×
Popoff, Alexandre

On the use of relational presheaves in transformational music theory. (English) Zbl 1535.00041

J. Math. Music 16, No. 1, 51-79 (2022).

Cited in 1 Document

MSC:

00A65 Mathematics and music
18A05 Definitions and generalizations in theory of categories
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)

Keywords:

binary relations; relational presheaf; category theory; transformational analysis; musical graphs

Software:

Opycleid; GAP
Cite Review PDF
Full Text: DOI

References:

[1] Akhmedov, Azer; Winter, Michael, Chordal and Timbral Morphologies Using Hamiltonian Cycles, Journal of Mathematics and Music, 8, 1, 1-24 (2014) · Zbl 1327.00014 · doi:10.1080/17459737.2014.893033
[2] Albini, Giovanni, and Antonini, Samuele. 2009. “Hamiltonian Cycles in the Topological Dual of the Tonnetz.” In Mathematics and Computation in Music, edited by Elaine Chew, Adrian Childs, and Ching-Hua Chuan, 1-10. Berlin, Heidelberg: Springer. · Zbl 1247.00015
[3] Albini, Giovanni, and Bernardi, Marco Paolo. 2019. “Graph Theory and Music: A Mathematical Tool for Musicians.” In ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics, edited by Luigi Cocchiarella, 1813-1824. Cham: Springer International Publishing.
[4] Amiot, Emmanuel.2013. “The Torii of Phases.” In Mathematics and Computation in Music, edited by Jason Yust, Jonathan Wild, and John Ashley Burgoyne, 1-18. Berlin, Heidelberg: Springer. · Zbl 1270.00023
[5] Amiot, Emmanuel.2017. “Strange Symmetries.” In Mathematics and Computation in Music, edited by Octavio A. Agustín-Aquino, Emilio Lluis-Puebla, and Mariana Montiel, 135-150. Cham: Springer International Publishing. · Zbl 1456.00053
[6] Arias, Juan Sebastián., Spaces of Gestures are Function Spaces, Journal of Mathematics and Music, 12, 2, 89-105 (2018) · Zbl 1406.00001 · doi:10.1080/17459737.2018.1496489
[7] Arias-Valero, Juan Sebastián; Lluis-Puebla, Emilio, Some Remarks on Hypergestural Homology of Spaces and Its Relation to Classical Homology, Journal of Mathematics and Music, 2, 3, 1-21 (2020) · Zbl 1471.00006 · doi:10.1080/17459737.2020.1722269
[8] Cannas, Sonia, Antonini, Samuele, and Pernazza, Ludovico. 2017. “On the Group of Transformations of Classical Types of Seventh Chords.” In Mathematics and Computation in Music, edited by Octavio A. Agustín-Aquino, Emilio Lluis-Puebla, and Mariana Montiel, 13-25. Cham: Springer International Publishing. · Zbl 1456.00063
[9] Catanzaro, Michael J., Generalized Tonnetze, Journal of Mathematics and Music, 5, 2, 117-139 (2011) · Zbl 1226.00017 · doi:10.1080/17459737.2011.614448
[10] Childs, Adrian P., Moving Beyond Neo-Riemannian Triads: Exploring a Transformational Model for Seventh Chords, Journal of Music Theory, 42, 2, 181-193 (1998) · doi:10.2307/843872
[11] Clough, John., A Rudimentary Geometric Model for Contextual Transposition and Inversion, Journal of Music Theory, 42, 2, 297-306 (1998) · doi:10.2307/843880
[12] Clough, John.2008. Flip-Flop Circles and Their Groups, Ned - new edition ed. 23-48. Boydell & Brewer. http://www.jstor.org/stable/.
[13] Coecke, Bob, Genovese, Fabrizio, Lewis, Martha, and Marsden, Dan. 2017. “Generalized Relations in Linguistics and Cognition.” In Logic, Language, Information, and Computation, edited by Juliette Kennedy and Ruy J. G. B. de Queiroz, 256-270. Berlin, Heidelberg: Springer. · Zbl 1408.91186
[14] Cohn, Richard., Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions, Music Analysis, 15, 1, 9-40 (1996) · doi:10.2307/854168
[15] Cohn, Richard., Neo-Riemannian Operations, Parsimonious Trichords, and Their “Tonnetz” Representations, Journal of Music Theory, 41, 1, 1-66 (1997) · doi:10.2307/843761
[16] Crans, Alissa S.; Fiore, Thomas M.; Satyendra, Ramon, Musical Actions of Dihedral Groups, The American Mathematical Monthly, 116, 6, 479-495 (2009) · Zbl 1229.00013 · doi:10.1080/00029890.2009.11920965
[17] Douthett, Jack.2008. Filtered Point-Symmetry and Dynamical Voice-Leading, Ned - new edition ed. 72-106. Boydell & Brewer. http://www.jstor.org/stable/.
[18] Douthett, Jack; Steinbach, Peter, Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition, Journal of Music Theory, 42, 2, 241-263 (1998) · doi:10.2307/843877
[19] Fiore, Thomas M.; Noll, Thomas, Commuting Groups and the Topos of Triads, 69-83 (2011), Berlin, Heidelberg: Springer, Berlin, Heidelberg · Zbl 1335.00121 · doi:10.1007/978-3-642-21590-2_6
[20] Fiore, Thomas M.; Noll, Thomas, Voicing Transformations of Triads, SIAM Journal on Applied Algebra and Geometry, 2, 2, 281-313 (2018) · Zbl 1391.00035 · doi:10.1137/16M1110054
[21] Fiore, Thomas M.; Noll, Thomas; Satyendra, Ramon, Morphisms of Generalized Interval Systems and PR-Groups, Journal of Mathematics and Music, 7, 1, 3-27 (2013) · Zbl 1327.00019 · doi:10.1080/17459737.2013.785724
[22] Fong, Brendan; Spivak, David I., An Invitation to Applied Category Theory: Seven Sketches in Compositionality (2019), Cambridge University Press · Zbl 1464.18001
[23] Forrest, David L., PL Voice Leading and the Uncanny in Pop Music, Music Theory Online, 23, 4 (2017) · doi:10.30535/mto.23.4.6
[24] Forte, Allen., The Structure of Atonal Music (1973), Yale University Press
[25] Gap, GAP - Groups, Algorithms, and Programming, Version 4.11.0 (2020), The GAP Group
[26] Gollin, Edward., Some Aspects of Three-Dimensional ‘Tonnetze’, Journal of Music Theory, 42, 2, 195-206 (1998) · doi:10.2307/843873
[27] Heine, Erik., Chromatic Mediants and Narrative Context in Film, Music Analysis, 37, 1, 103-132 (2018) · doi:10.1111/musa.12106
[28] Hook, Julian., Uniform Triadic Transformations, Journal of Music Theory, 46, 1-2, 57-126 (2002) · doi:10.1215/00222909-46-1-2-57
[29] Hook, Julian., Cross-Type Transformations and the Path Consistency Condition, Music Theory Spectrum, 29, 1, 1-40 (2007) · doi:10.1525/mts.2007.29.1.1
[30] Hook, Julian.2008. Signature Transformations, Ned - new edition ed., 137-160. Boydell & Brewer. http://www.jstor.org/stable/.
[31] Jedrzejewski, Franck., Hétérotopies Musicales (2019), Hermann
[32] Kelly, G. M., and Street, Ross. 1974. “Review of the Elements of 2-Categories.” In Category Seminar, edited by Gregory M. Kelly, 75-103. Berlin, Heidelberg: Springer. · Zbl 0334.18016
[33] Klumpenhouwer, Henry.1991. “A Generalized Model of Voice-Leading for Atonal Music.” PhD thesis, Harvard University.
[34] Kolman, Oren., Transfer Principles for Generalized Interval Systems, Perspectives of New Music, 42, 1, 150-189 (2004)
[35] Leinster, Tom.1998. “Basic Bicategories.” https://arxiv.org/abs/math/9810017. · Zbl 1295.18001
[36] Lewin, David., Transformational Techniques in Atonal and Other Music Theories, Perspectives of New Music, 21, 1-2, 312-381 (1982) · doi:10.2307/832879
[37] Lewin, David., Generalized Music Intervals and Transformations (1987), Yale University Press
[38] Lewin, David., Klumpenhouwer Networks and Some Isographies That Involve Them, Music Theory Spectrum, 12, 1, 83-120 (1990) · doi:10.2307/746147
[39] MacLane, Saunders., Categories for the Working Mathematician, 5 (1971), New York: Springer-Verlag, New York · Zbl 0705.18001
[40] Mannone, Maria, and Turchet, Luca. 2019. “Shall We (Math and) Dance?” In Mathematics and Computation in Music, edited by Mariana Montiel, Francisco Gomez-Martin, and Octavio A. Agustín-Aquino, 84-97. Cham: Springer International Publishing. · Zbl 1448.00018
[41] Mazzola, Guerino., The Topos of Music: Geometric Logic of Concepts, Theory, and Performance (2002), Basel: Birkhäuser Verlag, Basel · Zbl 1104.00003
[42] Mazzola, Guerino; Andreatta, Moreno, Diagrams, Gestures and Formulae in Music, Journal of Mathematics and Music, 1, 3, 199-200 (2007) · Zbl 1139.00006 · doi:10.1080/17459730701774269
[43] Noll, Thomas.2005. “The Topos of Triads.” In Colloquium on Mathematical Music Theory, Grazer Math. Ber., Vol. 347, edited by Karl-Franzens Graz, 103-135. University in Graz, Austria. · Zbl 1132.00309
[44] Noll, Thomas., Musical Intervals and Special Linear Transformations, Journal of Mathematics and Music, 1, 2, 121-137 (2007) · Zbl 1181.00022 · doi:10.1080/17459730701375026
[45] Pattison, Philippa., Algebraic Models for Social Networks (1993), Cambridge University Press · Zbl 0839.92029
[46] Peck, Robert W., Wreath Products in Transformational Music Theory, Perspectives of New Music, 47, 1, 193-210 (2009)
[47] Peck, Robert W.2011. “NthRoots of Pitch-Class Inversion.” In Mathematics and Computation in Music, edited by Carlos Agon, Moreno Andreatta, Gérard Assayag, Emmanuel Amiot, Jean Bresson, and John Mandereau, 196-206. Berlin, Heidelberg: Springer. · Zbl 1339.00017
[48] Piovan, Luis A., A Tonnetz Model for Pentachords, Journal of Mathematics and Music, 7, 1, 29-53 (2013) · Zbl 1327.00023 · doi:10.1080/17459737.2013.769637
[49] Plotkin, Richard., Chord Proximity, Parsimony, and Analysis with Filtered Point-Symmetry, Music Theory Online, 25, 2 (2019) · doi:10.30535/mto.25.2.3
[50] Plotkin, Richard; Douthett, Jack, Scalar Context in Musical Models, Journal of Mathematics and Music, 7, 2, 103-125 (2013) · Zbl 1327.00024 · doi:10.1080/17459737.2013.820804
[51] Popoff, Alexandre., Building Generalized Neo-Riemannian Groups of Musical Transformations As Extensions, Journal of Mathematics and Music, 7, 1, 55-72 (2013) · Zbl 1327.00025 · doi:10.1080/17459737.2013.768712
[52] Popoff, Alexandre., Opycleid: A Python Package for Transformational Music Theory, Journal of Open Source Software, 3, 32, 981 (2018) · doi:10.21105/joss.00981
[53] Popoff, Alexandre, Andreatta, Moreno, and Ehresmann, Andrée. 2015. “A Categorical Generalization of Klumpenhouwer Networks.” In Mathematics and Computation in Music, edited by Tom Collins, David Meredith, and Anja Volk, 303-314. Cham: Springer International Publishing. · Zbl 1321.00088
[54] Popoff, Alexandre; Agon, Carlos; Andreatta, Moreno; Ehresmann, Andrée, From K-Nets to PK-Nets: A Categorical Approach, Perspectives of New Music, 54, 2, 5-63 (2016) · doi:10.7757/persnewmusi.54.2.0005
[55] Popoff, Alexandre; Andreatta, Moreno; Ehresmann, Andrée, Relational Poly-Klumpenhouwer Networks for Transformational and Voice-Leading Analysis, Journal of Mathematics and Music, 12, 1, 35-55 (2018) · Zbl 1406.00003 · doi:10.1080/17459737.2017.1406011
[56] Popoff, Alexandre, Andreatta, Moreno, and Ehresmann, Andrée. 2019. “Groupoids and Wreath Products of Musical Transformations: A Categorical Approach from poly-Klumpenhouwer Networks.” In Mathematics and Computation in Music, edited by Mariana Montiel, Francisco Gomez-Martin, and Octavio A. Agustín-Aquino, 33-45. Cham: Springer International Publishing. · Zbl 1456.00095
[57] Reenan, Samuel; Bass, Richard, Types and Applications of \(####\) Seventh-Chord Transformations in Late Nineteenth-Century Music, Music Theory Online, 22, 2 (2016) · doi:10.30535/mto.22.2.3
[58] Rockwell, Joti., Birdcage Flights: A Perspective on Inter-Cardinality Voice Leading, Music Theory Online, 15, 5 (2009) · doi:10.30535/mto.15.5.4
[59] Sobociński, Paweł. 2012. “Relational Presheaves as Labelled Transition Systems.” In Coalgebraic Methods in Computer Science, edited by Dirk Pattinson and Lutz Schröder, 40-50. Berlin, Heidelberg: Springer. · Zbl 1328.68139
[60] Tymoczko, Dmitri., Geometrical Methods in Recent Music Theory, Music Theory Online, 16, 1 (2010) · doi:10.30535/mto.16.1.7
[61] Tymoczko, Dmitri., A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (2011), Oxford University Press · Zbl 1339.00022
[62] Tymoczko, Dmitri., In Quest of Musical Vectors, 256-282 (2016), World Scientific · Zbl 1368.00039 · doi:10.1142/9789813140103_0014
[63] Yust, Jason., Harmonic Qualities in Debussy’s “Les Sons Et Les Parfums Tournent Dans L”air Du Soir, Journal of Mathematics and Music, 11, 2-3, 155-173 (2017) · Zbl 1395.00023 · doi:10.1080/17459737.2018.1450457
[64] Yust, Jason., Generalized Tonnetze and Zeitnetze, and the Topology of Music Concepts, Journal of Mathematics and Music, 30, 2, 1-34 (2020) · Zbl 1466.00023 · doi:10.1080/17459737.2020.1725667
[65] Zelinka, Bohdan., Graphs of Semigroups, Časopis Pro Pěstování Matematiky, 106, 4, 407-408 (1981) · Zbl 0479.05033 · doi:10.21136/CPM.1981.108493
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.