Guerino Mazzola U & ETH Zürich [email protected]

www.encyclospace.org/talks

Gestures for the Science of Collaborative Arts

to be the case

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Facts

to refer to

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Processes

to make

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Gestures

to interact

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Collaboration

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Facts

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Processes

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Gestures

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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)

I C (3) II III IV V VI VII Pitch classes in Ÿ 12 truth values = chords

Schönberg‘s „harmonic strip“ N(X (3) ) (Harmonielehre 1911) V II IV VI VII III „ harmonic strip = Möbius strip“ is a logical fact!

I

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 a b c d e 0 2 3 4 6 8 10 12

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

0 2 3 4 6 10 12 c N(X MAX ) e 12 6 0 10 d 3 a 2 4

x dominates y iff Sp(y)



Sp(x)

b

22 20 18 16 14 12 10 8 6 4

w(x) =

S

M Œ Sp(x), m £ length(M) length(M) p

m = p = 2

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Facts

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Processes

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Gestures

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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 c = 0  (p) = 3p+7  {c,  (c),  2 (c),...} g   = {c, e, g} = major triad Classification of circle chords (cf.

Gruppen und Kategorien

) Ÿ 12 e



i

Diagram in a category C C = sets, groups, vector spaces, automata, etc.  = digraph

Nets of dodecaphonic series

• • D :   D D i ij t

C

= objects in

C

= morphisms in

C

Multiply iterated nets

a

il q

a

li p

l a

ll r

D

i

D

il q

D

li p

D

l

D

ll r

a

ij t

D

ij t D

j m a

jm s

a

jl k

D

j

D

jm s

D

m

D

jl k

C

F i f ij t

D

f il q f li p F j f jm s f jl k F m F l f ll r s j s s i s l G i g ij t

D *

g il q g li p G j g jm s g jl k G m m G l g ll r

F D

Perspectives of New Music (2006) Guerino Mazzola & Moreno Andreatta:

From a Categorical Point of View: K-nets as Limit Denotators

s

F D * F

s

Theorem:

Using topos theory, we can show that networks of networks of... are networks: Arbitrary 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

Gérard Milmeister (chief programmer) Julien Junod (counterpoint) Florian Thalmann (morphing) Karim Morsy (macro objects) http://www.rubato.org

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Facts

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Processes

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Gestures

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Collaboration

Paradigm ?

IRCAM talk Journal of Mathematics and Music (2007) Guerino Mazzola & Moreno Andreatta:

Diagrams, Gestures, and Formulas in Music.

Digraph

Transformations

Networks

pitch

gestures

time position thaw instrumentalize

instrumental interface

freeze (MIDI) √ h l

sonic events

e

score

analysis

Folie 2

Czeslaw Marek: Die Lehre des Klavierspiels

Every Noh play is a cross section of the life of one person, the shite .

Shite weaves a texture of fantasy using

curves .

The waki is a kind of cosubject and mirror person of the shite.

Waki describes reality 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  position t.

 time  1  2 +  1  2  1 0 — I  2  1   2  1

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   (t ) ) Z d 5 z parameter t  sequence of points: b 5 (t ) )  (t) = (  1 (t),...,  6 (t)) d 2 b 2 2 (t ) ) (t ) ) (t ) ) (t ) ) two base vectors of fingers d 2 , d 5 from carpus.

Gestural interpretation of Carl Czerny‘s op. 500

Zur Anzeige wird der QuickTime™ Dekompressor „H.264“ benötigt.

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The universe of gestures Paul Valéry ...

les ouvrages de la main prodigieuse de l‘ariste

Charles Alunni

égale et rivale de sa pensée - l‘une n‘est rien sans l‘autre Ce n‘est pas la règle qui gouverne l‘action diagrammatique, mais l‘action qui fait émerger la règle.

Jean Cavaillès Pierre Boulez Francis Bacon

Comprendre est attraper le geste et pouvoir continuer.

Mais on est passé depuis Adorno de l‘informel à l‘informulé, de l‘objet au geste.

Ludwig Wittgenstein Cecil Taylor

I think with my pen.

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?

Bertold Brecht Theodor W. Adorno

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.

...non verbis sed gestibus.

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 of continuous curves  : I  X

gesture

= morphism g:   of digraphs  X pitch „  -gesture in X“  g X Have category of gestures position time

What about real shapes?

  pitch

TipSpace

time position

knot circle Answer: Hypergestures! „loop of loops“

time space ET-dance Gesture Zur Anzeige wird der QuickTime™ Dekompressor „Animation“ benötigt.

space

Hypergesture impossible!



g h

X = torus

Fact:

For a a digraph   

@

X of  and topological space X, the set -gestures in X bears a canonical topology (essentially the compact-open topology).

X  

@

 

@

X „gestures of gestures of gestures of....“

Gestoids:

Transition from gestures to algebraic structures connecting gestures homotopic curves: homotopy classes of curves X X 0 0 1 0 1 1 composition of homotopy classes of curves is associative!

The homotopy classes of curves of a gesture g, together with their composed curves, give rise to a category, the

groupoid

Gr g of the gesture.

Ist objects are the vertices x,y,... of g, while the morphisms Gr g (x,y) are the homotopy classes c of composed curves from x to y.

The

gestoid

G g is the ¬ -linear category ø¬ Gr g .

The morphism G g (x,y) are the formal ¬ linear combinations S n a n c n of morphisms c n in Gr g (x,y).

Gestures = Musical String Theory g: i — i e i2 p t 1 — G g  ¬ p 1 (S 1 ) fundamental group p 1 (S 1 )  e i2 p nt Ÿ ~ n X = S 1 S n a n e i2 p nt ~ Fourier representation f(t) = S n a n e i2 p nt

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Facts

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Processes

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Gestures

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Collaboration

chick-sent-me-high-ee • 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“ „...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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