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.
Download ReportTranscript 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.
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) 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 • Motivation - performance - and music theory - gestural music and painting - French philosophy - Embodied AI • Speculum - the musical oniontology - classification of global compositions and networks • Gestures - categories of gestures - hypergestures - the Escher theorem and free jazz • Symbols - homotopy - gestoids - finitely generated abelian groups and networks • Motivation - performance - and music theory - gestural music and painting - French philosophy - Embodied AI • Speculum - the musical oniontology - classification of global compositions and networks • Gestures - categories of gestures - hypergestures - the Escher theorem and free jazz • Symbols - homotopy - gestoids - finitely generated abelian groups and networks 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. 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. 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 √ thaw Instruments/ playing action Musical sound score Musical theory/ notation 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 -): Ce n‘est pas la règle qui gouverne l‘action diagrammatique, mais l‘action qui fait émerger la règle. Jean Cavaillès (1903 - 1944): Comprendre est attraper le geste et pouvoir continuer. Embodied AI Stumpy: AI Lab, U Zurich Zur Anzeige w ird der QuickTime™ Dekompressor „YUV420 codec“ benötigt. • Motivation - performance - and music theory - gestural music and painting - French philosophy - Embodied AI • Speculum - the musical oniontology - classification of global compositions and networks • Gestures - categories of gestures - hypergestures - the Escher theorem and free jazz • Symbols - homotopy - gestoids - finitely generated abelian groups and networks The Oniontology of Music communication Processes Gestures signs Facts facts The category GloComA of global objective A-addressed compositions has objects KI, i.e., coverings of sets K by atlases I of local objective A-addressed compositions with manifold gluing conditions and manifold morphisms ff: KI LJ, including and compatible with atlas morphisms f: I J II V IV VI I VII III Theorem (global addressed geometric classification) Let A = locally free of finite rank over commutative ring R Consider the objective global compositions KI at A with (*): • the chart modules R.Ki are locally free of finite rank • the function modules G(Ki) are projective There is a subscheme Jn* of a projective R-scheme of finite type whose points w: Spec(S) Jn* parametrize the isomorphism classes of objective global compositions at address SRA with (*). processes KI can be reconstructed from the coefficient system of retracted functions on free global compositions res*nG(KI) nG(ADn*) Fact: This construction is a special case of a local network. Global compositions are classified by limits of powerset denotators. 3 7 Ÿ12 T4 T5.-1 Ÿ12 T11.-1 D Ÿ12 2 T2 Ÿ12 4 A global network Theorem Given address A in Mod, we have a verification functor |?|: ALfModred AGlob from the category ALfModred of reduced, A-addressed locally flat global networks to the category AGlob of A-addressed global compositions. Corollary There are non-interpretable global networks in ALfModred Local and Global Limit Denotators and the Classification of Global Compositions. COLLOQUIUM ON MATHEMATHICAL MUSIC THEORY H. Fripertinger, L. Reich (Eds.) Grazer Math. Ber., ISSN 1016–7692 Bericht Nr. 347 (2005), Global Networks in Computer Science? http://www.encyclospace.org/talks/networks.ppt 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 - performance - and music theory - gestural music and painting - French philosophy - Embodied AI • Speculum - the musical oniontology - classification of global compositions and networks • Gestures - categories of gestures - hypergestures - the Escher theorem and free jazz • Symbols - homotopy - gestoids - finitely generated abelian groups and networks 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 digraphs = „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. ? morphism g: D X of digraphs withvalues in a spatial digraph X of a topological space X (= digraph of continuous curves in X) (Local) Gesture = body skeleton pitch g X D time position A gesture morphism u: g h is a digraph morphism u, 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 (local) gestures Advantage: Digraphs have an inherent (intuitionistic) logic, because the category of digraphs is a topos. This is not only cheap, but free design! A global gesture (only bodies shown) 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) j = 6: the carpus 2(t) (t) 1 = Ÿ Zur Anzeige w ird der QuickTime™ Dekompressor „H.264“ benötigt. Stefan Müller Renate Wieland & Jürgen Uhde: Forschendes Üben Die Klangberührung ist das Ziel der zusammenfassenden Geste, der Anschlag ist sozusagen die Geste in der Geste. pitch tip space real forms? p D onset position Digraph(F, X )= „loop of loops “ knot circle topological space of (local) gestures of of digraph F with values in a spatial digraph . Notation: F @X X ET-dance gesture time space Zur Anzeige w ird der QuickTime™ Dekompressor „Animation“ benötigt. space hypergesture impossible! g E h morphism exists! g h Gestural maps are particular continuous maps (u,v): F @X G @Y canonically induced by a pair of maps u: G F (digraphs) v: X Y (continuous) The category HG = HG1 of hypergestures has 1) hypergestures as objects and 2) gestural maps as continuous maps. The category HGn of n-fold hypergestures has 1) objects: n-fold hypergestures g: Fn Fn-1@ ... F1@ X 2) (n-1)-fold gestural maps as continuous maps. Have chain of successively refined gestural categories G HG = HG1 HG2 ... HGn HGn+1 ... which represent the granularity of gestural relations, much as in differential geometry, where the categories of n-times differentiable manifolds do. E.g. gluing local gestures to global gestures in quasi-anatomic joints Proposition (Escher Theorem) Given a topological space X, a sequence of digraphs F1 , F2, ... Fn and a permutation of 1, 2,... n. Then there is a homeomorphism F1@ ... Fn@X F(1)@... F (n)@X h g k Comprendre est attraper le geste et pouvoir continuer. • Motivation - performance - and music theory - gestural music and painting - French philosophy - Embodied AI • Speculum - the musical oniontology - classification of global compositions and networks • Gestures - categories of gestures - hypergestures - the Escher theorem and free jazz • Symbols - homotopy - gestoids - finitely generated abelian groups and networks Gestoids: From Gestures to Symbols homotopic curves X X 0 0 1 1 0 1 composition of homotopic curves is associative The homotopy classes of curves of a gesture g define the Gestoid Gg of a gesture g. This consists of the linear combinations n ancn of homotopy classes cn of curves between given points x, y of gesture g. x y i— 1(X) Ÿn ? finitely generated abelian groups? g: Gg ¬ 1(S1) i ei2t 1 X = S1 fundamental group 1(S1) Ÿ ei2nt ~ n 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 action of Ÿn S3 Zn 1(S3) 0 1 1(Ln,1) Ÿn Ln,1 gestures =? string theory of music