Formulas Gestures

1. Music FormulasGestures Mathematics Alexander Grothendieck:„This is probably the mathematics of the new age“ Guerino Mazzola U Minnesota & Zürich mazzola@umn.edu   guerino@mazzola.ch     www.encyclospace.org        

2. Yoneda‘s Lemma in Music: Reinventing Points Nobuo Yoneda (1930-1996)

4. B f·g change of address g f space F A A@F Hom(A,F)

5. Sets cartesian products X x Y disjoint sums X È Y powersets XY characteristic maps c: X —> 2 no „algebra“ RMod@ = RModopp@Ens = {F:RModopp —> Sets} presheaves have all these properties RMod abelian category, direct sums etc. has „algebra“ no powersets no characteristic maps

6. CŸ12 (pitch classes mod. octave) C  Ÿ12 ~> Trans(C,C)  Ÿ12@Ÿ12 Gottlob Frege (@Ÿ12 = (Hom(-, Ÿ12)) A  RMod A@F   M A@M C  Ÿ12 F  RMod@ 2 C  2A@F= A@2F A@W = {sub-presheaves of @A} = {sieves in A} W C^ A@WF = {sub-presheaves of @A  F} = {F-sieves in A}

7. 1A f:B  A B@C^ = {(f:BA, c.f)| c  C}  B@A  B@F C f@C^ = C.f F @A applications of general caseto harmonic topologies, ToM ch 24

8. f @  h • Category RLoc of localcompositions (over R): • objects = F-sieves in A, i.e. K  @A  F • morphisms: • K  @A  F, L  @B  G • f: K  L : A  B (change of address) • such that there is h: F  G with: K  @A  F f/: K  L L  @B  G Full subcategories RObLocRLoc of objective local compositions K = C^ and RLocMod RObLoc of modular local compositions, C  A@M, M =R-module

9. Thomas Noll 1995:models Hugo Riemann‘s harmonyself-addressed tones O= { } Euclid‘s punctual address x:O® Ÿ12 x O x:Ÿ12 ® Ÿ12 z:Ÿ12 ® Ÿ12 zÎ Ÿ12@Ÿ12

10. dominant triad {g, b, d} tonic triad {c, e, g} Dt Tc f ƒe: Ÿ12 @Ÿ12 ® Ÿ12 [e] @Ÿ12 [e] „relative consonances“ Trans(Dt,Tc) = < f  Ÿ12@Ÿ12| f: Dt® Tc> Fuxian counterpoint: Trans(Dt,Tc) = Trans(Ke, Ke)|ƒe

11. thread (« Faden ») Pierre Boulezstructures Ia (1952)  analyzed by G. Ligeti The composition is a system of threads!

12. Ÿ12 Messiaen: modes et valeurs d‘intensité S 0 11 strong dichotomy of class 71 symmetry T7.11 dodecaphonic series A = Ÿ11, F =Ÿ12 (pitch classes) S: Ÿ11  Ÿ12, S = (S0, S1, ... S11) ei ~> Si,e1= (1, 0, ... 0), etc. e0 = 0

13. The yoga of Boulez‘s construction is acanonical system of address changes on addressŸ11Ÿ11 (affine tensor product) generating new series of series used in the composition.

14. 3, 2, 9, 8, 7, 6, 4, 1, 0, 10, 5, 11 4, 5, 10, 11, 0, 1, 3, 6, 7, 9, 2, 8 T7.11 A:ist. 11 B:ist. 11 A:ist. 10 B:ist. 10 A:ist. 9 B:ist. 9 A:ist. 8 B:ist. 8 A:ist. 7 B:ist. 7 A:ist. 6 B:ist. 6 A:ist. 5 B:ist. 5 A:ist. 4 B:ist. 4 A:ist. 3 B:ist. 3 A:ist. 2 B:ist. 2 A:ist. 1 B:ist. 1 A:ist. 0 B:ist. 0

15. part A part B Gérard Milmeister

16. fourth movement: Coherence/Opposition

17. II III IV V VI VII I global theory

18. II VI V IV I VII III K = {0, 2, 4, 5, 7, 9, 11}Ÿ12 J = {I, II,..., VII} triadic degrees in K covering KJ nerve n(KJ) = harmonic strip

19. The category RGlobMod ofglobal modular compositions: • objects: • - an address A, • - a covering I of a finite set G by subsets Gi, • - atlas (Ki)I, Ki  A@Mi , Mi = R-modules • - bijections gi: Gi® Ki • - gluing conditions: (gjgi-1)/IdA: Kij Kji • = A-addressed global modular composition GI • morphisms:...

20. Theorem (global addressed geometric classification) • Let A be a locally free module of finite rank over a commutative R. • Consider the A-addressed global modular compositions GI with the following properties (*): • the modules R.Gi generated by the charts Gi are locally free of finite rank • the modules of affine functions G(Gi) are projective • Then there exists a subscheme Jn* of a projective R-scheme of finite type whose points w: Spec(S) ® Jn* parametrize the isomorphism classes of SRA -addressed global modular compositions with properties (*). ToM, ch 15, 16

21. Cat  objective Yoneda f: X  Y Frege  @f: @X  @Y balance

22. 3 6 1 4 5 i (Gi)res  (i) Edgar Varèse res 2 4 6 3 5 1 (Gi) A@R Gi 2 resolution A GI

23. (Gi)res  (i) (GiGj)res  (i j) 3 6 1 4 5 i (Gi)res  (i) N = N = 2  pr(/) (N) = N N A@limnerf(AD)(F) A@R

24. F x A h F G    A B x y address change • Category ∫C of C-addressed points • objects of ∫C • x: @A  F, F = presheaf in C@~x F(A), write x: A  F A = address, F = space of x • morphisms of ∫C • x: A  F, y: B  G h/: x  y

25. hllr/llr xi: Ai Fi hilq/ilq xl: Al Fl hlip/lip hijt/ijt hjlk/jlk xj: Aj Fj hjms/jms PNM 2004 xm: Am Fm Applications: neural networs, automata, OO classes coordinateof x local network in C= diagram x of C-addressed points x:   ∫C

26. 3 7 T4   Ÿ12 Ÿ12 T5.-1 T11.-1 Ÿ12 Ÿ12 T2   2 4 Ÿ12 T4 Ÿ12 (3, 7, 2, 4)  0@lim(D) T5.-1 T11.-1 Ÿ12 T2 Ÿ12 A = 0 D Klumpenhouwer networks

27. Ÿ12 Ÿ12 Id/T11.-1 Ks s Ÿ12 Ÿ12 Ÿ11 Ÿ11 T11.-1/Id T11.-1/Id Ÿ11 Ÿ11 s Us UKs Id/T11.-1 network of dodecaphonic series

28. Musical Transformational Theory David LewinGeneralized Musical Intervals and Transformations Cambridge UP 1987/2007: If I am at s and wish to get to t, what characteristic gesture should I perform in order to arrive there? (Opposition to what he calls cartesian approach, of res extensae.) This attitude is by and large the attitude of someone inside the music, as idealized dancer and/or singer. No external observer (analyst, listener) is needed.

29. Gestures in Performance Theory Theodor W. AdornoTowards a Theory of Musical Reproduction(1946) Polity, 2006: Correspondingly the task of the interpreter would be to consider the notes until they are transformed into original manuscripts under the insistent eye of the observer; however not as images of the author‘s emotion—they are also such, but only accidentally—but as the seismographic curves, which the body has left to the music in its gestural vibrations. Robert S. HattenInterpreting Musical Gestures, Topics, and Tropes 2004, Indiana UP 2004, p.113 Given the importance of gesture to interpretation, why do we not have a comprehensive theory of gesture in music?

30. Free 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.

31. Gilles Châtelet (1944-1999) Le geste est élastique, il peut se ramasser sur lui-même, sauter au-delà de lui-même et retentir, alors que lafonction ne donne que la forme du transit d'un terme extérieur à un autre terme extérieur, alors que l'acte s'épuise dans son résultat. (...) Figuring Space, 2000 Henri Poincaré (1854-1912) Localiser un objet en un point quelconque signifie se représen-ter le mouvement (c'est-à-dire les sensations musculaires qui les accompagnent et qui n'ont aucun caractère géométrique) qu'il faut faire pour l'atteindre. La valeur de la science, 1905

32. a11 a12 a13 a21 a22 a23 a31 a32 a33 x y z a b c = in algebra, we compactify gestures to formulas rotation matrix formula a11x+a12y+a13z = a a21x+a22y+a23z = b a31x+a32y+a33z = c

33. the Fregean drama: morphisms/fonctions are the„phantoms“ (prisons?) of gestures. Y f(x) f(x) f(x) teleportation (x) (x x x X

34. S T K P X T Q „Two attempts of reanimation“ 1. Gabriel: formulas via digraphs = „quiver algebras“ => RK, quiver algebra => R[X], polynomial algebra mathematics of Lewin‘s musical transformation theory

35. ¬ x.eit — 0 -x x 2. Multiplication of complex numbers:from phantom to gesture: infinite factorization Robert Peck: imaginary rotation

36. Cat  objectve Yoneda f: X  Y Frege  morphic Yoneda? @f: @X  @Y Châtelet  @f: @X  @Y balance

37. Journal of Mathematics and Music 2007, 2009 Taylor & Francis MCM Proceedings 2011 Springer

38. Gesture = -addressed point g:  in spatial digraph X of topological space X (= digraph of continuous curves I  X I = [0,1]) pitch  X  body time skeleton position X g 

39. pitch time position tip space realistic forms? p 

40. Digraph(, X) = topological space of gestures with skeleton  and body in X notation: @X  knot circle  Hypergestures! „loop of loops“

41. time space space ET dance gesture

42. Proposition (Escher Theorem) For a topological space X, a sequence of digraphs 1 , 2, ...n and a permutation  of 1, 2,... n, there is a homeomorphism 1@ ...n@X  (1)@ ...(n)@X    

43. counterpoint

44. Escher Theorem for Musical Creativity

45. Gestoids:from gestures to formulas The homotopy classes of curves of a gesture gdefine the R-linear category Gestoid RGg of gesture g, R = commutative ring. It is generated by R-linear combinationsn ancnof homotopy classes cn of the gesture‘s curves joining given points x, y. x y

46. i— i ei2t — g: 1 X = S1 1(X)  Ÿn, n ≥ 0? Yes: All groups are fundamental groups! ¬ Gg  ¬ 1(S1) fundamental group 1(S1)  Ÿ ei2nt ~ n n anei2nt ~ Fourier formula f(t) = n an ei2nt

47. Diyah Larasati Bill Messing Schuyler Tsuda Dancing the Violent Body of Sound

48. X X u u a g f Z W W v v b Y Y How can we „gestify“ formulas? Category [f] of factorizations of morphism f inC: objects morphisms If C is topological, then [f] is canonically a topological category

49. X u1 u0 W1 W0 c = continuousfunctorfor chosen topology on [f] f v1 v0 Y Curve spaces? These are the „infinite factorizations“: Order category  = {0 ≤ x ≤ y ≤ 1} of unit interval I  curve space = @[f]

50. A -gesture in f is a -addressed point g:   f   f =@[f] [f] : c ~> c(0), c(1) f g   X Y= Gest[f] Gest[f]= Digraph / f ∏ X@Y • Gestures ? • spatial digraph X Y Y ZX Y X Zbicategories...