Transcript Operators on Vector Fields of Genealogical Stemmata for Musical Performance Guerino Mazzola U & ETH Zürich Internet Institute for Music Science [email protected] www.encyclospace.org.

Operators on Vector Fields
of Genealogical Stemmata
for Musical Performance
Guerino Mazzola
U & ETH Zürich
Internet Institute for Music Science
[email protected]
www.encyclospace.org
Contents
• Performance Fields
• Cell Hierarchies
• Algorithms and Calculations
• Initial Performances
• Operator Typology
Fields
H
h
X
x=
√(X)
Fields
√
E
L
T(E) = (d√E/dE)-1
[q /sec]
pE
I1
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
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 —Para, 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
p(I1) I2
√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)
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
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
Calculations
RUBATO® software:
Calculations via Runge-Kutta-Fehlberg methods for
numerical ODE solutions
Initials
X = X0 = X(0)
I
t1
X1 = X(t1)
Initials
X0
I
Xj
(tk+tj)/2
?
Xk
(tk+ti)/2
Xi
x=
Initials
X
q
Q
XQ
Q Space(I)
I
√(X)
xq
Q
Q
Q
Closure(Space(I)) Space(X)
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
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
Operator Types
Qw
Z
T
Zw
T
Tw
T
?
Z
T
The Lie Derivative Approach
√(X,Y) = (x(X),y(X,Y)) √l(X,Y) = (x(X),l(X).y(X,Y))
Space(X) Space(Y)
Space(X)
Y
Yl
X
X
Y
Typology
J(√) =
X
Yl
0
y/X y/Y
=
A 0
B C
A-1
J(√l)-1 =
X
x/X
-C-1 BA-1 - l-1C-1.yƒdl.A-1
L= ln(l), DY=(1,…,1),
eY = embedding of Y-tangent space
YL= Y — [LX(L)C-1y-(e-L -1)C-1DY]eY
0
l-1C-1
Typology
YL= Y —[LX(L)C-1y-(e-L -1)C-1DY]eY
y = U.Y + v
C-1 = U-1
L0
YL= Y — [LX(L)(Y+Const.)]eY
YL= Y — LYX (L)(R.Y+C)eY
eC.R: —Space(Y)  —Space(Y)
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
Typology
RUBATO®: Scalar operator
Linear action Qw on
ED-tangent bundle
Direction of field
changes
Numerical
integration control