Musical Gestures and their Diagrammatic Logic Musical Gestures and their Diagrammatic Logic Guerino Mazzola U & ETH Zürich U & ETH Zürich 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) 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)

musique mathématique LA VERITÉ DU BEAU DANS LA MUSIQUE Guerino Mazzola summer 2006

gestegeste formuleformule harmonie de gestes ~ composition de formules ~ musique mathématique

Ryukoku violin robot

Waseda wabot II

Musical Gestures Musical Gestures Gesture Categories Gesture Categories Diagram Logic Diagram Logic

Musical Gestures Musical Gestures Gesture Categories Gesture Categories Diagram Logic Diagram Logic

l h e sonic events score analysis instrumental interface √ thaw freeze (MIDI) instrumentalizeinstrumentalizegestualizegestualizeposition pitch timegestures

Ceslaw Marek: Lehre des KlavierspielsAtlantis-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-sub- ject 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.Shite weaves a texture of fantasy using curves. Waki describes reality using straight lines.Waki describes reality using straight lines.

— positionpitchtime 01  1111 2222  2 +  1  t.  2222 1111  1   2

√ gestures √ score he l HE L positionpitchE

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

C3 DIN8996

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

x z y   1 1 (t ) )   6 6 ) )   2 2 ) )   3 3 ) )   4 4 ) )   5 5 ) ) 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 parameter t  sequence of points:  (t) = (  1 (t),...,  6 (t)) two base vectors of fingers d 2, d 5 from carpus. e = time

Geometric constraints: six boxes

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

Use cubic polynomials for gestural coordinates, i.e., 76 variables of coefficients:  x j (t)= x j,3 t 3 + x j,2 t 2 + x j,1 t + x j,0  y j (t)= y j,3 t 3 + y j,2 t 2 + y j,1 t + y j,0  z j (t) = z j,3 t 3 + z j,2 t 2 + z j,1 t + z j,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 Musical Gestures Gesture Categories Gesture Categories Diagram Logic Diagram Logic

vx w y c a bd Quiver = category of quivers (= digraphs, diagram schemes, etc.) D = A V ht x = t(a) y = h(a) a E = B W h‘t‘ uq D Quiver ( D, E )

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: f  D E X  Y  g h u Gesture (g, h) category of (local) gestures (Local) Gesture = morphism g: D  of quivers with values in a spatial quiver of a metric space X (= quiver of continuous curves in X) XX  D position pitch time X g

A global gesture being covered by three local gestures

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

E g h Hypergesture impossible! g h Morphism exists!

Musical Gestures Musical Gestures Gesture Categories Gesture Categories Diagram Logic Diagram Logic

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

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

Heyting logic on set Sub(g) of subgestures of g h, k  Sub(g) h  k = h  k 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 Heyting logic on set Sub(g) of subgestures of g h, k  Sub(g) h  k = h  k 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

VII I III V II VI IV cd e f g a b C-major hypergesture Fingers = Quiver ( F, ) X  Fingers F =

V I  I VI IV =

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. Problems:

Graphics of sequence of physical gestural curves finger 3 finger 2 C4

A morphism between hypergestures g h u