1. Guerino Mazzola U & ETH Zürich Internet Institute for Music Science guerino@mazzola.ch www.encyclospace.org Operators on Vector Fields of Genealogical Stemmata for Musical Performance Operators on Vector Fields of Genealogical Stemmata for Musical Performance

2. Contents Performance Fields Cell Hierarchies Algorithms and Calculations Initial Performances Operator Typology Performance Fields Cell Hierarchies Algorithms and Calculations Initial Performances Operator Typology

3. Fields

4. √ 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)

5. 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

6. 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  — Para, 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

7. 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(I 1 )  I 2 p. √ I 1 = √ I 2.p  I 1 Tp.Z 1 = Z 2.p

8. 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)

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

10. (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))

11. 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

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

13. I t1t1 X = X 0 = X(0) X 1 = X(t 1 ) Initials

14. I X0X0 XiXi XkXk XjXj (t k +t i )/2 (t k +t j )/2 ? Initials

15. x = √ (X) X Q q XQXQ xqxq I Q Q  Space(I) Closure(Space(I))  Space(X) Q Q

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

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

18. 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“

19. Typology Operator Types TT T  T?  ww T TwTw QwQwQwQw √ (X,Y) = (x(X),y(X,Y)) The Lie Derivative Approach √ (X,Y) = (x(X), (X).y(X,Y)) Space(X)  Space ( Y) Space(X) YX YX

20. TypologyYX YX J( √ ) =  x/  X 0  y/  Y  y/  X A 0 CB = J( √ ) -1 = A -1 0  C -1 BA -1  -1 C -1.y ƒ d. A -1 -1 C -1  = ln( ),  Y=(1,…,1), e Y = embedding of Y-tangent space Y  =  Y — [L X (  )C -1 y-(e -  -1)C -1  Y]e Y

21. Typology Y  =  Y —[L X (  )Cy-(e -  -1)C  Y]e Y Y  =  Y —[L X (  )C -1 y-(e -  -1)C -1  Y]e Y y = U.Y + v C -1 = U -1  Y  =  Y — [L X (  )(Y+Const.)]e Y Y  =  Y — L Y X (  )(R.Y+C)e Y e C.R: — Space(Y)  — Space(Y)

22. 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

23. 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

24. Typology RUBATO ® : Scalar operator Linear action Q w on ED-tangent bundle Direction of field changes Numerical integration control