DANS CES MURSDEVOTED VOUÉS AUX IN THESE WALLS TO MERVEILLES THE MARVELS J’ACCUEILLE GARDE I RECEIVEET AND KEEPLES THEOUVRAGES WORKS DETHE LA MAIN PRODIGIEUSE OF PRODIGIOUS HAND DE OF L’ARTISTE THE ARTIST ÉGALEAND ET RIVALE DE HIS SA PENSÉE EQUAL RIVAL OF THOUGHT L’UNE N’EST RIEN SANS L’AUTRE ONE IS NOTHING WITHOUT THE OTHER (Paul Valéry, Palais Chaillot) Guerino.
Download ReportTranscript DANS CES MURSDEVOTED VOUÉS AUX IN THESE WALLS TO MERVEILLES THE MARVELS J’ACCUEILLE GARDE I RECEIVEET AND KEEPLES THEOUVRAGES WORKS DETHE LA MAIN PRODIGIEUSE OF PRODIGIOUS HAND DE OF L’ARTISTE THE ARTIST ÉGALEAND ET RIVALE DE HIS SA PENSÉE EQUAL RIVAL OF THOUGHT L’UNE N’EST RIEN SANS L’AUTRE ONE IS NOTHING WITHOUT THE OTHER (Paul Valéry, Palais Chaillot) Guerino.
DANS CES MURSDEVOTED VOUÉS AUX IN THESE WALLS TO MERVEILLES THE MARVELS J’ACCUEILLE GARDE I RECEIVEET AND KEEPLES THEOUVRAGES WORKS DETHE LA MAIN PRODIGIEUSE OF PRODIGIOUS HAND DE OF L’ARTISTE THE ARTIST ÉGALEAND ET RIVALE DE HIS SA PENSÉE EQUAL RIVAL OF THOUGHT L’UNE N’EST RIEN SANS L’AUTRE ONE IS NOTHING WITHOUT THE OTHER (Paul Valéry, Palais Chaillot) Guerino Mazzola U Minnesota & Zürich [email protected] [email protected] www.encyclospace.org Mathematical Theory of Gestures in Music LA VÉRITÉ DU BEAU DANS LA MUSIQUE Guerino Mazzola www.encyclospace.org/talks • Motivation • Ontology • Gestures • Symbols • Motivation • Ontology • Gestures • Symbols Gestures in Performance Theory Theodor W. Adorno („Zu einer Theorie der musikalischen Reproduktion“ 1946): Danach wäre die the Aufgabe Correspondingly task ofdes theInterpreten, interpreter Noten so betrachen, sie dem would be zu to consider thebis notes until they are insistenten Blick Originalmanuskripte transformed into in original manuscripts under sichinsistent verwandeln; aber als however not the eye ofnicht the observer; Bilder der of Seelenregung Autors — are as images the author‘sdes emotion—they sie sind auch nur akzidentiell also such, butdies, only aber accidentally—but as — the sondern als diecurves, seismographischen Kurven, seismographic which the body has die der Körper Musik selber in seinen left to the musicder in its gestural vibrations. gestischen Erschütterungen hinterlassen hat. Gestures in Performance Theory (Jürgen Uhde &) Renate Wieland („Forschendes Üben“ 2002): Originally affects were actions, Affekte waren ursprünglich ja related to an exterior object, along Handlungen, bezogen aufthe einprocess Objekt of interiorization they were from draussen, im Prozess derdetached Verinnerlichung their butvon they are still determined habenobject, sie sich ihrem Gegenstand by the coordinates space. is gelöst, aber immerof noch sind(...) sieThere bestimmt therefore something like von den Koordinaten desgestural Raumes.(space) (...) Es coordinates. gibt mithin etwas wie gestische (Raum-)Koordinaten. Musical Transformational Theory David Lewin („Generalized Musical Intervals and Transformations“ 1987): If I am at s and wish to get to t, what characteristic gesture should I perform in order to arrive there? 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? 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. pitch gestures time sound events h instrumentalize e l position instrumentalinterface √ McNeill • thaw Gestural semiotics: Adam Kendon, David Instruments/ Musical sound playing action • Presemiotic gestures: Susan Goldin-Meadow Advances in mathematical reasoning score May IRCAM CRISIS 18th 2002 are very likely to come first in gesture—and they do. Musical theory/ notation AK: Gesture DM: Gesture & Thought SG: Hearing Gesture Do new ideas always come first in gesture, regardless of domain? analysis Tellef Kvifte: Instruments and the electronic Age. Solum forlag, Oslo, 1988 Painting Francis Bacon 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 Gilles Deleuze (1925 - 1995): Francis Bacon. La logique de la sensation. Editions de la Différence, Paris 1981 „graph“ „diagramme“ „geste“ Charles Alunni (1951 -): It isn‘est not the that governs the diagrammatic action, Ce pasrule la règle qui gouverne l‘action but the action thatmais causes the rule emerge. diagrammatique, l‘action quitofait émerger la règle. Jean Cavaillès (1903 - 1944): Comprendre estisattraper geste et pouvoir Understanding catchinglethe gesture and continuer. being capable of continuing. Embodied AI Stumpy: AI Lab, U Zurich Zur Anzeige w ird der QuickTime™ Dekompressor „YUV420 codec“ benötigt. • Motivation • Ontology • Gestures • Symbols The Ontology of Music expression signs signification physical psychological content communication poiesis neutral level symbolic aesthesis The Oniontology of Music communication Processes Gestures signs Facts 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. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Koji Shibuya‘s Ryukoku violin robot • facts: complete classification of addressed global compositions Mathematical Music Theory • processes: relative classification of global networks via a functor to global compositions • gestures: no mathematical theory • Motivation • Ontology • Gestures • Symbols algebra compactifies gestures to symbolic formulas rotation Zur Anzeige w ird der QuickTime™ Dekompressor „Animation“ benötigt. matrix equation a11x+a12y+a13z = a a21x+a22y+a23z = b a31x+a32y+a33z = c a11 a12 a13 a21 a22 a23 a31 a32 a33 x a y = b z c „attempt at resuscitation“ Peter Gabriel: Symbolic formulas via directed graphs = „quiver algebras“ S T RK K P X Q T RK = R[X] polynomial algebra mathematics of transformational theory Graphs are only the „skeleton“ of gestures, the „flesh“ is missing. ? Gesture = morphism g: D X of directed graphs with values in a spatial directed graph X of a topological space X (= directed graph of continuous curves in X) body skeleton pitch g X D time position A gesture morphism u: g h is a morphism u of directed graphs, such that there is a continuous map f: X Y which defines a commutative diagram: D g X u E f h Y 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! e = time Z 6(t) Q u ic k T im e ™ a n d a T I F F ( Un c o m p r e s s e d ) Y y z decom pr essor a r e n e e d e d t o s e e t h is p ic t u r e . 5(t) (t) 4(t) (t) One hand product = 123456 of 6 gestural curves in space-time (x,y,z;e) of piano j = 1, 2, ... 5: tips of fingers 3(t) (t) xX 1(t) 2(t) (t) j = 6: the carpus Zur Anzeige w ird der QuickTime™ Dekompressor „H.264“ benötigt. Stefan Müller Renate Wieland & Jürgen Uhde: Forschendes Üben Die The Klangberührung sound contact is ist target das Ziel der the of the zusammenfassenden embracing gesture, Geste,the dertouch Anschlag is so ist to sozusagen speak the gesture die Geste der Geste. withininthe gesture. pitch tip space real forms? p D onset position Digraph(F, X )= „loop of loops “ knot circle topological space of gestures of of directed graph F with values in a spatial directd graph . Notation: F @X X ET-dance gesture time space Zur Anzeige w ird der QuickTime™ Dekompressor „Animation“ benötigt. space g E h hypergesture impossible! e Z Q u ic k T im e ™ x X a n d a T I F F ( Un c o m p r e s s e d ) yY z decom pr essor a r e n e e d e d t o s e e t h is p ic t u r e . 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 Zur Anzeige w ird der QuickTime™ Dekompressor „Animation“ benötigt. Zur Anzeige w ird der QuickTime™ Dekompressor „Animation“ benötigt. h g k Comprendre est attraper le geste et pouvoir continuer. • Motivation • Ontology • Gestures • Symbols Gestoids: From Gestures to Symbols homotopic curves X The Fundamental Group X 0 0 1 1 0 1 composition of homotopy classes of curves is associative (a.b).c = a.(b.c) have fundamental group 1(X) = {homotopy classes of curves} + composition Which groups are fundamental groups 1(X)? • Ÿ integers? • Ÿ12 (pitch class group!)? • finitely generated abelian groups? i— i ei2t g: 1 X = S1 1(S1) = Ÿ — The homotopy classes of curves of a gesture g define the Gestoid Gg of a gesture g. This consists of the linear combinations with complex coefficients n a n c n of homotopy classes cn of curves between given points x, y of gesture g. x y 1(S1) = Ÿ ei2nt i— ~ n i Gg = ¬ 1(S1) g: ei2t 1 X = S1 n an ei2nt ~ Fourier formula f(t) = n an ei2nt — Fourier ballet F 2F Zur Anzeige w ird der QuickTime™ Dekompressor „Animation“ benötigt. 3F 4F QED 1(X) = Ÿn? action of Ÿn S3 Zn 1(S3) 0 1 1(Ln,1) = Ÿn Ln,1 Musical Poetology Michael Cherlin („Schoenberg‘s Musical Imagination“ 2007) Metaleptic Schoenberg: Trope of metaphor of a metaphor ~ hypergesture. Search for a language for the musical drama, which is neither emotional program nor structural formalism but a thinking in music... He's a lead foot/ couch potato