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Mathematical Theory of Gestures in Music

DANS CES MURS VOUÉS AUX MERVEILLES J’ACCUEILLE ET GARDE LES OUVRAGES DE LA MAIN PRODIGIEUSE DE L’ARTISTE ÉGALE ET RIVALE DE SA PENSÉE L’UNE N’EST RIEN SANS L’AUTRE (Paul Valéry, Palais Chaillot). Mathematical Theory of Gestures in Music.

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Mathematical Theory of Gestures in Music

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  1. DANS CES MURS VOUÉS AUX MERVEILLES J’ACCUEILLE ET GARDE LES OUVRAGES DE LA MAIN PRODIGIEUSE DE L’ARTISTE ÉGALE ET RIVALE DE SA PENSÉE L’UNE N’EST RIEN SANS L’AUTRE (Paul Valéry, Palais Chaillot) Mathematical Theory of Gestures in Music IN THESE WALLS DEVOTED TO THE MARVELSI RECEIVE AND KEEP THE WORKS OF THE PRODIGIOUS HAND OF THE ARTIST EQUAL AND RIVAL OF HIS THOUGHT ONE IS NOTHING WITHOUT THE OTHER Guerino Mazzola U Minnesota & Zürich mazzola@umn.edu   guerino@mazzola.ch     www.encyclospace.org        

  2. LA VÉRITÉDU BEAU DANS LA MUSIQUE Guerino Mazzola www.encyclospace.org/talks

  3. Motivation • Ontology • Gestures • Symbols

  4. Motivation • Ontology • Gestures • Symbols

  5. Gestures in Performance Theory Theodor W. Adorno(„Zu einer Theorie der musikalischen Reproduktion“ 1946): Danach wäre die Aufgabe des Interpreten, Noten so zu betrachen, bis sie dem insistenten Blick in Originalmanuskripte sich verwandeln; nicht aber als Bilder der Seelenregung des Autors — sie sind auch dies, aber nur akzidentiell — sondern als die seismographischen Kurven, die der Körper der Musik selber in seinen gestischen Erschütterungen hinterlassen hat. Correspondingly the task of the interpreter would be to consider the notes until they are transformed into original manuscripts under the insistent eye of the observer; however not as images of the author‘s emotion—they are also such, but only accidentally—but as the seismographic curves, which the body has left to the music in its gestural vibrations.

  6. Gestures in Performance Theory (Jürgen Uhde &) Renate Wieland („Forschendes Üben“ 2002): Affekte waren ursprünglich ja Handlungen, bezogen auf ein Objekt draussen, im Prozess der Verinnerlichung haben sie sich von ihrem Gegenstand gelöst, aber immer noch sind sie bestimmt von den Koordinaten des Raumes. (...) Es gibt mithin etwas wie gestische (Raum-)Koordinaten. Originally affects were actions, related to an exterior object, along the process of interiorization they were detached from their object, but they are still determined by the coordinates of space. (...) There is therefore something like gestural (space) coordinates.

  7. Musical Transformational Theory David Lewin(„Generalized Musical Intervalsand Transformations“ 1987): If I am at s and wish to get to t, what characteristic gesture should I perform in order to arrive there?

  8. Music Theory Robert S. Hatten(„Interpreting Musical Gestures, Topics, and Tropes“ 2004) Given the importance of gesture to interpretation, why do we not have a comprehensive theory of gesture in music?

  9. Jazz Cecil Taylor The body is in no way supposed to get involved in Western music.I try to imitate on the piano the leaps in space a dancer makes.

  10. gestures soundevents pitch h time e l position instrumental-interface Instruments/playing action Musical sound √ thaw Musical theory/notation score Tellef Kvifte: Instruments andthe electronic Age. Solum forlag, Oslo, 1988 analysis instrumentalize • Gestural semiotics: Adam Kendon, David McNeill • Presemiotic gestures: Susan Goldin-MeadowAdvances in mathematical reasoning are very likely to come first in gesture—and they do. Do new ideas always come first in gesture, regardless of domain? IRCAM CRISIS May 18th 2002 AK: Gesture DM: Gesture & Thought SG: Hearing Gesture

  11. Francis Bacon Painting The marks are made, and you survey the thing like you would a sort of graph. And you see within this graph the possibilities of all types of fact being planted..David Sylvester: Interview with Francis Bacon: The Brutality of Fact. Thames and Hudson, New York 1975

  12. Jean Cavaillès (1903 - 1944):Comprendre est attraper le geste et pouvoir continuer. Gilles Deleuze (1925 - 1995): Francis Bacon. La logique de la sensation. Editions de la Différence, Paris 1981„graph“  „diagramme“  „geste“ Charles Alunni (1951 -):Ce n‘est pas la règle qui gouverne l‘action diagrammatique, mais l‘action qui fait émerger la règle. It is not the rule that governs the diagrammatic action, but the action that causes the rule to emerge. Understanding is catching the gesture and being capable of continuing.

  13. Embodied AI Stumpy: AI Lab, U Zurich „Cheap design“

  14. Motivation • Ontology • Gestures • Symbols

  15. expression signification content physical psychological symbolic poiesis neutral level aesthesis The Ontology of Music signs realities communication

  16. Facts realities communication Processes Gestures signs The Oniontology of Music

  17. Gesture Theory in Computer Music Research: • Frédéric Bevilacqua • Claude Cadoz • Antonio Camurri • Rolf Inge Godøy • Stefan Müller • Norbert Schnell • Koji Shibuya • McAgnus Todd • Marcelo Wanderley • etc.

  18. Koji Shibuya‘sRyukoku violin robot

  19. facts: complete classification of addressed global compositions • processes: relative classification of global networks via a functor to global compositions • gestures: no mathematical theory Mathematical Music Theory   

  20. Motivation • Ontology • Gestures • Symbols

  21. a11 a12 a13 a21 a22 a23 a31 a32 a33 x y z a b c = algebra compactifies gestures to symbolic formulas rotation matrix equation a11x+a12y+a13z = a a21x+a22y+a23z = b a31x+a32y+a33z = c

  22. S T K RK P X T Q RK = R[X]polynomial algebra „attempt at resuscitation“ Peter Gabriel: Symbolic formulas via directed graphs = „quiver algebras“ mathematics of transformational theory

  23. ? Graphs are only the „skeleton“ of gestures, the „flesh“ is missing.

  24. Gesture = morphism g: D  of directed graphs with values in a spatial directed graph of a topological space X (= directed graph of continuous curves in X) pitch   X X body time skeleton position X g D

  25.  X Y g D u  f E h A gesture morphism u:gh is a morphism u of directed graphs, such that there is a continuous map f: X  Y whichdefines a commutative diagram: G(g, h)category G of gestures Advantage: Directed graphs have an inherent (intuitionistic) logic, because the category of directed graphs is a topos. This is not only cheap, but free design!

  26. e = time y z   (t ) (t ) 6 6   (t ) (t ) 5 5   (t ) (t ) 4 4   (t ) (t ) 3 3     (t ) (t ) (t ) (t ) x 2 2 1 1 One hand  product  = 123456of 6 gestural curves in space-time (x,y,z;e) of piano j = 1, 2, ... 5: tips of fingers j = 6: the carpus

  27. Stefan Müller

  28. pitch onset position tip space real forms? p D Renate Wieland & Jürgen Uhde:Forschendes Üben Die Klangberührung ist das Ziel der zusammenfassenden Geste, der Anschlag ist sozusagen die Geste in der Geste. The sound contact is the target of the embracing gesture, the touch is so to speak the gesture within the gesture.

  29. Digraph(F, ) = topological space of gestures of of directed graph F with values in a spatial directd graph . Notation:F @X    X X knot circle Hypergestures! „loop of loops “

  30. time space space ET-dance gesture

  31. g E h hypergesture impossible!

  32. y z e x

  33. Proposition (Escher Theorem) Given a topological space X, a sequence of directed graphs F1 , F2, ...Fn and a permutation  of 1, 2,... n. Then there is a topological isomorphism F1@ ...Fn@X  F(1)@ ...F (n)@X     

  34. h g k Comprendre est attraper le geste et pouvoir continuer.

  35. Motivation • Ontology • Gestures • Symbols

  36. homotopic curves X Gestoids: From Gestures to Symbols AlgebraicTopology

  37. X 0 1 0 1 0 1 The Fundamental Group composition of homotopy classes of curves is associative (a.b).c = a.(b.c) have fundamental group1(X) = {homotopy classes of curves} + composition

  38. i— i ei2t — g: 1 X = S1 Which groups are fundamental groups 1(X)? • Ÿ integers? • Ÿ12 (pitch class group!)? • finitely generated abelian groups? 1(S1) = Ÿ

  39. x y The homotopy classes of curves of a gesture gdefine the Gestoid Gg of a gesture g. This consists of the linear combinations with complex coefficientsn ancnof homotopy classes cn of curves between given points x, y of gesture g. too abstract?

  40. i— i ei2t — g: 1 X = S1 1(S1) = Ÿ ei2nt ~ n Gg = ¬ 1(S1) n anei2nt ~ Fourier formula f(t) = n an ei2nt

  41. Fourier ballet F 2F 3F 4F QED

  42. Zn 1 1(X) = Ÿn? action of Ÿn S3 1(S3)  0 Ln,1 1(Ln,1) = Ÿn

  43. Musical Poetology Michael Cherlin(„Schoenberg‘s Musical Imagination“ 2007) Metaleptic Schoenberg: Trope of metaphor of a metaphor~hypergesture. Search for a language forthe musical drama, which isneither emotional programnor structural formalismbut a thinking in music... He's a lead foot/ couch potato

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