Rights statement: The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-030-01692-0_14
Accepted author manuscript, 1 MB, PDF document
Available under license: CC BY-NC: Creative Commons Attribution-NonCommercial 4.0 International License
Final published version
Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSN › Conference contribution/Paper
On Linear Algebraic Representation of Time-span and Prolongational Trees. / Tojo, Satoshi; Marsden, Alan Alexander; Hirata, Keiji.
Music Technology with Swing: 13th International Symposium, CMMR 2017, Matosinhos, Portugal, September 25-28, 2017, Revised Selected Papers. ed. / Mitsuko Aramaki; Matthew Davies; Richard Kronland-Martinet; Sølvi Ystad. Cham : Springer, 2018. p. 199-212 (Lecture Notes in Computer Science; Vol. 11265).Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSN › Conference contribution/Paper
}
TY - GEN
T1 - On Linear Algebraic Representation of Time-span and Prolongational Trees
AU - Tojo, Satoshi
AU - Marsden, Alan Alexander
AU - Hirata, Keiji
N1 - The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-030-01692-0_14
PY - 2018
Y1 - 2018
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 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.
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 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.
KW - Time-span tree
KW - Prolongational tree
KW - Generative Theory of Tonal Music
KW - Matrix Linear algebra
U2 - 10.1007/978-3-030-01692-0_14
DO - 10.1007/978-3-030-01692-0_14
M3 - Conference contribution/Paper
SN - 9783030016913
T3 - Lecture Notes in Computer Science
SP - 199
EP - 212
BT - Music Technology with Swing
A2 - Aramaki, Mitsuko
A2 - Davies, Matthew
A2 - Kronland-Martinet, Richard
A2 - Ystad, Sølvi
PB - Springer
CY - Cham
ER -