집합이론연구

 

 Set Theory

 

 

1998년 2학기

 

 

 

한양대학교 대학원 음악이론전공 박사과정

담당교수 : 박재성 (朴在成)  월요일 4:00-5:15    금요일 10:30-12:00

 

강의목표 및 교과과정

 

집합이론의 원리와 이론을 발전시킨 문헌들을 연구한다.  또한 집합이론을 작품 분석에 적용시키는 방법을 연구한다.

 

 

1.

강의

2.

토론 (과제물로 주어지는 논문에 관한 질문·답변 등에 적극적으로 참여하고, 또한 자신의 의견을 개진할 수 있어야 함).

 

 

평가방법 : 절대 평가와 상대 평가를 혼합 시켜 평가함

강의시간에서의 토론

50%

(과제로 주어지는 “서적-논문-작품의 내용에 관한 토론)

중간 리포트

20%

(강의 시간에 발표함)

기말 리포트

20%

(강의 시간에 발표하지 않음)

출결

10%

(결석은 원칙적으로 허용되지 않음. 단, 사전에 통보를 하여 담당교수의 인정을 받을 경우에는 감점되지 않음.

 

 

 

제 1 주

Introduction: Concepts and Basic Operations of Set Theory

 

Joseph N. Straus, Introduction to Post-Tonal Theory, pp. 1-58:

 

(1)

Chapter 1 "Basic Concepts and Definitions" (pp. 1-15): Octave equivalence; Pitch class; Enharmonic equivalence; Integer notation; Mod 12; Intervals; Pitch intervals; Ordered pitch-class intervals; Unordered pitch-class intervals; Interval class; Interval-class content; Exercises.

 

(2)

Analysis: Webern, "Wie bin ich froh!" (How happy I am!) from Three Songs, Op. 25 (pp. 16-22)

 

(3)

 Analysis: Schoenberg, "Nacht," from Pierrot Lunaire, Op. 21 (pp. 23-25)

 

(4)

Chapter 2 "Pitch-Class Sets" (pp. 26-46): Pitch-class sets; Normal form; Transposition; Inversion; Index number; Set class; Prime form; Exercise.

 

(5)

Analysis: Schoenberg, Book of the Hanging Gardens, Op. 15, No. 11 (pp. 47-52)

 

(6)

Analysis: Bartók, String Quartet No. 4, 1st Mov. (pp. 53-58)

Jeffrey L. Gillespie, "Motivic Transformations and Networks in Schoenberg's 'Nacht' from Pierrot Lunaire," Intégral 6 (1992): 34-65.

제 2 주

Joseph N. Straus, Introduction to Post-Tonal Theory:

 

(1)

Chapter 3 "Some Additional Relationships" (pp. 59-78): Common tones under transposition (Tn); Common tones under inversion (TnI); Degrees of symmetry; Z-relation; Complement relation; Subset and superset relations; Large-scale composing-out; Exercises

 

(2)

Analysis: Webern, Movements for String Quartet, Op. 5, No. 4 (pp. 79-83)

 

(3)

Analysis: Berg, "Schlafend trägt man mich," from Four Songs, Op. 2, No. 2 (pp. 84-88)

David W. Beach, "Pitch Structure and the Analytic Process in Atonal Music: An Interpretation of the Theory of Sets," Music Theory Spectrum 1 (1979): 7-22.

제 3 주

Allen Forte, The Structure of Atonal Music, pp. 24-46 ("The subsets of a pc set," IInvariant subsets under transposition," "InInvariant subsets under inversion.”

Stefan Kostka, Materials and Techniques of Twentieth-Century Music, pp. 183-205 ("Nonserial Atonality").

() Gary wittlich, "Sets and Ordering Procedures in Twentieth-Century Music," in Gary Wittlich, ed., Aspects of Twentieth-Century Music, pp. 455-70.

제 4 주

Robert D. Morris, Class Notes for Atonal Music Theory:

 

(1)

Chapter 1 "Some Definitions in Atonal Music Theory" (pp. 1-9): Pitch-Space (p-space); Pitch; Ordered Pitch Interval; Unordered Pitch Interval; Pitch-Class Space (pc-space); Pitch-Class (pc); Mod 12 Arithmetic; Ordered Pitch-Class Interval (interval); Interval Class (ic); PCset; Set Membership (); Set Union (); Set Intersection (); Set Inclusion (); Null Set; Universe (or universal set); Complmentary Set; (Inclusion Identities); Segment (pcseg); Row; Cardinality; Interval Seccession of a Segment (INT); Retrogression (R); Transposition (Tn); Inversion (I); Transposition with Inversion (TnI); TTOs; Invariance; Set-Class (SC); Abstract Inclusion; Abstract Complementation; Row-Class

 

(2)

Chapter 2 "Characteristics of P-Space" (pp. 10-11): General Properties of P-Space; Registers in P-Space; The Octave and Octave Doubling

Alan Chapman, "Some Intervallic Aspects of Pitch-Class Set Relations," Journal of Music Theory 25 (1981): 275-90.

() Allen Forte, The Structure of Atonal Music, pp. 7-22 ("Segments").

제 5 주

John Rahn, Basic Atonal Theory:

 

(1)

Chapter 2 "The Integer Model of Pitch" (pp. 19-39, Exercise included): Terms and Assumptions; Pitch and Its Intervals; Pitch-class and Its Intervals; Intervals (A Recapitulation); Normal Form

 

(2)

Analysis: Schoenberg, Five Pieces for Orchestra, "Farben," Op. 16, No. 3 (pp. 59-73)

Richard Chrisman, "Anton Webern's Six Bagatelles for String Quartet, Op. 9: The Unfolding of Intervallic Successions," Journal of Music Theory 23 (1979): 81-122.

제 6 주

Robert D. Morris, Class Notes for Atonal Music Theory:

 

(1)

Chapter 3 "Pitch and PC Operation" (pp. 12-19): Basic Operations; Transposition; The Identity Operator; Inversion; Operator Variables; Composite Operators; The Algebraic Representation of Operators; Canonical Operators (the TTOs); Types of Operators; Mathematical Expressions in Atonal Music Theory; The Cyclic Representation of PC Operators; The Tn Cycles

 

(2)

Chapter 4 "Interval-Class Content and Transposition" (pp. 20-25): Interval-Class Content of PCsets; Interval-Class Vectors; Computing IC Vectors; Tn Partial and Total Invariance; Mapping Tn Relations from PC- to P-Space

James Baker, "Coherence in Webern's Six Pieces for Orchestra, Op. 6," Music Theory Spectrum 4 (1982): 1-27. 안소영, "베베른의「관현악을 위한 6개의 소품집」, 작품 6에서 발견되는 응집성," 「音樂論壇」 제8집(1994), 119-55쪽.

강순미, <베베른>의 「관현악을 위한 6개의 소품집」, 작품 6에서의 집합구조에 관한 연구, 「音樂論壇」 제10집(1996), 41-77쪽.

제 7 주

John Rahn, Basic Atonal Theory: Chapter 3: "Basic Operations" (pp. 40-58, Exercise included): Transposition (Pitch Transposition, Pitch-Class Transposition, Identity and Inverse Operations); Inversion (Pitch Inversion, Pitch-Class Inversion); Compound Operations; Multiplicative Operations

Elizabeth West Marvin, "The Structural Role of Complementation in Webern's Orchestra Pieces (1913)," Music Theory Spectrum 5 (1983): 76-88. 안소영 역, "안톤 베베른의「관현악 소품집 (1913)」에서 발견되는 여집합의 구조적 기능." 「音樂論壇」 제7집(1993), 289-313쪽.

제 8 주

Robert D. Morris, Class Notes for Atonal Music Theory:

 

(1)

Chapter 5 "Inversion with Transposition in PC-Space" (pp. 26-32): TnI Notation and Identities; Inversional Center or Axis of Symmetry; Common PCs between PCsets Related by TnI; Even and Odd Indices of Inversion; TnI and Operator Cycles; Translating TnI from PC- to P-Space

 

(2)

Chapter 6 "Sets and Set-Classes in PC-Space" (pp. 33-38): The Distribution of PCsets; Invariance; Degree of Symmetry; Z-Related SCs; The Complement Theorem; Set-Class Inclusion (abstract inclusion); The ZC-Relation

John Rahn, Basic Atonal Theory: Chapter 4: "Set Types" (pp. 74-96, Exercise included): Types; Tn-types (Tn/TnI-types, Set Types in Music); Applications (How to Find the type of a Set, Recognizing Sets Related by Tn, Recognizing Sets Related by TnI, Names of Operations/Sets/Lines,); Symmetry (Degree of Symmetry, Inversional Symmetry, Transpositional Symmetry); Union and Disunion of Inversionally Symmetrical Sets

제 9 주

Robert D. Morris, Class Notes for Atonal Music Theory: Chapter 7 "Normal-Form Representative" (pp. 39-42): PCset Representation of a SC; Rahn/Morris Algorithm: Normal-Form Representative; Other Algorithms for Finding PCset Representatives of SCs; Normal Form (Rahn); Normal Order (Forte); Prime Form (Forte)

Charles H. Lord, "Intervallic Similarity Relations in Atonal Set Analysis," Journal of Music Theory 25 (1981): 91-111.

제 10 주

Robert D. Morris, Class Notes for Atonal Music Theory: Chapter 8 "Segments in P- and PC-Space" (pp. 43-46): The INT of a Segment; The BIPns of Pitch-Class Segment

John Rahn, Basic Atonal Theory: Chapter 5 "Common-Tone Theorems" (pp. 97-123, Exercise included): Multiplicity/Interval Content/Interval Vector (Multiplicity, Interval Content, Interval Vector, Invariance of Interval Content/Z-related Sets, Interval Content under Multiplicative Operations, Hexachord Theorem); Common-Tome Theorems for Transposition (The Theorems, Example--A Compositional Structure); Common-Tone Theorem for TnI TICS Vectors; Theorems about TICS Vectors of Tn and TnI-related Sets; Subset Content; Further Readings

제 11 주

Robert D. Morris, Class Notes for Atonal Music Theory: Chapter 9 "Notes on Forte's K- and Kh-Relations" (pp. 47-57): Notation; Inclusion Identities; Abstract Complementation; K- and Kh--Relations; Properties of K- and Kh-Relations; K- and Kh-Complexes; SC Inclusion Lattices; Further Observations about K- and Kh-Complexes; Making the Set-Complex Chart; Determining the Nexus Set

Allen Forte, The Structure of Atonal Music, pp. 93-100, Part 2 "Pitch-Class Set Complexes, 2.0~2.2 "The Set complex K; The subcomplex Kh."

제 12 주

George Perle, Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern, 5th ed., re. (Berkeley and Los Angeles: University of California Press, 1981).

제 13 주

Robert D. Morris, Class Notes for Atonal Music Theory:

 

(1)

Chapter 10 "M and MI Operations" (pp. 58-62): Aspects of the M/MI Operators; Uses of the M and MI Operators

 

(2)

Chapter 11 "Operator Cycles" (pp. 63-66): Cycles and Invariance under all PC Operations; Operator Cycle Table

Jonathan Dunsby and Arnold Whittall, "Pitch-class Sets," in Music Analysis: In Theory and Practice (New Haven and London: Yale University Press, 1988), pp. 131-53.

제 14 주

Joseph N. Straus, Introduction to Post-Tonal Theory:

 

(1)

Chapter 4 "Centricity and Some Important Referential Collections" (pp. 89-109): Tonality and centricity; The diatonic collection; The diatonic octad; The octatonic collection; Inversional axis; Exercises

 

(2)

Analysis: Stravinsky, Oedipus Rex, rehearsal nos. 167-70 (pp. 110-113)

 

(3)

Analysis: Bartók, Sonata, 1st movement, mm. 1-17 (pp. 114-17)

제 15 주

Richard Chrisman, "Identification and Correlation of Pitch-Sets," Journal of Music Theory 15/1-2 (1971): 58-83.

Richard Chrisman, "Describing Structural Aspects of Pitch-Sets Using Successive-Interval Arrays," Journal of Music Theory 21/1 (1977): 1-28.

제 16 주

Earl Henry, Music Theory, vol. II (Englewood Cliffs: Prentice-Hall, Inc., 1985), Chapter 13 "Atonality," pp. 339-62.

Allen Winold, Harmony: Patterns and Principles, vol. II (Englewood Cliffs: Prentice-Hall, Inc., 1986), Chapter 28 "Integer Notation and Analysis," pp. 241-263.

David Mancini, "Teaching Set Theory in the Undergraduate Core Curriculum," Journal of Music Theory Pedagogy 5/1 (1991): 95-107.

기말 리포트의 제출은 기말고사의 1주일 후.

 

 교재 및 참고문헌

 

Abraham, Gerald. "The Bartók of the Quartet," Music and Letters 26/4 (1945): 185-94.

Baker, James. "Coherence in Webern's Six Pieces for Orchestra, Op. 6." Music Theory Spectrum 4 (1982): 1-27.

Beach, David W. "Pitch Structure and the Analytic Process in Atonal Music: An Interpretation of the Theory of Sets." Music Theory Spectrum 1 (1979): 7-22.

Bernard, Jonathan W. "Space and Symmetry in Bartók." Journal of Music Theory 30/2 (1986): 185-201.

Castine, Peter. Set Theory Objects: Abstractions for Computer-Aided Analysis and Composition of Serial and Atonal Music. (European University Studies, Series XXXVI Musicology Vol. 121). Frankfurt am Main: Peter Lang, 1994.

Chapman, Alan. "Some Intervallic Aspects of Pitch-Class Set Relations" Journal of Music Theory 25 (1981): 275-90.

Chrisman, Richard. "Identification and Correlation of Pitch-Sets." Journal of Music Theory 15/1-2 (1971): 58-83.

______. "Describing Structural Aspects of Pitch-Sets Using Successive-Interval Arrays." Journal of Music Theory 21/1 (1977): 1-28.

______. "Anton Webern's Six Bagatelles for String Quartet, Op. 9: The Unfolding of Intervallic Successions." Journal of Music Theory 23 (1979): 81-122.

Dunsby, Jonathan, and Arnold Whittall. Music Analysis: In Theory and Practice. New Haven and London: Yale University Press, 1988.

Forte, Allen. The Structure of Atonal Music. New Haven and London: Yale University Press, 1973.

Gillespie, Jeffrey L. "Motivic Transformations and Networks in Schoenberg's 'Nacht' from Pierrot Lunaire." Intégral 6 (1992): 34-65.

Henry, Earl. Music Theory, vol. II. Englewood Cliffs: Prentice-Hall, Inc., 1985.

Kostka, Stefan. Materials and Techniques of Twentieth-Century Music. Englewood Cliffs: Prentice-Hall, Inc., 1990.

Lord, Charles H. "Intervallic Similarity Relations in Atonal Set Analysis." Journal of Music Theory 25 (1981): 91-111.

Mancini, David. "Teaching Set Theory in the Undergraduate Core Curriculum." Journal of Music Theory Pedagogy 5/1 (1991): 95-107.

Marvin, Elizabeth West. "The Structural Role of Complementation in Webern's Orchestra Pieces (1913)." Music Theory Spectrum 5 (1983): 76-88.

Morris, Robert D. Class Notes for Atonal Music Theory, 1991.

Nelson, Mark. "Folk Music and the 'Free and Equal Treatment of the Twelve Tones': Aspects of Béla Bartók's Synthetic Methods." College Music Symposium 27 (1987): 59-116.

Perle, George. Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern. 5th ed., re. Berkeley and Los Angeles: University of California Press, 1981.

Rahn, John. Basic Atonal Theory. New York & London: Longman, 1980.

Straus. Joseph N. Introduction to Post-Tonal Theory. Englewood Cliffs, N.J.: Prentice Hall, 1990.

Winold, Allen. Harmony: Patterns and Principles, vol. II. Englewood Cliffs: Prentice-Hall, Inc., 1986.

Wittlich, Gary E. “Sets and Ordering Procedures in Twentieth-Century Music.” In Aspects of Twentieth-Century Music.” Englewood Cliffs: Prentice-Hall, Inc., 1975.

 

 참고문헌 (국내)

 

강순미. "<베베른>의「관현악을 위한 6개의 작품」작품 6번에서의 집합 구조에 관한 연구." 「음악논단」 제10집(1996), 41-77쪽.

박재성. "집합이론의 역사적 배경과 기본 원리에 관한 연구." 「음악논단」 제7집(1993), 315-74쪽.

______. "무조음악의 의미와 무조음악의 선율선 분석을 위한 서열음정패턴(OIP)의 개념." 「음악논단」 제9집(1995), 89-165쪽.

______. "음고 조직을 이용한 대표음정패턴(RIP) 이론의 제시:서열음정패턴(OIP) 개념의 발전." 「음악논단」 제10집(1996), 151-277쪽.

______. "OIP 및 RIP 개념을 이용한 초기 무조음악 선율에서의 동기적 대응성:안톤 베베른의 "14개의 스테판 죠지 가곡" 중 작품 3의 분석." 「음악연구」17집(1998), 273-322쪽.

______. "<안톤 베베른>의 작품 4번에서의 음고 구조의 통일성 : 선율 구조의 분석." 「음악논단」제12집(1998), 109-178쪽.

서경선. "無調性 音樂의 構造的 音體系:Allen Forte의 The Structure of Atonal Music을 중심으로." 「음악논단」 제1집(1984):137-71쪽.

안소영 역. "안톤 베베른의「관현악 소품집 (1913)」에서 발견되는 여집합의 구조적 기능." 「음악논단」 제7집(1993), 289-313쪽.

______. "베베른의「관현악을 위한 여섯 개의 소품집」, 작품 6에서 발견되는 응집성." 「음악논단」 제8집(1994), 119-55쪽.

 

 

 분석작품

Bartók, Sonata, 1st movement

______, String Quartets

Berg, "Schlafend trägt man mich," from Four Songs, Op. 2, No. 2

Schoenberg, Book of the Hanging Gardens, Op. 15, No. 11

______, Five Pieces for Orchestra, "Farben," Op. 16, No. 3

______, "Nacht" from Pierrot Lunaire, Op. 21

Stravinsky, Oedipus Rex, rehearsal nos. 167-70

Webern, Orchestra Pieces (1913)

______, Movements for String Quartet, Op. 5, No. 4

______, Six Pieces for Orchestra, Op. 6

______, "Wie bin ich froh!" (How happy I am!) from Three Songs, Op. 25

 

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