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