Mathematical Theory

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Mathematical Theory of Gestures in Music
Guerino Mazzola U Minnesota Zürich mazzola_at_umn.e
du   guerino_at_mazzola.ch     www.encyclospace.o
rg        
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• Motivation - performance - and music theory -
  gestural music and painting - French
  philosophy - Embodied AI
• Speculum - the musical oniontology -
  classification of global compositions and
  networks
• Gestures - categories of gestures -
  hypergestures - the Escher theorem and free
  jazz
• Symbols - homotopy - gestoids - finitely
  generated abelian groups and networks

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• Motivation - performance - and music theory -
  gestural music and painting - French
  philosophy - Embodied AI
• Speculum - the musical oniontology -
  classification of global compositions and
  networks
• Gestures - categories of gestures -
  hypergestures - the Escher theorem and free
  jazz
• Symbols - homotopy - gestoids - finitely
  generated abelian groups and networks

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Gestures in Performance Theory
Theodor W. Adorno(Zu einer Theorie der
musikalischen Reproduktion 1946) Danach wäre
die Aufgabe des Interpreten, Noten so zu
betrachen, bis sie dem insistenten Blick in
Originalmanuskripte sich verwandeln nicht aber
als Bilder der Seelenregung des Autors sie
sind auch dies, aber nur akzidentiell sondern
als die seismographischen Kurven, die der Körper
der Musik selber in seinen gestischen
Erschütterungen hinterlassen hat.
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Gestures in Performance Theory
Jürgen Uhde Renate Wieland (Forschendes Üben
2002) Affekte waren ursprünglich ja Handlungen,
bezogen auf ein Objekt draussen, im Prozess der
Verinnerlichung haben sie sich von ihrem
Gegenstand gelöst, aber immer noch sind sie
bestimmt von den Koordinaten des Raumes. (...) Es
gibt mithin etwas wie gestische
(Raum-)Koordinaten.
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Musical Transformational Theory
David Lewin(Generalized Musical Intervalsand
Transformations 1987) If I am at s and wish to
get to t, what characteristic gesture should I
perform in order to arrive there?
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Music Theory
Robert S. Hatten(Interpreting Musical Gestures,
Topics, and Tropes 2004) Given the importance
of gesture to interpretation, why do we not have
a comprehensive theory of gesture in music?
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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.
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instrumentalize
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Painting
The marks are made, and you survey the thing like
you would a sort of graph. And you see within
this graph the possibilities of all types of fact
being planted..David Sylvester Interview with
Francis Bacon The Brutality of Fact. Thames and
Hudson, New York 1975
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Embodied AI
Stumpy AI Lab, U Zurich
Cheap design
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• Motivation - performance - and music theory -
  gestural music and painting - French
  philosophy - Embodied AI
• Speculum - the musical oniontology -
  classification of global compositions and
  networks
• Gestures - categories of gestures -
  hypergestures - the Escher theorem and free
  jazz
• Symbols - homotopy - gestoids - finitely
  generated abelian groups and networks

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The Oniontology of Music
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facts
The category GloComA of global objective
A-addressed compositions has objects KI,
i.e., coverings of sets K by atlases I of local
objective A-addressed compositions with manifold
gluing conditions and manifold morphisms ff KI
LJ, including and compatible with atlas
morphisms f I J
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• Theorem (global addressed geometric
  classification)
• Let A locally free of finite rank over
  commutative ring R
• Consider the objective global compositions KI at
  A with ()
• the chart modules R.Ki are locally free of
  finite rank
• the function modules G(Ki) are projective
• There is a subscheme Jn of a projective R-scheme
  of finite type whose points w Spec(S) Jn
  parametrize the isomorphism classes of objective
  global compositions at address S?RA with ().

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processes
KI can be reconstructed from the coefficient
system of retracted functions on free global
compositions resnG(KI) ? nG(ADn)
Fact This construction is a special case of a
local network. Global compositions are
classified by limits of powerset denotators.
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A global network
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Theorem Given address A in Mod, we have a
verification functor ? ALfModred ?
AGlob from the category ALfModred of reduced,
A-addressed locally flat global networks to the
category AGlob of A-addressed global compositions.
Corollary There are non-interpretable global
networks in ALfModred
Local and Global Limit Denotators and the
Classification of Global Compositions. COLLOQUIUM
ON MATHEMATHICAL MUSIC THEORY H. Fripertinger,
L. Reich (Eds.) Grazer Math. Ber., ISSN
10167692 Bericht Nr. 347 (2005), Global
Networks in Computer Science?http//www.encyclosp
ace.org/talks/networks.ppt
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• Gesture Theory in Computer Music Research
• Frédéric Bevilacqua
• Claude Cadoz
• Antonio Camurri
• Rolf Inge Godøy
• Stefan Müller
• Norbert Schnell
• Koji Shibuya
• McAgnus Todd
• Marcelo Wanderley
• etc.

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Mathematical Music Theory
?
?
?
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• Motivation - performance - and music theory -
  gestural music and painting - French
  philosophy - Embodied AI
• Speculum - the musical oniontology -
  classification of global compositions and
  networks
• Gestures - categories of gestures -
  hypergestures - the Escher theorem and free
  jazz
• Symbols - homotopy - gestoids - finitely
  generated abelian groups and networks

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algebra compactifies gestures to symbolic formulas
rotation
matrix equation
a11xa12ya13z a a21xa22ya23z
b a31xa32ya33z c
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attempt at resuscitation Peter Gabriel
Symbolic formulas via digraphs quiver algebras
mathematics of transformational theory
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Graphs are only the skeleton of gestures, the
flesh is missing.
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A gesture morphism u g ? h is a digraph morphism
u, such that there is a continuous map f X ? Y
whichdefines a commutative diagram
G(g, h)category G of (local) gestures
Advantage Digraphs have an inherent
(intuitionistic) logic, because the category of
digraphs is a topos. This is not only cheap, but
free design!
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A global gesture (only bodies shown)
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One hand ? product ? ?1??2??3??4??5??6 of
6 gestural curves in space-time (x,y,ze) of
piano j 1, 2, ... 5 tips of fingers j 6
the carpus
?1 Ÿ
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Stefan Müller
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Renate Wieland Jürgen UhdeForschendes
Üben Die Klangberührung ist das Ziel der
zusammenfassenden Geste, der Anschlag ist
sozusagen die Geste in der Geste.
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Hypergestures!
loop of loops
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ET-dance gesture
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hypergesture impossible!
morphism exists!
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Have chain of successively refined gestural
categoriesG ? HG HG1 ? HG2 ?... HGn ? HGn1
?... which represent the granularity of
gestural relations, much as in differential
geometry, where the categories ofn-times
differentiable manifolds do. E.g. gluing local
gestures to global gestures in quasi-anatomic
joints
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Comprendre est attraper le geste et pouvoir
continuer.
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• Motivation - performance - and music theory -
  gestural music and painting - French
  philosophy - Embodied AI
• Speculum - the musical oniontology -
  classification of global compositions and
  networks
• Gestures - categories of gestures -
  hypergestures - the Escher theorem and free
  jazz
• Symbols - homotopy - gestoids - finitely
  generated abelian groups and networks

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Gestoids From Gestures to Symbols
AlgebraicTopology
composition of homotopic curves is associative
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The homotopy classes of curves of a gesture
gdefine the Gestoid Gg of a gesture g. This
consists of the linear combinations ?n
ancnof homotopy classes cn of curves between
given points x, y of gesture g.
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?1(X) ? ??Ÿn ? finitely generated abelian
groups?
Gg ? ?1(S1) fundamental group?1(S1) ?
??Ÿ ei2?nt n
?n an ei2?nt
Fourier formula f(t) ? n an
ei2?nt
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Fourier ballet
F
2F
3F
4F
QED
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action of Ÿn
S3
?1(S3) ? 0
Ln,1
?1(Ln,1) ? Ÿn
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gestures ? string theory of music