Presentation is loading. Please wait.

Presentation is loading. Please wait.

Formulas Gestures Formulas GesturesMusic Mathematics Guerino Mazzola U Minnesota & Zürich

Similar presentations


Presentation on theme: "Formulas Gestures Formulas GesturesMusic Mathematics Guerino Mazzola U Minnesota & Zürich"— Presentation transcript:

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

2 Yonedas Lemma in Music: Reinventing Points Nobuo Yoneda (1930-1996)

3

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

5 R Mod @ = R Mod opp @Ens = {F: R Mod opp > Sets} presheaves have all these properties Sets cartesian products X x Y disjoint sums X Y powersets X Y characteristic maps X > no algebra R Mod abelian category, direct sums etc. has algebra no powersets no characteristic maps

6 2 C Ÿ 12 ~> Trans(C,C) Ÿ 12 @ Ÿ 12 C Ÿ (pitch classes mod. octave) C Ÿ 12 MA@M A@F A R Mod A R Mod F R Mod @ F R Mod @ C 2 A@F = A@2 F C^ A@ F = {sub-presheaves of @A F} = {F-sieves in A} A@ = {sub-presheaves of @A} = {sieves in A} Gottlob Frege (@ Ÿ 12 = (Hom(-, Ÿ 12 ))

7 F @A 1A1A1A1A f:B A C f@C^ = C.f B@C^ = {(f:B A, c.f)| c C} B@A B@F applications of general case to harmonic topologies, ToM ch 24

8 Category R Loc of local compositions (over R): objects = F-sieves in A, i.e. K @A F objects = F-sieves in A, i.e. K @A F morphisms: 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 L @B G f @ h f/ : K L Full subcategories R ObLoc R Loc of objective local compositions K = C^ and R LocMod R ObLoc of modular local compositions, C A@M, M = R-module

9 x: Ÿ 12 Ÿ 12 z: Ÿ 12 Ÿ 12 x O x: O Ÿ 12 Euclids punctual address O = { } z Ÿ 12 @ Ÿ 12 Thomas Noll 1995: models Hugo Riemanns harmony self-addressed tones

10 Trans(Dt,Tc) = Trans(Dt,Tc) = f Dt dominant triad {g, b, d} Tc tonic triad {c, e, g} relative consonances ƒ Ÿ 12 @ Ÿ 12 Ÿ 12 [ ] @ Ÿ 12 [ ] Fuxian counterpoint: Trans(Dt,Tc) = Trans(K, K )| ƒ Trans(Dt,Tc) = Trans(K, K )| ƒ

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

12 A = Ÿ 11, F = Ÿ 12 (pitch classes) S: Ÿ 11 Ÿ 12, S = (S 0, S 1,... S 11 ) e i ~> S i, e 1 = (1, 0,... 0), etc. e 0 = 0 Ÿ 12 S 011 dodecaphonic series Messiaen: modes et valeurs dintensité strongdichotomy of class 71 symmetry T 7.11

13 The yoga of Boulezs construction is a canonical system of address changes on address Ÿ 11 Ÿ 11 (affine tensor product) generating new series of series used in the composition.

14 B:ist. 11 A:ist. 11 B:ist. 10 A:ist. 10 B:ist. 9 A:ist. 9 B:ist. 8 A:ist. 8 B:ist. 7 A:ist. 7 B:ist. 6 A:ist. 6 B:ist. 5 A:ist. 5 B:ist. 4 A:ist. 4 B:ist. 3 A:ist. 3 B:ist. 0 A:ist. 0 B:ist. 1 A:ist. 1 B:ist. 2 A:ist. 2 3, 2, 9, 8, 7, 6, 4, 1, 0, 10, 5, 11 4, 5, 10, 11, 0, 1, 3, 6, 7, 9, 2, 8 T 7.11

15 part A part B Gérard Milmeister

16 fourth movement: Coherence/Opposition

17 I IVVIIIIIVIVII global theory

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

19 The category R GlobMod of global modular compositions: objects: objects: - an address A, - a covering I of a finite set G by subsets G i, - atlas (K i ) I, K i A@M i, M i = R-modules - bijections g i : G i K i - gluing conditions: (g j g i -1 )/Id A : K ij K ji = A-addressed global modular composition G I morphisms:... 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 G I with the following properties (*): the modules R.G i generated by the charts G i are locally free of finite rank the modules R.G i generated by the charts G i are locally free of finite rank the modules of affine functions (G i ) are projective the modules of affine functions (G i ) are projective Then there exists a subscheme J n* of a projective R-scheme of finite type whose points : Spec(S) J n* parametrize the isomorphism classes of S R A -addressed global modular compositions with properties (*). 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 G I with the following properties (*): the modules R.G i generated by the charts G i are locally free of finite rank the modules R.G i generated by the charts G i are locally free of finite rank the modules of affine functions (G i ) are projective the modules of affine functions (G i ) are projective Then there exists a subscheme J n* of a projective R-scheme of finite type whose points : Spec(S) J n* parametrize the isomorphism classes of S R A -addressed global modular compositions with properties (*). ToM, ch 15, 16

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

22 A@R 1 2 34 6 5 GiGiGiGi 6 534 1 res i 2 (G i ) res ( i ) (G i ) res ( i ) (G i ) (G i ) Edgar Varèse resolution A resolution A GIGIGIGI

23 6 5 3 4 1 A@R 2 (G i ) res ( i ) (G i ) res ( i ) i (G i G j ) res ( i j ) (G i G j ) res ( i j ) N = pr ( / ) (N ) = N pr ( / ) (N ) = N N A@lim nerf( A (F ) N = (G i ) res ( i ) (G i ) res ( i )

24 yx Category C of C-addressed points objects of C objects of C x: @A F, F = presheaf in C @ ~ x F(A), write x: A F A = address, F = space of x h FA GB address change morphisms of C morphisms of C x: A F, y: B G h/ : x y x: A F, y: B G h/ : x y F A x

25 x i : A i F i h il q / il q h jm s / jm s h li p / li p h jl k / jl k h ll r / ll r x j : A j F j x m : A m F m x l : A l F l h ij t / ij t local network in C = diagram x of C-addressed points x : C coordinate of x 2004 Applications: neural networs, automata, OO classes PNM

26 Ÿ 12 T4T4T4T4 T2T2T2T2 T 5.-1 T 11.-1 D3724 Ÿ 12 T4T4T4T4 T2T2T2T2 T 5.-1 T 11.-1 (3, 7, 2, 4) 0@lim( D ) Klumpenhouwer networks A = 0

27 network of dodecaphonic series Ÿ 12 s Us Ks UKs T 11.-1/Id Id/T 11.-1 Ÿ 11 s

28 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. Musical Transformational Theory

29 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 authors emotionthey are also such, but only accidentally but as the seismographic curves, which the body has left to the music in its gestural vibrations. Gestures in Performance Theory 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?

30 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. 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. Free Jazz

31 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 Gilles Châtelet (1944-1999) 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 Henri Poincaré (1854-1912)

32 a 11 x+a 12 y+a 13 z = a a 21 x+a 22 y+a 23 z = b a 31 x+a 32 y+a 33 z = c a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 xyz abc = rotation matrix formula in algebra, we compactify gestures to formulas

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

34 Two attempts of reanimation 1. Gabriel: formulas via digraphs = quiver algebras SP T Q K T X mathematics of Lewins musical transformation theory => R[X], polynomial algebra => RK, quiver algebra

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

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

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

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

39 p realistic forms? tip space positionpitchtime

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

41 space space time ET dance gesture

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

43 counterpoint

44 Escher Theorem for Musical Creativity

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

46 e i2 t i1 i X = S 1 ¬ G g ¬ 1 (S 1 ) fundamental group 1 (S 1 ) Ÿ e i2 nt ~ n ~ Fourier formula f(t) = n a n e i2 nt ~ Fourier formula f(t) = n a n e i2 nt n a n e i2 nt n a n e i2 nt g: 1 (X) Ÿ n, n 0? 1 (X) Ÿ n, n 0?Yes: All groups are fundamental groups! 1 (X) Ÿ n, n 0? 1 (X) Ÿ n, n 0?Yes: All groups are fundamental groups!

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

48 How can we gestify formulas? Category [f] of factorizations of morphism f in C: fXY W u v gXY W u v Z a b objectsmorphisms If C is topological, then [f] is canonically a topological category

49 Curve spaces? These are the infinite factorizations: Order category = {0 x y 1} of unit interval I fXY W0W0W0W0 u0u0u0u0 v0v0v0v0 W1W1W1W1 u1u1u1u1 v1v1v1v1 c = continuous functor for chosen topology on [f] curve space = @[f]

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

51 X = @X X: c ~> c(0), c(1) g: X The set of these categorical gestures is a topological category, denoted by @X denoted by @X.

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

53 Two homological constructions for categorical gestures: 1. Extension modules. In loc. cit. we have shown that gestures in factorization categories [f] in R Mod can be used to define the classical extension modules Ext n (W, Z) for R-modules W, Z. loc. cit.

54 2. Singular homology for gestures 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 = 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 = Observe that a singular n-chain c: I n X with values in a topological space X is also a 1-chain c: I I n-1 @X, etc. The n-chain R-module C n (R, X) is generated by iterated 1- chains: I n @X I@I@...I@X. 4 1 2 3 I0I0I0I0 0 I1I1I1I1 1 I2I2I2I2 2

55 Using the Escher Theorem, we have boundary homomorphisms n : C n (X. * ) C n-1 (X. * ) for any sequence * of digraphs, generalizing..., and 2 = 0, whence homology modules n : C n (X. * ) C n-1 (X. * ) for any sequence * of digraphs, generalizing..., and 2 = 0, whence homology modules H n = Ker( n )/Im( n+1 ). H n = Ker( n )/Im( n+1 ).

56 l h e sound objects score analysis instrumental interface instrumentalizeinstrumentalize position pitch timegestures thaw

57 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. Its the tamed gestures which make reference. Gilles Châtelet (1944 - 1999)

58 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, 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, 1905, p. 67]. 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....

59 g h hypergesture impossible! g h morphism exists!

60 hypergesture impossible! 4 1 2 3 I0I0I0I0 I1I1I1I1 I2I2I2I2 0 1 2

61 ŸnŸnŸnŸn L n,1 S3S3S3S3 1 action of Ÿ n 1 (L n,1 ) = Ÿ n 1 (L n,1 ) = Ÿ n 1 (S 3 ) 0 1 (S 3 ) 0

62 GL n + GL n - example: C = GL n (), [f] GL n () 2, usual topology example: C = GL n ( ), [f] GL n ( ) 2, usual topology I GL n 2 c c I = continuous

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


Download ppt "Formulas Gestures Formulas GesturesMusic Mathematics Guerino Mazzola U Minnesota & Zürich"

Similar presentations


Ads by Google