DANS CES MURS VOUS AUX MERVEILLES JACCUEILLE ET

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 & ETH Zürich guerino@mazzola. ch www. encyclospace. org Musical Gestures and their Diagrammatic Logic

LA VERITÉ DU BEAU DANS LA MUSIQUE Guerino Mazzola e u q i s mu e u q i t a m é h t ma summer 2006

formule ~ harmonie de gestes ~ e u q i s mu composition de formules geste e u q i t a m é h t ma

Ryukoku violin robot

Waseda wabot II

• Musical Gestures • Gesture Categories • Diagram Logic

• Musical Gestures • Gesture Categories • Diagram Logic

gestualize gestures pitch time instrumentalize e l position instrumental interface thaw sonic events h √ freeze (MIDI) score analysis

Ceslaw Marek: Lehre des Klavierspiels Atlantis-Verlag Zürich 1972/77

Folie 2

Every No play is a cross section of the life of one person, the shite. The shite is an appearance (demon, etc. ) and a subject = one of the five elements (fire, water, wood, earth, metal) The waki is A kind of co-subject and mirror person of the shite.

The No gestures are reduced to the kata units and made symbolic. This enables a richer communication than with common gestures. Important: • Shite weaves a texture of fantasy using curves. • Waki describes reality using straight lines.

pitch 1 0 time — position 2 2 1 t. 1 2 2 + 1 1

pitch E position √gestures H h √score E L l e

Ph. D thesis of Stefan Müller (Mazzola G & Müller S: ICMC 2003) Symbolic score (a) Without fingering annotation (b) with fingering annotation

C 3 DIN 8996

Independent symbolic gesture curves for fingers 2 et 3 Curve parameter t on horizontal axis

One hand product = 1 2 3 4 5 6 of 6 gestural curves in space-time (x, y, z; e) of piano j = 1, 2, . . . 5: tips of fingers, j = 6: the carpus, 6 = root e = time y z 6(t) parameter t sequence of points: (t) = ( 1(t), . . . , 6(t)) 5(t) 4(t) 3(t) x 1(t) 2(t) two base vectors of fingers d 2, d 5 from carpus.

Geometric constraints: six boxes

Have masses mj and maximal forces Kj for fingers/carpus j. d 2 space 3 /de 2 The Newton condition for fingers or carpus j is mj d 2 spacej /de 2(t) < Kj for all 0 ≤ t ≤ 1.

Use cubic polynomials for gestural coordinates, i. e. , 76 variables of coefficients: xj(t) = xj, 3 t 3 + xj, 2 t 2 + xj, 1 t + xj, 0 yj(t) = yj, 3 t 3 + yj, 2 t 2 + yj, 1 t + yj, 0 zj(t) = zj, 3 t 3 + zj, 2 t 2 + zj, 1 t + zj, 0 e(t) = e 3 t 3 + e 2 t 2 + e 1 t + e 0 Geometric and physical constraints polynomial inequalities: P(t) > 0 for all 0 ≤ t ≤ 1. These inequalities are guaranteed by Sturm chains.

Symbolic gestural curve Physical gestural curve

fingers 2, 3: geometric constraints fingers 2, 3: physical constraints

Gestural interpretation of Carl Czerny‘s op. 500

• Musical Gestures • Gesture Categories • Diagram Logic

Quiver = category of quivers (= digraphs, diagram schemes, etc. ) D=A h t u x = t(a) V q d v a y = h(a) w c b x E=B h‘ t‘ a W y Quiver(D, E) D

morphism g: D X of quivers with values in a spatial quiver X of a metric space X (= quiver of continuous curves in X) (Local) Gesture = A gesture morphism u: g h is a quiver morphism u, such that there is a continuous map f: X Y which defines a commutative diagram: D g X u E pitch f h Y Gesture(g, h) category of (local) gestures g X D time position

A global gesture being covered by three local gestures

Quiver(F, X ) = Hyperg estures! metric space of (local) gestures of of quiver F with values in a spatial quiver X. r s t Uhde: Renate Wieland & Jürgen Forschendes Üben Die Klangberührung ist das Ziel der zusammenfassenden Geste, der Anschlag ist sozusagen die Geste in der Geste. F E

Hypergesture impossible! g E h Morphism exists! g h

• Musical Gestures • Gesture Categories • Diagram Logic

The category Quiver is a topos D E 1= D+E 0=Ø Alexander Grothendieck DE Quiver( ≈ Quiver(E , D E) Quiver( , DE) , D) ≈ Quiver(E , D)

Subobject classifier = T In particular: The set Sub(D) of subquivers of a quiver D is a Heyting algebra: have „Quiver logic“. Ergo: v w x y Local/global gestures, ANNs, Klumpenhouwer-nets, and global networks enable logical operators ( , , , )

Heyting logic on set Sub(g) of subgestures of g h, k Sub(g) h k=h k h k (complicated) h = h Ø tertium datur: h ≤ h u: g 1 g 2 Sub(u): Sub(g 2) Sub(g 1) homomorphism of Heyting algebras = contravariant functor Sub: Gesture Heyting

C-major hypergesture c b a IVV d III VI I g Fingers II VII e f Fingers = Quiver(FX , ) F=

V I IV = VI I

Problems: • Investigate the possible role and semantics of gestural logic in concrete contexts such as local/global musical/robot gestures and more specific environments. . . (and more generally: Quiver logic for ANNs, Klumpenhouwer-nets, global networks). • Investigate a (formal) propositional/predicate language of gestures with values in Heyting algebras of quivers.