Transcript Formulas Music Gestures Mathematics Alexander Grothendieck: „This is probably the mathematics of the new age“ Guerino Mazzola U Minnesota & Zürich [email protected] [email protected] www.encyclospace.org.

Formulas
Music
Gestures
Mathematics
Alexander
Grothendieck:
„This is probably
the mathematics
of the new age“
Guerino Mazzola
U Minnesota & Zürich
[email protected]
[email protected]
www.encyclospace.org
Yoneda‘s Lemma in Music:
Reinventing Points
Nobuo Yoneda (1930-1996)
B
f·g
change of address g
f
A
space F
A@F
Hom(A,F)
Sets
cartesian products X x Y
disjoint sums X Y
powersets XY
characteristic maps c:X —> 2
no „algebra“
@=
RMod
= {F: R
opp@Ens
RMod
Modopp —> 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
(@Ÿ12 = (Hom(-, Ÿ12))
A  RMod
F  RMod@
A@F
A@M
C  ŸM
12
C  2A@F = A@2F
C^ A@WF
Gottlob Frege
~> Trans(C,C)  Ÿ12@Ÿ12
2 = {sub-presheaves of @A}
A@W
W
= {sieves in A}
= {sub-presheaves of @A  F}
= {F-sieves in A}
B@C^ = {(f:BA, c.f)| c  C}  B@A  B@F
F
C
f@C^ = C.f
@A
1A
f:B  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
Thomas Noll 1995:
models Hugo Riemann‘s harmony
self-addressed tones
x: O Ÿ12
O={
}
O
x
Euclid‘s
punctual address
x: Ÿ12 Ÿ12
z: Ÿ12 Ÿ12
z Ÿ12@Ÿ12
dominant triad {g, b, d}
tonic triad {c, e, g}
„relative consonances“
Dt
Tc
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
Pierre Boulez
structures Ia (1952)
 analyzed by G. Ligeti
thread (« Faden »)
The composition is a
system of threads!
dodecaphonic series
Messiaen: modes et valeurs d‘intensité
Ÿ12
S
strong
dichotomy
of class 71
symmetry T7.11
0
A = Ÿ11, F = Ÿ12 (pitch classes)
S: Ÿ11  Ÿ12, S = (S0, S1, ... S11)
ei ~> Si,
e1 = (1, 0, ... 0), etc.
e0 = 0
11
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,
4,
2,
5,
9,
10,
8,
11,
7,
0,
6,
1,
4,
3,
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
1,
6,
0, 10, 5,
7, 9, 2,
11
8
T7.11
Gérard Milmeister
part A
part B
fourth movement: Coherence/Opposition
I
II
III
IV
global theory
V
VI
VII
K = {0, 2, 4, 5, 7, 9, 11}  Ÿ12
J = {I, II,..., VII} triadic degrees in K
covering KJ
nerve n(KJ) = harmonic strip
II
VI
V
IV
I
VII
III
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
Cat 
f: X  Y
Frege 
objective Yoneda
@f: @X  @Y
3
resolution A
6
1
4
5
i
G(Gi)res  G(i)
Edgar Varèse
res
2
GI
4
6
3
A@R
G(Gi)
Gi
1
2
5
3
6
1
4
5
i
N = G(Gi)res  G(i)
2
N = G(Gi Gj)res  G(i  j)

pr(/) (N) = N
A@R
N A@limnerf(A)(F)
Category ∫C of C-addressed points
• objects of ∫C
x: @A  F, F = presheaf in C@
~
x  F(A), write
x: A  F
F
• morphisms of ∫C
x: A  F, y: B  G
h/: x  y
address change
F
A
x
h

G

x

A
A = address, F = space of x
B
y
local network in C = diagram x of C-addressed points
xi: Ai Fi
x:   ∫C
hilq/ilq
hlip/lip
hllr/llr
xl: Al Fl
hijt/ijt
xj: Aj Fj
PNM
Applications:
neural networs,
automata,
OO classes
2004
hjlk/jlk
hjms/jms
xm: Am Fm
coordinate
of x
3
Ÿ12
A=0
T5.-1
Ÿ12
2
7
T4
D
Ÿ12
T11.-1
Ÿ12
T2
4
Ÿ12
Ÿ12
(3, 7, 2, 4)  0@lim(D)
T5.-1
Ÿ12
Klumpenhouwer networks
T4
T11.-1
T2
Ÿ12
network of dodecaphonic series
Ÿ12
Ÿ12
Id/T11.-1
s
Ÿ11
Ÿ11
T11.-1/Id
T11.-1/Id
Ÿ11
s
Us
Ÿ12
Ks
Ÿ11
Id/T11.-1
UKs
Ÿ12
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)
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
Henri Poincaré (1854-1912)
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.
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
a
y = b
z
c
the Fregean drama:
morphisms/fonctions are the
„phantoms“ (prisons?)
of gestures.
Y
f(x)
f(x)
f(x)
(x)
(x
x
x
X
teleportation
„Two attempts of reanimation“
1. Gabriel: formulas via digraphs = „quiver algebras“
S
T
K
=> RK, quiver algebra
P
Q
T
X
=> 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
0
—
x
balance
Cat 
f: X  Y
Frege 
objectve Yoneda
@f: @X  @Y
Châtelet 
morphic Yoneda?
@f: @X  @Y
Journal of Mathematics and Music
2007, 2009 Taylor & Francis
MCM Proceedings 2011
Springer

-addressed point g:   X

in spatial digraph X
of topological space X
(= digraph of continuous curves I  X
I = [0,1])
Gesture =
body
skeleton
pitch
g
X

time
position
pitch
tip space
realistic forms?
p

time
position
 = topological space of gestures
Digraph(, X)
with skeleton  and body in X
notation:  @X
„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.
x
y
1(X)  Ÿn, n ≥ 0?
Yes:
All groups are fundamental
groups!
i—
i
ei2t
g:
¬ Gg  ¬ 1(S1)
1
X = S1
fundamental group
1(S1)  Ÿ
ei2nt
~ n
n an ei2nt
~ Fourier formula f(t) = n an ei2nt
—
Diyah Larasati
Dancing the Violent
Body of Sound
Bill Messing
Schuyler Tsuda
How can we „gestify“ formulas?
Category [f] of factorizations of morphism f in C:
objects
morphisms
X
u
u
f
W
v
a
g
Z
W
v
Y
If
X
b
Y
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
X
u1
u0

c = continuous
functor
for chosen
topology on [f]
W1
W0
curve space = @[f]
v0
f
Y
v1
Gestures ?
• spatial digraph

f = @[f]
A G-gesture in f is a G-addressed point

g: G  f
[f] : c ~> c(0), c(1)
X
g
G

f
Y
Gest[f] = Digraph / f
∏
X Y =
X@Y
Gest[f]
Y Z  X Y  X Z
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:
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
I
0
I
𝜎
I
1
𝜎1
2
0
𝛾1
𝛾2
𝜎
2
𝛾4
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 1chains: In@X  I@[email protected]@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).