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SUM Classes and Quotient Generalized Interval Systems

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Mathematics and Computation in Music (MCM 2022)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13267))

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Abstract

The present paper develops algebraic properties of the SUM-class system first developed by Richard Cohn and explored by Robert Cook and Joseph Straus, in the context of David Lewin’s Generalized Interval System (GIS) concept. Motivated by his observation that harmonic triads whose pitch classes sum to a given value modulo 12 share certain voice-leading properties, Cohn defined SUM classes for the 24 consonant (major and minor) triads, and defined transformations on these equivalence classes. We present the SUM-class system as a quotient GIS structure, and explore the dual quotient GIS implied by Lewin’s theory for non-commutative GISs, and we generalize to other types of pitch-class sets (other set-classes).

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References

  1. Cohn, R.: Square dances with cubes. J. Music Theory 42(2), 283–296 (1998)

    Article  Google Scholar 

  2. Cohn, R.: Audacious Euphony: Chromaticism and the Triad’s Second Nature. Oxford University Press, Oxford (2012)

    Book  Google Scholar 

  3. Cook, R.: Transformational approaches to romantic harmony and the late works of César Franck. Ph.D. dissertation, University of Chicago (2001)

    Google Scholar 

  4. Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press New Haven (1987). Reprinted Oxford University Press, Oxford (2007)

    Google Scholar 

  5. Clampitt, D.: Pairwise well-formed scales: structural and transformational properties. Ph.D. dissertation, SUNY at Buffalo (1997)

    Google Scholar 

  6. Cohn, R.: Introduction to Neo-Riemannian theory: a survey and a historical perspective. J. Music Theory 42(2), 167–180 (1998)

    Article  Google Scholar 

  7. Cohn, R.: Weitzmann’s regions, my cycles, and Douthett’s dancing cubes. Music Theory Spectr. 22(1), 89–103 (2000)

    Article  Google Scholar 

  8. Orvek, D.: Generalized transformational voice-leading systems. M.A. thesis, Ohio State University (2019)

    Google Scholar 

  9. Straus, J.: Sum class. J. Music Theory 62(2), 279–338 (2018)

    Article  Google Scholar 

  10. Kochavi, J.: Some structural features of contextually-defined inversion operators. J. Music Theory 42(2), 307–320 (1998)

    Article  Google Scholar 

  11. Lewin, D.: Musical Form and Transformation: 4 Analytical Essays. Yale University Press, New Haven (1993). Reprinted Oxford University Press, Oxford (2007)

    Google Scholar 

  12. Hook, J.: Cross-type transformations and the path consistency condition. Music Theory Spectr. 29(1), 1–39 (2007)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Indiana University, Bloomington, IN, USA

    David Orvek

  2. Ohio State University, Columbus, OH, USA

    David Clampitt

Authors
  1. David Orvek
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  2. David Clampitt
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Corresponding author

Correspondence to David Clampitt .

Editor information

Editors and Affiliations

  1. Georgia State University, Atlanta, GA, USA

    Mariana Montiel

  2. Technological University of the Mixteca, Huajuapan de León, Mexico

    Octavio A. Agustín-Aquino

  3. Universidad Politécnica de Madrid, Madrid, Spain

    Francisco Gómez

  4. Life University, Marietta, GA, USA

    Jeremy Kastine

  5. National Autonomous University of Mexico, Mexico City, Distrito Federal, Mexico

    Emilio Lluis-Puebla

  6. Georgia State University, Atlanta, GA, USA

    Brent Milam

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Orvek, D., Clampitt, D. (2022). SUM Classes and Quotient Generalized Interval Systems. In: Montiel, M., Agustín-Aquino, O.A., Gómez, F., Kastine, J., Lluis-Puebla, E., Milam, B. (eds) Mathematics and Computation in Music. MCM 2022. Lecture Notes in Computer Science(), vol 13267. Springer, Cham. https://doi.org/10.1007/978-3-031-07015-0_9

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  • DOI: https://doi.org/10.1007/978-3-031-07015-0_9

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-07014-3

  • Online ISBN: 978-3-031-07015-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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