Transcript DANS CES MURS VOUÉS AUX MERVEILLES J’ACCUEILLE ET GARDE LES OUVRAGES DE LA MAIN PRODIGIEUSE DE L’ARTISTE ÉGALE ET RIVALE DE SA PENSÉE L’UNE.

DANS CES MURS VOUÉS AUX MERVEILLES
J’ACCUEILLE ET GARDE LES OUVRAGES
DE LA MAIN PRODIGIEUSE DE L’ARTISTE
ÉGALE ET RIVALE DE SA PENSÉE
L’UNE N’EST RIEN SANS L’AUTRE
(Paul Valéry, Palais Chaillot)
Guerino Mazzola
U & ETH Zürich
[email protected]
www.encyclospace.org
Musical Gestures
and their
Diagrammatic Logic
LA VERITÉ
DU BEAU
DANS
LA MUSIQUE
Guerino Mazzola
summer 2006
composition
de formules
~
formule
~
harmonie
de gestes
geste
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Ryukoku
violin robot
Waseda
wabot II
• Musical Gestures
• Gesture Categories
• Diagram Logic
• Musical Gestures
• Gesture Categories
• Diagram Logic
gestualize
gestures
pitch
time
instrumentalize
e
l
position
instrumental
interface
thaw
sonic
events
h
√
freeze (MIDI)
score
analysis
Ceslaw Marek:
Lehre des
Klavierspiels
Atlantis-Verlag
Zürich 1972/77
Folie 2
Every No play is a cross section
of the life of one person, the shite.
The shite is an appearance (demon, etc.)
and a subject = one of the five elements
(fire, water, wood, earth, metal)
The waki is
A kind of co-subject and
mirror person
of the shite.
The No gestures are reduced to the kata units
and made symbolic.
This enables a richer communication than with
common gestures.
Important:
• Shite weaves a texture of fantasy using
curves.
• Waki describes reality using
straight lines.
pitch
1

0
time
—
position

2
2
1
t.
 1  2
2 + 1
1
pitch
E
position
√gestures
H
h
√score
E
L
l
e
PhD thesis of Stefan Müller
(Mazzola G & Müller S: ICMC 2003)
Symbolic score
(a) Without
fingering
annotation
(b) with
fingering
annotation
C3
DIN8996
Independent
symbolic
gesture curves
for fingers
2
et
3
Curve parameter t
on horizontal axis
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
 6(t)
d5
Y
y
z
parameter t 
sequence of points:
Q uickTim e™ and a TI FF ( Uncom pr essed) decom pr essor ar e needed t o see t his pict ur e.
b5
(t) = (1(t),...,6(t))
 5(t)
(t)
d2
b2
 4(t)
 3(t)
(t)
xX
 1(t)
 2(t)
two base vectors
of fingers
d2, d5
from carpus.
Geometric constraints: six boxes
Have masses mj and
maximal forces Kj
for fingers/carpus j.
d2 space3 /de2
The Newton condition for fingers or carpus j is
mj d2 spacej /de2(t) < Kj
for all 0 ≤ t ≤ 1.
Use cubic polynomials for gestural coordinates, i.e., 76
variables of coefficients:
xj(t) = xj,3 t3 + xj,2 t2 + xj,1 t + xj,0
 yj(t) = yj,3 t3 + yj,2 t2 + yj,1 t + yj,0
 zj(t) = zj,3 t3 + zj,2 t2 + zj,1 t + zj,0
 e(t) = e3 t3 + e2 t2 + e1 t + e0
Geometric and physical constraints 
polynomial inequalities: P(t) > 0 for all 0 ≤ t ≤ 1.
These inequalities are guaranteed by Sturm chains.
Symbolic gestural curve
Physical gestural curve
fingers 2, 3: geometric constraints
fingers 2, 3: physical constraints
Gestural interpretation of Carl Czerny‘s op. 500
Zur Anzeige wird der QuickTime™
Dekompressor „H.264“
benötigt.
• Musical Gestures
• Gesture Categories
• Diagram Logic
Quiver = category of quivers
(= digraphs, diagram schemes, etc.)
D=A
h
t
u
x = t(a)
V
d
v
a
q
y = h(a)
w
c
b
x
E=B
h‘
t‘
a
W
y
Quiver(D, E)
D

morphism g: D  X
of quivers withvalues in a
spatial quiver X of a metric space X
(= quiver of continuous curves in X)
(Local) Gesture =
A gesture morphism u: g  h is a quiver morphism u,
such that there is a continuous map f: X  Y which
defines a commutative diagram:
D
g

X

u
E
pitch
f

h
Y
Gesture(g, h)
category of (local) gestures
g
X
D
time
position
A global gesture
being covered
by three
local gestures

Quiver(F,X ) =
metric space of (local) gestures of
of quiver F with values in a

spatial quiver X
.
r
s
t Uhde:
Renate Wieland & Jürgen
Forschendes Üben
Anzeige wird der QuickTime™
DieZur
Klangberührung
ist das Ziel
Dekompressor „Animation“
der zusammenfassenden
Geste,
benötigt.
der Anschlag ist sozusagen
die Geste in der Geste.
F
E
Hypergesture impossible!
g
E
h
Morphism exists!
g
h
• Musical Gestures
• Gesture Categories
• Diagram Logic
The category Quiver is a topos
DE
1=
D+E
0=Ø
Alexander Grothendieck
DE
Quiver(
≈
Quiver(E 
, DE)
Quiver( , DE)
, D)
≈
Quiver(E  , D)
Subobject classifier
 =
T
In particular:
The set Sub(D) of
subquivers
of a quiver D
is a Heyting algebra:
have „Quiver logic“.
Ergo:
v
w
x
y
Local/global gestures,
ANNs,
Klumpenhouwer-nets,
and global networks
enable logical
operators (, , ,)
Heyting logic on set Sub(g) of subgestures of g
h, k  Sub(g)
hk=hk
hk=hk
h  k (complicated)
h = h  Ø
tertium datur: h ≤  h
u: g1  g2
Sub(u): Sub(g2)  Sub(g1)
homomorphism of Heyting algebras
= contravariant functor
Sub: Gesture  Heyting
C-major hypergesture
c
b
IVV
a
d
III
VI
I
g
Fingers
II
VII
e
f

Fingers = Quiver(F,X
)
F=
V

I
IV
=
VI
I
Problems:
• Investigate the possible
role and semantics of gestural logic in concrete contexts
such as
local/global musical/robot gestures
and more specific environments...
(and more generally: Quiver logic for ANNs,
Klumpenhouwer-nets,
global networks).
• Investigate a (formal)
propositional/predicate language of gestures
with values in Heyting algebras of quivers.