Guerino Mazzola

1

• Guerino Mazzola
• U ETH Zürich
• Internet Institute for Music Science
• guerino_at_mazzola.ch
• www.encyclospace.org

2

• Performance Fields
• Cell Hierarchies
• Algorithms and Calculations
• Initial Performances
• Operator Typology

Contents
3
Fields
4
x v(X)
X
Fields
T(E) (dvE/dE)-1 q /sec
5
P-Cells
Z(X) J(v )(X)-1 D performance field, defined
on cube F the frame of Z X0 I initial
set X0 ÚXZ(t) ÚXZ integral curve through X
D (1,1,,1) Const. x0 vI(X0)
initial performance x x0 - t.D
6

• A Performance Cell C is a 5-tuple as follows
• a closed frame F aE,bE aH,bH ... Í
  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 Í Para, the symbolic kernel,
  such that every integral curve ÚXZ through X K
  hits I
• an initial performance map vII para
  (para e,h,l, physical parameters) such
  that for any X K and two points
• a ÚXZ(a), b ÚXZ(b),
• vI(b) - vI(a) (a-b).D

P-Cells
7
The category Cell of cells has these morphisms p
C1 C2

• we have Para2 Í Para1 p Para1 Para2
  is the projection such that
• p(F1) Í F2
• p(I1) Í I2
• p.vI1 vI2 .pI1
• Tp.Z1 Z2.p

P-Cells
8
Morphisms induce compatible performances
P-Cells
9
Product fields Tempo-Intonation field
P-Cells
10
Parallel fields Articulation field
P-Cells
11

• 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

P-Cells
12
RUBATO software Calculations via
Runge-Kutta-Fehlberg methods for numerical ODE
solutions
Calculations
13
Initials
14
Initials
15
Initials
Closure(Space(I)) Í Space(X)
16
l
Typology
Stemma
17
Big Problem Describe Typology of shaping
operators!
Emotions, Gestures, Analyses
Typology
18
Tempo Operators
Typology
Deformation of the articulation field hierarchy
Zw Qw(E,D).Z
Qw J(vw)-1 w-tempo
19
Operator Types
Typology
The Lie Derivative Approach
v(X,Y) (x(X),y(X,Y))
vl(X,Y) (x(X),l(X).y(X,Y))
20
Typology
L ln(l), DY(1,,1), eY embedding of
Y-tangent space
YL Y LX(L)C-1y-(e-L -1)C-1DYeY
21
YL Y LX(L)C-1y-(e-L -1)C-1DYeY y U.Y
v C-1 U-1 L 0 YL Y LX(L)(YConst.)eY
YL Y LYX (L)(R.YC)eY eC.R Space(Y)
Space(Y)
Typology
22

• 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 Î S_at_S
• Given the total field Y, define the operator
• YL,Dir Y LYZ(L).Dir.eS

Typology
23
Theorem For the deformation types
Typology
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
24
RUBATO Scalar operator
Typology