Formulas Gestures Music

Mathematics

Guerino MazzolaU Minnesota & Zürichmazzola@umn.edu   guerino@mazzola.ch     www.encyclospace.org        

Alexander Grothendieck:

„This is probably the mathematics of the new age“

Yoneda‘s Lemma in Music:Reinventing Points

Nobuo Yoneda (1930-1996)

A@F

f

change of address g

f·g

space FA

B

Hom(A,F)

RMod@ = RModopp@Ens= {F: RModopp —> Sets}

presheaveshave all these properties

Setscartesian products X x Ydisjoint sums X È Ypowersets XY

characteristic maps : c X —> 2no „algebra“

RModabelian category,direct sums etc.

has „algebra“no powersets

no characteristic maps

2

C Ÿ12 ~> Trans(C,C) Ÿ12@Ÿ12

C Ÿ12 (pitch classes mod. octave)

C Ÿ12MA@MA@F   A RMod F RMod@

C 2A@F = A@2F W

C^ A@WF = {sub-presheaves of @A F}= {F-sieves in A}

A@ W = {sub-presheaves of @A}= {sieves in A}

Gottlob Frege

(@Ÿ12 = (Hom(-, Ÿ12))

F

@A1A f:B A

C f@C^ = C.f

B@C^ = {(f:BA, c.f)| c C} B@A B@F

applications of general caseto 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 Gf: 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 RObLoc RLoc of objective local compositions K = C^ and

RLocMod RObLoc of modular local compositions, C A@M, M = R-module

x: Ÿ12 ® Ÿ12

z: Ÿ12 ® Ÿ12

xO

x: O ® Ÿ12

Euclid‘s punctual address

O = { }

z Î Ÿ12@Ÿ12

Thomas Noll 1995:models Hugo Riemann‘s harmonyself-addressed tones

Trans(Dt,Tc) = < f Ÿ12@Ÿ12 | f: Dt ® Tc >

f

Dt

dominant triad {g, b, d}

Tc

tonic triad {c, e, g}

„relative consonances“

ƒ : e Ÿ12 @Ÿ12 ® Ÿ12 [e] @ Ÿ12 [e]

Fuxian counterpoint:

Trans(Dt,Tc) = Trans(Ke, Ke)|ƒe

Pierre Boulezstructures Ia (1952) analyzed by G. Ligeti

thread (« Faden »)

The composition is a system of threads!

A = Ÿ11, F = Ÿ12 (pitch classes)

S: Ÿ11 Ÿ12, S = (S0, S1, ... S11)ei ~> Si, e1 = (1, 0, ... 0), etc.e0 = 0

Ÿ12

S

0 11

dodecaphonic series

Messiaen: modes et valeurs d‘intensité

strongdichotomyof class 71

symmetry T7.11

The yoga of Boulez‘s construction is acanonical system of address changes on address

Ÿ11 Ÿ11 (affine tensor product) generating new series of series

used in the composition.

B:ist. 11A:ist. 11

B:ist. 10A:ist. 10

B:ist. 9A:ist. 9

B:ist. 8A:ist. 8

B:ist. 7A:ist. 7

B:ist. 6A:ist. 6

B:ist. 5A:ist. 5

B:ist. 4A:ist. 4

B:ist. 3A:ist. 3

B:ist. 0A:ist. 0

B:ist. 1A:ist. 1

B:ist. 2A:ist. 2

3, 2, 9, 8, 7, 6, 4, 1, 0, 10, 5, 114, 5, 10, 11, 0, 1, 3, 6, 7, 9, 2, 8 T7.11

part A part B

Gérard Milmeister

fourth movement: Coherence/Opposition

I IV VII III VI VII

global theory

I

IV

II

VIV

III

VII

K = {0, 2, 4, 5, 7, 9, 11} Ÿ12

J = {I, II,..., VII} triadic degrees in Kcovering KJ

nerve n(KJ) = harmonic strip

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:...

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 arelocally 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

f: X YCat

Frege @f: @X @Y

balance

objective Yoneda

A@R 1

2

3

4

6

5Gi

6

5

3

41

res

i

2

(Gi)res (i)

(Gi)

Edgar Varèse

resolution A

GI

6

5

3

41

A@R

2

(Gi)res (i)i

(Gi Gj)res (i j)N =

pr(/) (N) = N

N A@limnerf(AD)(F)

N = (Gi)res (i)

yx

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

h

F

A

G

B

address change

• morphisms of ∫C

x: A F, y: B Gh/: x y

FA x

xi: Ai Fi hilq/il

q

hjms/jm

s

hlip/li

p

hjlk/jl

k

hllr/ll

r

xj: Aj Fj

xm: Am Fm

xl: Al Fl

hijt/ij

t

local network in C = diagram x of C-addressed points

x: ∫C

coordinateof x

2004

Applications: neural networs, automata, OO classes

PNM

Ÿ12

Ÿ12

Ÿ12

Ÿ12

T4

T2

T5.-1 T11.-1D

3 7

2 4

Ÿ12

Ÿ12

Ÿ12

Ÿ12

T4

T2

T5.-1 T11.-1(3, 7, 2, 4) 0@lim(D)

Klumpenhouwer networks

A = 0

network of dodecaphonic series

Ÿ12

Ÿ12

Ÿ12

Ÿ12

s

Us

Ks

UKs

T11.-1/IdT11.-1/Id

Id/T11.-1

Id/T11.-1

Ÿ11 Ÿ11

Ÿ11 Ÿ11

s

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.

Musical Transformational 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.

Gestures in Performance Theory

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?

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.

Free Jazz

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)

a11x+a12y+a13z = aa21x+a22y+a23z = ba31x+a32y+a33z = c

a11 a12 a13

a21 a22 a23

a31 a32 a33

xyz

abc

=

rotation matrix formula

in algebra, we compactify gestures to formulas

X

Y

f(x)

x

f(x)

f(x) (x) x (x

teleportation

the Fregean drama: morphisms/fonctions are the

„phantoms“ (prisons?) of gestures.

„Two attempts of reanimation“

1. Gabriel: formulas via digraphs = „quiver algebras“

S

P

T

Q

K

TX

mathematics of Lewin‘s musical transformation theory

=> R[X], polynomial algebra

=> RK, quiver algebra

¬

2. Multiplication of complex numbers:from phantom to gesture: infinite factorization

x-x

—0

x.eit

Robert Peck: imaginary rotation

f: X YCat

Frege @f: @X @Y

balance

objectve Yoneda

@f: @X @YChâtelet

morphic Yoneda?

Journal of Mathematics and Music2007, 2009 Taylor & Francis

MCM Proceedings 2011Springer

position

pitch

time

Xg

body

skeleton

Gesture = -addressed point g: in spatial digraph Xof topological space X(= digraph of continuous curves I X

I = [0,1])

X

p

realistic forms?tip space

position

pitch

time

circle

knot

„loop of loops“

Hypergestures!

Digraph(, X) = topological space of gestures with skeleton and body in X notation: @X

space

space

time

ET dance gesture

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

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 ancn

of homotopy classes cn of the gesture‘s curves joining given points x, y.

Gestoids: from gestures to formulas

y

x

ei2t

—

i—

1

i

X = S1

¬ Gg ¬ 1(S1)

fundamental group 1(S1) Ÿei2nt ~ n

~ Fourier formula f(t) = n an ei2nt n an ei2nt

g:

1(X) Ÿn, n ≥ 0?

Yes: All groups are fundamental groups!

Dancing the Violent Body of Sound

Diyah Larasati Bill Messing Schuyler Tsuda

How can we „gestify“ formulas? Category [f] of factorizations of morphism f inC:

f

X

Y

W

u

v

g

X

Y

W

u

v

Z

a

b

objects morphisms

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 = continuousfunctorfor chosen topology on [f]

curve space = @[f]

Gestures ?• 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@YY 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 transformationsbetween 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:

1. 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.

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 ↑= • → •

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.

𝛾4

𝛾

1 𝛾

2 𝛾3

I0

𝜎0

I1

𝛾

1

I2

𝛾

2

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).