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Spelled Heptachords

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6726)

Abstract

This paper develops a theory of spelled pitch classes (spcs) and spelled pitch − class sets (spc sets), incorporating pitch spelling into the techniques of pitch-class set theory. The symmetries of spc space are transposition and inversion along the line of fifths. Because of the inextricable link between pitch spelling and diatonic scales, spelled heptachords—seven-note spc sets that include each letter name exactly once—occupy a privileged position in this theory. Spelled heptachords may be regarded as inflected diatonic scales, and possess a number of structural characteristics not shared by other spc sets. The 66 equivalence classes of spelled heptachords without enharmonic doublings or voice crossings are enumerated. A diatonic musical structure together with a spelled heptachord determine an spc structure in which the notes of the diatonic structure are inflected by the corresponding accidentals from the heptachord; spc structures arising in this way show promise as powerful tools in analysis of chromatic harmony.

Keywords

Pitch-class set theory Diatonic set theory Pitch spelling Heptachords Chromaticism 

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References

  1. 1.
    Forte, A.: The Structure of Atonal Music. Yale University Press, New Haven (1973)Google Scholar
  2. 2.
    Brinkman, A.R.: A Binomial Representation of Pitch for Computer Processing of Musical Data. Music Theory Spectrum 8, 44–57 (1986)CrossRefGoogle Scholar
  3. 3.
    Agmon, E.: A Mathematical Model of the Diatonic System. Journal of Music Theory 33, 1–25 (1989)CrossRefGoogle Scholar
  4. 4.
    Douthett, J., Hook, J.: Formal Diatonic Intervallic Notation. In: Chew, E., Childs, A., Chuan, C.-H. (eds.) MCM 2009. Communications in Computer and Information Science, vol. 38, pp. 104–114. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Regener, E.: Pitch Notation and Equal Temperament: A Formal Study. University of California Press, Berkeley (1973)Google Scholar
  6. 6.
    Pople, A.: Using Complex Set Theory for Tonal Analysis: An Introduction to the Tonalities Project. Music Analysis 23, 153–194 (2004)CrossRefGoogle Scholar
  7. 7.
    Hook, J.: Enharmonic Systems: A Theory of Key Signatures, Enharmonic Equivalence and Diatonicism. Journal of Mathematics and Music 1, 99–120 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Junod, J., Audétat, P., Agon, C., Andreatta, M.: A Generalisation of Diatonicism and the Discrete Fourier Transform as a Mean for Classifying and Characterising Musical Scales. In: Chew, E., Childs, A., Chuan, C.-H. (eds.) MCM 2009. Communications in Computer and Information Science, vol. 38, pp. 166–179. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Temperley, D.: The Line of Fifths. Music Analysis 19, 289–319 (2000)CrossRefGoogle Scholar
  10. 10.
    Hook, J.: Signature Transformations. In: Douthett, J., Hyde, M.M., Smith, C.J. (eds.) Music Theory and Mathematics: Chords, Collections, and Transformations, pp. 137–160. University of Rochester Press, Rochester (2008)Google Scholar
  11. 11.
    Tymoczko, D.: Voice Leadings as Generalized Key Signatures. Music Theory Online 11(4) (2005)Google Scholar
  12. 12.
    Riley, M.: The ‘Harmonic Major’ Mode in Nineteenth-Century Theory and Practice. Music Analysis 23, 1–26 (2004)CrossRefGoogle Scholar
  13. 13.
    Hook, J.: An Integrated Transformational Theory of Diatonic and Chromatic Harmony. Presented at: Society for Music Theory, Los Angeles (2006) Google Scholar
  14. 14.
    Tymoczko, D.: A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. Oxford University Press, New York (2011)Google Scholar
  15. 15.
    Clough, J., Myerson, G.: Variety and Multiplicity in Diatonic Systems. Journal of Music Theory 29, 249–270 (1985)CrossRefGoogle Scholar
  16. 16.
    Audétat, P., Junod, J.: The Diatonic Bell, http://www.cloche-diatonique.ch/

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Indiana UniversityBloomingtonUSA

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