Musical Gestures and their Diagrammatic Logic

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

2. LA VERITÉDU BEAU DANS LA MUSIQUE Guerino Mazzola musique mathématique summer 2006

3. compositionde formules musique ~ ~ mathématique harmoniede gestes formule geste

5. Ryukoku violin robot

6. Waseda wabot II

7. Musical Gestures • Gesture Categories • Diagram Logic

8. Musical Gestures • Gesture Categories • Diagram Logic

9. gestualize gestures sonicevents pitch h time e l position instrumentalinterface √ thaw freeze (MIDI) score analysis instrumentalize

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

11. Folie 2

12. Every No play is a cross sectionof 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.

13. 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 usingcurves. • Waki describes reality usingstraight lines.

14. 1 pitch  0 — time position 2 1  2 2  1 1 t. 2 + 1

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

16. PhD thesis of Stefan Müller (Mazzola G & Müller S: ICMC 2003) Symbolic score (a) Withoutfingering annotation (b) with fingeringannotation

17. C3 DIN8996

18. Independent symbolicgesture curvesfor fingers 2et 3 Curve parameter ton horizontal axis

19. 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, 6 = root parameter t  sequence of points: (t) = (1(t),...,6(t)) two base vectors of fingersd2, d5from carpus.

20. Geometric constraints: six boxes

21. Have masses mj and maximal forces Kjfor fingers/carpus j. d2 space3 /de2 The Newton condition for fingers or carpus j is mj d2 spacej /de2(t) < Kj for all 0 ≤ t ≤ 1.

22. Use cubic polynomials for gestural coordinates, i.e., 76 variables of coefficients: xj(t) = xj,3 t3 + xj,2 t2 + xj,1 t + xj,0  yj(t) = yj,3 t3 + yj,2 t2 + yj,1 t + yj,0  zj(t) = zj,3 t3 + zj,2 t2 + zj,1 t + zj,0  e(t) = e3 t3 + e2 t2 + e1 t + e0 Geometric and physical constraints polynomial inequalities: P(t) > 0 for all 0 ≤ t ≤ 1. These inequalities are guaranteed by Sturm chains.

23. Symbolic gestural curve Physical gestural curve

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

25. Gestural interpretation of Carl Czerny‘s op. 500

27. Musical Gestures • Gesture Categories • Diagram Logic

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

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

30. A global gesture being coveredby threelocal gestures

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

32. g E h g h Hypergesture impossible! Morphism exists!

33. Musical Gestures • Gesture Categories • Diagram Logic

34. 1 = Alexander Grothendieck Quiver(, DE) Quiver(, DE) ≈ Quiver(E  , D) ≈ Quiver(E  , D) The category Quiver is a topos D  E D+E 0 = Ø DE

35.  = v w x y T In particular:The set Sub(D) of subquiversof a quiver Dis a Heyting algebra: have „Quiver logic“. Ergo: Local/global gestures, ANNs, Klumpenhouwer-nets, and global networksenable logicaloperators (, , ,) Subobject classifier

36. Heyting logic on set Sub(g) of subgestures of g h, k  Sub(g)h  k = h  kh  k = h  k h  k (complicated) h = h  Ø tertium datur: h ≤  h u: g1  g2Sub(u): Sub(g2)  Sub(g1) homomorphism of Heyting algebras = contravariant functor Sub: Gesture Heyting

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

38. V IV I VI I  =

39. Investigate the possible role and semantics of gestural logic inconcrete 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 gestureswith values in Heyting algebras of quivers. Problems: