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Mathematics and the Twelve-Tone System: Past, Present, and Future

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Mathematics and Computation in Music (MCM 2007)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 37))

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Abstract

Certainly the first major encounter of non-trivial mathematics and non-trivial music was in the conception and development of the twelve-tone system from the 1920s to the present. Although the twelve-tone system was formulated by Arnold Schoenberg, it was Milton Babbitt whose ample but non-professional background in mathematics made it possible for him to identify the links between the music of the Second-Viennese school and a formal treatment of the system. To be sure, there were also important inroads in Europe as well,1 but these were not often marked by the clarity and rigor introduced by Babbitt in his series of seminal articles from 1955 to 1973 (Babbitt 1955, 1960, 1962, 1973).

This paper was originally published in Perspectives of New Music 45(2): 76–107. The editors of these proceedings are grateful to the editors and publishers of Perspectives of New Music for their kind permission to reprint this article in this collection.

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References

  • Alegant, B.: The 77 Partitions of the Aggregate. Ph.D. dissertation, University of Rochester (1993)

    Google Scholar 

  • Alegant, B., Lofthouse, M.: Having Your Cake and Eating It, Too: The Property of Reflection in Twelve-Tone Rows (Or, Further Extensions on the Mallalieu Complex). Perspectives of New Music 40(2), 233–274 (2002)

    Google Scholar 

  • Alphonce, B.: The Invariance Matrix, Ph.D. dissertation. Yale University (1974)

    Google Scholar 

  • Babbitt, M.: Some Aspects of Twelve-Tone Composition. The Score and IMA Magazine 12, 53–61 (1955)

    MathSciNet  Google Scholar 

  • Babbitt, M.: Twelve-Tone Invariants as Compositional Determinants. Musical Quarterly 46, 245–259 (1960)

    Google Scholar 

  • Babbitt, M.: Set Structure as a Compositional Determinant. Journal of Music Theory 5(2), 72–94 (1961)

    Article  Google Scholar 

  • Babbitt, M.: Twelve-Tone Rhythmic Structure and the Electronic Medium. Perspectives of New Music 1(1), 49–79 (1962)

    Article  Google Scholar 

  • Babbitt, M.: Since Schoenberg. Perspectives of New Music 12(1–2), 3–28 (1973)

    Article  Google Scholar 

  • Babbitt, M.: The Function of Set Structure in the Twelve-Tone System. Ph.D. dissertation. Princeton University, Princeton (1992)

    Google Scholar 

  • Batstone, P.: Multiple Order Functions in Twelve-Tone Music. Perspectives of New Music 10(2) 11(1), 60–71, 92–111 (1972)

    Article  Google Scholar 

  • Bazelow, A.R., Brickle, F.: A Partition Problem Posed by Milton Babbitt (Part I). Perspectives of New Music 14(2), 280–293 (1976)

    Article  Google Scholar 

  • Benjamin, B.: Meta-Variations: Studies in the Foundations of Musical Thought (I). Perspectives of New Music 8(1), 1–75 (1969)

    Article  Google Scholar 

  • Benjamin, B.: Sketch of a Musical System (Meta-Variations, Part II). Perspectives of New Music 8(2), 49–112 (1970a)

    Article  Google Scholar 

  • Benjamin, B.: The Construction of Musical Syntax (I). Perspectives of New Music 9(1), 23–42 (1970b)

    Article  Google Scholar 

  • Benjamin, B.: Musical Syntax (II). Perspectives of New Music 9(2) 10(1), 232–270 (1971)

    Article  Google Scholar 

  • Benjamin, B.: Meta-Variations, Part IV: Analytic Fallout (I). Perspectives of New Music 11(1), 146–223 (1972)

    Article  MathSciNet  Google Scholar 

  • Benjamin, B.: Meta-Variations, Part IV: Analytic Fallout (II). Perspectives of New Music 11(2), 156–203 (1973)

    Article  Google Scholar 

  • Forte, A.: A Theory of Set-Complexes for Music. Journal of Music Theory 8(2), 136–183 (1966)

    Article  Google Scholar 

  • Forte, A.: The Structure of Atonal Music. Yale University Press, New Haven (1973)

    Google Scholar 

  • Forte, A.: Pitch-Class Set Genera and the Origin of Modern Harmonic Species. Journal of Music Theory 32(2), 187–334 (1988)

    Article  MathSciNet  Google Scholar 

  • Fripertinger, H.: Enumeration in Music Theory (1992), http://www.unigraz. at/~fripert/musical_theory.html

    Google Scholar 

  • Gamer, C.: Deep Scales and Difference Sets in Equal Tempered Systems. Proceedings of the American Society of University Composers 2, 113–122 (1967a)

    Google Scholar 

  • Gamer, C.: Some Combinatorial Resources in Equal Tempered Systems. Journal of Music Theory 11(1), 32–59 (1967b)

    Article  Google Scholar 

  • Haimo, E., Johnson, P.: Isomorphic Partitioning and Schoenberg’s Fourth String Quartet. Journal of Music Theory 28, 47–72 (1984)

    Article  Google Scholar 

  • Hanson, H.: The Harmonic Materials of Twentieth-Century Music. Appleton-Century-Crofts, New York (1960)

    Google Scholar 

  • Hauer, J.M.: Vom Melos zur Pauke: Eine Einfürung in die Zwölftonmusik. Universal Edition, Vienna (1925)

    Google Scholar 

  • Howe, H.S.: Some Combinatorial Properties of Pitch-Structures. Perspectives of New Music 4(1), 45–61 (1965)

    Article  Google Scholar 

  • Kowalski, D.: The Array as a Compositional Unit: A Study of Derivational Counterpoint as a Means of Creating Hierarchical Structures in Twelve-Tone Music. Ph.D. dissertation. Princeton University, Princeton (1985)

    Google Scholar 

  • Lansky, P.: Affine Music. PhD dissertation. Princeton University, Princeton (1973)

    Google Scholar 

  • Lewin, D.: Intervallic Relations Between Two Collections of Notes. Journal of Music Theory 3(2), 298–301 (1959)

    Article  Google Scholar 

  • Lewin, D.: The Intervallic Content of a Collection of Notes, Intervallic Relations Between a Collection of Notes and its Complement: An Application to Schoenberg’s Hexachordal Pieces. Journal of Music Theory 4(1), 98–101 (1960)

    Article  Google Scholar 

  • Lewin, D.: A Theory of Segmental Association in Twelve-Tone Music. Perspectives of New Music 1(1), 276–287 (1962)

    Article  MathSciNet  Google Scholar 

  • Lewin, D.: On Partial Ordering. Perspectives of New Music 14(2)-15(1), 252–259 (1976)

    Article  MathSciNet  Google Scholar 

  • Lewin, D.: Some New Constructs Involving Abstract Pcsets, and Probabilistic Applications. Perspectives of New Music 18(2), 433–444 (1980a)

    Article  Google Scholar 

  • Lewin, D.: On Extended Z-Triples. Theory and Practice 7, 38–39 (1982)

    Google Scholar 

  • Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press, New Haven (1987)

    Google Scholar 

  • Lewin, D.: Klumpenhouwer Networks and Some Isographies that Involve Them. Music Theory Spectrum 12(1), 83–120 (1990)

    Article  Google Scholar 

  • Lewin, D.: Musical Form and Transformation: 4 Analytic Essays. Yale University Press, New Haven (1993)

    Google Scholar 

  • Lewin, D.: Special Cases of the Interval Function Between Pitch-Class Sets X and Y. Journal of Music Theory 45(1), 1–30 (2001)

    Article  Google Scholar 

  • Martino, D.: The Source Set and Its Aggregate Formations. Journal of Music Theory 5(2), 224–273 (1961)

    Article  Google Scholar 

  • Mazzola, G., Müller, S.: Stefan Goller, contributors. The Topos of Music: Geometric Logic of Concepts, Theory, and Performance. Birkhäuser, Basel (2002)

    Google Scholar 

  • Mead, A.: Some Implications of the Pitch-Class/Order-Number Isomorphism Inherent in the Twelve-Tone System: Part One. Perspectives of New Music 26(2), 96–163 (1988)

    Article  Google Scholar 

  • Mead, A.: Some Implications of the Pitch-Class/Order-Number Isomorphism Inherent in the Twelve-Tone System: Part Two. Perspectives of New Music 27(1), 180–233 (1989)

    Article  Google Scholar 

  • Messiaen, O.: Technique de mon langage musical. Leduc, Paris (1944)

    Google Scholar 

  • Morris, R.D.: More on 0,1,4,2,9,5,11,3,8,10,7,6. Theory Only 2(7), 15–20 (1976)

    Google Scholar 

  • Morris, R.D.: On the Generation of Multiple Order Number Twelve-tone Rows. Journal of Music Theory 21, 238–263 (1977)

    Article  Google Scholar 

  • Morris, R.D.: A Similarity Index for Pitch-Class Sets. Perspectives of New Music 18(1–2), 445–460 (1980)

    Google Scholar 

  • Morris, R.D.: Combinatoriality without the Aggregate. Perspectives of New Music 21(1–2), 432–486 (1983)

    Google Scholar 

  • Morris, R.D.: Set-Type Saturation Among Twelve-Tone Rows. Perspectives of New Music 22(1–2), 187–217 (1985)

    Google Scholar 

  • Morris, R.D.: Composition with Pitch-Classes: A Theory of Compositional Design. Yale University Press, New Haven (1987)

    Google Scholar 

  • Morris, R.D.: Compositional Spaces and Other Territories. Perspectives of New Music 33(1–2), 328–359 (1995a)

    Google Scholar 

  • Morris, R.D.: Equivalence and Similarity in Pitch and their Interaction with Pcset Theory. Journal of Music Theory 39(2), 207–244 (1995b)

    Article  Google Scholar 

  • Morris, R.D.: K, Kh, and Beyond. In: Baker, J., Beach, D., Bernard, J. (eds.) Music Theory in Concept and Practice. University of Rochester Press, Rochester (1997)

    Google Scholar 

  • Morris, R.D.: Class Notes for Advanced Atonal Theory, vol. 2. Frog Peak Music (2001)

    Google Scholar 

  • Morris, R.D., Alegant, B.: The Even Partitions in Twelve-Tone Music. Music Theory Spectrum 10, 74–103 (1988)

    Article  Google Scholar 

  • Morris, R., Starr, D.: The Structure of All-Interval Series. Journal of Music Theory 18(2), 364–389 (1974)

    Article  Google Scholar 

  • O’Connell, W.: Tone Spaces. Die Reihe 8, 34–67 (1963)

    Google Scholar 

  • Perle, G.: Twelve-Tone Tonality. University of California Press, Berkeley (1977)

    Google Scholar 

  • Polansky, L.: Morphological Metrics: An Introduction to a Theory of Formal Distances. In: Proceedings of the International Computer Music Conference. Compiled by James Beauchamp, Computer Music Association, San Francisco (1987)

    Google Scholar 

  • Polansky, L., Bassein, R.: Possible and Impossible Melodies: Some Formal Aspects of Contour. Journal of Music Theory 36, 259–284 (1992)

    Article  Google Scholar 

  • Quinn, I.: General Equal-Tempered Harmony. Perspectives of New Music 44(2) 45(1), 144–159, 4–63 (2006)

    Google Scholar 

  • Rahn, J.: On Pitch or Rhythm: Interpretations of Orderings of and In Pitch and Time. Perspectives of New Music 13, 182–204 (1974)

    Article  Google Scholar 

  • Rahn, J.: Basic Atonal Theory. Longman Press, New York (1980a)

    Google Scholar 

  • Rahn, J.: Relating Sets. Perspectives of New Music 18(2), 483–502 (1980b)

    Article  Google Scholar 

  • Regener, E.: On Allen Forte’s Theory of Chords. Perspectives of New Music 13(1), 191–212 (1974)

    Article  Google Scholar 

  • Roger, J.: Toward a System of Rotational Arrays. Perspectives of New Music 7(1), 80–102 (1968)

    Article  Google Scholar 

  • Schoenberg, A.: Composition with Twelve Tones (I). In Style and Idea: Selected Writings. Univ. California Press, Berkeley (1975)

    Google Scholar 

  • Scotto, C.G.: Can Non-Tonal Systems Support Music as Richly as the Tonal System? D.M.A. dissertation, University of Washington (1995)

    Google Scholar 

  • Starr, D.V.: Sets, Invariance, and Partitions. Journal of Music Theory 22(1), 1–42 (1978)

    Article  MathSciNet  Google Scholar 

  • Starr, D.V.: Derivation and Polyphony. Perspectives of New Music 23(1), 180–257 (1984)

    Article  Google Scholar 

  • Starr, D., Morris, R.: A General Theory of Combinatoriality and the Aggregate. Perspectives of New Music 16(1) 16(2), 3–35, 50–84 (1977–1978)

    Article  Google Scholar 

  • Stockhausen, K.: how time passes..... Die Reihe 3, 4–27 (1959)

    Google Scholar 

  • Verdi, L.: The History of Set Theory for a European Point of View. Perspectives of New Music 45(1), 154–183 (2007)

    Google Scholar 

  • Winham, G.: Composition with Arrays. Ph.D. dissertation. Princeton University, Princeton (1970)

    Google Scholar 

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Authors and Affiliations

  1. Eastman School of Music, University of Rochester, Rochester, USA

    Robert Morris

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  1. Robert Morris
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  1. Abteilung III Akustik und Musiktechnologie, Staatliches Institut für Musikforschung Preußischer Kulturbesitz, Tiergartenstr. 1, 10785, Berlin, Germany

    Timour Klouche

  2. Departament de Teoria i Composicio i Direccio, Escola Superior de Musica de Catalunya, C. Padilla 155-Edifici L’Auditori, 08013, Barcelona, Spain

    Thomas Noll

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Morris, R. (2009). Mathematics and the Twelve-Tone System: Past, Present, and Future. In: Klouche, T., Noll, T. (eds) Mathematics and Computation in Music. MCM 2007. Communications in Computer and Information Science, vol 37. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04579-0_27

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  • DOI: https://doi.org/10.1007/978-3-642-04579-0_27

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