Guerino Mazzola U & ETH Zürich Internet Institute for Music Science Performance and Interpretation Performance and Interpretation

Fields

√ H E L E e pEpE pepe √E√E √ E (I 1 ) √ E (I k ) I1I1 IkIk X T(E) = (d √ E /dE) -1 [ q /sec]he l x = √ (X)

P-Cells X x = √ (X) √  = (1,1,…,1) = Const. x 0 = √ I (X 0 ) initial performance x = x 0  t.  Z(X)= J( √ )(X) -1  performance field performance field, defined on F frame cube F = the frame of Z I = initial set X 0 Œ I = initial set X 0 = Ú X Z(t) Ú X Z = integral curve through X X0X0 x0x0 F

P-Cells Product fields: Tempo-Intonation field E H S(H) EH EH Z(E,H)=(T(E),S(H)) T(E)

(e(E),d(E,D) = e(E+D)-e(E)) T(E) P-Cells Parallel fields: Articulation field E D ED E Z(E,D) =  T(E,D) = (T(E),2T(E+D)  T(E))

P-Cells Performance Cell A Performance Cell C is a 5-tuple as follows frame a closed frame F = [a E,b E ] ¥ [a H,b H ] ¥...  — Para, Para = {E,H,L,…} = finite set of symbolic parameters performance field a Lipschitz-continuous performance field Z, defined on a neighbourhood of F initial set a  polyhedral initial set I, i.e., a finite union of possibly degenerate simplexes of any dimension in — Para symbolic kernel a finite set K  F, the symbolic kernel, such that every integral curve Ú X Z through X Œ K hits I initial performance an initial performance map √ I : I  — para (para = {e,h,l,…} physical parameters) such that for any X Œ K and two points a = Ú X Z(  ), b = Ú X Z(  ),  √ I (b)  √ I (a) = (  ).  K I F √I√I Z

P-Cells Cell p: C 1  C 2 The category Cell of cells has these morphisms p: C 1  C 2 : K1K1 I1I1 F1F1 √I1√I1 Z1Z1 K2K2 I2I2 F2F2 √I2√I2 Z2Z2 C1C1C1C1 C2C2C2C2 p we have Para 2  Para 1 p: — Para 1  — Para 2 is the projection such that p(F 1 )  F 2 p(K 1 )  K 2 p(I 1 )  I 2 c p. √ I 1 = √ I 2.p  I 1 Tp.Z 1 = Z 2.p

P-Cells Morphisms induce compatible performances K1K1 I1I1 F1F1 √I1√I1 Z1Z1 K2K2 I2I2 F2F2 √I2√I2 Z2Z2 K2K2K2K2 √2√2 √2(K2)√2(K2)√2(K2)√2(K2) pp K1K1K1K1 √1√1 √1(K1)√1(K1)√1(K1)√1(K1)

Root Fundament P-Cells Work with Basis Basis parameters E, H, L, and corresponding fields T(E), S(H), I(L) Pianola Pianola parameters D, G, C cell hierarchy 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 I I ¥ S T ¥ I T ¥ S  T ¥ S  T ¥ I T ¥ I ¥ S  T ¥ I ¥ S TT

Typology T TT T Z(  T, ) Stemma mother daughter granddaughter

Emotions, Gestures, Analyses Typology Big Problem: Describe Typology of shaping operators! w(E,H,…) H E

Os X  RUBATO ®    Java Classes for Modules, Forms, and Denotators examples

Calculations RUBATO ® software: Calculations via Runge-Kutta-Fehlberg methods for numerical ODE solutions

Typology Tempo Operators T(E)w(E)T w (E) = w(E).T(E) Deformation of the articulation field hierarchy TT TwTw T TwTw  ww T TwTw? Q w (E,D) = w(E) 0 w(E+D)—w(E) w(E+D)  w = Q w (E,D).Z Q w = J( √ w ) -1 „w-tempo“

Typology RUBATO ® : Scalar operator Linear action Q w 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 In the given hierarchy, choose a hierarchy space Z select a weight  on Z select a weight  on Z choose any subspace S of the root space choose any subspace S of the root space select an affine directional endomorphism Dir  S  S select an affine directional endomorphism Dir  S  S Given the total field Y, define the operator Y ,Dir  =  Y — L Y Z (  Dir.e S

Typology Theorem: For the deformation types  ww T TwTw QwQwQwQw TT T  T? there is a suitable data set (Z,S, ,Dir) for the respective cell hierarchies such that the deformations are defined by directed Lie derivative operators Y ,Dir  =  Y — L Y Z (  Dir.e S T ¥ I T  T? I 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 alt tenor bass score sum of all

Inverse Theory Lie type Restriction Affine transport

Inverse Theory Stefan Müller: EspressoRubette

Inverse Theory Lie type Restriction Restriction Sum Affine transport

Affinetransportparameters Lie operator parameters:weights,directions Output fields Z. fiber(Z.) Inverse Theory 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