Formulas Gestures Music
Mathematics
Guerino MazzolaU Minnesota & Zü[email protected] [email protected] 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@[email protected]@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).
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