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SUM Classes and Quotient Generalized Interval Systems

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


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).


  • Generalized Interval System
  • Group homomorphism
  • Quotient group
  • SUM class

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  • DOI: 10.1007/978-3-031-07015-0_9
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Correspondence to David Clampitt .

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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.

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  • Print ISBN: 978-3-031-07014-3

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