-matrix. A set of equivalent contours is called a cseg.9 See Example 4.

The other criterion for equivalence involves transforma- tional relations. Two different contours or csegs are trans- formationally equivalent if they are related by identity, ret- rograde, inversion, and/or retrograde-inversion, to within the

8The INT1 of a contour corresponds to the INTi of pc-set theory; INT, of the ordered pc set X lists the successive ordered pc intervals between the pcs of X. See Morris, Composition. Friedmann's equivalent term CAS stands for Contour Adjacency Series. See Friedmann, "Methodology."

9All equivalent contours reduce to the same contour when normalized. This normalized contour can be used to name the cseg. In Example 4. the three equivalent contours reduce to .

CAS.8 An array of contour intervals, called a COM-matrix, has the function of the interval-vector in pc-set theory. The contour interval from the ath member to the bth member of a contour is located in the intersection of the COM-matrix's ath row and bth column.

Contours need not be of pitches in time. As Example 2 shows, the contour can be interpreted as pitches, dynamics, or chord densities in time. As we will see later, contours need not even be temporal.

Some of the properties of a COM-matrix are illustrated in Example 3. The matrix's main diagonal is the series of zeros descending from its upper left hand corner. The CAS or INT, is always given on the diagonal to the right of the main di- agonal. These features and others, like the plus/minus sym- metry around the main diagonal, show that the COM-matrix is exactly analogous to a row table or T-matrix in atonal theory.

Given the wealth of all possible contours, it is useful to define equivalence relations in order to group contours into types or equivalence classes. The most important kinds of equivalence are two. The first is founded on similitude. This kind of similarity is based on identical COM-matrices, since two contours that have the same overall shape share the same COM-matrix. A set of equivalent contours is called a cseg.9 See Example 4.

The other criterion for equivalence involves transforma- tional relations. Two different contours or csegs are trans- formationally equivalent if they are related by identity, ret- rograde, inversion, and/or retrograde-inversion, to within the

8The INT1 of a contour corresponds to the INTi of pc-set theory; INT, of the ordered pc set X lists the successive ordered pc intervals between the pcs of X. See Morris, Composition. Friedmann's equivalent term CAS stands for Contour Adjacency Series. See Friedmann, "Methodology."

9All equivalent contours reduce to the same contour when normalized. This normalized contour can be used to name the cseg. In Example 4. the three equivalent contours reduce to .

CAS.8 An array of contour intervals, called a COM-matrix, has the function of the interval-vector in pc-set theory. The contour interval from the ath member to the bth member of a contour is located in the intersection of the COM-matrix's ath row and bth column.

Contours need not be of pitches in time. As Example 2 shows, the contour can be interpreted as pitches, dynamics, or chord densities in time. As we will see later, contours need not even be temporal.

Some of the properties of a COM-matrix are illustrated in Example 3. The matrix's main diagonal is the series of zeros descending from its upper left hand corner. The CAS or INT, is always given on the diagonal to the right of the main di- agonal. These features and others, like the plus/minus sym- metry around the main diagonal, show that the COM-matrix is exactly analogous to a row table or T-matrix in atonal theory.

Given the wealth of all possible contours, it is useful to define equivalence relations in order to group contours into types or equivalence classes. The most important kinds of equivalence are two. The first is founded on similitude. This kind of similarity is based on identical COM-matrices, since two contours that have the same overall shape share the same COM-matrix. A set of equivalent contours is called a cseg.9 See Example 4.

The other criterion for equivalence involves transforma- tional relations. Two different contours or csegs are trans- formationally equivalent if they are related by identity, ret- rograde, inversion, and/or retrograde-inversion, to within the

8The INT1 of a contour corresponds to the INTi of pc-set theory; INT, of the ordered pc set X lists the successive ordered pc intervals between the pcs of X. See Morris, Composition. Friedmann's equivalent term CAS stands for Contour Adjacency Series. See Friedmann, "Methodology."

9All equivalent contours reduce to the same contour when normalized. This normalized contour can be used to name the cseg. In Example 4. the three equivalent contours reduce to .

CAS.8 An array of contour intervals, called a COM-matrix, has the function of the interval-vector in pc-set theory. The contour interval from the ath member to the bth member of a contour is located in the intersection of the COM-matrix's ath row and bth column.

Contours need not be of pitches in time. As Example 2 shows, the contour can be interpreted as pitches, dynamics, or chord densities in time. As we will see later, contours need not even be temporal.

Some of the properties of a COM-matrix are illustrated in Example 3. The matrix's main diagonal is the series of zeros descending from its upper left hand corner. The CAS or INT, is always given on the diagonal to the right of the main di- agonal. These features and others, like the plus/minus sym- metry around the main diagonal, show that the COM-matrix is exactly analogous to a row table or T-matrix in atonal theory.

Given the wealth of all possible contours, it is useful to define equivalence relations in order to group contours into types or equivalence classes. The most important kinds of equivalence are two. The first is founded on similitude. This kind of similarity is based on identical COM-matrices, since two contours that have the same overall shape share the same COM-matrix. A set of equivalent contours is called a cseg.9 See Example 4.

The other criterion for equivalence involves transforma- tional relations. Two different contours or csegs are trans- formationally equivalent if they are related by identity, ret- rograde, inversion, and/or retrograde-inversion, to within the

8The INT1 of a contour corresponds to the INTi of pc-set theory; INT, of the ordered pc set X lists the successive ordered pc intervals between the pcs of X. See Morris, Composition. Friedmann's equivalent term CAS stands for Contour Adjacency Series. See Friedmann, "Methodology."

9All equivalent contours reduce to the same contour when normalized. This normalized contour can be used to name the cseg. In Example 4. the three equivalent contours reduce to .

CAS.8 An array of contour intervals, called a COM-matrix, has the function of the interval-vector in pc-set theory. The contour interval from the ath member to the bth member of a contour is located in the intersection of the COM-matrix's ath row and bth column.

Contours need not be of pitches in time. As Example 2 shows, the contour can be interpreted as pitches, dynamics, or chord densities in time. As we will see later, contours need not even be temporal.

Some of the properties of a COM-matrix are illustrated in Example 3. The matrix's main diagonal is the series of zeros descending from its upper left hand corner. The CAS or INT, is always given on the diagonal to the right of the main di- agonal. These features and others, like the plus/minus sym- metry around the main diagonal, show that the COM-matrix is exactly analogous to a row table or T-matrix in atonal theory.

Given the wealth of all possible contours, it is useful to define equivalence relations in order to group contours into types or equivalence classes. The most important kinds of equivalence are two. The first is founded on similitude. This kind of similarity is based on identical COM-matrices, since two contours that have the same overall shape share the same COM-matrix. A set of equivalent contours is called a cseg.9 See Example 4.

The other criterion for equivalence involves transforma- tional relations. Two different contours or csegs are trans- formationally equivalent if they are related by identity, ret- rograde, inversion, and/or retrograde-inversion, to within the

8The INT1 of a contour corresponds to the INTi of pc-set theory; INT, of the ordered pc set X lists the successive ordered pc intervals between the pcs of X. See Morris, Composition. Friedmann's equivalent term CAS stands for Contour Adjacency Series. See Friedmann, "Methodology."

9All equivalent contours reduce to the same contour when normalized. This normalized contour can be used to name the cseg. In Example 4. the three equivalent contours reduce to .

CAS.8 An array of contour intervals, called a COM-matrix, has the function of the interval-vector in pc-set theory. The contour interval from the ath member to the bth member of a contour is located in the intersection of the COM-matrix's ath row and bth column.

Contours need not be of pitches in time. As Example 2 shows, the contour can be interpreted as pitches, dynamics, or chord densities in time. As we will see later, contours need not even be temporal.

Some of the properties of a COM-matrix are illustrated in Example 3. The matrix's main diagonal is the series of zeros descending from its upper left hand corner. The CAS or INT, is always given on the diagonal to the right of the main di- agonal. These features and others, like the plus/minus sym- metry around the main diagonal, show that the COM-matrix is exactly analogous to a row table or T-matrix in atonal theory.

Given the wealth of all possible contours, it is useful to define equivalence relations in order to group contours into types or equivalence classes. The most important kinds of equivalence are two. The first is founded on similitude. This kind of similarity is based on identical COM-matrices, since two contours that have the same overall shape share the same COM-matrix. A set of equivalent contours is called a cseg.9 See Example 4.

The other criterion for equivalence involves transforma- tional relations. Two different contours or csegs are trans- formationally equivalent if they are related by identity, ret- rograde, inversion, and/or retrograde-inversion, to within the

8The INT1 of a contour corresponds to the INTi of pc-set theory; INT, of the ordered pc set X lists the successive ordered pc intervals between the pcs of X. See Morris, Composition. Friedmann's equivalent term CAS stands for Contour Adjacency Series. See Friedmann, "Methodology."

9All equivalent contours reduce to the same contour when normalized. This normalized contour can be used to name the cseg. In Example 4. the three equivalent contours reduce to .

CAS.8 An array of contour intervals, called a COM-matrix, has the function of the interval-vector in pc-set theory. The contour interval from the ath member to the bth member of a contour is located in the intersection of the COM-matrix's ath row and bth column.

Contours need not be of pitches in time. As Example 2 shows, the contour can be interpreted as pitches, dynamics, or chord densities in time. As we will see later, contours need not even be temporal.

Some of the properties of a COM-matrix are illustrated in Example 3. The matrix's main diagonal is the series of zeros descending from its upper left hand corner. The CAS or INT, is always given on the diagonal to the right of the main di- agonal. These features and others, like the plus/minus sym- metry around the main diagonal, show that the COM-matrix is exactly analogous to a row table or T-matrix in atonal theory.

Given the wealth of all possible contours, it is useful to define equivalence relations in order to group contours into types or equivalence classes. The most important kinds of equivalence are two. The first is founded on similitude. This kind of similarity is based on identical COM-matrices, since two contours that have the same overall shape share the same COM-matrix. A set of equivalent contours is called a cseg.9 See Example 4.

The other criterion for equivalence involves transforma- tional relations. Two different contours or csegs are trans- formationally equivalent if they are related by identity, ret- rograde, inversion, and/or retrograde-inversion, to within the

8The INT1 of a contour corresponds to the INTi of pc-set theory; INT, of the ordered pc set X lists the successive ordered pc intervals between the pcs of X. See Morris, Composition. Friedmann's equivalent term CAS stands for Contour Adjacency Series. See Friedmann, "Methodology."

9All equivalent contours reduce to the same contour when normalized. This normalized contour can be used to name the cseg. In Example 4. the three equivalent contours reduce to .

CAS.8 An array of contour intervals, called a COM-matrix, has the function of the interval-vector in pc-set theory. The contour interval from the ath member to the bth member of a contour is located in the intersection of the COM-matrix's ath row and bth column.

Contours need not be of pitches in time. As Example 2 shows, the contour can be interpreted as pitches, dynamics, or chord densities in time. As we will see later, contours need not even be temporal.

Some of the properties of a COM-matrix are illustrated in Example 3. The matrix's main diagonal is the series of zeros descending from its upper left hand corner. The CAS or INT, is always given on the diagonal to the right of the main di- agonal. These features and others, like the plus/minus sym- metry around the main diagonal, show that the COM-matrix is exactly analogous to a row table or T-matrix in atonal theory.

Given the wealth of all possible contours, it is useful to define equivalence relations in order to group contours into types or equivalence classes. The most important kinds of equivalence are two. The first is founded on similitude. This kind of similarity is based on identical COM-matrices, since two contours that have the same overall shape share the same COM-matrix. A set of equivalent contours is called a cseg.9 See Example 4.

The other criterion for equivalence involves transforma- tional relations. Two different contours or csegs are trans- formationally equivalent if they are related by identity, ret- rograde, inversion, and/or retrograde-inversion, to within the

8The INT1 of a contour corresponds to the INTi of pc-set theory; INT, of the ordered pc set X lists the successive ordered pc intervals between the pcs of X. See Morris, Composition. Friedmann's equivalent term CAS stands for Contour Adjacency Series. See Friedmann, "Methodology."

9All equivalent contours reduce to the same contour when normalized. This normalized contour can be used to name the cseg. In Example 4. the three equivalent contours reduce to .

CAS.8 An array of contour intervals, called a COM-matrix, has the function of the interval-vector in pc-set theory. The contour interval from the ath member to the bth member of a contour is located in the intersection of the COM-matrix's ath row and bth column.

Contours need not be of pitches in time. As Example 2 shows, the contour can be interpreted as pitches, dynamics, or chord densities in time. As we will see later, contours need not even be temporal.

Some of the properties of a COM-matrix are illustrated in Example 3. The matrix's main diagonal is the series of zeros descending from its upper left hand corner. The CAS or INT, is always given on the diagonal to the right of the main di- agonal. These features and others, like the plus/minus sym- metry around the main diagonal, show that the COM-matrix is exactly analogous to a row table or T-matrix in atonal theory.

Given the wealth of all possible contours, it is useful to define equivalence relations in order to group contours into types or equivalence classes. The most important kinds of equivalence are two. The first is founded on similitude. This kind of similarity is based on identical COM-matrices, since two contours that have the same overall shape share the same COM-matrix. A set of equivalent contours is called a cseg.9 See Example 4.

The other criterion for equivalence involves transforma- tional relations. Two different contours or csegs are trans- formationally equivalent if they are related by identity, ret- rograde, inversion, and/or retrograde-inversion, to within the

8The INT1 of a contour corresponds to the INTi of pc-set theory; INT, of the ordered pc set X lists the successive ordered pc intervals between the pcs of X. See Morris, Composition. Friedmann's equivalent term CAS stands for Contour Adjacency Series. See Friedmann, "Methodology."

9All equivalent contours reduce to the same contour when normalized. This normalized contour can be used to name the cseg. In Example 4. the three equivalent contours reduce to .

CAS.8 An array of contour intervals, called a COM-matrix, has the function of the interval-vector in pc-set theory. The contour interval from the ath member to the bth member of a contour is located in the intersection of the COM-matrix's ath row and bth column.

Contours need not be of pitches in time. As Example 2 shows, the contour can be interpreted as pitches, dynamics, or chord densities in time. As we will see later, contours need not even be temporal.

Some of the properties of a COM-matrix are illustrated in Example 3. The matrix's main diagonal is the series of zeros descending from its upper left hand corner. The CAS or INT, is always given on the diagonal to the right of the main di- agonal. These features and others, like the plus/minus sym- metry around the main diagonal, show that the COM-matrix is exactly analogous to a row table or T-matrix in atonal theory.

Given the wealth of all possible contours, it is useful to define equivalence relations in order to group contours into types or equivalence classes. The most important kinds of equivalence are two. The first is founded on similitude. This kind of similarity is based on identical COM-matrices, since two contours that have the same overall shape share the same COM-matrix. A set of equivalent contours is called a cseg.9 See Example 4.

The other criterion for equivalence involves transforma- tional relations. Two different contours or csegs are trans- formationally equivalent if they are related by identity, ret- rograde, inversion, and/or retrograde-inversion, to within the

8The INT1 of a contour corresponds to the INTi of pc-set theory; INT, of the ordered pc set X lists the successive ordered pc intervals between the pcs of X. See Morris, Composition. Friedmann's equivalent term CAS stands for Contour Adjacency Series. See Friedmann, "Methodology."

9All equivalent contours reduce to the same contour when normalized. This normalized contour can be used to name the cseg. In Example 4. the three equivalent contours reduce to .

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Theory and Analysis of Musical Contour 207 Theory and Analysis of Musical Contour 207 Theory and Analysis of Musical Contour 207 Theory and Analysis of Musical Contour 207 Theory and Analysis of Musical Contour 207 Theory and Analysis of Musical Contour 207 Theory and Analysis of Musical Contour 207 Theory and Analysis of Musical Contour 207 Theory and Analysis of Musical Contour 207 Theory and Analysis of Musical Contour 207 Theory and Analysis of Musical Contour 207

Example 1. Contours: notation and concepts Example 1. Contours: notation and concepts Example 1. Contours: notation and concepts Example 1. Contours: notation and concepts Example 1. Contours: notation and concepts Example 1. Contours: notation and concepts Example 1. Contours: notation and concepts Example 1. Contours: notation and concepts Example 1. Contours: notation and concepts Example 1. Contours: notation and concepts Example 1. Contours: notation and concepts

contour of X :< 0 2 3 1 > contour of X :< 0 2 3 1 > contour of X :< 0 2 3 1 > contour of X :< 0 2 3 1 > contour of X :< 0 2 3 1 > contour of X :< 0 2 3 1 > contour of X :< 0 2 3 1 > contour of X :< 0 2 3 1 > contour of X :< 0 2 3 1 > contour of X :< 0 2 3 1 > contour of X :< 0 2 3 1 > graph of X graph of X graph of X graph of X graph of X graph of X graph of X graph of X graph of X graph of X graph of X

INT1 or CAS (contour adjacency series) of X : + + - > INT1 or CAS (contour adjacency series) of X : + + - > INT1 or CAS (contour adjacency series) of X : + + - > INT1 or CAS (contour adjacency series) of X : + + - > INT1 or CAS (contour adjacency series) of X : + + - > INT1 or CAS (contour adjacency series) of X : + + - > INT1 or CAS (contour adjacency series) of X : + + - > INT1 or CAS (contour adjacency series) of X : + + - > INT1 or CAS (contour adjacency series) of X : + + - > INT1 or CAS (contour adjacency series) of X : + + - > INT1 or CAS (contour adjacency series) of X : + + - >

Example 3. COM-matrix of < 0 3 1 2 > Example 3. COM-matrix of < 0 3 1 2 > Example 3. COM-matrix of < 0 3 1 2 > Example 3. COM-matrix of < 0 3 1 2 > Example 3. COM-matrix of < 0 3 1 2 > Example 3. COM-matrix of < 0 3 1 2 > Example 3. COM-matrix of < 0 3 1 2 > Example 3. COM-matrix of < 0 3 1 2 > Example 3. COM-matrix of < 0 3 1 2 > Example 3. COM-matrix of < 0 3 1 2 > Example 3. COM-matrix of < 0 3 1 2 >

main diagonal

0 + + +

- 0 - -

- + 0 +

- >+ 0 INTI of contour = < +,-, + >

main diagonal

0 + + +