- TojoMarsdenHirataLNCS2018AAM
**Rights statement:**The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-030-01692-0_14Accepted author manuscript, 1.38 MB, PDF document

Available under license: CC BY-NC: Creative Commons Attribution-NonCommercial 4.0 International License

- https://link.springer.com/chapter/10.1007%2F978-3-030-01692-0_14
Final published version

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.** / Tojo, Satoshi; Marsden, Alan Alexander; Hirata, Keiji.

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

Tojo, S, Marsden, AA & Hirata, K 2018, On Linear Algebraic Representation of Time-span and Prolongational Trees. in M Aramaki, M Davies, R Kronland-Martinet & S Ystad (eds), *Music Technology with Swing: 13th International Symposium, CMMR 2017, Matosinhos, Portugal, September 25-28, 2017, Revised Selected Papers.* Lecture Notes in Computer Science, vol. 11265, Springer, Cham, pp. 199-212. https://doi.org/10.1007/978-3-030-01692-0_14

Tojo, S., Marsden, A. A., & Hirata, K. (2018). On Linear Algebraic Representation of Time-span and Prolongational Trees. In M. Aramaki, M. Davies, R. Kronland-Martinet, & S. Ystad (Eds.), *Music Technology with Swing: 13th International Symposium, CMMR 2017, Matosinhos, Portugal, September 25-28, 2017, Revised Selected Papers *(pp. 199-212). (Lecture Notes in Computer Science; Vol. 11265). Springer. https://doi.org/10.1007/978-3-030-01692-0_14

Tojo S, Marsden AA, Hirata K. On Linear Algebraic Representation of Time-span and Prolongational Trees. In Aramaki M, Davies M, Kronland-Martinet R, Ystad S, editors, Music Technology with Swing: 13th International Symposium, CMMR 2017, Matosinhos, Portugal, September 25-28, 2017, Revised Selected Papers. Cham: Springer. 2018. p. 199-212. (Lecture Notes in Computer Science). https://doi.org/10.1007/978-3-030-01692-0_14

@inproceedings{a2f625d531504c509c400091752b2642,

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 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 {\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 to model compositional processes with simple algebraic operations.",

keywords = "Time-span tree, Prolongational tree, Generative Theory of Tonal Music, Matrix Linear algebra",

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

note = "The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-030-01692-0_14",

year = "2018",

doi = "10.1007/978-3-030-01692-0_14",

language = "English",

isbn = "9783030016913",

series = "Lecture Notes in Computer Science",

publisher = "Springer",

pages = "199--212",

editor = "Mitsuko Aramaki and Matthew Davies and Richard Kronland-Martinet and S{\o}lvi Ystad",

booktitle = "Music Technology with Swing",

}

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 -