Transcript Performance and Interpretation Guerino Mazzola U & ETH Zürich Internet Institute for Music Science [email protected] www.encyclospace.org Fields.

Performance
and
Interpretation
Guerino Mazzola
U & ETH Zürich
Internet Institute for Music Science
[email protected]
www.encyclospace.org
Fields
H
h
X
x=
√(X)
Fields
√
E
L
Zur Anzei ge wird der Qui ckTi me™
Dekompressor “GIF”
benöti gt.
I1
T(E) = (d√E/dE)-1
[q /sec]
pE
Ik
E
e
l
√E
pe
√E(I1) √E(Ik)
e
P-Cells
X
X0
√
F
Z(X) = J(√ )(X)-1 D
performance field, defined on
cube F = the frame of Z
X0 ΠI = initial set
X0 = ÚXZ(t)
ÚXZ = integral curve through X
x=
√(X)
x0
D = (1,1,…,1)
= Const.
x0 = √I(X0)
initial performance
x = x0 - t.D
Product fields: Tempo-Intonation field
P-Cells
Z(E,H)=(T(E),S(H))
H
EH
E
H
S(H)
T(E)
E
P-Cells
Parallel fields: Articulation field
ED
Z(E,D) = T(E,D) = (T(E),2T(E+D)-T(E))
E
D
(e(E),d(E,D) = e(E+D)-e(E))
E
T(E)
P-Cells
A Performance Cell C is a 5-tuple as follows
• a closed frame F = [aE,bE] ¥[aH,bH] ¥... —Para,
Para = {E,H,L,…} = finite set of symbolic parameters
• a Lipschitz-continuous performance field Z,
defined on a neighbourhood of F
• a polyhedral initial set I, i.e., a finite union of
possibly degenerate simplexes of any dimension in —Para
• a finite set K F, the symbolic kernel, such that
every integral curve ÚXZ through X Œ K hits I
• an initial performance map √I:I —para
(para = {e,h,l,…} physical parameters) such that
for any X ΠK and two points
√
I
a = ÚXZ(a), b = ÚXZ(b),
√I(b) - √I(a) = (a-b).D
F
K
I
Z
P-Cells
The category Cell of cells has these morphisms p: C1 C2 :
• we have Para2 Para1
p: —Para1 —Para2
√I1
C1
K1
is the projection such that
I1
• p(F1) F2
Z1
• p(K1) K2
p
• p(I1) I2c
√I2
• p.√I1 = √I2 .pI1
• Tp.Z1 = Z2.p
F1
C2
F2
K2
I2
Z2
Morphisms induce compatible performances
P-Cells
√I1
F1
K1
I1
√1
K1
√1(K1)
Z1
p
√I2
F2
K2
I2
Z2
K2
p
√2
√2(K2)
P-Cells
Work with
Basis parameters E, H, L,
and corresponding fields T(E), S(H), I(L)
Pianola parameters D, G, C
A cell hierarchy is a Diagram D in Cell such that
• there is exactly one root cell
• the diagram cell parameter sets are closed
under union and non-empty intersection
T ¥ S
T ¥ I ¥ S
T
Root
T ¥ I
T¥ S
T¥ I¥ S
T
S
T¥ I
I¥ S
I
Fundament
Typology
mother
T
daughter
T
l
granddaughter
Z(T,l)
Tl
Stemma
Typology
Big Problem:
Describe Typology of shaping operators!
Emotions, Gestures, Analyses
w(E,H,…)
H
E
examples
Java Classes for
Modules,
Forms, and Denotators
RUBATO®
L
L
S
S
Os X
Calculations
RUBATO® software:
Calculations via Runge-Kutta-Fehlberg methods for
numerical ODE solutions
Tempo Operators
Typology
T(E)
w(E)
Tw(E) = w(E).T(E)
Deformation of the articulation field hierarchy
T
Tw
Z
T
Tw
T
Qw(E,D) =
w(E)
0
w(E+D)—w(E) w(E+D)
Qw = J(√w)-1 „w-tempo“
?
Zw
Tw
Zw = Qw(E,D).Z
Typology
RUBATO®: Scalar operator
Linear action Qw on
ED-tangent bundle
Direction of field
changes
Numerical
integration control
Typology
The directed Lie derivative operator construction:
• In the given hierarchy, choose a hierarchy
space Z
• select a weight Lon Z
• choose any subspace S of the root space
• select an affine directional endomorphism
Dir:S  S
Given the total field Y, define the operator
YL,Dir = Y — LYZ(L).Dir.eS
Typology
Theorem:
For the deformation types

T
Qw
Zw
T
Tw
T
?
Z
T
T¥ I
T
I
?
Z
T
there is a suitable data set (Z,S,L,Dir) for the
respective cell hierarchies such that the deformations
are defined by directed Lie derivative operators
YL,Dir = Y — LYZ(L).Dir.eS
method of characteristics
performance
J.S. Bach: Die Kunst der Fuge — Contrapunctus III
Joachim Stange-Elbe
Metrical and Motivic Weights
act on agogics, dynamics, and articulation
sopran
score
alt
sum of all
tenor
bass
Inverse Theory
Restriction
Lie type
Affine transport
Inverse Theory
Inverse Theory
Restriction
Restriction
Lie type
Lie type
Affine transport
Sum
Inverse Theory
Lie operator
parameters:
weights,
directions
Output
fields Z.
Affine
transport
parameters
fiber(Z.)
Roberto Ferretti
Inverse Theory
The Topos of Music
Geometric Logic of
Concepts, Theory, and Performance
in collaboration with
Moreno Andreatta, Jan Beran, Chantal Buteau,
Roberto Ferretti, Anja Fleischer, Harald Fripertinger,
Jörg Garbers, Stefan Göller, Werner Hemmert,
Mariana Montiel, Stefan Müller, Andreas Nestke,
Thomas Noll, Joachim Stange-Elbe, Oliver Zahorka
www.encyclospace.org