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Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSN › Conference contribution/Paper › peer-review

Published

**On Linear Algebraic Representation of Time-span and Prolongational Trees.** / Marsden, Alan Alexander; Tojo, Satoshi; Hirata, Keiji.

Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSN › Conference contribution/Paper › peer-review

Marsden, AA, Tojo, S & Hirata, K 2017, On Linear Algebraic Representation of Time-span and Prolongational Trees. in R Kronkland-Martinet, S Ystad & M Aramaki (eds), *Proceedings of the 13th International Symposium on Computer Music Multidisciplinary Research.* Les éditions de PRISM, Marseille, pp. 126-136. <http://cmmr2017.inesctec.pt/wp-content/uploads/2017/09/11_CMMR_2017_paper_59.pdf>

Marsden, A. A., Tojo, S., & Hirata, K. (2017). On Linear Algebraic Representation of Time-span and Prolongational Trees. In R. Kronkland-Martinet, S. Ystad, & M. Aramaki (Eds.), *Proceedings of the 13th International Symposium on Computer Music Multidisciplinary Research *(pp. 126-136). Les éditions de PRISM. http://cmmr2017.inesctec.pt/wp-content/uploads/2017/09/11_CMMR_2017_paper_59.pdf

Marsden AA, Tojo S, Hirata K. On Linear Algebraic Representation of Time-span and Prolongational Trees. In Kronkland-Martinet R, Ystad S, Aramaki M, editors, Proceedings of the 13th International Symposium on Computer Music Multidisciplinary Research. Marseille: Les éditions de PRISM. 2017. p. 126-136

@inproceedings{bf6ea83c2f37472f8f09256491ef32f1,

title = "On Linear Algebraic Representation of Time-span and Prolongational Trees",

abstract = "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 intuitive and visible, such a graphic representation cannot be treated in mathematical formalization. Especially in the GTTM, the conjunction height of two branches is often arbitrary, contrary to the notion of hierarchy. As even a tree is a kind of graph, and a graph is often represented by a matrix, we show the linear algebraic representation of a tree, specifying the conjunction heights. Thereafter, we explain the {\textquoteleft}reachability{\textquoteright} 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 with the objective of modelling compositional processes with simple algebraic operations.",

author = "Marsden, {Alan Alexander} and Satoshi Tojo and Keiji Hirata",

year = "2017",

month = sep,

day = "25",

language = "English",

pages = "126--136",

editor = "R. Kronkland-Martinet and S. Ystad and M. Aramaki",

booktitle = "Proceedings of the 13th International Symposium on Computer Music Multidisciplinary Research",

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TY - GEN

T1 - On Linear Algebraic Representation of Time-span and Prolongational Trees

AU - Marsden, Alan Alexander

AU - Tojo, Satoshi

AU - Hirata, Keiji

PY - 2017/9/25

Y1 - 2017/9/25

N2 - 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 intuitive and visible, such a graphic representation cannot be treated in mathematical formalization. Especially in the GTTM, the conjunction height of two branches is often arbitrary, contrary to the notion of hierarchy. As even a tree is a kind of graph, and a graph is often represented by a matrix, we show the linear algebraic representation of a tree, specifying the 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 with the objective of modelling compositional processes with simple algebraic operations.

AB - 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 intuitive and visible, such a graphic representation cannot be treated in mathematical formalization. Especially in the GTTM, the conjunction height of two branches is often arbitrary, contrary to the notion of hierarchy. As even a tree is a kind of graph, and a graph is often represented by a matrix, we show the linear algebraic representation of a tree, specifying the 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 with the objective of modelling compositional processes with simple algebraic operations.

M3 - Conference contribution/Paper

SP - 126

EP - 136

BT - Proceedings of the 13th International Symposium on Computer Music Multidisciplinary Research

A2 - Kronkland-Martinet, R.

A2 - Ystad, S.

A2 - Aramaki, M.

PB - Les éditions de PRISM

CY - Marseille

ER -