On Linear Algebraic Representation of Time-span and Prolongational Trees

  • Satoshi TojoEmail author
  • Alan Marsden
  • Keiji Hirata
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11265)


In constructive music theory, such as Schenkerian analysis and the Generative Theory of Tonal Music (GTTM), the hierarchical importance of pitch events is conveniently represented by a tree structure. Although a tree is easy to recognize and has high visibility, such an intuitive representation can hardly be treated in mathematical formalization. Especially in GTTM, the conjunction height of two branches is often arbitrary, contrary to the notion of hierarchy. Since a tree is a kind of graph, and a graph is often represented by a matrix, we show the linear algebraic representation of trees, specifying conjunction heights. Thereafter, we explain the ‘reachability’ between pitch events (corresponding to information about reduction) by the multiplication of matrices. In addition we discuss multiplication with vectors representing a sequence of harmonic functions, and suggest the notion of stability. Finally, we discuss operations between matrices to model compositional processes with simple algebraic operations.


Time-span tree Prolongational tree Generative Theory of Tonal Music Matrix Linear algebra 



This work is supported by JSPS Kaken 16H01744 and BR160304.


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Japan Advanced Institute of Science and TechnologyNomiJapan
  2. 2.Lancaster UniversityBailrigg, LancasterUK
  3. 3.Future University HakodateHakodate, HokkaidoJapan

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