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					                                       Gestures for
                                    the Science of
                             Collaborative Arts


Guerino Mazzola
U & ETH Zürich
guerino@mazzola.ch
www.encyclospace.org/talks
to be the case   • Facts

   to refer to   • Processes

     to make     • Gestures

   to interact   • Collaboration
• Facts

• Processes

• Gestures

• Collaboration
Point of departure: free jazz vs. maths




Splitting objective facts from subjective action
Mathematical Music Theory (MaMuTh)
    Books:
•   Gruppen und Kategorien in der Musik (1985)
•   Geometrie der Töne (1990)
•   The Topos of Music (2002)
•   La vérité du beau dans la musique (2006)

  Software:
• presto®
  (Atari 1988)

• Rubato®
  (NEXTSTEP 1983, Mac OS X 1998, Java 2002)
MaMuTh Models
•   Tonal Modulation (Muzzulini, Straub, Noll, Radl)
•   Fux Counterpoint (Muzzulini, Hichert, Wieser, Brunner)
•   Riemann Function Theory (Noll)
•   Riemann/GTTM Metric and Rhythms (Volk-Fleischer)
•   Reti Motif Theory (Straub, Buteau, Nestke)
•   Ruwet-Nattiez Paradigmatic Composition (Blum, Milmeister)
•   Classification and Enumeration Theory (Fripertinger, Noll)
•   Performance Theory (Zahorka, Stange-Elbe, Müller)
•   Denotator Concept Framework
    (Zahorka, Montiel, Göller, Milmeister)
          C(3)


         I          II   III   IV   V   VI   VII




Pitch classes in Ÿ12
truth values = values
chords = truth chords
Schönberg‘s „harmonic strip“ N(X(3))
(Harmonielehre 1911)

             II

                                   VI
            V              IV


                                             I


VII

                   III


       „ harmonic strip = Möbius strip“
                        is a logical fact!
John Keats (Ode on Grecian Urn, 1819):
„Beauty is truth, truth beauty,“—that is
all Ye know on earth, and all ye need to
know.




Alexander Grothendieck (1960...):

Topos Theory
Unification of logic and geometry
„Truth values are point sets“
n/16
           0   2   3   4   6   8   10   12


       a
       b
       c
       d
       e
Nerve N(XMAX)           a
of covering             b
                        c
MAX = {a, b, c, d, e}   d
                        e


             c


                            a


                                b
e
                    d
                    0       2   3        4   6           10   12



                        c


                                    0            3
                                                     a
N(XMAX)                                                  4
                                6                    2
                                                                   b
          e   12                    10
                                    d


                   x dominates y iff Sp(y)  Sp(x)
w(x) = S M Œ Sp(x), m   £ length(M) length(M)p

22
20                                         m=p=2
18
16
14
12
10
 8
 6
 4
• Facts

• Processes

• Gestures

• Collaboration
Transformational Theory, K-nets (Lewin et al.)
            sets of notes




                 Conference „Autour de la Set Theory“
                        IRCAM/Paris Oct. 15-16 2003
                      (J. Rahn, R. Morris, A. Forte,...)
                    Circle Chords
                                                  c

                                                               Ÿ12
c=0
                                            
 (p) = 3p+7

                                                      
                                                               e
{c, (c), 2(c),...}                                  
                                        g

= {c, e, g}

= major triad
Classification of circle chords (cf. Gruppen und Kategorien)
         Diagram in a category C
         C = sets, groups, vector spaces, automata, etc.
          = digraph
                                                   Nets of dodecaphonic series
         D:   C
         • Di = objects in C                            Multiply iterated nets
         • Dijt = morphisms in C

    i                           q       allr       Di                        q        Dllr
                          ail                                          Dil

                          alip      l                                  Dlip      Dl
        aijt                                            Dijt

               j           ajlk                D               Dj       Djlk

                   ajms                                         Djms
               m                                               Dm                      C
                  Fi                   filq          fllr                   Gi                  gilq          gllr
                                                                  si
                                         flip   Fl                                                glip   Gl
                               D                                       sl                D*
                       fijt                                                      gijt
                                        fjlk                                                     gjlk
                              Fj                                                        Gj
                                                            sj
                                fjms                                                     gjms

                              Fm                                                        Gm
                                                                 sm


                                                                 s
                              FD                                                        FD*
                                                            Fs

Perspectives of New Music (2006)
                                         Theorem: Using topos theory, we can show that
Guerino Mazzola & Moreno Andreatta:      networks of networks of... are networks: Arbitrary
From a Categorical Point of View:
K-nets as Limit Denotators               iterations possible without conceptual changes.
                        Theorem: Isomorphism classes of global
                        compositions are described by local
                        networks of affine functions.

                        Corollary: The theory of global
                        compositions is absorbed by the theory of
                        global networks.



                                                  global compositions


                                                                        global networks
COLLOQUIUM ON MATHEMATHICAL MUSIC THEORY
H. Fripertinger, L. Reich (Eds.)
Grazer Math. Ber., ISSN 1016–7692 Bericht Nr. 347 (2005),
Guerino Mazzola: Local and Global Limit Denotators and the
Classification of Global Compositions
                     http://www.rubato.org
 Gérard Milmeister
(chief programmer)

      Julien Junod
    (counterpoint)

 Florian Thalmann
       (morphing)

     Karim Morsy
   (macro objects)
• Facts

• Processes

• Gestures

• Collaboration
                                             Paradigm




                      May 18th 2002
                       IRCAM talk
                                   ?                    Transformations
                                                        Networks




                                             Digraph
Journal of Mathematics and Music (2007)
Guerino Mazzola & Moreno Andreatta:
Diagrams, Gestures, and Formulas in Music.
pitch       gestures                                   h        sonic
                                                               events
                            instrumentalize
               time                                              e
 position                                                  l


                             instrumental
                               interface

             thaw      freeze (MIDI)               √



                                score




                                        analysis
                       Folie 2




Czeslaw Marek:
Die Lehre des Klavierspiels
Every Noh play is a cross section       The waki is a kind of cosubject
of the life of one person, the shite.    and mirror person of the shite.
Shite weaves a texture of fantasy                Waki describes reality
using curves.                                      using straight lines.
Noh = expression of a space-time

The semiotic of gestures is reversed:

Gestures serve mainly the expression
of movements within this space-time,
the (usual) semantic is of second order.
pitch
                                               1
                                      
                                                       I

                                               0
                       time
                                               —
       position



 
                                          2        1  2

                                 2
                  1

                       2 + 1                 1
 t.
One hand  product  = 123456
of 6 gestural curves in space-time (x,y,z;e) of piano
j = 1, 2, ... 5: tips of fingers,
j = 6: the carpus, 6 = root
                                                                                                                           e = time

                                   Z                                                                                                           Y
                                                                                                                                               y        parameter t 
                                                                                                   z                                                    sequence of points:
     6(t)               c m
                                    d5                                                                 s c
                      Q ui kTi e™ and a TI FF ( Uncom pr essed) dec om pr essor ar e needed t o see t hi pi t ur e.




                                                                                                                      b5
                                                                                                                                       5(t)
                                                                                                                                                        (t) = (1(t),...,6(t))
             d2
                                       b2                                                                                                       4(t)
                                                                                                                                                3(t)

                                                                                                                                                        two base vectors
              1(t)                                                                                                           2(t)                     of fingers
x
X
                                                                                                                                                        d2, d5
                                                                                                                                                        from carpus.
Gestural interpretation of Carl Czerny‘s op. 500
Zur Anzeige wird der QuickTime™
     Dekompressor „H.264“
             benötigt.
The universe of gestures
• Paul Valéry  ...les ouvrages de la main prodigieuse de l‘ariste
                        égale et rivale de sa pensée - l‘une n‘est rien sans l‘autre
•   Charles Alunni Ce n‘est pas la règle qui gouverne l‘action
                        diagrammatique, mais l‘action qui fait émerger la règle.
•   Jean Cavaillès      Comprendre est attraper le geste et pouvoir continuer.
•   Pierre Boulez       Mais on est passé depuis Adorno de l‘informel à l‘informulé,
                        de l‘objet au geste.
•   Ludwig Wittgenstein I think with my pen.
•   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.
•   Robert S. Hatten    Given the importance of gesture to interpretation, why do we not
                        have a comprehensive theory of gesture in music?
•   Francis Bacon       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.
•   Bertold Brecht      ...non verbis sed gestibus.
•   Theodor W. Adorno Der Ausdruck ist bereits die Rationalisierung des Gestischen,
                        d.h. dessen Objektivierung durch Signifikation.
Hugues de Saint-Victor (Parisian theologian XIIth century):

Gestus est motus et figuratio membrorum corporis,
ad omnem agendi et habendi modum.


Gesture is the movement and figuration of the body's
limbs, according to whatever modality of action and attitude
For a topological space X, we have
           
the digraph X of continuous curves : I  X

                                   
gesture = morphism       g:   X       „-gesture in X“
          of digraphs

                         pitch




                     g
                            X
                                             time

                             position

Have category of gestures
                                  pitch
                                                 TipSpace
What about real shapes?
                              

                          
                                                            time

                                      position
                    Answer: Hypergestures!



         „loop of loops“




knot



circle
        ET-dance Gesture
time




space




                           Zur Anzeige wird der QuickTime™
                              Dekompre ssor „An imation“
                                        benötigt.




space
Hypergesture impossible!




            g

    
            h

                           X = torus
Fact:
For a a digraph  and topological space X, the set
    
@X of -gestures in X bears a canonical topology
(essentially the compact-open topology).




                                               
                                           @@X

                                           „gestures
                                            of gestures
X
                                            of gestures
                                            of....“
    Gestoids: Transition from gestures to algebraic
    structures connecting gestures

                                 homotopic
                                 curves:
                                 homotopy classes
                                 of curves
         X




                                         composition
                                         of homotopy
                                         classes of curves
X                    0       1
     0           1                       is associative!
             0           1
The homotopy classes of curves of a gesture g,
together with their composed curves, give rise
to a category, the groupoid Grg of the gesture.
Ist objects are the vertices x,y,... of g, while the
morphisms Grg(x,y) are the homotopy classes c
of composed curves from x to y.

The gestoid Gg is the ¬-linear category ø¬Grg.
The morphism Gg(x,y) are the formal ¬-linear
combinations

                        Sn ancn

of morphisms cn in Grg(x,y).
                                       i—
     Gestures
                                        i
         =                                    ei2pt
Musical String Theory

             g:                                       1   —


                                                X = S1
  Gg  ¬ p1(S1)

  fundamental group
  p1(S1)  Ÿ
  ei2pnt ~ n

  Sn an ei2pnt ~ Fourier representation f(t) = Sn an ei2pnt
• Facts

• Processes

• Gestures

• Collaboration
                        • Jazz: Collaboration as a performing art
                          R.A.R. MacDonald & G.B. Wilson: Construction of jazz: how jazz musicians present their collabortive
                          musical practice. Musicae Scientiae X, Nr.1, Spring 2006


                        • Flow (Mihaly Csikszentmihalyi)
                          is being described as „being completely involved in
                          an activity for its own sake. The ego falls away. Time
                          flies. Every action, movement, and thought follows
                          inevitably from the previous one, like playing jazz.“
                          Jazz improvisation as flow:
                          „...the music will work itself“
chick-sent-me-high-ee
                          „...goes into autopilot“
                          „...stream of consciousness“
                          B..J. Kenney M. Gellrich: Improvisation. In: R. Parncutt and G.E. McPherson (eds.) The science and
                          psychology of musical performance. Creative Strategies for teaching and
                          learning. New York: OUP 2002
• Francis Bacon (relating to the diagram debate):
  In the way I work I don’t in fact know very often what
  the paint will do, and it does many things which are very
  much better than I could make it do. Is that an accident?
   ...
   What has never yet been analysed is why this particular
   way of painting is more poignant than illustration. I
   suppose because it has a life completely of its own. (...)
   So the artist may be able to open up or rather, should I
   say, unlock the valves of feeling.
   D. Sylvester: Interview with Francis Bacon: The Brutality of Fact. Thames and Hudson, New York 1975
Thesis: Gestures help
• articulate/communicate
• organize/structure
• intensify/deepen
the flow in collaborative arts
Thank you!

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posted:10/22/2012
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