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

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

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).

Keywords

Music Theory Compositional Space Array Column Combinatorial Array Cardinality Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Alegant, B.: The 77 Partitions of the Aggregate. Ph.D. dissertation, University of Rochester (1993)Google Scholar
  2. 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
  3. Alphonce, B.: The Invariance Matrix, Ph.D. dissertation. Yale University (1974)Google Scholar
  4. Babbitt, M.: Some Aspects of Twelve-Tone Composition. The Score and IMA Magazine 12, 53–61 (1955)MathSciNetGoogle Scholar
  5. Babbitt, M.: Twelve-Tone Invariants as Compositional Determinants. Musical Quarterly 46, 245–259 (1960)Google Scholar
  6. Babbitt, M.: Set Structure as a Compositional Determinant. Journal of Music Theory 5(2), 72–94 (1961)CrossRefGoogle Scholar
  7. Babbitt, M.: Twelve-Tone Rhythmic Structure and the Electronic Medium. Perspectives of New Music 1(1), 49–79 (1962)CrossRefGoogle Scholar
  8. Babbitt, M.: Since Schoenberg. Perspectives of New Music 12(1–2), 3–28 (1973)CrossRefGoogle Scholar
  9. Babbitt, M.: The Function of Set Structure in the Twelve-Tone System. Ph.D. dissertation. Princeton University, Princeton (1992)Google Scholar
  10. Batstone, P.: Multiple Order Functions in Twelve-Tone Music. Perspectives of New Music 10(2) 11(1), 60–71, 92–111 (1972)CrossRefGoogle Scholar
  11. Bazelow, A.R., Brickle, F.: A Partition Problem Posed by Milton Babbitt (Part I). Perspectives of New Music 14(2), 280–293 (1976)CrossRefGoogle Scholar
  12. Benjamin, B.: Meta-Variations: Studies in the Foundations of Musical Thought (I). Perspectives of New Music 8(1), 1–75 (1969)CrossRefGoogle Scholar
  13. Benjamin, B.: Sketch of a Musical System (Meta-Variations, Part II). Perspectives of New Music 8(2), 49–112 (1970a)CrossRefGoogle Scholar
  14. Benjamin, B.: The Construction of Musical Syntax (I). Perspectives of New Music 9(1), 23–42 (1970b)CrossRefGoogle Scholar
  15. Benjamin, B.: Musical Syntax (II). Perspectives of New Music 9(2) 10(1), 232–270 (1971)CrossRefGoogle Scholar
  16. Benjamin, B.: Meta-Variations, Part IV: Analytic Fallout (I). Perspectives of New Music 11(1), 146–223 (1972)CrossRefMathSciNetGoogle Scholar
  17. Benjamin, B.: Meta-Variations, Part IV: Analytic Fallout (II). Perspectives of New Music 11(2), 156–203 (1973)CrossRefGoogle Scholar
  18. Forte, A.: A Theory of Set-Complexes for Music. Journal of Music Theory 8(2), 136–183 (1966)CrossRefGoogle Scholar
  19. Forte, A.: The Structure of Atonal Music. Yale University Press, New Haven (1973)Google Scholar
  20. Forte, A.: Pitch-Class Set Genera and the Origin of Modern Harmonic Species. Journal of Music Theory 32(2), 187–334 (1988)CrossRefMathSciNetGoogle Scholar
  21. Fripertinger, H.: Enumeration in Music Theory (1992), http://www.unigraz. at/~fripert/musical_theory.htmlGoogle Scholar
  22. 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
  23. Gamer, C.: Some Combinatorial Resources in Equal Tempered Systems. Journal of Music Theory 11(1), 32–59 (1967b)CrossRefGoogle Scholar
  24. Haimo, E., Johnson, P.: Isomorphic Partitioning and Schoenberg’s Fourth String Quartet. Journal of Music Theory 28, 47–72 (1984)CrossRefGoogle Scholar
  25. Hanson, H.: The Harmonic Materials of Twentieth-Century Music. Appleton-Century-Crofts, New York (1960)Google Scholar
  26. Hauer, J.M.: Vom Melos zur Pauke: Eine Einfürung in die Zwölftonmusik. Universal Edition, Vienna (1925)Google Scholar
  27. Howe, H.S.: Some Combinatorial Properties of Pitch-Structures. Perspectives of New Music 4(1), 45–61 (1965)CrossRefGoogle Scholar
  28. 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
  29. Lansky, P.: Affine Music. PhD dissertation. Princeton University, Princeton (1973)Google Scholar
  30. Lewin, D.: Intervallic Relations Between Two Collections of Notes. Journal of Music Theory 3(2), 298–301 (1959)CrossRefGoogle Scholar
  31. 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)CrossRefGoogle Scholar
  32. Lewin, D.: A Theory of Segmental Association in Twelve-Tone Music. Perspectives of New Music 1(1), 276–287 (1962)CrossRefMathSciNetGoogle Scholar
  33. Lewin, D.: On Partial Ordering. Perspectives of New Music 14(2)-15(1), 252–259 (1976)CrossRefMathSciNetGoogle Scholar
  34. Lewin, D.: Some New Constructs Involving Abstract Pcsets, and Probabilistic Applications. Perspectives of New Music 18(2), 433–444 (1980a)CrossRefGoogle Scholar
  35. Lewin, D.: On Extended Z-Triples. Theory and Practice 7, 38–39 (1982)Google Scholar
  36. Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press, New Haven (1987)Google Scholar
  37. Lewin, D.: Klumpenhouwer Networks and Some Isographies that Involve Them. Music Theory Spectrum 12(1), 83–120 (1990)CrossRefGoogle Scholar
  38. Lewin, D.: Musical Form and Transformation: 4 Analytic Essays. Yale University Press, New Haven (1993)Google Scholar
  39. Lewin, D.: Special Cases of the Interval Function Between Pitch-Class Sets X and Y. Journal of Music Theory 45(1), 1–30 (2001)CrossRefGoogle Scholar
  40. Martino, D.: The Source Set and Its Aggregate Formations. Journal of Music Theory 5(2), 224–273 (1961)CrossRefGoogle Scholar
  41. Mazzola, G., Müller, S.: Stefan Goller, contributors. The Topos of Music: Geometric Logic of Concepts, Theory, and Performance. Birkhäuser, Basel (2002)Google Scholar
  42. 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)CrossRefGoogle Scholar
  43. 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)CrossRefGoogle Scholar
  44. Messiaen, O.: Technique de mon langage musical. Leduc, Paris (1944)Google Scholar
  45. 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
  46. Morris, R.D.: On the Generation of Multiple Order Number Twelve-tone Rows. Journal of Music Theory 21, 238–263 (1977)CrossRefGoogle Scholar
  47. Morris, R.D.: A Similarity Index for Pitch-Class Sets. Perspectives of New Music 18(1–2), 445–460 (1980)Google Scholar
  48. Morris, R.D.: Combinatoriality without the Aggregate. Perspectives of New Music 21(1–2), 432–486 (1983)Google Scholar
  49. Morris, R.D.: Set-Type Saturation Among Twelve-Tone Rows. Perspectives of New Music 22(1–2), 187–217 (1985)Google Scholar
  50. Morris, R.D.: Composition with Pitch-Classes: A Theory of Compositional Design. Yale University Press, New Haven (1987)Google Scholar
  51. Morris, R.D.: Compositional Spaces and Other Territories. Perspectives of New Music 33(1–2), 328–359 (1995a)Google Scholar
  52. Morris, R.D.: Equivalence and Similarity in Pitch and their Interaction with Pcset Theory. Journal of Music Theory 39(2), 207–244 (1995b)CrossRefGoogle Scholar
  53. 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
  54. Morris, R.D.: Class Notes for Advanced Atonal Theory, vol. 2. Frog Peak Music (2001)Google Scholar
  55. Morris, R.D., Alegant, B.: The Even Partitions in Twelve-Tone Music. Music Theory Spectrum 10, 74–103 (1988)CrossRefGoogle Scholar
  56. Morris, R., Starr, D.: The Structure of All-Interval Series. Journal of Music Theory 18(2), 364–389 (1974)CrossRefGoogle Scholar
  57. O’Connell, W.: Tone Spaces. Die Reihe 8, 34–67 (1963)Google Scholar
  58. Perle, G.: Twelve-Tone Tonality. University of California Press, Berkeley (1977)Google Scholar
  59. 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
  60. Polansky, L., Bassein, R.: Possible and Impossible Melodies: Some Formal Aspects of Contour. Journal of Music Theory 36, 259–284 (1992)CrossRefGoogle Scholar
  61. Quinn, I.: General Equal-Tempered Harmony. Perspectives of New Music 44(2) 45(1), 144–159, 4–63 (2006)Google Scholar
  62. Rahn, J.: On Pitch or Rhythm: Interpretations of Orderings of and In Pitch and Time. Perspectives of New Music 13, 182–204 (1974)CrossRefGoogle Scholar
  63. Rahn, J.: Basic Atonal Theory. Longman Press, New York (1980a)Google Scholar
  64. Rahn, J.: Relating Sets. Perspectives of New Music 18(2), 483–502 (1980b)CrossRefGoogle Scholar
  65. Regener, E.: On Allen Forte’s Theory of Chords. Perspectives of New Music 13(1), 191–212 (1974)CrossRefGoogle Scholar
  66. Roger, J.: Toward a System of Rotational Arrays. Perspectives of New Music 7(1), 80–102 (1968)CrossRefGoogle Scholar
  67. Schoenberg, A.: Composition with Twelve Tones (I). In Style and Idea: Selected Writings. Univ. California Press, Berkeley (1975)Google Scholar
  68. 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
  69. Starr, D.V.: Sets, Invariance, and Partitions. Journal of Music Theory 22(1), 1–42 (1978)CrossRefMathSciNetGoogle Scholar
  70. Starr, D.V.: Derivation and Polyphony. Perspectives of New Music 23(1), 180–257 (1984)CrossRefGoogle Scholar
  71. 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)CrossRefGoogle Scholar
  72. Stockhausen, K.: how time passes..... Die Reihe 3, 4–27 (1959)Google Scholar
  73. Verdi, L.: The History of Set Theory for a European Point of View. Perspectives of New Music 45(1), 154–183 (2007)Google Scholar
  74. Winham, G.: Composition with Arrays. Ph.D. dissertation. Princeton University, Princeton (1970)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Robert Morris
    • 1
  1. 1.Eastman School of MusicUniversity of RochesterRochesterUSA

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