1. Formulas Gestures Music Mathematics
Alexander Grothendieck: „This is probably the mathematics of the new age“
Guerino Mazzola
U Minnesota & Zürich  

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

4. 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.
Hom(A,F)

5. Sets
cartesian products X x Y
disjoint sums X È Y
powersets XY
characteristic maps c: X —> 2
no „algebra“ =
= {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  Ÿ ~> Trans(C,C) 
= (Hom(-, Ÿ12))
C  Ÿ12 M
A  RMod
F 
C  =
= {sub-presheaves
= {sieves in A} W 2
Gottlob Frege
C^ = {sub-presheaves  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 case to harmonic topologies, ToM ch 24

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

9. 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

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

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

12. 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

13. 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.

14. 3, 2, , , 7, , , , , 10, 5,
4, 5, , , 0, , , , , 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. Gérard Milmeister
part A
part B

16. fourth movement: Coherence/Opposition

17. I II
III IV V VI
VII
global theory

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

19. 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  , 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. balance
objective Yoneda
f: X  Y
Cat 
Frege 

22. 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
(Gi)

23. (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

24. Category ∫C of C-addressed points objects of ∫C
 F, F = presheaf in ~ 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

25. 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

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

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

28. 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.

29. 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?

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) 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

32. 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 =

33. 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

34. 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

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

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

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

38. 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 

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

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

41. ET dance gesture
time
space
space

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
...  ... 

43. counterpoint

44. Escher Theorem for Musical Creativity

45. 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

46. 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

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

48. 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

49. 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 =

50. f = @[f] [f] : c ~> c(0), c(1) 
Gestures ?
spatial digraph
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] ∏
Y Z  X Y  X Z bicategories...

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

52. 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
...  ... 

53. 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.

54. Singular homology for gestures𝜎1 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 → etc. The n-chain R-module Cn(R, X) is generated by iterated 1-chains:  
Replacing I by the topological category  and X by a topological category, a n-chain can be interpreted as a hypergesture in the n-fold hypergesture category over the line digraph ↑= • → •

55. 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).

56. 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

57. Gilles Châtelet ( )
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.

58. 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. ...

59. hypergesture impossible! h
morphism exists! g h

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

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

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

63. 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 ?