Formulas Gestures Music 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        

Yoneda‘s Lemma in Music: Reinventing Points Nobuo Yoneda (1930-1996) Look differently at spaces! Take points as being affine homomorphisms from the zero space A=0. Here is the generic notation and visualization: Apoint is an A-parametrized set of points in F.

A@F Hom(A,F) f·g change of address g f space F A B Look differnetly at spaces! Take points as being affine homomorphisms from the zero space A=0. Here is the generic notation and visualization: Apoint is an A-parametrized set of points in F. A@F Hom(A,F)

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

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

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 case to harmonic topologies, ToM ch 24

Category RLoc of local compositions (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 @  h 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

x: O ® Ÿ12 x O O = { } x: Ÿ12 ® Ÿ12 z Î Ÿ12@Ÿ12 z: Ÿ12 ® Ÿ12 Thomas Noll 1995: models Hugo Riemann‘s harmony self-addressed tones x O x: O ® Ÿ12 x: Ÿ12 ® Ÿ12 Euclid‘s punctual address O = { } z: Ÿ12 ® Ÿ12 z Î Ÿ12@Ÿ12

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

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

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

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

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

Gérard Milmeister part A part B

fourth movement: Coherence/Opposition

I II III IV V VI VII global theory

J = {I, II,..., VII} triadic degrees in K covering KJ nerve n(KJ) = harmonic strip I IV II VI V III VII

The category RGlobMod of global 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:...

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

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

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

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

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 F A x x h  y morphisms of ∫C x: A  F, y: B  G h/: x  y F A  G B  address change

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

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

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

Musical Transformational Theory David Lewin Generalized 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.

Gestures in Performance Theory Theodor W. Adorno Towards 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. Hatten Interpreting 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?

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.

Gilles Châtelet (1944-1999) Henri Poincaré (1854-1912) Le geste est élastique, il peut se ramasser sur lui-même, sauter au-delà de lui-même et retentir, alors que la fonction 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 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

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

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

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

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

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

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

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

realistic forms? tip space position pitch time  p

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

ET dance gesture time space space

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 

counterpoint

Escher Theorem for Musical Creativity

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

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

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

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

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

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

Categorical gestures and homological constructions More generally: For any topological category X we have a curve space = @X, whose elements, the categorical curves, are continuous functors  → X instead of continuous curves. @X is canonically a topological category, morphisms = continuous natural transformations between categorical curves. Categorical gestures are gestures g with values in the spatial digraph X = @X X: c ~> c(0), c(1)  g:  → X  The set of these categorical gestures is a topological category, denoted by @X.

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

Two homological constructions for categorical gestures: Extension modules. In loc. cit. we have shown that gestures in factorization categories [f] in RMod can be used to define the classical extension modules Extn(W, Z) for R-modules W, Z. loc. cit.

Singular homology for gestures 𝜎1 I0 𝜎0 I2 𝜎2 𝛾4 𝛾1 𝛾2 𝛾3 Observe that a singular n-chain c: In → X with values in a topological space X is also a 1-chain c: I → In-1@X, etc. The n-chain R-module Cn(R, X) is generated by iterated 1-chains: In@X  I@I@...I@X.  Replacing I by the topological category  and X by a topological category, a n-chain can be interpreted as a hypergesture in ↑@↑@... ↑@X, the n-fold hypergesture category over the line digraph ↑= • → •

Using the Escher Theorem, we have boundary homomorphisms ∂n: Cn(X.*) → Cn-1(X.*) for any sequence * of digraphs, generalizing ↑↑... ↑, and ∂2 = 0, whence homology modules Hn = Ker(∂n)/Im(∂n+1).

instrumental interface position pitch time gestures l h e sound objects instrumental interface instrumentalize thaw  More precisely, performance is a transformation P from the symbolic reality of the score to the physical reality of sounds (a quarter note is not determined in its physical duration, only the metronome yields the relation between symbolic reality and physical reality). The mathematical analysis of tempo and pitch transformations shows that the performance transformation P can be described by a performance vector field, as shown to the left in the symbolic parameter space. Essentially, performance is calculated via integration of such performance vector fields, much as the time is calculated via integration of the tempo curve. In this language, performance operators are built in order to produce new performance fields from the old ones inherited from the relative mothers. score analysis

Gilles Châtelet (1944 - 1999) Figuring Space. Kluwer 2000: The gesture is elastic: it can crouch on itself, leap beyond itself and rever- berate, whereas the function gives only the form of the transit from one external term to another external term, whereas the act exhausts itself in its result. The gesture is therefore involved with the implicit pole of the relation. It‘s the tamed gestures which make reference.

space ~ spatium ~ ex pati point ~ pungere (to prick) Giuseppe Longo 1999/Henri Poincaré (Géométrie et Cognition) La Géométrie des Grecs était une "science des figures"; avec Riemann, et après Descartes, elle est devenue une "science de l'espace". Poincaré est allé plus loin, en soulignant le rôle du mouvement dans l'espace : « un être immobile n'aurait jamais pu acquérir la notion d'espace puisque, ne pouvant corriger par ses mouvements les effets des changements des objets extérieurs, il n'aurait eu aucune raison de les distinguer des changements d'état» [Poincaré, La Science et l'Hypothèse 1902, p. 78] ...« localiser un objet en un point quelconque signifie se représenter 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» [Poincaré, La valeur de la Science 1905, p. 67]. space ~ spatium ~ ex pati point ~ pungere (to prick) Paul Klee (trajectory of eyes ~ grazing cows) Embodied AI—> Rolf Pfeifer et al. ...

hypergesture impossible!  h morphism exists! g h

I1 I0 I2 𝜎0 𝜎1 𝛾1 𝛾2 𝜎2 𝛾4 𝛾3 hypergesture impossible!

action of Ÿn S3 Ÿn 1(S3)  0 1 Ln,1 1(Ln,1) = Ÿn

example: C = GLn(—), [f]  GLn(—)2, usual topology  c cI = continuous I GLn+ GLn-

Gest[h]  Gest[f] Gest[h°f] Y Z  X Y  X Z bicategories... f: X  Y, h: Y  Z Gest[h]  Gest[f] Gest[h°f] p q* Y Z  X Y  X Z bicategories... h  f h°f geometric morphism, q* is logical f p q* q*  h  Study geometric morphisms Q*, Q*, e.g. when do they stem from composition h°f ?